Regularized Curve Lengthening from the Strong FCH Flow
RREGULARIZED CURVE LENGTHENING FROM THE STRONGFCH FLOW
YUAN CHEN AND KEITH PROMISLOW
Abstract.
We show that level sets of nearly circular bilayer interfaces evolv-ing under the mass preserving L -gradient flow of the strong scaling of the func-tionalized Cahn Hilliard gradient flow satisfy a regularized curve-lengtheningflow in the sharp interface limit. We employ energy estimates and a rigorouscenter-unstable Galerkin reduction with an asymptotically large number ofmodes to show that perturbations of circular bilayer equilibrium that add in-terfacial mass excite a motion against mean curvature, regularized by a higherorder Willmore expression. For sufficiently small added mass the expand-ing interface undergoes a non-circular transient as it absorbs the perturbativemass, relaxing to a circular bilayer with a larger radius after the added massis consumed. For mass perturbations beyond a threshold, the interfacial dy-namics show sensitivity to initial perturbations that is exponentially large inthe interfacial width parameter. Contents
1. Introduction 21.1. Background and motivation 21.2. Main results 51.3. Notation 92. Coordinates and preliminary estimates 102.1. The whiskered coordinates 102.2. Perturbed interfaces 123. Quasi-steady states 213.1. Construction of Φ p p p t q Date : July 5, 2019.2010
Mathematics Subject Classification.
Primary 35K25, 35K55; Secondary 35Q92.
Key words and phrases.
Functionalized Cahn-Hilliard, Interfacial dynamics, Curvelengthening.K.P. acknowledges support from NSF grants DMS-1409940 and DMS-1813203. a r X i v : . [ m a t h . A P ] J u l YUAN CHEN AND KEITH PROMISLOW
Acknowledgement 73Appendix 73References 761.
Introduction
Background and motivation.
The functionalized Cahn-Hilliard(FCH) freeenergy was introduced in [18] to model the free energy of mixtures of amphiphilicmolecules and solvent. Amphiphilic molecules are formed by chemically bondingtwo components whose individual interactions with the solvent are energetically fa-vorable and unfavorable, respectively. When blended with the solvent, amphiphilicmolecules have a propensity to phase separate, forming amphiphilic rich domainsthat are thin, generically the thickness of two molecules, in at least one direction.The FCH free energy is given in terms of the volume fraction u ´ b ´ of the am-phiphilic molecule over a domain Ω as(1.1) F p u q : “ ż Ω ε ˆ ∆ u ´ ε W p u q ˙ ´ ε p ´ ´ η | ∇ u | ` η ε W p u q ¯ d x, where W : R ÞÑ R is a smooth tilted double well potential with local minimaat u “ b ˘ with b ´ ă b ` , W p b ´ q “ ą W p b ` q , and W p b ´ q ą
0. The state u ” b ´ corresponds to pure solvent, while u ” b ` denotes a maximum packingof amphiphilic molecules. The system parameters η ą η characterize keystructural properties of the amphiphilic molecules. The small positive parameter ε ! p “ p “
1, in which the amphiphilic structure terms dominate the Willmore residual andrefer the interested reader to [19] for a detailed discussion of physical meaning of theparameters. A key feature of the functionalized Cahn-Hilliard energy (1.1) is thatits approximate minima include vast families of saddle points of a Cahn-Hilliardtype energy. Within the FCH the competitors for minima include codimensionone bilayers, codimension two pores, and codimension three micelles that are thebuilding blocks of many biologically relevant organelles.We address mass-preserving gradient flows of the FCH energy (1.1) and rigor-ously derive the interface motion for nearly circular codimension one interfaces. Thechemical potential associated to F is denoted F “ F p u q , and given by a rescalingof the variational derivative(1.2) F p u q : “ ε δ F δu “ p ε ∆ ´ W p u qqp ε ∆ u ´ W p u qq ` ε p p η ε ∆ u ´ η W p u qq . The strong, p “
1, scaling of the FCH under the mass-preserving L gradient flowtakes the form(1.3) B t u “ ´ Π F p u q , subject to periodic boundary conditions on Ω Ă R . Here Π is the zero-massprojection given by(1.4) Π f : “ f ´ (cid:104) f (cid:105) L , EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW where we have introduced the mass function(1.5) (cid:104) f (cid:105) L : “ | Ω | ż Ω f d x. The existence of a family of almost radially symmetric equilibrium bilayer solutionsof (1.3) was established in [10]. For a prescribed total system mass(1.6) ż Ω p u ´ b ´ q d x “ εM, the radial equilibrium are parameterized uniquely, up to translation, by the radius R of the bilayer, so long as R P p R ´ , R ` q , where 0 ă R ´ ă R ` are independentof ε and R ` is less than the domain width. The radial symmetry is broken bycorrections that are exponentially small in ε that transform the radially symmet-ric profiles into equilibrium solutions, Φ ˚ p x ; R q satisfying the periodic boundaryconditions. These “radial-periodic” equilibrium admit the expansionΦ ˚ p x q “ φ p z p x qq ` ε σ ˚ W p b ´ q ` O ´ ε ¯ , in the L p Ω q norm. Here z is ε scaled distance to a circular interface Γ ˚ and φ isthe leading order bilayer profile defined as the unique non-trivial solution of(1.7) B z φ “ W p φ q , that is homoclinic to the left (solvent) well, b ´ , of W . The equilibrium solutiondoes not return to the pure solvent phase u “ b ´ in the far field, rather in the bulkthere is an order- ε perturbation of the amphiphilic density, σ ˚ “ σ ˚ ` εσ ˚ , whoseleading order satisfies(1.8) σ ˚ : “ ´ η ` η m m , in terms of the bilayer mass per unit length and L p R q norm of its derivative,(1.9) m : “ ż R p φ ´ b ´ q d z, m “ } φ } L p R q . We establish that perturbations of the radially-periodic equilibrium evolve atleading order along a manifold of bilayer profiles(1.10) Φ p x ; Γ , σ q “ φ p z q ` εσ p W p b ´ qq ` O ´ ε ¯ , where z denotes ε scaled signed distance to perturbed interface Γ and σ is a spatiallyconstant perturbation to the bulk density. It is instructive to examine the leadingorder reduction of the FCH energy at Φ p¨ ; Γ , σ q ,(1.11) F Γ ,σ : “ F p Φ p¨ ; Γ , σ qq “ m ż Γ | κ | d s ´ ε p ´ m p η ` η q| Γ | ` ε ´ σ | Ω | p W p b ´ qq , which can be viewed as the sum of a Canham-Helfrich energy, [6, 22], and a bulkterm parametrized by σ . The energy reflects a balance between curvature, curvelength, and bulk terms that drives the dynamics which we analyze in this paper.The FCH gradient flow constrains the total mass, (1.6), and within the family(1.10) the curve length | Γ | is slaved at leading order to the bulk density parameter σ through the linear relation(1.12) | Γ | “ Mm ´ | Ω |p W p b ´ qq ¨ σm . YUAN CHEN AND KEITH PROMISLOW
For the p “ M and | Ω | , the reduced energy (1.11) can be reformulated as(1.13) F Γ ,σ “ m ż Γ | κ | d s ` ε ´ | Ω | p W p b ´ qq p σ ´ σ ˚ q . Since ε !
1, the energetic cost of bulk disequilibria, σ ‰ σ ˚ , dominates the contri-butions from curvature and curve length. If the bulk density is high, σ ´ σ " ? ε ,then the system can dissipate total energy by absorbing amphiphilic material fromthe bulk and lengthening the interface, even if that requires a departure from cir-cularity and an increase in interfacial curvature. We provide a rigorous analysis ofprecisely this evolution.Previous work on the FCH type models has focused on the formal analysis ofvarious gradient flows. These include the evolution of closed codimension one bi-layers in the weak ( p “
2) scaling of the FCH, [14], and the competitive dynamicsof coexisting codimension one bilayers and codimension two pores in the weak, [15]and the strong FCH, [4], respectively. In particular the results contained hereinare a rigorous partial justification of the regularized curve lengthening results con-tained in [4]. The rigorous existence and linear stability of the constant curvatureequilibrium of the strong FCH, including circular the bilayers studied herein, wasestablished in [10]. Conditions that characterize pearling in bilayers and filements,[28] and the existence of pearled bilayer structures in the super-critical regime,[30], have been demonstrated. A Γ convergence result was established in [31] foran FCH type energy in a weak scaling, p “
2, with the important distinction that η “ η ă W are equal depth, W p b ´ q “ W p b ` q . Their signconvention adds a positive Cahn-Hilliard term to the positive quadratic Willmoretype term while the equal-depth wells allow for heteroclinic connections (fronts)as a basis for minimizers, as opposed to the homoclinic connections (bilayers) con-sidered in the present work. The authors recover a Canham-Helfrich type limit ofsurface area plus surface integral of curvature squared. The studies [11, 12, 13]considered phase field asymptotic expansion and numerical implementations of thismodel. They view the Cahn-Hilliard term as a Lagrange multiplier that constrainsthe total surface area. Restoring the sign convention to the normal (1.1) and break-ing the equal depth assumption on W makes a classical Γ-convergence result from(1.1) to (1.11) unlikely as even though the FCH can be uniformly bounded frombelow, there exist energy bounded sequences that are not related to codimensionone interfaces. Indeed, energy bounded sequences can converge to higher codimen-sional structures with potentially lower limiting energy than similar mass sequencesconverging to codimension-one structures, see [16].(a) t “ t “
367 (c) t “
930 (d) t “ EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Figure 1) Numerical simulation of the strong FCH mass preserving L gradientflow on Ω “ r´ π, π s from a circular equilibrium perturbed by addition of aspatial constant that raises the bulk density to u “ εσ ˚ { W p b ´ q , double theequilibrium value. Left to right, color coded contours of the evolving interface atindicated times, show a meandering transient followed by relaxation to a circularequilibrium with larger radius. System parameters are ε “ . η “ .
45 and η “ . . Main results.
We consider the mass-preserving L gradient flow of the strongscaling of the FCH free energy, (1.2)-(1.3), in two space dimensions subject to pe-riodic boundary condition on a periodic domain Ω “ r´ L, L s and suitable initialcondition. Our main result is a rigorous derivation of the sharp interface evolutionof approximately circular codimension-one interfaces. This is a step toward a val-idation of the formal results obtained in [4], where the authors applied multiscaleanalysis to the H ´ gradient flow of the strong FCH free energy. They showedthat on the ε ´ time scale the curvature κ of a codimension one interface Γ evolvesunder the H ´ gradient flow according to(1.14) B t κ “ ´p ∆ s ` κ q V, where the ε -scaled normal velocity(1.15) V “ ν p σ p t q ´ σ ˚ q κ ` εk b ∆ s κ, is proportional to curvature with a time-dependent coefficient that can be positiveor negative depending upon the initial data. Here, ν and k b are fixed positiveconstants and ∆ s is the Laplace-Beltrami operator associated to Γ. The spatiallyconstant bulk density σ “ σ p t q is determined by conservation of mass, and thecritical value σ ˚ , is given in (1.8). When the bulk density is above this critical value, σ ´ σ ˚ ą
0, the interface absorbs mass from the bulk, and moves against curvature,in a meandering or buckling motion that is regularized by the higher order diffusion, k b ∆ s . This regime is called regularized curve lengthening (RCL), and the lowerorder diffusion plays an essential role in the local existence. Conversely, when σ ´ σ ˚ ă
0, the interface releases mass to the bulk and contracts under a meancurvature driven flow (MCF). In both cases conservation of total mass drives σ toequilibrium. We focus on the RCL regime, as the MCF has been studied by manyauthors, in particular we point to the seminal work [23] by Huisken and the levelset approach developed in [17].In the absence of a maximum principle for the fourth-order system (1.3), ourapproach is based on detailed energy estimates and modulation methods. A signif-icant complication arises from the fact that the number of weakly damped modeswithin the system scales with ε ´ in R . This requires control of a large numberof modulational parameters which form a de facto Galerkin approximation for theone dimensional PDE that underlies the interfacial motion. A key step in the anal-ysis arises from the preconditioning of these modes through an implicitly definedparameterization of the evolving interface.We consider initial data of the form(1.16) u “ Φ p x ; Γ , σ p qq ` v , YUAN CHEN AND KEITH PROMISLOW where Φ is defined through (3.10) with radial initial interface Γ and bulk density σ p q , while v is a perturbation with negligible mass satisfying(1.17) } v } L p Ω q ` ε } v } H p Ω q À ε { . We show that the interface evolution Γ p t q can be described through a finite collec-tion of parameters p p t q . We write the unperturbed and perturbed interfaces as Γ and Γ p and represent them as images of maps γ and γ p that are related througha Galerkin-type expansion(1.18) γ p p s q : “ γ p s q ` ? πR p p E ` p E q ` ˜ p ˜Θ ` N ´ ÿ i “ p i ˜Θ i p ˜ s q ¸ n p s q , where p “ p p , p , . . . , p N ´ q are time dependent “meander-mode” parameters, t E , E u are the canonical unit vectors in R , n is the normal to Γ , t ˜Θ i u i ě arethe Laplace-Beltrami eigenmodes of the perturbed interface . In particular s denotesarc length along the interface Γ while ˜ s “ ˜ s p s, p q measures arc length along Γ p tothe unique point on Γ p that lies along the normal n p s q from the (p , p q translateof γ p s q . Fixing N „ ρ { ε ´ sufficiently large, with the spectral cut-off ρ ą ε , we dynamically decompose the solution u as(1.19) u “ Φ p p x ; σ q ` v K , ż Ω v K “ , where the p -perturbed profile Φ p is defined in (3.10) of Lemma 3.2, v K is orthogonalto the space Z ˚ introduced in (4.54), that approximates the first N meandereigenmodes of the linearization of F about Φ p , and σ P R is the time dependentbulk density parameter determined by p , see Corollary 3.6.In addition to the meander modes, there is a large but finite-dimensional pearlingslow space Z ˚ defined in (4.54) whose modes may undergo weak damping or slowgrowth. We decompose v K as(1.20) v K “ Q p q q ` w, w P Z K˚ , where Q “ Q p q q is the projection of v K onto Z ˚ and q P R N are time dependent”pearling-mode” parameters. We impose the condition(1.21) η ˆ λ ´ S m m ˙ ă η ˆ λ ` S m m ˙ , on the amphiphilic structure parameters η ą η P R , which guarantees thatbilayers Φ p p σ q with bulk density parameter σ sufficiently enough to σ ˚ are pearlingstable. Here λ ă φ , and S defined in (4.19) depends only upon the particular choice of double well, W . A detailed investigation of the onset of the pearling instability was conductedin [10, 28].We show the meander-mode parameters p evolve according to the Galerkin pro-jection of the curvature driven motion of the perturbed interface Γ p with curvature κ p given by(1.22) B t κ p “ ´p ∆ s p ` κ p q V p , with unscaled normal velocity(1.23) V p “ ε m m p σ ´ σ ˚ q κ p ` ε W p κ p q . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Here m , m ą W p κ p q is the Willmore expressionthat takes the form(1.24) W p κ p q “ ∆ s p κ p ` κ p ` ακ p ` R , where the spatial independent parameter α “ α p σ q is defined in (3.14) and Rincorporates the remaining terms that are relatively small in L p d s p q . Our analysisapproximates the geometric flow (1.22) by a partially uncoupled system for themeander-mode parameters, which we further decompose as p “ p p , p , p , ˆ p q withˆ p : “ p p , . . . , p N ´ q . Specifically, p controls the radius of the underlying circleand satisfies(1.25) p “ ´ ε c R p p ´ p ˚ q ` O ´ ε | ln ε | ´ | p ´ p ˚ | , ε { } ˆ p } V ¯ ` O ` ε ρ ´ ` } v K } L ` ε } v K } H ˘˘ relaxing to an equilibrium value p ˚ determined by the total system mass, (1.33).The key decay rate c is a time independent constant given by (3.32). The V normsof p are introduced in Definition 2.2. The translational modes, p and p uncoupleand move at most negligibly(1.26) p k “ O ´ ε | p ´ p ˚ |} ˆ p } V , ε ρ } ˆ p } V , ε ρ ´ ` } v K } L ` ε } v K } H ˘¯ , k “ , . The remainder of the meander-modes, ˆ p , control the deviation of the interface fromcircularity and satisfy a coupled system that is transiently forced by the motion ofthe radial parameter, p , (1.27) ˆ p “ ´ ε „ c Θ R p p ´ p ˚ q ` D ` I ` U T ˘ ` εR D ˆ p ` O ˆ ε | ln ε | | p ´ p ˚ |} D ˆ p } l ˙ ` O ` ε ρ } D ˆ p } l , ε ρ ´ ` } v K } L ` ε } v K } H ˘˘ . Here D is an p N ´ q ˆ p N ´ q diagonal matrix with entries(1.28) D kk “ β k ´ , k “ , . . . , N ´ , and } D k ˆ p } l „ } ˆ p } V k . The p N ´ q ˆ p N ´ q matrix U is given in (5.72) anddiscussed in greater detail in Corollary 5.7. In particular, its p l, k q componentsadmit the leading order expansion(1.29) U lk „ ż I s Θ l Θ k d s, l, k “ , ¨ ¨ ¨ , N ´ , where I “ r , | Γ |s , s is arc-length and Θ j are the associated Laplace-Beltramieigenmodes. Theorem 1.1.
Let ε ą be sufficiently small. Consider the strong FCH gra-dient flow (1.2) - (1.3) with p “ subject to periodic boundary condition and withparameters satisfying the pearling stability condition (1.21) . Then there exists achoice of spectral cut-off ρ ą in (4.5) and smooth function σ “ σ p p q , defined inCorollary 3.6, such that for all ε P p , ε s and all initial data u p q “ u in the formof (1.16) satisfying (1.17) with bulk density σ p q satisfying (1.30) | σ p q ´ σ ˚ | ! ε { p| ln ε |q { , YUAN CHEN AND KEITH PROMISLOW the solution u can be decomposed as (1.19) - (1.20) for all t ě . Moreover, theseexists ν ą independent of ε , such that (1.31) } u ´ Φ p } L p Ω q ` ε } u ´ Φ p } H p Ω q À min ! ε { ρ ´ , ερ ´ e ´ νε t { ) , while the meander parameters, p “ p p t q , given by (1.25) - (1.27) satisfy the estimates (1.32) } ˆ p } V ` ε } ˆ p } V À ε { e ´ νε t { , ż e νε t } ˆ p p t q} V d t À ε ´ . In particular the center coordinates p p , p q converge to an equilbrium p p ˚ , p ˚ q alonga transient path with total length bounded by ? ε | ln ε | . Equation (1.27) is the heart of the result. It encodes the competition betweenthe transient evolution arising from the relaxation of p to its equilibrium value,and the dissipation arising from the Willmore expression p β k ´ q{ R ą
0. Thetransient terms, multiplied by p ´ p ˚ , are initially dominant for moderate valuesof k , but decay to lower order at the end of transient period demarcated at t “ T „ ε ´ . Indeed, since the Laplace-Beltrami eigenvalues t β k u satisfy β “ β “ β k „ k for large k , we see that the D Willmore term dominates for large k ě ε ´ for all time, damping out high wave numbers, and dominates for moderate valuesof k after the decay of the transient. The coupling matrix U contributes at leadingorder to the transients and reflects the impact of the change of interfacial lengthon the representation of the interfacial shape in the associated Laplace-Beltramimodes. The Laplace-Beltrami and cubic curvature terms in the leading Willmoreexpression (1.24) contribute at leading order to the regularizing terms in (1.27),interestingly the ακ p term cancels with a corresponding term in the second orderexpansion of the equilibrium bulk density, σ ˚ , see (5.107) and (5.115).With initial data in the form (1.16), a key restriction is that the added massmust be small in the scaled variables, see (1.30). Conservation of mass links theequilibrium value of the radial perturbation parameter to the initial perturbationof the bulk density(1.33) p ˚ ´ p p q „ c R π ¯ B p σ p q ´ σ ˚ q , where the bulk mass parameter ¯ B is defined in (3.3). The constraint (1.17) yieldsthe initial non-circularity bound(1.34) } ˆ p p q} V ! ε. Estimates on } ˆ p } V are required to prevent self-intersection of the curve Γ p and tocontrol the error terms in (1.25)-(1.27). Indeed, the estimate (5.162) and (5.168)afford a maximal value of } ˆ p p t q} V {} ˆ p p q} V that depends exponentially upon an ε scaling of the mass perturbation,(1.35) } ˆ p p T q} V „ e νε | p ˚ ´ p p q| T } ˆ p p q} V , where T „ ε ´ } ln ε | { denotes the end of the transient growth period r , T s .To keep } ˆ p p T q} V ! ε { we impose the condition | p ˚ ´ p p q| ! | ε ln ε | { , thattranslates through (1.33) to (1.30). The result permits radial expansions that are O p? ε q in the original coordinates. This is large compared to the O p ε q width of thebilayer, but small compared to the O p q domain size. Theorem 1 can accommodate EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW larger radial expansion if the initial non-circular perturbation, as measured by } ˆ p p q} V is exponentially small. This is precisely the situation presented in Figure1 in which the initial data is a perfectly circular bilayer equilibrium perturbed onlyby a spatially constant offset and computational noise.It’s natural to compare these results with interface dynamics results derived forthe Cahn-Hilliard equation. For the Cahn-Hilliard, much of the initial work, no-tably [29] and [1], focused on formal and rigorous derivations of the Mullins-Sekerkaflow in the ε Ñ R . Extension of these results to incorporate interactionof radial droplets with a domain boundary were considered by [5] and others. TheFCH gradient flows differs from Cahn-Hilliard flows in that it conserves the totalvolume of the interfacial material, rather than of the materials that comprise thebulk regions. Thus, while it is natural to consider ε small in the FCH system, thelimit ε Ñ O p ε q , (1.6). On a technical level, the large number of unstable or weaklystable modes present in the FCH system presents challenges, as is reflected in thesingularly perturbed nature of the RCL system. The Cahn-Hilliard equations en-joy structure-structure interactions compatible with the Lifschitz-Slyosov-Wagner(LSW) theory of coarsening introduced in [26] and [33] which reduces the interac-tions of finite numbers of near-circular interfaces to systems of ODEs for the radiiand center positions. Formal reductions similar to LSW theory for the FCH, in-cluding interactions of constant curvature codimension one and two structures werederived in [15].The remainder of this article is organized as follows. In Section 2, we present thelocal coordinates and estimates on the variation of the interface. In Section 3 weconstruct the approximate profile near the perturbed interface Γ p and extend it tothe whole domain. In Section 4 we present the spectrum and establish the coercivityof the linearized operator Π L p . The analysis of the nonlinear flow and derivationof the regularized curve lengthening and shortening are derived in Section 5.1.3. Notation.
We present some general notation.(1) The symbol C generically denotes a positive constant whose value dependsonly on the system parameters η , η , the domain Ω, and radius of initialcircle R . In particular its value is independent of ε and ρ , so long as theyare sufficiently small. The value of C may vary line to line without remark.In addition, A À B indicates that quantity A is less than quantity B upto a multiplicative constant C as above. The notation A ^ B denotes theminimum of A and B . The expression f “ O p a q indicates the existence ofa constant C , as above, and a norm | ¨ | for which | f | ď C | a | . (2) The quantity ν is a positive number, independent of ε , that denote anexponential decay rate. It may vary from line to line.(3) If a function space X p Ω q is comprised of functions defined on the wholespatial domain Ω, we will drop the symbol Ω. (4) We use E as the characteristic function of an index set E Ă N , i.e. E p x q “ x P E ; E p x q “ x R E . We denote the usual Kronecker delta by δ ij “ " , i “ j , i ‰ j (5) For a vector q “ p q j q j , and matrix Q “ p Q ij q ij we denote the norms } q } l k “ ˜ÿ j | q j | k ¸ { k , and } Q } l k “ ˜ÿ i,j | Q ij | k ¸ { k , for k P N ` , and } q } l “ max j | q j | . We writeq j “ O p a q e j , Q ij “ O p a q E ij , where e “ p e j q j is a vector with } e } l “ E is a matrix with operatornorm } E } l ˚ “ } q } l “ O p a q or } Q } l ˚ “ O p a q respectively.See (2.58)-(2.59) of Notation 2.9 for usage.(6) The matrix e θ R denotes rotation through the angle θ with the generator R . More explicitly, R “ ˆ ´
11 0 ˙ , e θ R “ ˆ cos θ ´ sin θ sin θ cos θ ˙ . Coordinates and preliminary estimates
In this section we introduce the local whiskered coordinates near the perturbedinterfaces and establish bounds connecting the variation of the interface to theparameters p .2.1. The whiskered coordinates.
We consider a smooth curve Γ Ă R whichdivides Ω “ Ω ` Y Ω ´ into an exterior Ω ` and an interior Ω ´ . The interface Γ isgiven parametrically as Γ “ (cid:32) γ p s q : s P I Ă R ( , with tangent vector T p s q P R (2.1) T p s q : “ γ {| γ | . Denoting the outer normal to Γ by n p s q then we have the relations(2.2) T “ κ n , n “ ´ κ T , where κ “ κ p s q is the curvature of Γ at γ p s q . By the implicit function theorem,there exists an open set N containing Γ such that for each x P N we may write(2.3) x “ γ p s q ` r n p s q for some r “ r p x q and s p x q . In this neighborhood, we define the scaled signed distance z “ r { ε and ”whiskers”of length (cid:96) : w (cid:96) p s q : “ (cid:32) γ p s q ` εz n p s q : r P r´ (cid:96), (cid:96) s ( . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Dressings.
For a fixed value of (cid:96) ą (cid:96) intersect each other nor with B Ω. We introducethe (cid:96) -reach of Γ, sometimes denoted the reach of Γ,(2.4) Γ (cid:96) “ ď s P I w (cid:96) p s q . Definition 2.1 (Dressing) . Given an surface Γ which is far from self intersectionwith length (cid:96) and a function f p z q : R Ñ R which tends to a constant f and whosederivatives tend to zero at an ε independent exponential rate as z Ñ ˘8 , then wedefine the dressed function, f p x q , of f with respect to Γ as f p x q “ f p z p x qq χ p ε | z p x q|{ (cid:96) q ` f p ´ χ p ε | z p x q|{ (cid:96) qq . Here χ : R Ñ R is a smooth cut-off function satisfying: χ p r q “ if r ď and χ p r q “ if r ě . From this definition the dressed function satisfies f p x q “ f p z p x qq , if | z p x q| ď (cid:96) { ε ; f , if | z p x q| ě (cid:96) { ε. Definition 2.2 (Dressed operator) . Let
L : D Ă L p R q ÞÑ L p R q be a self-adjointdifferential operator with smooth coefficients whose derivatives of all order decay tozero at an exponential rate at . We define the space D to consist of the functions f as in Definition 2.1, and the dressed operator L d : D X D ÞÑ L p Ω q and its r ’thpower, r P N , (2.5) L rd f : “ p L r f q d . If r ă then we assume that f P R p L q and the inverse L ´ f decays exponentiallyto a constant at ˘8 . A function f “ f p x q P L p Ω q is said to be localized near the interface Γ if thereexists ν ą x P Γ (cid:96) , | f p x p s, z qq| À e ´ ν | z | . The Jacobian.
Let J p s, z q be the Jacobian matrix with respect to change ofvariables, i.e. x ÞÑ p s, z q , and the Jacobian J p s, z q “ det J p s, z q . In two dimension, n (cid:107) γ and then J “ ´ γ p s q ` εz n p s q ε n p s q ¯ “ ´ γ p s q n p s q ¯ ˆ ´ εzκ p s q ε ˙ , where κ p s q “ ´ γ p s q ¨ n p s q| γ p s q| . If we define ˜J and J as(2.6) ˜J p s, z q “ ε p ´ εzκ p s qq , J p s q “ | γ p s q| , then the Jacobian is decomposed as: J p s, z q “ ˜J p s, z q J p s q . We also denote by G “ J T J the metric tensor. The reader is referred to the appendix of [24] for moredetails.Let f, g P L p Ω q with support in Γ (cid:96) , we introduce the inner product by changeof variable(2.7) (cid:104) f, g (cid:105) L “ ż R (cid:96) ż I f p s, z q g p s, z q J p s, z q d s d z “ ż I (cid:104) f, g (cid:105) ˜J J p s q d s, where we have introduced the whisker weighted inner product(2.8) (cid:104) f, g (cid:105) ˜J p s q : “ ż R (cid:96) f p s, z q g p s, z q ˜J p s, z q d z, and R a : “ r´ a, a s for a P R ` . The J weighted inner product over I is denoted by(2.9) (cid:104) f, g (cid:105) J “ ż I f p s, z q g p s, z q J p s q d s. Moreover, if f is localized near the interface Γ, then ż Ω f d x “ ż R (cid:96) ż I f p x p s, z qq J d s d z ` O p e ´ ν(cid:96) { ε q . Laplacian.
The ε -scaled Laplacian can be expressed in the local whiskeredcoordinates of Γ as(2.10) ε ∆ x “ J ´ B z p J B z q ` ε ∆ g “ B z ` ε H B z ` ε ∆ g . Here H is the extended curvature(2.11) H p s, z q : “ ´ κ p s q ´ εzκ p s q “ B z J ε J , and ∆ g is the induced Laplacian under metric tensor G , which can be decomposedinto∆ g : “ ? det G B s p G ? det G B s q “ ∆ s ` εzD s, , with G “ ˆ | γ ` εz n | ε ˙ . Here G is the p , q -component of the matrix G ´ , the inverse of G ; ∆ s is theLaplace-Beltrami operator on the surface Γ and D s, is a relatively bounded per-turbation of ∆ s . In particular, since | γ ` εz n | “ | γ || ´ εzκ | , we have(2.12) ∆ s “ | γ p s q| B s ˆ | γ p s q| B s ˙ , D s, “ a p s, z q ∆ s ` b p s, z qB s , in which coefficients a, b are given explicitly by(2.13) a p s, z q “ | ´ εzκ | ´ , b p s, z q “ | γ | B s a p s, z q , and hence obey(2.14) max i,j ` | ∇ ks B lz a | ` | ∇ k ´ s B lz b | ˘ ď Cε l ÿ i ď k | z i B is κ | . Perturbed interfaces.
In the sequel we consider the case that Γ is a circlewith radius R ą
0, so that the curvature is a constant(2.15) κ “ ´ { R , and the whiskered coordinates are defined on the whole plane minus the center ofthe circle. Let t β i u be scaled eigenvalues of the Laplace-Beltrami operator ´ ∆ s : H p I q Ñ L p I q associated with eigenfunctions t Θ i u , more explicitly,(2.16) ´ ∆ s Θ i “ β i Θ i L R . The ground state eigenmode is spatially constant:(2.17) Θ “ { a πR , β “ EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW and for k “ , ¨ ¨ ¨ ,(2.18)Θ k ´ “ ? πR cos ˆ ksR ˙ , Θ k “ ? πR sin ˆ ksR ˙ ; and β k ´ “ β k “ k. Here we have normalized Θ i in L p I q . To control the smoothness of the perturbedinterface we introduce the weighted perturbation space V kr p N q of parameters p ,and perturbed interface Γ p , a p -variation of Γ . Definition 2.3 (Perturbed interfaces) . (a) Given N , r ą , we say an N -vector p “ p p , ¨ ¨ ¨ , p N ´ q T P R N lies in the weighted space V kr “ V kr p N q if } p } k V kr p N q : “ N ´ ÿ j “ β krj | p j | k ă 8 . By abuse of notation we apply the same norm to ˆ p , starting the sum with j “ . When k “ , we omit the superscript k and denote the space by V r .(b) For each p P V , we define the p -variation of Γ , denoted by Γ p , as (2.19) Γ p : “ ! γ p : “ γ p , p s q ` ¯ p p ˜ s q n p s q , s P I ) . Here the leading order involves scaling and translations along x , x directions,which is given by (2.20) γ p , : “ γ p s q ` p Θ n p s q ` p p E ` p E q Θ , vector n p s q is the outer normal vector of the circle Γ parameterized by s and (2.21) ¯ p p ˜ s q : “ N ´ ÿ i “ p i ˜Θ i p ˜ s q , ˜Θ i p ˜ s q : “ Θ i ˆ πR ˜ s | Γ p | ˙ , where ˜ s “ ˜ s p s q P I p “ r , | Γ p |s is the arc length parametrization of γ p obtainedby solving (2.22) d˜ s d s “ | γ p | , ˜ s p q “ . The definition of Γ p is implicit in p through (2.19) and (2.21), we show thatthis is well posed in Lemma 2.5. The p Θ -term scales the circle from radius R to p R ` p Θ q , where Θ is the constant ground state Laplace-Beltrami eigenmodedefined in (2.17). The parameters p and p serve to translate the interface, and arethe only terms that are not normal to the original interface. However the projectionof these two terms onto n satisfies(2.23) Θ E ¨ n “ ? πR cos sR “ Θ , Θ E ¨ n “ ? πR sin sR “ Θ . We introduce the notation(2.24) ˆ p : “ p p , ¨ ¨ ¨ , p N ´ q T , for the meander-modes that control the distortion of Γ p from circularity. Theseterms contribute to ¯ p . The following embeddings are direct results of H¨older’sinequality and the asymptotic form of β j introduced in (2.18), details are omitted. Lemma 2.4.
Suppose that ˆ p P V with N ď ε ´ , then } ˆ p } V ` } ˆ p } V À } ˆ p } V , } ˆ p } V r ď N { } ˆ p } V r , } ˆ p } V À ε ´ } ˆ p } V À ε ´ } ˆ p } V . In addition, for any vector a P l p R m q we have the dimension dependent bound (2.25) } a } l ď m { } a } l . To justify the implicit construction of Γ p we impose the following assumptionswhich hold throughout the sequel:(2.26) N ď ε ´ , } ˆ p } V À , } ˆ p } V À } ˆ p } V ! , | p | ! . Lemma 2.5.
Suppose assumptions in (2.26) hold, then the system (2.22) combinedwith (2.19) has a unique solution and the interface Γ p is well defined.Proof. Using the definition of γ p in (2.19), taking derivative with respect to s yields(2.27) γ p “ γ ` p Θ n p s q ` d˜ s d s ¯ p p ˜ s q n p s q ` ¯ p p ˜ s q n p s q . Here and below, primes of ¯ p denote derivatives with respect to ˜ s . Since Γ is acircle, its outer normal vector can be written explicitly as below(2.28) n p s q “ ˆ cos sR , sin sR ˙ “ R γ p s q , n p s q “ R γ p s q , which combined with (2.27) and (2.22) implies(2.29) γ p “ ˆ ` p Θ R ` ¯ pR ˙ γ ` ¯ p | γ p | n p s q . Since γ lies in the tangent space while n is the outer normal, γ ¨ n “
0. Conse-quently, taking the absolute value of the identity, squaring and solving for | γ p | , weobtain(2.30) | γ p | “ a ´ p ¯ p q ˆ ` p Θ R ` ¯ pR ˙ . Taking derivative with respect to ˜ s , we deduce(2.31) B ˜ s | γ p | “ ¯ p R a ´ p ¯ p q ` ˆ ` p Θ R ` ¯ pR ˙ ¯ p ¯ p p ´ p ¯ p q q { , which implies(2.32) ˇˇ B ˜ s | γ p | ˇˇ À } ˆ p } V p ` } ˆ p } V q . Hence by classical ODE theory, the system (2.22) is solvable provided that } ˆ p } V is bounded and } ˆ p } V is sufficiently small. (cid:3) It follows from (2.16) and (2.21) that the scaled ˜Θ j satisfy(2.33) ´ ˜Θ j “ β p ,j ˜Θ j , β p ,j “ πβ j {| Γ p | , and functions in t ˜Θ j u are orthogonal to each other on L p d˜ s q , in particular,(2.34) ż I p ˜Θ j ˜Θ k | γ p | d s “ ż I p ˜Θ j ˜Θ k d˜ s “ | Γ p | πR δ jk . We remark that β p ,j equals β j { R when p “ , rather than β j . This difference innormalization simplifies our presentation. EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW The following Lemma estimates the length of Γ p and it’s variation with respectto p j . Lemma 2.6.
Under assumptions in (2.26) , the length of Γ p admits approximation: (2.35) | Γ p | “ π p R ` p Θ q ` O ´ } ˆ p } V ¯ . Moreover, there exists a unit vector e “ p e j q N ´ j “ such that (2.36) B p ln | Γ p | “ Θ R ` p Θ ` O ´ } ˆ p } V ¯ , B p j ln | Γ p | “ O ´ } ˆ p } V ¯ e j t j ě u . Proof.
The length of Γ p is given by(2.37) | Γ p | : “ ż I | γ p | d s. First, using Taylor expansion one can derive from (2.30) that(2.38) | γ p | “ ` p Θ R ` ¯ p p ˜ s q R ` O ` } ˆ p } V ˘ . In addition, from (2.22) we have d˜ s “ | γ p | d s , applying (2.38) implies(2.39) ż I ¯ p p ˜ s q d s “ ż I p ¯ p p ˜ s q| γ p | d˜ s “ O ` } ˆ p } V ˘ . The approximation (2.35) then follows from Lemma 2.4 which affords the estimate } ˆ p } V À } ˆ p } V .Returning to (2.37) with | γ p | given in (2.30), we take the derivative with respectto p j for j ě
3, arriving at(2.40) B p j | Γ p | “ ż I a ´ p ¯ p q B p j ¯ pR d s ´ ż I ¯ p B p j ¯ p p ´ p ¯ p qq { ˆ ` p Θ R ` ¯ pR ˙ d s. The second integral is higher order. Indeed, by the definition of ¯ p and definition of˜Θ l in (2.21) we have(2.41) B p j ¯ p “ ˜Θ j t j ě u ´ B p j | Γ p || Γ p | N ´ ÿ l “ p l ˜ s ˜Θ l ; B p j ¯ p “ ˜Θ j t j ě u ´ B p j | Γ p || Γ p | N ´ ÿ l “ p l ˜ s ˜Θ l . Plugging (2.41) into the right hand side of (2.40), and applying Lemma 2.9 andTaylor expansion we obtain(2.42) B p j | Γ p | “ R ż ˜Θ j | γ p | d˜ s ` O ´ } ˆ p } V ¯ ` e j ` |B p j | Γ p || ˘ . Observe the mass of ˜Θ j for j ě L p d˜ s q through (2.34). The first integralabove is order of } ˆ p } V by (2.38). This finishes the proof of the second estimatein (2.36) for j ě
3. Since the mass of ˜Θ j in L p d˜ s q is not zero when j “
0, thederivation that leads to (2.42) produces an extra term,(2.43) B p | Γ p | “ π Θ ` O ´ } ˆ p } V ¯ . Extracting the leading order from the last term on the right hand-side, the estimateof B p p ln | Γ p |q follows from the approximation of | Γ p | given in (2.35). Since rigidtranslation does not change length, | Γ p | is independent of p , p . (cid:3) From the definition of β p ,j given in (2.33) and the results of Lemma 2.6 we havethe estimate(2.44) β p ,j “ ˆ R ` p Θ ` O ´ } ˆ p } V ¯˙ β j . The following Lemma gives an approximate form of the curvature of Γ p . Lemma 2.7 (Curvature of Γ p ) . Suppose assumptions (2.26) hold, then the curva-ture of Γ p , defined by (2.45) κ p : “ γ p ¨ n p {| γ p | with n p “ e ´ π R { γ p L | γ p | . admits the following approximation (2.46) κ p p s q “ κ p , ` ` O p} ˆ p } V qp R ` p Θ q N ´ ÿ j “ p ´ β j q p j ˜Θ j ` O ` } ˆ p } V ˘ , κ p , “ ´ R ` p Θ . Here κ p , n p depend only on p and ˆ p , and satisfy the following bounds | κ p | À ` } ˆ p } V ; | n p ´ n | À } ˆ p } V . Proof.
Combining identities (2.29) and (2.38) indicates that | n p ´ n | “ O p} ˆ p } V q by the definition of n p in (2.45). To obtain an approximation of the curvature κ p we take one more derivative of (2.29), using γ “ ´ n p s q{ R and (2.28) we derive(2.47) γ p “ ´ R ` p Θ R n ´ R N ´ ÿ j “ p j ˜Θ j n ` N ´ ÿ j “ p j ˜Θ j | γ p | n ` N ´ ÿ j “ p j ˜Θ j | γ p | n ` R N ´ ÿ j “ p j ˜Θ j | γ p | γ “ ´ R ` p Θ R n ´ R N ´ ÿ j “ ` ` β j ˘ p j ˜Θ j n ` R N ´ ÿ j “ p j ˜Θ j | γ p | γ ` O ` } ˆ p } V ˘ . Here we also employed the identity (2.33), the approximation of β p ,j in (2.44), andapproximation of | γ p | in (2.38). Since e ´ π R { γ “ n and e ´ π R { n “ ´ γ , theestimate (2.29) implies(2.48) e ´ π R { γ p “ R ` p Θ R n ` R N ´ ÿ j “ p j ˜Θ j n ´ N ´ ÿ j ě p j ˜Θ j | γ p | γ , and hence(2.49) e ´ π R { γ p ¨ γ p “ ´ R « p R ` p Θ q ´ ´ R ` p Θ ` O p} ˆ p } V q ¯ ˆ N ´ ÿ j “ p ` β j q p j ˜Θ j ff ` O ` } ˆ p } V ˘ . In addition, from (2.38), we see that(2.50) 1 | γ p | “ ˜ ´ N ´ ÿ j “ p j ˜Θ j R ` p Θ ¸ R | R ` p Θ | ` O ` } ˆ p } V ˘ . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW The approximation (2.46) follows from applying (2.49) and (2.50) in the definitionof κ p given in (2.45). (cid:3) In what follows we introduce p s p , z p q , the whiskered coordinates correspondingto the perturbed interface Γ p with whiskered length 2 (cid:96) . On the interface Γ p ,where z p “
0, we have the relation s p “ s . These coordinates change with theparameters p following the relations given in Lemma 2.10. Moreover, the geometricquantities like n p , γ p and κ p on Γ p have natural extensions to the reach of Γ p . Inthe sequel all whiskered coordinates refer to p s p , z p q . And ˜ s p denotes arc-lengthreparameterization of s p . Notation 2.1.
In order to streamline the presentation of the calculations, variousterms will be denoted by h ´ z p , γ p k q p ¯ where h is a smooth function of z p and s p thatdepends upon the first k derivatives of γ p and possesses the following properties:(1) h “ h ´ z p , γ p k q p ¯ is defined in the (cid:96) -reach Γ (cid:96) p ,(2) Recalling the notation (2.20), we assume the decomposition of h (2.51) h ´ z p , γ p k q p ¯ “ h ´ z p , γ p k q p , ¯ ` ” h ´ z p , γ p k q p ¯ ´ h ´ z p , γ p k q p , ¯ı , has leading order term h ´ z p , γ p k q p , ¯ independent of s p ; moreover, if h isalso independent of z p , then h ´ γ p k q p , ¯ is a constant depending only on p .(3) If h as above is independent of z p , then we denote it by h ´ γ p k q p ¯ , andfurther assume that the decomposition (2.51) obeys (2.52) } ε l B ls p ´ h ´ γ p k q p ¯ ´ h ´ γ p k q p , ¯¯ } L p d s p q À } ˆ p } V k , and for l ě , (2.53) ››› ε l ´ B ls p ´ h ´ γ p k q p ¯ ´ h ´ γ p k q p , ¯¯››› L p d s p q À } ˆ p } V k ` , as long as | γ p k q p | À . Lemma 2.8.
If function f “ f p z p , s p q defined in the (cid:96) -reach Γ (cid:96) p has s p depen-dence only through | γ p | , κ p , n p ¨ n and ε k B ks p derivatives of these, then there exists h “ h p z p , γ p q in the sense of Notation 2.1 such that (2.54) f p z p , s p q “ h p z p , γ p q . Moreover, if f p z p , s p q is independent of z p , then h “ h p γ p q and under assumption (2.26) we have (2.55) } h p γ p q ´ h p γ p , q} L p d s p q À } ˆ p } V , and for l ě , (2.56) ››› ε l ´ B ls p ` h p γ p q ´ h p γ p , ˘››› L p d s p q À } ˆ p } V . Proof.
Estimates in (2.52), and hence (2.55)-(2.56), can be derived directly bythe approximations of | γ p | , κ p and n p in (2.38), (2.46). We only need to verify h p z p , γ p , q is independent of s p . This is true since | γ p , | “ p R ` p Θ q{ R by(2.30), κ p , admits form in (2.46) and(2.57) n ¨ n p , “ , for n p , “ e ´ π R { γ p , | γ p , | . Here we also used (2.48) by taking p j “ j ě (cid:3) Lemma 2.9.
Recalling the notation of section 1.3, if f P L p d˜ s p q , then there existsa unit vector e “ p e i q such that (2.58) ż I p f ˜Θ i d˜ s p “ O p} f } L p d˜ s p q q e i . If in addition f P L on I p , then for any vector a “ p a j q P l , we have (2.59) ˇˇˇˇˇÿ j ż I p f ˜Θ i a j ˜Θ j d˜ s p ˇˇˇˇˇ À } a } l } f } L e i , and there exists a matrix E “ p E ij q with l ÞÑ l norm one, such that (2.60) ż I p f ˜Θ i ˜Θ j d˜ s p “ O p} f } L q E ij . Proof.
The estimates follow from Plancherel and classic applications of Fouriertheory. (cid:3)
The following Lemma estimates the p -variation of the whiskered coordinatesassociated to Γ p . In particular it estimates the difference between p s p , z p q and p s, z q in terms of p . Lemma 2.10.
Let p s p , z p q be the whiskered coordinates subject to Γ p with whiskeredlength (cid:96) , then under assumption (2.26) for j “ or j P r , N ´ s there existfunctions h k “ h k p z p , γ p qp k “ , q in the sense of Notation 2.1 such that B s p B p j “ h ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ ` h ε ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ ; B z p B p j “ ´ ε ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ n ¨ n p ; while for j “ , , we have (2.61) B s p B p j “ h p z p , γ p q E j ¨ γ p ; B z p B p j “ ´ E j ¨ n p ε ? πR . Moreover, we have the estimate (2.62) | s p ´ s | À } p } V , | z p ´ z | ď ε ´ } p } V . Proof.
Any x P Γ (cid:96) p can be expressed in the whiskered coordinates of both Γ p andΓ . Equating these yields(2.63) γ p s q ` εz n p s q “ γ p p s p q ` εz p n p p s p q . Taking the derivative of (2.63) with respect to p j , the j -th component of the vector p , yields(2.64) 0 “ B γ p B p j p s p q ` γ p B s p B p j ` ε B z p B p j n p p s p q ` εz p B n p B p j p s p q ` εz p n p B s p B p j . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW The vectors γ p and n p are perpendicular to each other since γ p lies in the tangentspace while n p is the normal vector. Taking the dot product of (2.64) with γ p andrearranging, we obtain(2.65) B s p B p j “ p ´ εz p κ p q| γ p | ˆ ´ B γ p B p j ¨ γ p ` εz p B γ p B p j ¨ n p ˙ . Here we used that γ p ¨ n p is zero so that B p j p γ p ¨ n p q “
0, and definition of κ p givenin Lemma 2.7. Taking the dot product of identity (2.64) with n p p s p q and arguingas above we find(2.66) B z p B p j “ ´ ε B γ p B p j ¨ n p . From the definition of γ p in (2.19) and (2.21), for j ‰ , B γ p B p j “ ˜Θ j p s p q n p s p q ´ B p j | Γ p || Γ p | N ´ ÿ l “ ˜ s p p l ˜Θ l n p s p q ; B γ p B p j “ ˜ ˜Θ j p s p q ´ B p j | Γ p || Γ p | N ´ ÿ l “ ˜ s p p l ˜Θ l ¸ | γ p | n p s p q` ˜ Θ j p s p q ´ B p j | Γ p || Γ p | N ´ ÿ l “ ˜ s p p l ˜Θ l ¸ n p s p q . Since γ p is independent of p and p , we have B p j γ “ j “ ,
2. In addition,from (2.19) we have(2.68) B p j γ p “ E j { a πR , j “ , . The expressions for the derivatives of s p and z p with respect to p j follow by plugging(2.67) or (2.68) into (2.65)-(2.66). The estimates (2.62) follow directly from meanvalue theorem. (cid:3) This Lemma plays a key role in section 5.2. As a notational convenience we use˜Θ “ Θ . Lemma 2.11.
Recalling the notation of Section 1.3, if one of k or j equals zero,then the following results hold, (2.69) N ´ ÿ l “ p l ż I ˜Θ k ˜Θ j ˜Θ l d s p “ $’’’’&’’’’% p j Θ ` O ´ } ˆ p } V ¯ e j , if k “ , j ě k Θ ` O ´ } ˆ p } V ¯ e k , if j “ , k ě O ´ } ˆ p } V ¯ e , if k “ j “ . If both k and j are larger than , then we have (2.70) N ´ ÿ l “ p l ż I ˜Θ k ˜Θ j ˜Θ l d s p “ O ´ } ˆ p } V ¯ E jk . Proof.
Since d˜ s p “ | γ p | d s p by (2.22), we have(2.71) N ´ ÿ l “ p l ż I ˜Θ l ˜Θ j ˜Θ k d s p “ N ´ ÿ l “ p l ż I p ˜Θ l ˜Θ j ˜Θ k d˜ s p | γ p | . From the approximation of | γ p | given in (2.38), the right hand side of the identityabove yields the leading order approximation(2.72) N ´ ÿ l “ p l ż I ˜Θ l ˜Θ j ˜Θ k d s p “ R R ` p Θ N ´ ÿ l “ p l ż I p ˜Θ l ˜Θ j ˜Θ k d˜ s p ` O ´ } ˆ p } V ¯ E jk . If both k and j are larger than 1, the estimate (2.70) follows by Notation 2.9; if k “ j “
0, then (2.69) follows since the integration of ˜Θ l for l ě I p .This leaves the case k “ j “ k ` j ‰ . Without loss of generality weconsider j “ k ‰
0. Since ˜Θ “ Θ is constant we have(2.73) N ´ ÿ l “ p l ż I p ˜Θ l ˜Θ j ˜Θ k d˜ s p “ Θ N ´ ÿ l “ p l ż I p ˜Θ l ˜Θ k d˜ s p . Applying identity (2.34) with | Γ p | approximated by (2.35) to the right-hand side ofthe identity (2.73), using the result in the right-hand side of (2.72) completes thecase j “
0, and finishes the proof of the Lemma. (cid:3)
Lemma 2.12.
Under assumptions (2.26) , curvature of Γ p admits the followingprojection identities: (2.74) ż I p κ p ˜Θ k d˜ s p “ ´ c πR δ k ´ β k ´ R p R ` p Θ q p k t k ě u ` O ´ } ˆ p } V ¯ e k ; ż I p κ p ˜Θ k d˜ s p “ ´ p R ` p Θ q c πR δ k ´ p β k ´ q R p R ` p Θ q p k t k ě u ` O ´ } ˆ p } V } ˆ p } V ¯ e k ; ż I p ∆ s p κ p ˜Θ k d˜ s p “ p β k ´ q β k R p R ` p Θ q p k t k ě u ` O ´ } ˆ p } V } ˆ p } V ¯ e k t k ě u . Proof.
The expansion (2.46) of κ p in Lemma 2.7 can be written as:(2.75) κ p p s p q “ κ p , ` Q ` Q , κ p , : “ ´ R ` p Θ , where Q denotes terms linear in p and Q contains the terms quadratic or higherin p . More Explicitly,(2.76) Q “ p R ` p Θ q N ´ ÿ j “ ` ´ β j ˘ p j ˜Θ j ; Q “ O p} ˆ p } V qp R ` p Θ q N ´ ÿ j “ ` ´ β j ˘ p j ˜Θ j ` O ` } ˆ p } V ˘ . From Lemma 2.4 we have } ˆ p } V À } ˆ p } V , and the quadratic term Q contributes(2.77) ż I p Q ˜Θ k d˜ s p “ O ´ } ˆ p } V ¯ e k ; EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW while the projection of the linear term takes the form(2.78) ż I p Q ˜Θ k d˜ s p “ p ´ β k q p k R p R ` p Θ q t k ě u ` O ´ } ˆ p } V ¯ e k . Finally, the zero-order term of κ p , contributes(2.79) ż I p κ p , ˜Θ k d˜ s p “ ˆ ´ c πR ` O ´ } ˆ p } V ¯˙ δ k . Combining (2.75) with the identities (2.77)-(2.79) yields the first line of (2.74). Forthe second line of this equation, we expand(2.80) κ p “ κ p , ` κ p , Q ` ˜ Q , where ˜ Q denotes quadratic terms in p and under assumption (2.26) satsfies | ˜ Q | À| ¯ p | . Since κ p , is independent of s p , the second line of (2.74) follows from (2.78),(2.79), the definition of κ p , in (2.75), and H¨older’s inequality. For the third line,integration by parts and utilizing (2.33) imply(2.81) ż I p ∆ s p κ p ˜Θ k d˜ s p “ ´ β p ,k ż I p Q ˜Θ k d˜ s p ` ż I p ∆ s p Q ˜Θ k d˜ s p . The first term on the right-hand side is dominant, and can be estimated by (2.78)and (2.35). The second term on the right-hand side is higher order term, can bebounded by(2.82) ˇˇˇˇˇż I p ∆ s p Q ˜Θ k d˜ s p ˇˇˇˇˇ À } ˆ p } V } ˆ p } V e k . The result follows. (cid:3) Quasi-steady states
In this section we construct the family of quasi-equilibrium profiles Φ p param-eterized by the bulk density parameter, σ and the meander parameters p throughthe interface Γ p . The bulk density parameter is tuned to match the mass of Φ p tothe conserved system mass.3.1. Construction of Φ p . Given a circle Γ , the p -variation Γ p of Γ has a 2 (cid:96) -reach, Γ (cid:96) p , for some positive (cid:96) provided that p is sufficiently small, independent of ε . As introduced in section 2.1, the whiskered coordinates in Γ (cid:96) p are denoted by p s p , z p q . The construction of the local profile begins with φ defined on L p R q asthe nontrivial solution of(3.1) B z φ ´ W p φ q “ , that is homoclinic to the left well b ´ of W . In particular φ is unique up totranslation, even, and converges to b ´ as z tends to ˘8 at the exponential rate a W p b ´ q ą . The linearization L of (3.1) about φ ,(3.2) L : “ ´B z ` W p φ p z qq , is a Sturm-Liouville operator on the real line whose coefficients decay exponentiallyfast to constants at z “ 8 . The following Lemma follows from classic results anddirect calculations, see for example Chapter 2.3.2 of [25]. Lemma 3.1.
The spectrum of L is real, and uniformly positive except for twopoint spectra: λ ă and λ “ . Moreover, it holds that L φ “ , L φ “ ´ W p φ q ˇˇ φ ˇˇ , L ` zφ ˘ “ ´ φ . The ground state eigenfunction ψ of L is even and positive, with ground stateeigenvalue λ ă . The operator L has an inverse that is well defined on the L perp of its kernel, span t φ u , and both L and its inverse preserve evenness andoddness. Some care must be taken to distinguish between functions in L p R q and theirdressings that reside in L . To simplify notation we will drop the d superscript onthe dressed function and use L p to denote the dressed operator as introduced inDefinition 2.2). As an example, since 1 is orthogonal to φ we may define B k “ L ´ k , and its dressing subject to Γ p ,(3.3) B d p ,k p x q : “ p L ´ k q d “ L ´ k p , , defined on all of Ω. Recalling the mass function, (1.5) we introduce(3.4) ¯ B d p ,k : “ (cid:10) B d p ,k (cid:11) L . With this notation we suppress the d superscript and define the first dressed cor-rection φ to the pulse profile(3.5) φ p σ q “ φ p z p ; σ q : “ σB p , ` η d ´ p , ` z p φ ˘ , which depends upon the bulk density and meander parameters, σ and p . In Corol-lary 3.6 σ will be slaved to p to fix the total mass of Φ p . As a function on R , φ is smooth and is even with respect to z p . The second order correction φ isthe product of whisker independent dressed functions and the whisker dependentcurvature κ p “ κ p p s p q , see (2.45). As such φ is not the dressing of a function ofone variable, indeed we define it on each whisker s p , as the solution of(3.6)L φ p z, s p q “ ´ L ˆ zκ p φ ` W p φ q φ ˙ ´ ˆ κ p φ ` p´ η ` W p φ q φ q L φ ` η d W p φ q φ ˙ ´ κ p ´ φ p σ ˚ q ` p´ η ` W p φ q φ p σ ˚ qq φ ¯ . However, each s p dependent term decays exponentially to zero in z p , we may dress φ around Γ p to obtain(3.7) φ : “ ´ L ´ p , ˆ z p κ p φ ` W p φ q φ ˙ ´ L ´ p , ˆ κ p φ ` p´ η ` W p φ q φ q L p , φ ` η d W p φ q φ ˙ ´ κ p L ´ p , ´ p , φ p σ ˚ q ` p´ η ` W p φ q φ p σ ˚ qq φ ¯ . This construction requires that the right-hand side of (3.6) lies in the range of L .As will be seen below, this motivates the choice of σ “ σ ˚ in the φ terms in theodd part of φ . Indeed, the right-hand side of the first line of (3.7) is even in z , andhence orthogonal in L p R q to φ . The terms on the second line are odd in z , wedenote them by p L φ q odd . As κ p is a function only of s p , and L is self-adjoint, EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW with L φ “
0, we return to the L p R q and calculate, for each value of s p ,(3.8) ż R ` L φ ˘ odd φ d z “ ´ κ p ˆ ´ η m ` ż R W p φ q| φ | φ p σ ˚ q d z ˙ , where m is defined in (1.9). In light of Lemma 3.1, we have L p zφ q “ ´ p φ q “ W p φ q| φ | . Using the definition of φ and integration by parts, we have(3.9) 2 ż R W p φ q| φ | φ p σ ˚ q d z p “ ż R L p zφ q φ p σ ˚ q d z “ ´ m σ ˚ ` η d m . From the definition, (1.8), of σ ˚ we see that the terms on the right-hand side of(3.8) cancel, and we deduce the bounded invertibility of L in (3.7). We are inposition to introduce the profile. Lemma 3.2.
Let the bulk density parameter σ and meander parameters p satisfy(2.26). Then for φ , φ , and φ defined in (3.1), (3.5) , and (3.7) respectively, and φ ě “ φ ě p z p ; σ ˚ q and φ p ,e : “ φ e p x ´ p E ´ p E ; κ p ˚ , σ ˚ q , the higher order andexponential corrections given in Lemma 3.3, the quasi-equilibrium profile (3.10)Φ p p x ; σ q : “ φ p z p q ` εφ p z p ; σ q ` ε φ p s p , z p ; σ, σ ˚ q ` ε φ ě p z p ; σ ˚ q ` e ´ (cid:96)ν { ε φ p ,e , has the following residual (3.11) F p Φ p q “ F m ` e ´ (cid:96)ν { ε F e with F m “ εσ ` ε F ` ε F ` ε F ě . Here the expansion terms in the main residual F m have the form (3.12) F “ κ p p σ ´ σ ˚ q f p z p q ; F “ φ ∆ s p κ p ` f p z p , γ p q , F ě “ ` f , p z p , γ p q ` ε ∆ g ˘ ∆ g f , p z p , γ p q ` f , p z p , γ p q , where f , f , f , , f , and f , are smooth functions which decay exponentially fastto a constant as | z p | Ñ 8 . In addition, F is odd with respect to z p and F , F obey the following projection properties: (3.13) ż R (cid:96) F φ d z p “ m p σ ˚ ´ σ q κ p ` O p e ´ (cid:96)ν { ε q ; ż R (cid:96) F φ d z p “ m ˜ ´ ∆ s p κ p ´ κ p ` ακ p ¸ ` O p e ´ (cid:96)ν { ε q . Here α “ α p σ ; η , η q is given explicitly by (3.14) α “ m ˜ ż R W p φ q φ φ ´ L ´ ´ p´ η ` W p φ q φ p σ ˚ qq φ ¯ ` φ p σ ˚ q ¯ d z ` ż R ˆ η φ φ ´ η d W p φ q φ ´ L ˆ W p φ q φ ˙ ´ W p φ q φ L φ ˙ φ z d z ` η d ż R W p φ q φ L ´ ` φ p σ ˚ q ` p´ η ` W p φ q φ p σ ˚ qq φ ˘ d z ¸ . Proof.
For brevity of notation, we drop the subscript p in the proof. Withoutloss of generality, we may assume the support of the exponential correction φ e lies outside of Γ (cid:96) . Within Γ (cid:96) the ε -scaled Laplacian takes the form (2.10) and thevariational derivative F p Φ q can be written asF p Φ q “ “ B z ` ε H B z ` ε ∆ g ´ W p Φ q ` εη ‰ “ B z Φ ` ε H B z Φ ` ε ∆ g Φ ´ W p Φ q ‰ ` εη d W p Φ q . (3.15)The components of the profile Φ were chosen to make the residual Π F p Φ q smallto O p ε q . We expand F p Φ q in powers of ε , and introduce φ ě : “ φ ` εφ ` ε φ ě and φ ě : “ φ ` εφ ě . Taylor expanding the k -th derivative of W p Φ q around φ for k “ , W p k q p Φ q “ W p k q p φ q ` εW p k ` q p φ q φ ` ε ˆ W p k ` q p φ q φ ` W k ` p h k q φ ˙ `` ε W p k ` q p φ qp φ ` εφ ě q φ ě ` ε W p k ` q p φ q φ ě . Similarly the expansion of the extended curvature H to third order takes the formH “ ´ κ ´ εzκ “ ´ κ ´ εz κ ´ εzκ “ ´ κ ´ εzκ ´ ε z κ ´ ε z κ ´ εzκ . The whiskered coordinate expression (3.15) of F p Φ q admits the expansion(3.17) F p Φ q “ ε ` L φ ` η d W p φ q ˘ ` ε F ` ε F ` ε F ě . Using the identities from Lemma 3.1, F and F reduce toF “ L ˆ κφ ` L φ ` zκ φ ` W p φ q φ ˙ ` ` κ B z ´ η ` W p φ q φ ˘ ˆ p κφ ` L φ q ` η d W p φ q φ ;F “ L φ ě ` L ˆ κ B z φ ` W p φ q φ φ ` z κ φ ` zκ φ ` W p q p φ q φ ˙ ` p κ B z ´ η ` W p φ q φ q ˆ L φ ` κφ ` κ zφ ` W p φ q φ ˙ ´ ∆ s κφ ` ˆ W p q p φ q φ ` W p φ q φ ˙ p κφ ` L φ q` κ zφ ` zκ B z L φ ` η d ˆ W p φ q φ ` W p φ q φ ˙ . Within Γ (cid:96) p using the expressions for φ , φ in (3.5) and (3.7) we see that the O p ε q term in (3.17) reduces to the constant σ . Using the definition of φ given in (3.7),the term F further reduces toF “ κ L p φ ´ φ p σ ˚ qq ` κ B z L p φ ´ φ p σ ˚ qq ´ W p φ qp φ ´ φ p σ ˚ qq κφ , and the final expression for F in (3.12) follows from (3.5) with the reductions forF and F obtained from similar calculations. In particular F ě takes the exact EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW form: ´ pB z ` ε H B z ` ε ∆ g ´ W p Φ q ` εη q ˆ W p φ q p φ ě ` φ φ ě q ` ∆ g φ ě ` W p q p h q p φ φ ě ` εφ φ ě ` ε φ ě q` p z κ φ ` zκ φ ` κ B z φ q zκ ` κ B z φ ě ´ εzκ ˙ ´ p H B z ` ε ∆ g ´ W p h q φ ě ` η qˆ ˆ L φ ` W p φ q φ φ ` z κ φ ` zκ φ ` κ B z φ ` W p q p φ q φ ˙ ´ ˆ ∆ g ´ zκ B z ´ W p q p h q φ ě ´ W p φ q φ ě ˙ ´ κφ ` zκ φ ` L φ ` W p φ q φ ¯ ` ˆ z κ ´ εzκ B z ` W p q p h q p φ ` εφ ě q φ ě ´ W p φ q φ ě ˙ p κφ ` L φ q , where the h terms denote remainders from Taylor expansion.The projection of F onto φ is similar to the calculation of (3.8) and (3.9) andomitted. To estimate the projection of F in L p R (cid:96) q to the function φ “ φ p z q , itsuffices to consider the odd part of F . Indeed, since φ , φ are all even functionswith respect to z , we haveF odd “ L ` κ B z φ even ` z κ φ ` W p φ q φ φ odd ˘ ´ ∆ s κφ ` ` ´ η ` W p φ q φ ˘ ˆ p L φ odd ` κφ q ` κ B z ˆ L φ even ` ` zκ φ ` W p φ q φ ˙ ` κ W p q p φ q φ φ ` κW p φ q φ even φ ` κ zφ ` η d W p φ q φ odd . Integrating by parts, using properties of L from Lemma 3.1 and re-organizating,we obtain ż R (cid:96) F φ d z “ ´ ∆ s κm ´ η κ ż R (cid:96) L φ φ z d z ` I ` I ` I ` O p e ´ (cid:96)νε q where I : “ κ ż R (cid:96) L φ φ z d z ; I : “ ż R (cid:96) W p φ q φ φ L φ odd d z ; I : “ η d ż R (cid:96) W p φ q φ φ odd d z. Recalling the definition of α in (3.14), the projection of F in (3.13) then followsfrom the identities: I “ ´ κ m ` η κ ż R (cid:96) L φ φ z d z ´ κ ż R (cid:96) W p φ q φ L φ φ z d z ´ η d κ ż R (cid:96) W p φ q φ φ z d z ` κ ż R (cid:96) W p φ q φ φ d z ; I “ ´ κ ż R (cid:96) W p φ q φ φ L ´ pp´ η ` W p φ q φ p σ ˚ qq φ q´ κ ż R (cid:96) W p φ q φ φ φ p σ ˚ q d z ; I “ ´ η d κ ż R (cid:96) W p φ q φ L ´ ` φ p σ ˚ q ` p´ η ` W p φ q φ p σ ˚ qq φ ˘ d z. Outside of Γ (cid:96) , the profile Φ reduces to a constant plus an error correction(3.18) Φ “ b ´ ` εφ ` ε φ ` ε φ ` e ´ (cid:96)ν { ε φ e , The corrections φ e and transition from Γ (cid:96) to Γ (cid:96) induce the exponentially smallterms F e in (3.11). (cid:3) The profile Φ p constructed in Lemma 3.2 is designed to capture the transientdepartures from circularity, and then return to an exact equilibrium as t Ñ 8 and(3.19) p p t q Ñ p ˚ “ p p ˚ , p ˚ , p ˚ , , ¨ ¨ ¨ , q , σ p t q Ñ σ ˚ “ σ ˚ ` O p ε q . The existence of exact equilibrium for circular interface was established in [10] andis quoted here.
Lemma 3.3.
Let Γ ˚ be a circle centered at origin with finite curvature κ ˚ thatis strictly contained within the periodic domain Ω . Let z ˚ denote ε -scaled distanceto Γ ˚ . Then for each ε sufficiently small there exists a unique constant σ ˚ “ σ ˚ ` εσ ˚ě p κ ˚ , ε q , a uniformly (in ε ) bounded function φ ě “ φ ě p z ˚ ; κ ˚ , ε q whichdecays exponentially fast to a constant as z ˚ Ñ 8 , and a uniformly (in ε ) smoothfunction φ e “ φ e p x ; κ ˚ , ε, Ω q and a ν ą independent of ε sufficiently small suchthat, Φ ˚ p x q : “ φ p z ˚ p x qq ` εφ p z ˚ p x q ; σ ˚ q ` ε φ p z ˚ p x q ; σ ˚ , κ ˚ q ` ε φ ě p z ˚ p x q ; κ ˚ q` e ´ (cid:96)ν { ε φ e p x ; κ ˚ , σ ˚ q , are exact equilibrium of (1.3) subject to periodic boundary conditions on Ω . Trans-lates of periodic extensions of Φ ˚ are also exact equilibrium. Establishing the long time convergence requires Lipschitz estimates on the resid-uals of the quasi-equilibrium as σ ´ σ ˚ , p ´ p ˚ , and ˆ p (recall (2.24)) tend to zero.While Π F p Φ p ˚ , σ ˚ q “
0, for notational convenience we did not expand σ ˚ in ordersof ε in Lemma 3.2, hence Π F k p Φ p ˚ , σ ˚ q ‰ k “ , ,
4. Recalling the notationof the dressing process introduced in section 2, we have the following estimates.
Lemma 3.4.
Under assumptions (2.26) , the components of the residual F p Φ p q given in (3.11) satisfy } F } L À ε { | σ ´ σ ˚ | ; } F ´ F } L `} F ě ´ F } L À ε { ´ } ˆ p } V ` ¯ ; } F e } L À and the overall residual satisfies the Lipschitz estimate } Π F p Φ p q} L À ε { | σ ´ σ ˚ | ` ε { ´ } ˆ p } V ` | p ´ p ˚ | ¯ ` ε { } ˆ p } V . Proof.
The L -bounds of the difference between F k and its bulk value for k “ , , , F , F given in Lemma 3.2. In light of Lemma2.7 and assumption (2.26), we have(3.20) | κ p | À } ˆ p } V À . The L -estimate of Π F p Φ p q follows from comparing it to the zero residual of Φ ˚ .Indeed we write(3.21) } Π F p Φ p q} L “ } Π F p Φ p q ´ Π F p Φ ˚ q} L EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW where Φ ˚ is the equilibrium solution associated with bulk density state σ ˚ andinterface(3.22) Γ ˚ : “ Γ p ˚ ` Θ ´ p p ´ p ˚ q E ` p p ´ p ˚ q E ¯ . obtained by translating Γ p ˚ to place its center at p p , p q . The triangle inequalityand the expansion of F “ F p Φ p q from Lemma 3.2, yield the estimate(3.23) } Π p F p Φ p q ´ F p Φ ˚ qq} L ď ε } Π p F ´ F p Φ ˚ qq} L ` ε } Π p F ´ F p Φ ˚ qq} L ` ε } Π p F ě ´ F ě p Φ ˚ qq} L ` e ´ (cid:96)ν { ε } F e ´ F e p Φ ˚ q} L . We use the form of the F k p Φ ˚ q residuals to establish that they are Lipschitz in | p ´ p ˚ | . We observe from Lemma 3.2 that F admits the general form F “ κ p f p z p qp σ ´ σ ˚ q , while F p Φ ˚ q “ κ ˚ f p z ˚ qp σ ˚ ´ σ ˚ q . We deduce that } F ´ F p Φ ˚ q} L ď| σ ´ σ ˚ | ż Ω κ p f p z p q d x ` | σ ˚ ´ σ ˚ | ż Ω | κ p f p z p q ´ κ ˚ f p z ˚ q| d x. The integrals contribute a factor of ε since the integrands are bounded and their sup-port is localized near the interface Γ p and Γ ˚ . We decompose the second integrandas κ p p f p z p q ´ f p z ˚ qq ` p κ p ´ κ ˚ q f p z ˚ q which we bound by | z p ´ z ˚ | ` | κ p ´ κ ˚ | in its support set. Using the estimates of Lemmas 2.10, 2.7 and 2.4, recalling | σ ˚ ´ σ ˚ | À ε we arrive at the bound } Π F ´ Π F p Φ ˚ q} L À ε | σ ´ σ ˚ | ` ε ´ | p ´ p ˚ | ` } ˆ p } V ¯ . The L bounds of F ´ F p Φ ˚ q and F ě ´ F ě p Φ ˚ q involve higher derivatives of thecurvatures which are controlled with the aid of (3.20), specifically } ∆ s p κ p ´ ∆ s ˚ κ ˚ } L loc À ε { } ˆ p } V ; } ε ∆ s p κ p } L ` } κ p } L À ` } ˆ p } V À . The term F e incorporates error from φ e and from the dressing process. However φ e in Lemma 3.2 cancels with the corresponding term in Φ ˚ , and the residual is duesolely to the dressing which contributes(3.24) } F e ´ F e p Φ ˚ q} L À } ˆ p } V . Reporting the term-wise bounds into (3.23) completes the proof. (cid:3)
In Section 5 we require bounds on the mass of F p Φ p q ´ F m , where F m is thebulk value of F m defined in (3.11). Lemma 3.5.
Imposing assumptions (2.26) , then there exist smooth functions C k “ C k p p q for k “ , such that ż Ω p F p Φ p q ´ F m q d x “ C p p q ε ` C p p q ε p σ ´ σ ˚ q ` O ´ ε } ˆ p } V , ε } ˆ p } V ¯ . Proof.
We expand F p Φ p q in Lemma 3.2, subtract F m and integrate,(3.25) ż Ω p F p Φ p q ´ F m q d x “ ε ż Ω F p Φ p q d x ` ε ż Ω p F p Φ p q ´ F q d x ` ε ż Ω p F ě p Φ p q ´ F q d x ` e ´ (cid:96)ν { ε ż Ω F e d x. Recalling form of F in (3.12), where f p z p q is odd in z p , we deduce(3.26) ε ż Ω F p Φ p q d x “ ´ ε p σ ´ σ ˚ q ż R (cid:96) ż I κ p z p f p z p q| γ p | d s p d z p , “ Cε p σ ´ σ ˚ q ż I h p γ p q d s p , where the function h “ h p γ p q following Notation 2.1. Employing the decomposition(2.51) and estimate (2.55) implies(3.27) ε ż Ω F p Φ p q d x “ C p p q ε p σ ´ σ ˚ q ` O ´ ε | σ ´ σ ˚ |} ˆ p } V ¯ . From the form of F given in (3.12), the oddness of φ with respect to z p implies(3.28) ε ż Ω p F p Φ p q ´ F q d x “ ε ż R (cid:96) ż I ` φ ∆ s p κ p ` p f p z p , γ p q ´ f q ˘ J p d s p d z p “ ε ˆ ε ż I h p γ p q ∆ s p κ p d s p ` ż I h p γ p q d s p ˙ “ O ´ ε } ˆ p } V ¯ ` ε ´ C p p q ` O p} ˆ p } V q ¯ . Similar estimates show that(3.29) ε ż Ω p F ě p Φ p q ´ F q d x ` e ´ (cid:96)ν { ε ż Ω F e d x “ O ´ ε } ˆ p } V ¯ ` ε ´ C p p q ` O p} ˆ p } V q ¯ . Combining (3.27)-(3.29) with (3.26) yields (3.25). (cid:3)
The bulk density parameter.
The profile Φ p is constructed to contain thetotal mass, and combined with the mass conservation of the flow (1.3) we have(3.30) 0 “ ż Ω p u ´ Φ p q d x “ ż Ω p u ´ Φ p q d x. From the expansion (3.10) of Φ p with φ “ φ p σ q given by (3.5) and (3.30) wededuce(3.31) σ p p q “ B p , ż Ω „ ε ´ u ´ φ p z p q ´ ε φ p s p , z p q ´ ε φ ě p z p q ´ e ´ (cid:96)ν { ε φ p ,e ¯ ´ η d ` L ´ p , p z p φ q ˘ d x. The following result approximates the dependence of σ on p p , ˆ p q . Corollary 3.6.
If we assume u “ Φ p σ p qq ` v with } v } L ď ε , then the bulkdensity parameter σ explicitly given by (3.31) admits the following approximation: σ p p q “ σ p q ` ε ´ ¯ B ż Ω v d x ´ c m m Θ p ` Cε p ` ε C p p q p ` O ´ } ˆ p } V ¯ . Here c takes the form (3.32) c “ πm ` Cε ¯ B m ą C and C p p q denote computational constants and smooth functions of p . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Proof.
We replace u with Φ p σ p qq ` v in the right-hand side of (3.31) and set p “ in (3.10) to expand Φ p σ p qq . Remarking that the mass of φ p ,e “ φ e p x ´ p E ´ p E q is independent of translation and hence of p , the exponential termscancel, and setting aside v , the dominant contribution comes from(3.33) T p p q : “ ż Ω p φ p z q ´ φ p z p qq d x ` ε ż Ω ´ φ p z ; σ p qq ´ η d ` L ´ p , p z p φ q ˘¯ d x ` ε ż Ω p φ p z q ´ φ p s p , z p qq d x ` ε ż Ω p φ ě p z q ´ φ ě p z p qq d x. The φ and φ ě integrals can be dealt with similarly. Addressing the φ integral,we rewrite(3.34) ż Ω p φ p z q ´ φ p z p qq d x “ ż Ω p φ p z q ´ b ´ q d x ´ ż Ω p φ p z p q ´ b ´ q d x. Up to exponentially small terms φ p z q´ b ´ is localized within Γ (cid:96) , changing to localcoordinates(3.35) ż Ω p φ p z q´ b ´ q d x “ ε ż I ż R (cid:96) p φ p z q´ b ´ q| γ |p ´ εzκ q d s d z “ ε | Γ |p m ` Ce ´ (cid:96)ν { ε q , where we used the even symmetry of φ . A similar argument shows that ż Ω p φ p z p q ´ b ´ q d x “ ε | Γ p |p m ` Ce ´ (cid:96)ν { ε q , which combined with (3.34) and (3.35) implies(3.36) ż Ω p φ p z q ´ φ p z p qq d x “ p m ` Ce ´ (cid:96)ν { ε q p| Γ | ´ | Γ p |q . The φ ě integral is bounded similarly and we deduce that(3.37) ż Ω p φ p z q ´ φ p z p qq d x ` ε ż Ω p φ ě p z q ´ φ ě p z p qq d x “ ε ´ ` Cε ¯ m p| Γ | ´ | Γ p |q . The φ integral is more complicated. In the sense of Notation 2.9, from (3.7) the φ term admits the expression φ p s p , z p q “ h p κ p , z p ; σ q where h is a smooth, localizedfunction of κ p , z p . We rewrite the φ integrand as φ p s, z ; σ p qq ´ φ p s p , z p ; σ q “ ´ h p κ , z ; σ p qq ´ h p κ , z ; σ q ¯ ` ´ h p κ , z ; σ q ´ h p κ , z p ; σ q ¯ ` ´ h p κ , z p ; σ q ´ h p κ p , z p ; σ q ¯ . Integrating this expression, the first term is dealt with by Lipschitz estimates, thesecond term is treated as in (3.36), and for the third term we change to localcoordinates and integrate out the z p dependence, obtaining(3.38) ż Ω p φ p z q ´ φ p s p , z p qq d x “ Cε pp σ ´ σ p qq ` p| Γ | ´ | Γ p |qq` ε ˜ h p κ q| Γ p | ´ ε ż I ˜ h p κ p q J p , d s p . In light of the approximation of κ p in Lemma 2.7 and of J p , “ | γ p | given in (2.38),it holds that(3.39) ż I ˜ h p κ p q J p , d s p “ ˜ h p κ p , q| Γ p | ` O ´ } ˆ p } V ¯ . Since κ p , “ ´ {p R ` p Θ q and κ “ ´ { R , we may further simplify this equality ż I ˜ h p κ p q J p , d s p ´ ˜ h p κ q| Γ | “ C p p q p ` O ´ } ˆ p } V ¯ , which when substituted into the right hand side of equality (3.38) yields(3.40) ż Ω p φ p z q ´ φ p s p , z p qq d x “ Cε p σ ´ σ p qq ` εC p p q p ` Cε ` | Γ | ´ | Γ p | ˘ ` O ´ ε } ˆ p } V ¯ . For the φ term, we recall its definition (3.5) and replacing p z p , σ q by p z, σ p qq respectively, we have ż Ω ´ φ p z q ´ η d ` L ´ p , p z p φ q ˘¯ d x “ σ p q ¯ B ` Cε p| Γ | ´ | Γ p |q , where the extra factor of ε arises from the localization of L ´ p , p z p φ q near Γ p .Reporting each of the estimates into (3.33), and in turn substituting T into (3.31)yields(3.41) σ p p q “ ¯ B ¯ B p , σ p q ` ε ´ ¯ B p , ż Ω v d x ´ p| Γ p | ´ | Γ |q ¯ B p , p m ` Cε q ` ε C p p q p ` Cε p σ p p q ´ σ p qq ` O ´ ε } ˆ p } V ¯ . Since the far field values B p , “ B , we may use the even parity of B p , withrespect to z p to simplify(3.42) ¯ B p , ´ ¯ B “ ż Ω p B p , ´ B p , q d x ´ ż Ω p B ´ B q d x, “ ε ż R (cid:96) p B ´ B q d z p| Γ p | ´ | Γ |q , “ Cε p| Γ p | ´ | Γ |q . Combining (3.42) with the identity (3.41) for σ p p q yields(3.43) p ´ Cε qp σ p p q ´ σ p qq “ ´ ε ´ ¯ B p , ż Ω v d x ´ m ` Cε ¯ B p , ´ | Γ p | ´ | Γ | ¯ ` ε C p p q p ` O ´ ε } ˆ p } V ¯ . From (2.35) we estimate the difference in length of Γ p and Γ (3.44) | Γ p | ´ | Γ | “ π p Θ ` O ´ } ˆ p } V ¯ . We estimate the difference between ¯ B ´ p , and ¯ B ´ from (3.42) and (3.44)(3.45) 1¯ B p , ´ B “ ´ Cε p| Γ p | ´ | Γ |qp ¯ B ` Cε p| Γ p | ´ | Γ |qq ¯ B “ πC ¯ B ε p ` ε C p p q ` O ´ ε } ˆ p } V ¯ . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Dividing both sides of (3.43) by p ´ Cε q , which combined with (3.44)-(3.45)implies the Lemma for c in the form of (3.32). (cid:3) Remark 3.7.
Denoting the equilibrium value of p by p ˚ , by comparing the twovalue of σ we obtain, (3.46) σ ˚ ´ σ p p q “ c m m Θ p p ´ p ˚ q ` O ´ ε | p ´ p ˚ | , } ˆ p } V ¯ . From Lemma 2.10 and the expression for σ given in (3.31), we deduce the fol-lowing. Corollary 3.8.
Under assumption (2.26) , using Notation 2.1 the sensitivity of σ to p j can be written as B σ B p j “ (cid:68) h p γ p q , ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p (cid:69) J p , ` ε (cid:68) h p γ p q , ε ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ (cid:69) J p , for j ‰ , ; while for j “ , , we have B σ B p j “ (cid:10) h p γ p q , E j ¨ n p (cid:11) J p , ` ε (cid:10) h p γ p q , E j ¨ γ p (cid:11) J p , This Corollary above permits reductions of the p j derivatives of the profile Φ p . Corollary 3.9.
For j “ , , there exist functions h k “ h k p z p , γ p qp k “ , , ¨ ¨ ¨ , q enjoying the properties of Notation 2.1 for which (3.47) B Φ p B p j “ ´ ε ` φ ` εφ ` ε h ˘ E j ¨ n p ? πR ` εh E j ¨ γ p ` εh ´ (cid:104) h , E j ¨ n p (cid:105) J p , ` (cid:10) h , ε E j ¨ γ p (cid:11) J p , ¯ , where n p p s p q is the normal to Γ p at location s p , and E , E P R denote theCartesian unit vectors. For j P Σ zt , u , we have the expansions (3.48) B Φ p B p j “ ´ ε ` φ ` εφ ˘ n p ¨ n ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ ` εh ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ ` εh ε ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ ` εh ˆ (cid:68) h , ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ (cid:69) J p , ` (cid:68) h , ε ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ (cid:69) J p , ˙ , for another set of functions h k “ h k p z p , γ p qp k “ , , ¨ ¨ ¨ , q with the properties ofNotation 2.1.Proof. From (3.10) the p j partial derivative admits the expansion B Φ p B p j p s p , z p q “p φ ` εφ ` ε B z p φ ` ε φ q B z p B p j ` ε B s p φ B s p B p j ` ε B σ p φ ` εφ q B σ B p j , (3.47) and (3.48) follow from Lemma 2.10 and Corollary 3.8. (cid:3) Linear stability
The nonlinear stability analysis hinges upon the properties of the linearization ofthe flow (1.3) about the quasi-steady bilayer profiles Φ p constructed in Lemma 3.2.While these solutions are not equilibrium, we exploit coercivity properties of thelinearized operator to show that it controls the nonlinearities along the transientflow from initial profile to final circular equilibrium. The linearization takes theform Π L p where(4.1) L p : “ δ F δu ˇˇˇ u “ Φ p “p ε ∆ ´ W p Φ p q ` εη qp ε ∆ ´ W p Φ p qq´ p ε ∆Φ p ´ W p Φ p qq W p Φ p q ` εη d W p Φ p q , denotes the second variational derivative of F at Φ p and η d : “ η ´ η . Associated toeach p is the reach Γ (cid:96) p , (2.4), of the interface Γ p , defined in (2.19). When restrictedto functions with support within the reach, the cartesian Laplacian admits theexpansion (2.10) in the local coordinates p s p , z p q , which induces the expansion(4.2) L p “ L p , ` ε L p , ` ε L p , ě . The leading order operator takes the form(4.3) L p , : “ ` L p , ´ ε ∆ s p ˘ “ L p . where we have introduced L p : “ L p , ´ ε ∆ s p . Much of the structure of the FCHflow stems from L p , and its balancing of L p , , the Γ p dressing of L defined in(3.2) through Definition 2.2, against the Laplace-Beltrami operator of Γ p . The nextcorrection takes the form(4.4) L p , “p κ p B z p ` W p φ q φ ´ z p ε D s p , ´ η q L p ` L p p κ p B z p ` W p φ q φ ´ z p ε D s p , q ` W p φ q ` κ p ` φ ˘ ` L p , φ ˘ ` η d W p φ q . The second and higher order correction term, L p , ě , is relatively compact withrespect to L p , and its precise form is not material. We use the expansion (4.2) toconstruct approximate slow spaces that characterize the small spectrum of L p inthe sense that the operator is uniformly coercive on their compliment. A key issueis the asymptotically large size of these slow spaces.4.1. Approximate slow spaces.
The approximate slow space Z is a tensor prod-uct of functions of z p and s p that exploits the balance of the operator L p , viewedas acting on the tensor product space L p R q ˆ L p I q . As both L p , and L p areself-adjoint, it is sufficient to establish coercivity of L p . The eigenfunctions of L p , and ∆ s p “ B s p are addressed in Lemma 3.1 and (2.33) respectively. Definition 4.1. (a) Let ψ k p z q be the normalized eigenfunction of the linearized op-erator L as an operator in L p R q associated with eigenvalue λ k . We introducethe dressed and scaled version ˜ ψ k as ˜ ψ k p z p q : “ ε ´ { ψ k p z p q . (b) For k “ , , we introduce the disjoint index sets: (4.5) Σ k “ Σ k p Γ p , ρ q “ t j ˇˇ Λ kj : “ p λ k ` ε β p ,j q ď ρ u , and Σ : “ Σ X Σ , and the preliminary slow space Z “ Z p Γ p , ρ q Ă L , Z p Γ p , ρ q : “ span ! Z I p j q j p : “ ˜ ψ I p j q p z p p x qq ˜Θ j p ˜ s p p x qq ˇˇˇ j P Σ k and I p j q “ k P t , u ) . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW The Laplace-Beltrami eignevalues β p ,j , introduced in (2.33), admit the approxi-mations (2.44). The exponential decay of ψ k to zero away from the interface impliesthat the corrections arising from dressing are exponentially minor, in particularthere exist ν ą L p , Z I p i q i p “ Λ I p i q i Z I p i q i p ` O p e ´ (cid:96)ν { ε q , for all i P Σ. These functions are compressed by an factor of ρ by L p , . To estimate the sizes of Σ and Σ , subject to condition (2.26), we recall theidentity (2.44) which implies β p ,k „ Ck . The ground-state eigenvalue λ ă k lies in Σ p ρ q if and only if(4.7) ε ´ b ´ λ ´ ρ { À k À ε ´ b ´ λ ` ρ { , ùñ N : “ | Σ p ρ q| „ ε ´ ρ { ;and λ “
0, so k lies in Σ p ρ q if and only if(4.8) 0 ď k À ε ´ ρ { , ùñ N : “ | Σ p ρ q| „ ε ´ ρ { . The lower bound of elements in Σ , ε ´ a ´ λ ´ ρ { , is of order ε ´ while theupper bound of Σ is of order ε ´ ρ { . We deduce that Σ and Σ are disjoint for ρ suitably small. In this case N : “ | Σ p ρ q| “ | Σ p ρ q| ` | Σ p ρ q| „ ε ´ ρ { . Using the formalism of Notation 2.1 we have the following estimates.
Lemma 4.2.
Assume (2.26) holds, and h “ h p γ p k q p q is a function satisfying Nota-tion 2.1. If | γ p k q p | À then there exists a matrix E “ p E ij q which is bounded as amap from l p R N q to l p R N q such that (4.9) ˇˇˇˇ (cid:68) h p γ p k q p q ˜Θ i , ˜Θ j (cid:69) J p , ˇˇˇˇ À p δ ij ` } ˆ p } V k q E ij hold for i, j P Σ “ Σ Y Σ , for k “ , . Moreover for all i, j such that I p i q ‰ I p j q we have (4.10) ˇˇˇˇ (cid:68) h p γ p q ˜Θ i , ˜Θ j (cid:69) J p , ˇˇˇˇ À ε } ˆ p } V E ij . Proof.
We assume h is independent of ˜ s p at leading order and use decomposition(2.51) to rewrite the integrals(4.11) (cid:68) h p γ p k q p q ˜Θ i , ˜Θ j (cid:69) J p , “ h p γ p k q p , q ż I p ˜Θ i ˜Θ j d˜ s p ` ż I p ´ h p γ p k q p q ´ h p γ p k q p , q ¯ ˜Θ i ˜Θ j d˜ s p The first term on the right hand side is zero due to the orthogonality of t ˜Θ i u inthe space L p d˜ s p q , see (2.34). For the second term we bound the L -norm of thedifference as(4.12) ˇˇˇ h p γ p k q p q ´ h p γ p k q p , q ˇˇˇ À } ˆ p } V k . and deduce from Lemma 2.9 that these terms are O p} ˆ p } V k q E ij for a matrix E asabove; the estimate (4.9) follows. For (4.10), when I p i q ‰ I p j q we have β i ‰ β j . Integrating by parts twice we transfer the highest derivative of ˜Θ i to ˜Θ j and generate some lower derivative terms from the product rule with h . We write theresult in the form(4.13) ż I p h ` γ p ˘ ˜Θ i ˜Θ j d˜ s p “ ´ ż I p h ` γ p ˘ ˜Θ i ˜Θ j d˜ s p ´ ż I p B ˜ s p h ˜Θ i ˜Θ j d˜ s p “ ż I p h ` γ p ˘ ˜Θ i ˜Θ j | γ p | d s p ` ż I p B ˜ s p h ´ ˜Θ i ˜Θ j ´ ˜Θ i ˜Θ j ¯ d˜ s p , and applying (2.33) we obtain(4.14) 4 π β p ,j ´ β p ,i | Γ p | ż I p h ` γ p ˘ ˜Θ i ˜Θ j d˜ s p “ ż I p B ˜ s p h ´ ˜Θ i ˜Θ j ´ ˜Θ i ˜Θ j ¯ d˜ s p . Following the steps use to derive (4.11) and (4.12), we decompose |B ˜ s p h | as in (2.51),apply Lemma 2.9 by bounding L -norm of the differences and use the point-wisebounds | ˜Θ j | À β p ,j and | Θ i | À β p ,i . Under assumption (2.26) we may divide bothsides of the inequality (4.14) by β p ,j ´ β p ,i „ p β j ´ β i q{ R to obtain(4.15) ż I p h ` γ p ˘ ˜Θ i ˜Θ j d˜ s p À } ˆ p } V | β i ´ β j | E ij À } ˆ p } V | β i ´ β j | E ij . Moreover, β k „ Ck , and since, without loss of generality i P Σ and j P Σ of k (4.7) we estimate(4.16) | β i ´ β j | ě Cε ´ . The bound (4.10) follows from (4.15). (cid:3)
The estimates of Lemma 4.2 are central to controlling the action of the operator L p when restricted to the asymptotically large slow space Z . A benefit of conduct-ing our analysis in R is that single derivatives of the Laplace-Beltrami eigenmodesbehave well. Indeed, from (2.18) we have(4.17) ˜Θ i “ " ´ β p ,i ˜Θ i ` , i odd ,β p ,i ˜Θ i ´ , i even , which furthermore implies(4.18) ˜Θ p k q i P span t ˜Θ i , ˜Θ i u . Recalling the definition (3.3) of B p , we introduce the p -independent system pa-rameter(4.19) S : “ ż R W p φ p z qq B p z q | ψ p z q| d z. Lemma 4.3.
Assume (2.26) holds with } ˆ p } V À ρ { . The basis functions ofthe slow space ! Z I p k q k p , k P Σ ) are approximately orthonormal to each other in L . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW More precisely there exist matrices E with l ˚ -norm one for which we have the esti-mates for (cid:68) Z I p i q i p , Z I p j q j p (cid:69) L (4.20) (cid:68) Z I p i q i p , Z I p j q j p (cid:69) L “ $’&’% ˆ ` p Θ R ` O p} ˆ p } V q ˙ δ ij ` O p ε { } ˆ p } V q E ij , I p i q “ I p j q ; O ´ ε { } ˆ p } V ^ ε } ˆ p } V ¯ E ij , I p i q ‰ I p j q . Moreover, the linear operator L p restricted to the preliminary slow space Z p Γ p , ρ q act as M ij : “ (cid:68) L p Z I p i q i p , Z I p j q j p (cid:69) L , obeying approximation M ij “ $’’’’’’’&’’’’’’’% p ` p Θ { R q ` Λ i ` ε p σS ` η d λ q ˘ ` O p ερ { q i “ j and I p i q “ ; p ` p Θ { R q Λ i ` O p ερ { q if i “ j and I p i q “ ; O ´ ε , ε { } ˆ p } V ¯ E ij i ‰ j and I p i q “ I p j q ; O ´ ε , ε { } ˆ p } V ^ ε } ˆ p } V ¯ E ij I p i q ‰ I p j q .Proof. Using the localization of the basis functions, we establish the approximateorthonormality (4.20) by integrating over Γ (cid:96) p . Recalling that J p “ J p , ˜J p with˜J p “ ε ´ ε z p κ p p s p q , we write(4.21) (cid:68) Z I p i q i p , Z I p j q j p (cid:69) L “ ż R (cid:96) ż I ψ I p i q ψ I p j q ˜Θ i ˜Θ j J p , d s p d z p ´ ε (cid:68) κ p ˜Θ i , ˜Θ j (cid:69) J p , ż R (cid:96) ψ I p i q ψ I p j q z p d z p . The orthogonality of t Θ i u given in (2.34) shows that the first term on the right-hand side contributes the main δ ij term in (4.20). The claim second term on theright hand side can be bounded by(4.22) ˇˇˇˇ ε (cid:68) κ p ˜Θ i , ˜Θ j (cid:69) J p , ż R (cid:96) ψ I p i q ψ I p j q z p d z p ˇˇˇˇ À ε { } ˆ p } V E ij . Indeed, if I p i q “ I p j q (4.22) holds obviously since | ψ I p i q | z p is odd. On the otherhand, if I p i q ‰ I p j q we use estimate (4.9) from Lemma 4.2 to bound the projectionof κ p to ˜Θ i ˜Θ j in L p d˜ s p q , and (4.22) follows from(4.23) } ˆ p } V À N { } ˆ p } V À ε ´ { } ˆ p } V , where we used Lemma 2.4 and (4.8). Returning this estimate to (4.21) establishes(4.22). If we use (4.10) instead of (4.9), we are able to bound(4.24) ˇˇˇˇ ε (cid:68) κ p ˜Θ i , ˜Θ j (cid:69) J p , ż R (cid:96) ψ I p i q ψ I p j q z p d z p ˇˇˇˇ À ε } ˆ p } V E ij , for I p i q ‰ I p j q . Combining (4.22) and (4.24) with (4.21) implies the approximate orthogonality(4.20).
To establish the estimates of L p on Z we apply the expansion (4.2) of L p to theinner product:(4.25) (cid:68) L p Z I p i q i p , Z I p j q j p (cid:69) L “ (cid:68) L p , Z I p i q i p , Z I p j q j p (cid:69) L ` ε (cid:68) L p , Z I p i q i p , Z I p j q j p (cid:69) L ` ε (cid:68) L p , ě Z I p i q i p , Z I p j q j p (cid:69) L . Recalling (4.6) and employing the approximate orthogonality identity (4.20), weobtain the leading order (cid:68) L p , Z I p i q i p , Z I p j q j p (cid:69) L can be approximated by(4.26) $’&’% ˆ ` p Θ R ` O p} ˆ p } V q ˙ Λ I p j q j δ ij ` O ´ ε { } ˆ p } V , e ´ (cid:96)ν { ε ¯ E ij , I p i q “ I p j q ; O ´ ε { } ˆ p } V ^ ε } ˆ p } V , e ´ (cid:96)ν { ε ¯ , I p i q ‰ I p j q . Estimates on L p , restricted to Z are more complicated. Recalling (4.4), directcalculations establish(4.27) L p , p ψ I p i q ˜Θ i q “ Λ I p i q i ´ κ p ψ I p i q ˜Θ i ` W p φ q φ ψ I p i q ˜Θ i ´ z p ε D s p , ˜Θ i ψ I p i q ´ η ψ I p i q ˜Θ i ¯ ` L p ´ κ p ψ I p i q ˜Θ i ` W p φ q φ ψ I p i q ˜Θ i ´ z p ε D s p , ˜Θ i ψ I p i q ¯ ` W p φ qp κ p φ ` L p , φ q ψ I p i q ˜Θ i ` η d W p φ q ψ I p i q ˜Θ i . Since D s p , and L p incorporate derivatives with respect to ˜ s p scaled with ε , weapply (4.17)-(4.18) and we separate into cases for ˜Θ i and ˜Θ i . We also exploit theeven and odd parity of functions with respect to z p . We define functions h p z p , γ p q and h p z p , γ p q , denoting higher order terms, that enjoy the properties of Notation2.1. With these steps the identity (4.27) is rewritten as(4.28) L p , p ψ I p i q ˜Θ i q “ ` g K p z p , γ p q ` εh p z p , γ p q ˘ ˜Θ i ` ` g K p γ p , z p q ` εh p γ p , z p q ˘ ε ˜Θ i ` g ˚ p z p q ˜Θ i where the functions g K k “ g K k p z p , γ p q have opposite z p parity of ψ I p i q . Hence theysatisfy(4.29) ż R (cid:96) g K k p z p , γ p q ψ I p i q d z p “ , k “ , . The function g ˚ “ g ˚ p z p q is given explicitly by(4.30) g ˚ p z p q : “ Λ I p i q i ` W p φ q φ ψ I p i q ´ η ψ I p i q ˘ ` ` L p , ` ε β p ,i ˘ ` W p φ q φ ψ I p i q ˘ ` W p φ q L p , φ ψ I p i q ` η d W p φ q ψ I p i q . From (4.28) we decompose the p i, j q -th component of the bilinear form of L p , restricted to Z as(4.31) (cid:68) L p , Z I p i q i p , Z I p j q j p (cid:69) L “ I ` I ` I ` I , EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW where we have defined(4.32) I : “ ż R (cid:96) g ˚ p z p q ψ I p j q d z p (cid:68) ˜Θ i , ˜Θ j (cid:69) J p , , I : “ ż R (cid:96) ż I ` g K p z p , γ p q ` εh p γ p , z p q ˘ ψ I p j q ˜Θ i ˜Θ j J p , d s p d z p , I : “ ż R (cid:96) ż I ` g K p z p , γ p q ` εh p γ p , z p q ˘ ψ I p j q ε ˜Θ i ˜Θ j J p , d s p d z p , I : “ ´ ε ż R (cid:96) ż I L p , p ψ I p i q ˜Θ i q ψ I p j q ˜Θ j z p κ p J p , d s p d z p . In light of orthogonality (4.29), we see that I , I , I are higher order terms. Indeed,with the aids of Lemma 4.2, (4.17)-(4.18), and uniform bounds on εβ p ,i , a directcalculation establishes(4.33) I ` I ` I “ O p ε q E ij . From the orthogonality of t ˜Θ i u given in (2.34), the term I , is zero unless i “ j .As g ˚ “ g ˚ p z p q defined in (4.30) decays exponentially in z p , we may decompose(4.34) ż R (cid:96) g ˚ p z p q ψ I p i q d z p “ I ` I ` I ` Ce ´ (cid:96)ν { ε , where we have introduced the sub-terms I “ Λ I p i q i ż R ` W p φ q φ ψ I p i q ´ η ψ I p i q ˘ ψ I p i q d z ` ż R ` L ` ε β p ,i ˘ ` W p φ q φ ψ I p i q ˘ ψ I p i q d z, I “ ż R W p φ q L φ ψ I p i q ψ I p i q d z, I “ η d ż R W p φ q ψ I p i q ψ I p i q d z. Proceeding term by term, we integrate by parts in the second integral of I ,rewriting it as(4.35) I “ Λ I p i q i ż R ` W p φ q φ ψ I p i q ´ η ψ I p i q ˘ ψ I p i q d z, where Λ I p i q i ď ρ { for i P Σ in light of (4.5). Recalling the definition (3.5) of φ ,we separate I ,(4.36) I “ I , ` I , : “ σ ż R W p φ q B | ψ I p i q | d z ` η d ż R W p φ q zφ | ψ I p i q | d z. From the definition of L we observe that W p φ q φ ψ k “ λ k ψ k ´ L ψ k , and L p zψ k q “ zλ k ψ k ´ ψ k , which together with the self-adjointness of L on L p R q yield(4.37) I , “ η d ż R ´ λ I p i q ψ I p i q ψ I p i q z ´ ψ I p i q L p zψ I p i q q ¯ d z “ η d } ψ I p i q } L p R q . When I p i q “ ψ I p i q “ φ { m . Recalling the identify W p φ q| φ | “´ L φ from Lemma 3.1 yields(4.38) ż R W p φ q B | ψ | d z “ ´ m ż R B L φ d z “ ´ m ż R φ d z “ , and hence I , “ I p i q “
1. Combining the identity (4.37) with (4.36) weobtain(4.39) I “ σS δ I p i q ` η d } ψ I p i q } L p R q , where S was introduced in (4.19). Finally, from the definitions of L and ψ I p i q , I reduces to(4.40) I “ η d ż R p L ` B z q ψ I p i q ψ I p i q d z “ η d λ I p i q ´ η d } ψ I p i q } L p R q , where ψ I p i q has been normalized in L p R q . Combining estimates (4.35), (4.39) and(4.40) with (4.34) yields for some bounded constant C ,(4.41) ż R (cid:96) g ˚ p z p q ψ I p i q p z p q d z p “ p σS ` η d λ q δ I p i q ` C Λ I p i q i ` Ce ´ (cid:96)ν { ε , which combined with (2.34)-(2.35) and I defined in (4.32) furthermore implies(4.42) I “ „ˆ ` p Θ R ˙ p σS ` η d λ q δ I p i q ` O p ρ { q ` O p} ˆ p } V q δ ij . Applying } ˆ p } V À ρ { and combining estimates (4.42) and (4.33) with (4.31) imply(4.43) (cid:68) L p , Z I p i q i p , Z I p j q j p (cid:69) L “ $&% ˆ ` p Θ R ˙ ” p σS ` η d λ q δ I p i q ` O p ρ { q ı , i “ j ; O p ε q E ij , i ‰ j ;To address the L p , ě bilinear form we employ (4.18) and arrive at the general form:(4.44) L p , ě Z I p i q i p “ ε ´ { ´ h p z p , γ p q ˜Θ i ` h p z p , γ p q ε ˜Θ i ¯ , where the functions h and h enjoy the properties of Notation 2.1 and localizednear Γ p . Integrating out z p first, and then from Lemma 4.2, (4.17), and the uniformbounds on εβ i we deduce(4.45) ˇˇˇ (cid:68) L p , ě Z I p i q i p , Z I p j q j p (cid:69) ˇˇˇ À E ij . The conclusion follows from (4.25), the estimates (4.26) and (4.43)-(4.45). (cid:3)
Coercivity estimates for the constrained bilinear form L p ˇˇ Z were derived in [10]for the weak functionalization provided with ρ „ ? ε . However, these results wouldlead to an ε dependent estimate, (see Lemma 4.9 ) which is too small to close theenergy estimates. In Lemma 4.6 we will require ρ “ o p q , independent of ε . Toachieve this, we introduce the modified slow spaces with improved orthogonality. EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Modified approximate slow spaces.
The preliminary approximate slowspace Z is not sufficiently invariant under L p to close the energy estimates inSection 5. We introduce modified spaces. Lemma 4.4.
For i P Σ , there exists ϕ k,i “ ϕ k,i p z p , γ p q , enjoying the properties ofNotation 2.1 and localized near Γ p such that (4.46) ż R (cid:96) ϕ k,i p z p , γ p q ψ I p i q p z p q d z p “ . The modified basis functions (4.47) Z I p i q i p , ˚ : “ ´ ˜ ψ I p i q ` ε ˜ ϕ ,i ¯ ˜Θ i ` ε ˜ ϕ ,i ε ˜Θ i “ ε ´ { ”` ψ I p i q ` εϕ ,i ˘ ˜Θ i ` εϕ ,i ε ˜Θ i ı , are L p is invariant up to order ε in L p Ω q , satisfying (4.48) L p Z I p i q i p , ˚ “ ´ Λ I p i q i ` εδ I p i q p σS ` η d λ q ` Cερ { ¯ Z I p i q i p , ˚ ` ε { ´ h ˜Θ i ` h ε ˜Θ i ¯ ` ε { ÿ k “ ´ ε k ´ B ks p h ,k ˜Θ i ` ε k ´ B ks p h ,k ε ˜Θ i ¯ . Here the functions h “ h p z p , γ p q enjoy the properties of Notation 2.1 and are local-ized near Γ p .Proof. To establish the Lemma it suffices to construct ϕ k,i in the interior regionas the dressing process incorporates only exponentially small errors. Using theexpansion (4.2) of L p , we compute(4.49) L p Z I p i q i p , ˚ “ L p , Z I p i q i p ` ε ¨ ε ´ { ´ L p , p ψ I p i q ˜Θ i q ` L p , p ϕ ,i ˜Θ i q ` L p , p ϕ ,i ε ˜Θ i q ¯ ` ε ¨ ε ´ { ´ L p , p ϕ ,i ˜Θ i q ` L p , p ϕ ,i ε ˜Θ i q ` ε { L p , ě Z I p i q i p , ˚ ¯ . The first term is calculated as in (4.6). Since L p , “ L p , from (2.33) we see that(4.50) L p , p ϕ k,i ˜Θ i q “ ` L p , ` ε β p ,i ˘ ϕ k,i ˜Θ i ` ´ L p ´ ` L p , ` ε β p ,i ˘ ¯ ´ ϕ k,i ˜Θ i ¯ . If ϕ k,i “ ϕ k,i p z p , γ p q in the sense of Notation 2.1 then the second term will be smallin the sense of (4.53). It remains to determine ϕ k,i for which the ε -order term in(4.49) equals Λ I p i q i p ϕ ,i Θ i ` ϕ ,i ε Θ i q to leading order. From (4.28), for k “ , ϕ k,i p¨ , γ p q as the L p R q solutions to(4.51) ´` L ` ε β p ,i ˘ ´ ` λ ` ε β p ,i ˘ ¯ ϕ k,i “ ´ g K k p z, γ p q` δ k ´´ δ I p i q p σS ` η d λ q ` C Λ I p i q i ` Ce ´ (cid:96)ν { ε ¯ ψ I p i q ´ g ˚ p z q ¯ . The definition is well posed since (4.29) and (4.41) imply that the right-hand sideof the identity is orthogonal to ψ I p i q in L p R q . Dressing these functions on Γ p , we extend ϕ k,i to Ω. Applying (4.50) and (2.33) implies L p , p ψ I p i q i ˜Θ i q ` L p , p ϕ ,i ˜Θ i q ` L p , p ϕ ,i ˜Θ i q“ ´ δ I p i q p σS ` η d λ q ` Cρ { ¯ ψ I p i q ˜Θ i ` ´ Λ I p i q i ` C } ˆ p } V ¯ p ϕ ,i ˜Θ i ` ϕ ,i ˜ ε Θ i q` ´ L p ´ ` L p , ` ε β p ,i ˘ ¯ p ϕ ,i ˜Θ i ` ϕ ,i ε ˜Θ i q` ε ´ h p z p , γ p q ˜Θ i ` h p z p , γ p q ε ˜Θ i ¯ . Returning this expansion to (4.49), we obtain(4.52) L p Z I p i q i p , ˚ “ ´ Λ I p i q i ` εδ I p i q p σS ` η d λ q ` Cερ { ¯ Z I p i q i p , ˚ ` ε L p , ´ ˜ ϕ ,i ˜Θ i ` ˜ ϕ ,i ε ˜Θ i ¯ ` ε L p , ě Z I p i q i p , ˚ ` ε ¨ ε ´ { ´ h p z p , γ p q ˜Θ i ` h p z p , γ p q ε ˜Θ i ¯ ` ε ´ L p ´ ` L p , ` ε β p ,i ˘ ¯ p ˜ ϕ ,i ˜Θ i ` ˜ ϕ ,i ε ˜Θ i q . Expanding the operators L p , and L p , ě , and using (4.17), we write the second andthird terms as L p , ´ ˜ ϕ ,i ˜Θ i ` ˜ ϕ ,i ε ˜Θ i ¯ ` L p , ě Z I p i q i p , ˚ “ ε ´ { ´ h p z p , γ p q ˜Θ i ` h p z p , γ p q ε ˜Θ i ¯ , where h and h are new general functions. The conclusion follows from this iden-tity, (4.52), and the relation(4.53) ´ L p ´ ` L p , ` ε β p ,i ˘ ¯ p ˜ ϕ ,i Θ i ` ˜ ϕ ,i ε Θ i q“ ÿ k “ ´ ε k B ks p h ,k p z p , γ p q Θ i ` ε k B ks p h ,k p z p , γ p q ε Θ i ¯ . (cid:3) The spans of the modified basis functions from (4.47) define the modified ap-proximate slow spaces(4.54) Z ˚ : “ Z ˚ Y Z ˚ with Z k ˚ “ span ! Z I p i q i p , ˚ , i P Σ k ) . We define the L p Ω q orthogonal projection on the subspace Z k ˚ by Π Z k ˚ for k “ , Z ˚ the bilinear form of the full linearized operator Π L p ˇˇ Z ˚ ,induces an N ˆ N matrix M ˚ with entries(4.55) M ˚ ij “ (cid:68) Π L p Z I p i q i p , ˚ , Z I p j q j p , ˚ (cid:69) L . By construction, ϕ k,i p k “ , q is perpendicular to ψ I p i q as (4.46), so as we estab-lished (4.20) it is easy to verify that under assumption (2.26)(4.56) (cid:68) Z I p i q i p , ˚ , Z I p j q j p , ˚ (cid:69) L “ $&% ´ ` p Θ R ` O ´ } ˆ p } V ¯¯ δ ij ` O ´ ε { } ˆ p } V ¯ E ij , I p i q “ I p j q ,O ´ ε { } ˆ p } V ^ ε } ˆ p } V ¯ E ij , I p i q ‰ I p j q . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW From the definition of the zero-mass projection Π , the identity (4.55) can bewritten as(4.57) M ˚ ij “ (cid:68) L p Z I p i q i p , ˚ , Z I p j q j p , ˚ (cid:69) L ´ | Ω | ż Ω L p Z I p i q i p , ˚ d x ż Ω Z I p j q j p , ˚ d x. To estimate M ˚ ij , we need to control the mass of Z I p j q j p , ˚ and its image under L p . Corollary 4.5.
Under assumption (2.26) , for j P Σ , then there exist unit vectors e “ p e j q j P Σ such that (4.58) ż Ω Z I p j q j p , ˚ d x “ O p ε { q e j . and (4.59) ż Ω L p Z I p j q j p , ˚ d x “ O ´ ε { p δ j ` } ˆ p } V q ¯ e j . Furthermore, if j P Σ , then there exists a unit vector e “ p e j q j P Σ such that (4.60) ż Ω Z I p j q j p , ˚ d x “ O ´ ε { } ˆ p } V ¯ e j . Proof.
With Z I p j q j p , ˚ introduced in (4.47), we have ż Ω Z j p , ˚ d x “ ε { ż R (cid:96) ψ I p j q p z p q d z p ż I p ˜Θ j d˜ s p ` ε { ż R (cid:96) ż I p ´ ϕ ,j ˜Θ j ` ϕ ,j ε ˜Θ j ¯ p ´ εz p κ p q d˜ s p d z p . The integration of ˜Θ j with respect to ˜ s p is zero for j P Σ zt u while for j “ ψ I p j q “ ψ has odd parity in z p . We deduce that the first term on the right-handside is zero. After integrating with respect to z p , the second integral takes the form(4.61) ε { ˆ (cid:68) h p γ p qq , ˜Θ j (cid:69) J p , ` (cid:68) h p γ p qq , ε ˜Θ j (cid:69) J p , ˙ . The estimate (4.58) follows from (2.58). To derive (4.59) we employ Lemma 4.4and the estimate (4.60). The error bound involves the V -norm instead of the V -norm of ˆ p because there is an additional higher derivative acting on h “ h p γ p q asshown in (4.48). For the final estimate, follows from the decomposition (2.51) andestimate (2.55) since 0 R Σ , (4.60). (cid:3) Applying the orthogonality and mass estimates (4.56) and (4.58) to (4.57) yieldsthe expansion(4.62) M ˚ ij “ $’’’’’’&’’’’’’% p ` p Θ { R q Λ i ` ε p σS ` η d λ q ` O p ερ { q if i “ j, I p i q “ p ` p Θ { R q Λ i ` O p ερ { q if i “ j, I p i q “ O ´ ε , ε { } ˆ p } V ¯ E ij if i ‰ j, I p i q “ I p j q ; O ´ ε , ε { } ˆ p } V ^ ε } ˆ p } V ¯ E ij if I p i q ‰ I p j q ; We decompose M ˚ into a block structure corresponding to the pearling and mean-dering spaces,(4.63) M ˚ “ ˆ M ˚ p , q M ˚ p , q M ˚ p , q M ˚ p , q ˙ , M ˚ ij p k, l q “ M ˚ ij for i P Σ k , j P Σ l . The matrix E is norm-one as an operator from l p R N q to l p R N q , hence the N ˆ N subblock matrix M ˚ p , q is diagonally dominant if } ˆ p } V ! ε { and ρ is suitablysmall independent of ε . In particular, under the pearling stability condition (4.64) σS ` η d λ ą M ˚ p , q is positive definite. This result is formulated in the following Lemma. Lemma 4.6.
Let assumption (2.26) and the pearling stability condition (4.64) hold,uniformly in ε and ρ . If in addition } ˆ p } V À ε { ρ { and ρ ! , then for all q P l p R N q we have q T M ˚ p , q q ě ε p σS ` η d λ q} q } l . We define the pearling mode component Q associate to a vector q “ p q j q j P l via the relation(4.65) Q : “ ÿ j P Σ q j Z j p , ˚ , then (4.60) and (4.59) imply(4.66) ż Ω Q d x “ O ´ ε } ˆ p } V } q } l ¯ , ż Ω L p Q d x “ O ´ ε } ˆ p } V } q } l ¯ . Recalling the a ^ b notation from Section 1.3, the pearling mode component satisfiesthe following bounds. Lemma 4.7.
With the same assumptions as Lemma 4.6, we have the estimates } Q } L „ } q } l , and } Π Z ˚ Π L p Q } L À ´ ε ` ε { } ˆ p } V ^ ε } ˆ p } V ¯ } q } l . Moreover, if we assume w P Z K˚ and v K “ w ` Q is mass free, then (cid:104) Π L p Q, w (cid:105) L À ´ ε } q } l ` ´ ε { } ˆ p } V ^ ε } ˆ p } V ¯ } q } l ¯ } w } L ` ε } ˆ p } V } ˆ p } V } q } l , (cid:104) L p Q, w (cid:105) L À ´ ε } q } l ` ´ ε { } ˆ p } V ^ ε } ˆ p } V ¯ } q } l ¯ } w } L , } Π Z ˚ Π L p w } L À ´ ε ` ´ ε { } ˆ p } V ^ ε } ˆ p } V ¯¯ } w } L ` ε } ˆ p } V } q } l , } Π Z ˚ L p w } L À ´ ε ` ´ ε { } ˆ p } V ^ ε } ˆ p } V ¯¯ } w } L . Proof.
The equivalence of the L and l norms, } Q } L „ } q } l , follows directly fromthe orthogonality relation (4.56) which requires the condition } ˆ p } V ! ε { . For thesecond estimate we remark that } Π Z ˚ Π L p Q } L “ ˜ ÿ i P Σ (cid:10) Π L p Q, Z i p , ˚ (cid:11) L ¸ { “ } M ˚ p , q q } l . Applying (4.62) for the case I p i q ‰ I p j q yields the bounds. EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW For the remainder we take w P Z K˚ with the sum w ` Q mass-free, then from(4.66) we have(4.67) (cid:104) Π L p Q, w (cid:105) L “ (cid:104) L p Q, w (cid:105) L ´ | Ω | ż Ω L p Q d x ż Ω Q d x, “ (cid:104) L p Q, w (cid:105) L ` O ´ ε } ˆ p } V } ˆ p } V } q } l ¯ . To estimate the projection of L p Q onto w we apply Lemma 4.4, obtaining(4.68) (cid:104) L p Q, w (cid:105) L “ ε ÿ i P Σ q i ÿ k “ ε ´ { (cid:68) ε k B ks p h ˜Θ i ` ε k B ks p h ε ˜Θ i , w (cid:69) L ` O ` ε } w } L } q } l ˘ . The h terms can be bounded in two ways,(4.69) } ε k B ks p h } L À } ˆ p } V À ε ´ { } ˆ p } V , } ε k ´ B ks p h } L À } ˆ p } V À } ˆ p } V . Applying H¨older’s inequality to the first term on the right hand side of (4.68) yieldsthe estimate of the projection of L p Q into w . The estimate of the projection ofΠ L p Q onto w follows from (4.67).To estimate the projection of Π L p w onto the modified slow space Z ˚ , we usethe definition of Π (4.70) (cid:68) Π L p w, Z I p j q j p , ˚ (cid:69) L “ (cid:68) L p w, Z I p j q j p , ˚ (cid:69) L ´ | Ω | ż Ω Z I p j q j p , ˚ d x ż Ω L p w d x, and apply the estimate (4.58) and identity (4.74) from Lemma 4.9 below to deduce(4.71) ›› Π Z ˚ Π L p w ›› À } Π Z ˚ L p w } L ` ε p} w } L ` ε } ˆ p } V } q } l q . To bound the first term on the right-hand side of inequality (4.71), we write(4.72) (cid:68) L p w, Z I p j q j p , ˚ (cid:69) L “ (cid:68) w, L p Z I p j q j p , ˚ (cid:69) L , which combined with Lemma (4.4) and the orthogonality of w and Z ˚ implies (cid:68) w, L p Z I p j q j p , ˚ (cid:69) L “ ε (cid:68) w, ε ´ { p h ˜Θ j ` h ε ˜Θ j q (cid:69) L ` ε { ÿ k “ (cid:68) w, ε k ´ B ks p h ,k ˜Θ i ` ε k ´ B ks p h ,k ε ˜Θ i (cid:69) L , where the functions h “ h p z p , γ p q enjoy the properties of Notation 2.1. The esti-mate for projection of L p w onto Z ˚ follows from (4.69), and the conclusion followsfrom (4.71). (cid:3) The proof for the previous Lemma also implies the following result.
Corollary 4.8.
Under the assumptions of Lemma 4.6, for all v in L we have } Π Z K˚ L p Π Z ˚ v } L ` } Π Z K˚ Π L p Π Z ˚ v } L À ´ ε ` ´ ε { } ˆ p } V ^ ε } ˆ p } V ¯¯ } v } L . Coercivity.
In this section we establish the coercivity of the linearized op-erator Π L p on the space orthogonal to the modified approximate slow space Z ˚ .This is an extension of Theorem 2 . Lemma 4.9.
Suppose w P Z K˚ and ρ ą is suitably small independent of ε , thenfor ε small enough there is a positive constant C such that (4.73) (cid:104) L p w, w (cid:105) L ě Cρ ` ε } w } H ` } w } L ˘ and } L p w } L ě Cρ (cid:104) L p w, w (cid:105) L . Moreover, if v K “ w ` Q is mass free, then we have the estimate (4.74) ˇˇ (cid:104) L p w (cid:105) L ˇˇ “ | Ω | ˇˇˇˇż Ω L p w d x ˇˇˇˇ À ε { } w } L ` ε { } ˆ p } V } q } l , and in addition (4.75) Cε } ˆ p } V } q } l ` (cid:104) Π L p w, L p w (cid:105) L ě } L p w } L . Proof.
To establish (4.73), we introduce(4.76) L p , : “ ´ ε ∆ ` W p Φ p q ´ εη , and rewrite the linearized operator L p defined by (4.1) in the form L p “ p L p , q ` ε R where R “ ´ εη ´ W p Φ p q ε ´ ε ∆Φ p ´ W p Φ p q ¯ ` η d W p Φ p q . Since R is bounded in L -norm, then it follows that (cid:104) L p w, w (cid:105) L ě (cid:68) p L p , q w, w (cid:69) L ´ ε } R } L } w } L , and moreover for some C ą ε , } L p w } L ě ››› p L p , q w ››› L ´ ε C } R } L } w } L . Imposing the condition ε ! ρ “ o p q , then the coercivity estimates (4.73) for L p follow from Theorem 2.5 of [24] by replacing the preliminary approximate slowspace Z with the modified approximation Z ˚ . It remains to obtain estimates (4.74)and (4.75). From the definition of Π ,(4.77) (cid:104) Π L p w, L p w (cid:105) L “ } L p w } L ´ | Ω | ˆż Ω L p w d x ˙ . To estimate the mass term we turn to the definition, (4.1), of L p which implies(4.78) ż Ω L p w d x “ ż Ω ” ` ε ∆ ´ W p Φ p q ` εη ˘ ` ε ∆ ´ W p Φ p q ˘ w ´ ` ε ∆Φ p ´ W p Φ p q ˘ W p Φ p q w ` εη d W p Φ p q w ı d x. Since w satisfies periodic boundary conditions, both ∆ w and ∆ w has no masswhich allows us to rewrite (4.78) as(4.79) ż Ω L p w d x “ I ` I ` I , EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW where the terms I k p k “ , , q are defined by(4.80) I : “ ´ ε ż Ω W p Φ p q ∆ w d x, I : “ ż Ω ` W p Φ p q ˘ w d x, I : “ ´ ż Ω “` ε ∆Φ p ´ W p Φ p q ˘ W p Φ p q ´ ε p η d ´ η q W p Φ p q ‰ w d x. We address these terms one by one. For the first term we integrate by parts andadd a zero term I “ ´ ε ż Ω ` ∆ W p Φ p q ˘ w d x “ ´ ε ż Ω ∆ ` W p Φ p q ´ W p φ q ˘ w d x. Since ε ∆ p W p Φ p q ´ W p φ qq is bounded in L and exponentially localized nearthe interface Γ p we obtain(4.81) | I | À ε { } w } L . By the definition of Φ p , the quantity ε ∆Φ p ´ W p Φ p q is order of ε in L we deducethat the part of the integrand in the backets in I is order of ε in L , hence(4.82) | I | À ε } w } L . Finally, to bound I we decompose it into near and far-field parts(4.83) I “ ż Ω ” ` W p Φ p q ˘ ´ ` W p φ q ˘ ı w d x ` ` W p φ q ˘ ż Ω w d x. The mass of w balances with the mass of Q . From (4.60) and Lemma 4.7 we deducethat(4.84) | I | À ε { } w } L ` ε { } ˆ p } V } q } l . Combining estimates for I k p k “ , , q in (4.81)-(4.84) yields (4.74). We deduce(4.75) from these results combined with (4.77). (cid:3) Nonlinear flow
In a periodic domain in two space dimensions the bilayer dressings of circularcodimension-one interfaces form a three parameter family of equilbrium charac-terized by the interfacial radius and the location of the center of the circle. Theequilibrium solutions have a unique bulk density value, σ ˚ . We develop the re-duction of the FCH mass-preserving gradient flow based upon perturbations to theinterfacial shape parameterized by the meander modes, p “ p p t q P R N , evanes-cent excitations of bilayer width perturbations parameterized by the pearling modes q “ q p t q P R N , and a dynamic bulk density parameter σ “ σ p t q P R which is slavedto p through conservation of total mass. For system parameters corresponding tothe pearling stable regime and initial data that are proximal to the bilayer dressingof a circular codimension-one bilayer interface, we capture the flow via a Galerkin-type reduction of the normal velocity given at leading order via a closed-form ODEfor the meander modes. The bulk density parameter exerts the most dramaticimpact on the flow, switching it from a curve lengthening regime regularized bysurface diffusion when σ ą σ ˚ to a more familiar curve shortening regime when σ ă σ ˚ . In the curve lengthening regime we show that evolution generically leadsto a transient departure from circularity whose duration and size are controlledby σ p q ´ σ ˚ . For sufficiently small perturbations the transient relaxes back to a larger circular interface whose radius is determined by the total mass of the initialdata. In the curve shortening regime, the flow restores circularity monotonically,converging to a circular equilibrium with smaller radius. Section 5.1 presents thedecomposition of the full solution into the bilayer dressing ansatz parameterizedby the meander and the pearling modes plus an infinite dimensional but stronglydamped residual w “ w p x, t q . The dynamics of the meandering parameters arederived in Section 5.2, and the proof of the main theorem is presented in Section5.3.5.1. Decomposition of the flow.
We fix an initial circular interface Γ and de-compose solutions u of the FCH gradient flow (1.3) as(5.1) u p x, t q “ Φ p p x ; σ q ` v K p x, t ; q q , v K P p Z ˚ q K , ż Ω v K d x “ , where Φ p is a bilayer dressing of a perturbed interface Γ p introduced in Lemma 3.2and Definition 2.3 respectively. Here p “ p p t q are the coordinates in Z ˚ , the mean-dering approximate slow space introduced in (4.54). The bulk density parameter σ is defined by (3.31) to render v K mass free. Substituting the ansatz (5.1) into theequation (1.3) leads to a system for p and v K :(5.2) B t Φ p ` B t v K “ ´ Π F p Φ p q ´ Π L p v K ´ Π N p v K q , where N p v K q is the nonlinear term and defined by(5.3) N p v K q : “ F p Φ p ` v K q ´ F p Φ p q ´ L p v K . In what follows, we make the a priori assumptions that subsume those of 2.26,(5.4) | p p t q| ď | ln ε | ´ , } ˆ p } V À ε { ρ { , } p } l À ε { | ln ε | ` ε } ˆ p } V , These assumptions may be evoked without explicit mention. Moreover, from Lemma2.4 they have the following consequences(5.5) } ˆ p } V À } ˆ p } V À ε { , } ˆ p } V À N { } ˆ p } V ď , } p } l À ε { ρ { . The time derivative of Φ p satisfies the chain rule(5.6) B t Φ p “ p ¨ ∇ p Φ p , where pB p j Φ p q j P Σ are calculated in (3.47) for j “ , j ě
3. To sim-plify these expressions we break them into dominant and remainder terms. Usingthe form (4.47) of the basis elements of the meandering space Z ˚ , the definition ofthe rescaled eigenfunctions ˜ ψ “ ε ´ { φ { m and the Cartesian basis vectors (2.23),and the asymptotics of the logarithmic derivatives of the curve length, (2.36), weobtain the reformulation(5.7) B Φ p B p j ´ m ε { Z j p , ˚ “ φ ε p n p ´ n q ¨ m j ´ φ ε n p ¨ n B p j p ln | Γ p |q ˜ s p ¯ p ` R j , where m j “ E j for j “ ,
2, and m j “ ˜Θ j n for j P Σ zt , u , and R “p R , . . . R N q is the remainder which admits the upper bound,(5.8) } R } L À ε { p ` } ˆ p } V q . The N constraints implicit in the requirement v K K Z ˚ yield an ODE systemfor the evolution of p . This system is realized by projecting equation (5.2) onto Z ˚ .The orthogonal projection drives the growth of v K and it is essential to obtain anupper bound. This information is contained in (5.7) and Lemma 2.7 which bounds EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW the difference between n p and n . Combining this with the localization of theright-hand side of (5.7) to the interface we see that the evolution normal to Z ˚ issmall compared to the evolution of p , ››› Π p Z ˚ q K pB t Φ p q ››› L À ´ ε ´ { } ˆ p } V ` ε { p ` } ˆ p } V q ¯ } p } l . From (5.5) we assume } ˆ p } V !
1, and since Z K˚ Ă p Z ˚ q K we have the results(5.9) } Π p Z ˚ q K B t Φ p } L ď } Π p Z ˚ q K B t Φ p } L À ´ ε ´ { } ˆ p } V ` ε { ¯ } p } l . The operator Π L p is coercive on the perp of Z ˚ which includes the pearling space Z ˚ , and its dynamic parameters q “ p q i p t qq i P Σ . This motivates the decompositionof the perturbation v K as(5.10) v K “ Q p x, t q ` w p x, t q with w P Z K˚ p Γ p , ρ q , and Q p x, t q as defined in (4.65) with q j “ (cid:68) v K , Z j p , ˚ (cid:69) L . The decompositions (5.1)and (5.10) are well-posed initially by the following Lemma. Lemma 5.1.
Consider initial data u “ Φ p σ q ` v with | σ | À and } v } L ` ε } v } H À ε . Then for ε small enough, there exist unique constants p p q and σ “ σ p p p qq such that the following decomposition holds, (5.11) u “ Φ p p q p σ q` v K “ Φ p p q p σ q` Q ` w , with ż Ω v K d x “ , w P Z K˚ p Γ p p q , ρ q , where the pearling mode component and parameters are defined by Q “ ÿ j P Σ q j p q Z j p p q , ˚ with q j p q “ (cid:68) v K , Z j p p q , ˚ (cid:69) L . Moreover, for k “ , , we have the bounds: } v K } H k À ε ´ k ´ { } p p q} V ` ε ´ { } p p q} V k ` } v } H k , } q p q} l À } v K } L ; } w } H k À } v K } H k ` ε ´ k } v K } L , } p p q} V k À ε p ´ k q{ } v } L The proof of this Lemma is postponed to the appendix. In the remainder ofthis section we obtain the energy estimates for w and q separately, which requirean L -bound of the nonlinear term N p v K q . The ansatz is defined along a movingframe near Γ p and while the flow is very slow in the sense of (5.4) and (5.5), thisadds extra terms to time derivatives. Also the slow space Z ˚ “ Z ˚ p Γ p , ρ q changesas times varies in a speed can be bounded by the flow speed } p } l . Indeed,(5.12) B t Z I p j q j p , ˚ “ ε ´ ÿ i P Σ p i ˜Θ i ˜ ψ I p j q ˜Θ j ` O p} p } l q , which combined with the l - l estimate (2.25) and the scaling of N from (4.8) yield(5.13) }B t Z I p j q j p , ˚ } L À ε ´ } p } l . More significantly, the approximate space Z ˚ is not the invariant under the actionof the linearization, and this produces several linear terms that present an issuewhen closing the energy estimates. Indeed this motivates the construction of themodified approximate slow space Z ˚ from Z . Energy estimate of w . We derive two H estimates on w . The first, (5.15) issharper in terms of ε and plays an important role in the proof of the slow conver-gence toward equilibrium in Step w tends to zero as q Ñ and p Ñ p ˚ and is used to prove the conver-gence to equilibrium in Step v K , to write (5.2) as an evolution for w ,(5.14) B t w ` Π L p w “ ´B t Φ p ´ B t Q ´ Π F p Φ p q ´ Π L p p Q p x, t qq ´ Π N p v K q . Lemma 5.2.
Under the a priori assumptions (5.4) , the function w P Z K˚ , obeys (5.15) dd t (cid:104) L p w, w (cid:105) L ` } L p w } L À } p } l ` ε ρ ´ p} q } l ` } q } l q ` ε ` } N p v K q} L . In addition, as p relaxes to its equilibrium p ˚ , the error is controlled by the long-time estimates (5.16)dd t (cid:104) L p w, w (cid:105) L ` } L p w } L À} p } l ` ε ρ ´ ` } q } l ` } q } l ˘ ` ε p| p ´ p ˚ | ` } ˆ p } V q` ε } ˆ p } V ` } N p v K q} L . Proof.
Since the linearized operator L p depends on time through p , we have(5.17) 12 dd t (cid:104) L p w, w (cid:105) L “ (cid:104) B t w, L p w (cid:105) L ` (cid:104) B t p L p q w, w (cid:105) L . Considering the last term on the right-hand side, the definition (4.1) of L p providesthe expansion B t p L p q “ ´ ` ε ∆ ´ W ` εη ˘ ` W B t Φ p ˘ ´ W B t Φ p ` ε ∆ ´ W ˘ ´ ` ε ∆Φ p ´ W ˘ W p q B t Φ p ´ W ` ε ∆ ´ W ˘ B t Φ p ` εη d W B t Φ p , where W is evaluated at Φ p . Since Φ p is uniformly bounded in L and in L after action by powers of ε ∆, we identify the upper bound on the bilinear formgenerated by B t p L p q (5.18) (cid:104) B t p L p q w, w (cid:105) L À ` } p ¨ ∇ p Φ p } L ` } p ¨ ∇ p p ε ∆Φ p q} L ˘ ` } w } L ` } ε ∆ w } L ˘ . From (3.48), the local-coordinate expansion (2.10) of ε ∆, and the l - l estimate(2.25), we have } p ¨ ∇ p Φ p } L ` } p ¨ ∇ p p ε ∆Φ p q} L À ε ´ } p } l À ε ´ N { } p } l . Recalling (4.8) we have N À ε ´ ρ { ď ε ´ , and evoking the a priori assumption } p } l À ε { and the coercivity estimate (4.73), we obtain the upper bound on thebilinear term(5.19) (cid:104) B t p L p q w, w (cid:105) L À ερ ´ } L p w } L ď ε { } L p w } L . Returning to (5.17), substituting (5.14) for B t w , using the coercivity estimate (4.75)and the a priori assumption } ˆ p } V À ε { from (5.4) on the first term on the right-hand side, and bounding the second term via the bilinear estimate above, leadsto(5.20) dd t (cid:104) L p w, w (cid:105) L ` } L p w } L ď Cε } q } l ´ (cid:104) B t Φ p ` B t Q ` Π L p Q, L p w (cid:105) L ´ (cid:10) Π F p Φ p q ` Π N p v K q , L p w (cid:11) L . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Considering the terms on the right-hand side of (5.20), we apply H¨older’s inequalityto the last term(5.21) ˇˇ (cid:10) Π N p v K q , L p w (cid:11) L ˇˇ À } N p v K q} L } L p w } L . We separate B t Φ p into its projection onto Z ˚ and its complement Z K˚ in L , writing(5.22) (cid:104) B t Φ p , L p w (cid:105) L “ (cid:10) Π Z ˚ B t Φ p , L p w (cid:11) L ` (cid:68) Π Z K˚ B t Φ p , L p w (cid:69) L . Since L p is self-adjoint in L and w P Z K˚ , we may re-write the first term on theright-hand side,(5.23) (cid:104) B t Φ p , L p w (cid:105) L “ (cid:68) Π Z K˚ L p Π Z ˚ B t Φ p , w (cid:69) L ` (cid:68) Π Z K˚ B t Φ p , L p w (cid:69) L . Employing H¨older’s inequality, estimate (5.9) and Corollary 4.8, we obtain(5.24) ˇˇ (cid:104) B t Φ p , L p w (cid:105) L ˇˇ Àp ε ` ε { } ˆ p } V q}B t Φ p } L } w } L ` p ε { ` ε ´ { } ˆ p } V q} p } l } L p w } L . The identity (5.6) and the estimate (5.7) allow control of the time derivative of Φ p , }B t Φ p } L À ε ´ { } p } l . and hence to factor } p } l out of the right-hand side of (5.24). Using the a prioriassumption } ˆ p } V À ε { , the coercivity Lemma 4.9 shows the second term onright-hand side of (5.24) dominates the first. These arguments establish the bound(5.25) ˇˇ (cid:104) B t Φ p , L p w (cid:105) L ˇˇ À } p } l } L p w } L . For the Π L p Q term we project onto Z ˚ and its complement, use Q P Z ˚ , Corollary4.8, and finally the coercivity of Lemma 4.9 to establish(5.26) ˇˇ (cid:104) Π L p Q, L p w (cid:105) L ˇˇ “ ˇˇˇ (cid:68) Π Z K˚ Π L p Q, L p w (cid:69) L ` (cid:68) Π Z K˚ L p Π Z ˚ Π L p Q, w (cid:69) L ˇˇˇ À p ε ` ε { } ˆ p } V q} q } l p} w } L ` } L p w } L qÀ ερ ´ } q } l } L p w } L For the third term on right-hand side of (5.20) requires an investigation of B t Q .Applying Lemma 2.10 yields(5.27) B t Q “ ÿ j P Σ q j Z j p , ˚ ` ÿ j P Σ q j B t Z j p , ˚ . Note that the third term can be written as(5.28) (cid:104) B t Q, L p w (cid:105) L “ (cid:68) Π Z K˚ L p B t Q, w (cid:69) L . By employing (5.13) and Corollary 4.8 we may bound(5.29) } Π Z K˚ L p B t Q } L À p ε ` ε { } ˆ p } V q} q } l ` ε ´ } p } l } q } l . Here by the l - l estimate and scaling of N from (4.7), we have(5.30) } q } l ď ε ´ { } q } l . Using Holder’s inequality, the a priori assumptions and the coercivity to bound } w } L by } L p w } L we deduce from (5.28) - (5.30)(5.31) ˇˇ (cid:104) B t Q, L p w (cid:105) L ˇˇ À ερ ´ } L p w } L p} q } l ` } q } l q . It remains to bound the F p Φ p q term on the right-hand side of (5.20). Subtractingoff the main far-field value of F p Φ p q , using the definition of Π , Lemma 3.5, and(4.74) we find(5.32) ˇˇ (cid:104) Π F p Φ p q , L p w (cid:105) L ˇˇ “ ˇˇˇˇ (cid:104) F p Φ p q ´ F m , L p w (cid:105) L ` | Ω | ż Ω p F p Φ p q ´ F m q d x ż Ω L p w d x ˇˇˇˇ , ď ˇˇ (cid:104) F p Φ p q ´ F m , L p w (cid:105) L ˇˇ ` Cε { ´ } w } L ` ε } ˆ p } V } q } l ¯ . From Lemma 3.2 we have the expansionF ´ F m “ ε F ` ε p F ´ F q ` ε p F ě ´ F q ` e ´ (cid:96)ν { ε F e . Using Lemma 3.4 to bound the L -norm of the F , F ě and F e terms yields(5.33) ˇˇ (cid:104) F p Φ p q ´ F m , L p w (cid:105) L ˇˇ À ε { | σ ´ σ ˚ |} L p w } L ` ε ˇˇ (cid:104) F ´ F , L p w (cid:105) L ˇˇ ` ε { p} ˆ p } V ` q} L p w } L . The estimate of F is more delicate in order to establish a better bound. FromLemma 3.2, F admits the form:(5.34) F ´ F “ ´ φ ∆ s p κ p ` f p z p , γ p q ´ f where f ´ f is localized near the interface and enjoys the properties of Notation2.1. Applying H¨olders inequality, bounding the localized function f ´ f in L ,and using (2.52) we obtain(5.35) ˇˇ (cid:104) F ´ F , L p w (cid:105) L ˇˇ ď ˇˇ (cid:10) φ ∆ s p κ p , L p w (cid:11) L ˇˇ ` Cε { p ` } ˆ p } V q} L p w } L . The remaining task is to use the form of κ p and the orthogonality of w to Z ˚ to obtain an optimal estimate for the first term on the right-hand side of (5.35).Indeed, employing (2.46) from Lemma 2.7 and bounding the error terms, the highorder curvature derivative term is bounded by ˇˇ (cid:10) φ ∆ s p κ p , L p w (cid:11) L ˇˇ ď ε { m p R ` p Θ q (cid:42) N ´ ÿ j “ p ´ β j q β j p j Z j p , L p w (cid:43) ` Cε { ´ } ˆ p } V } ˆ p } V ` } ˆ p } V } ˆ p } V ¯ } L p w } L . Bounding the Z ˚ projection of L p w from Lemma 4.7 and using the a prior estimatesand the expansion (4.47) of Z j p , ˚ we obtain(5.36) ˇˇ (cid:10) φ ∆ s p κ p , L p w (cid:11) L ˇˇ À ε } ˆ p } V } L p w } L ` ε { p ε ` ε { } ˆ p } V q} w } L . Combining estimates (5.36) with (5.35) and using the coercivity of Lemma 4.9 weobtain(5.37) ˇˇ (cid:104) F ´ F , L p w (cid:105) L ˇˇ À ε } ˆ p } V } L p w } L ` ε { } L p w } L . By Lemma 2.4 we have(5.38) } ˆ p } V ď ε ´ } ˆ p } V À ε ´ { . Substituting the bound above into the previous estimate (5.37), and then that resultinto the right-hand side of (5.33) and combining the resulting estimate with (5.32)yields(5.39) ˇˇ (cid:104) Π F p Φ p q , L p w (cid:105) L ˇˇ À ε { | σ ´ σ ˚ |} L p w } L ` ε { } L p w } L ` ε } q } l . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Combining the estimates (5.21), (5.25)-(5.31) and (5.39) with (5.20) and usingYoung’s inequality, yields the first estimate (5.15). The derivation of the secondestimate (5.16) differs only in the use of Young’s inequality ˇˇ (cid:104) Π F p Φ p q , L p w (cid:105) L ˇˇ ď } Π F p Φ p q} L } L p w } L , and then bounding the L -norm of Π F p Φ p q from Lemma 3.4. (cid:3) Estimates of modulation parameters q p t q . We derive l estimates of q and q , and hence L estimates of Q and B t Q defined in (4.65) and (5.27). We rewrite(5.14) as an evolution for Q ,(5.40) B t Q ` Π L p Q “ ´B t Φ p ´ B t w ´ Π F p Φ p q ´ Π L p w ´ Π N p v K q . Lemma 5.3.
Under assumptions (5.4) , if the pearling stability condition (4.64) holds, then there exists C ą independent of ε such that the pearling parameters q “ p q k p t qq k P Σ obey } q } l À } q } l ` } N p v K q} L ` ε } w } L ` ε | ln ε | } ˆ p } V ` ε } ˆ p } V ; B t } q } l ` Cε } q } l À ε } w } L ` ε ´ } N p v K q} L ` ε | ln ε | } ˆ p } V ` ε } ˆ p } V . Proof.
Taking the L -inner product of equation (5.40) with Q yields(5.41) (cid:104) B t Q, Q (cid:105) L ` (cid:104) Π L p Q, Q (cid:105) L “ ´ (cid:104) B t w, Q (cid:105) L ´ (cid:104) B t Φ p , Q (cid:105) L ´ (cid:104) Π F p Φ p q , Q (cid:105) L ´ (cid:104) Π L p w, Q (cid:105) L ´ (cid:10) Π N p v K q , Q (cid:11) L . Using (5.27) we rewrite the first term on the left-hand side as(5.42) (cid:104) B t Q, Q (cid:105) L “ ÿ i,j P Σ q i q j (cid:68) Z i p , ˚ , Z j p , ˚ (cid:69) L ` ÿ i,j P Σ q i q j (cid:68) B t Z i p , ˚ , Z j p , ˚ (cid:69) L . The partial orthogonality of the basis t Z j p , ˚ u j P Σ , from (4.56), and the a prioriestimate } ˆ p } V À ε { ρ { yield(5.43) ÿ i,j P Σ q i q j (cid:68) Z i p , ˚ , Z j p , ˚ (cid:69) L ě R ` p Θ R B t } q } l ´ Cε { } ˆ p } V } q } l } q } l ě R ` p Θ R B t } q } l ´ Cερ { } q } l } q } l . Applying H¨older’s inequality to the second term on the right-hand side of (5.42),the estimate (5.13) on B t Z j p , ˚ , the a priori estimate } p } l À ε { ρ { , the estimate(5.30) yields the bound(5.44) ÿ i,j P Σ q i q j (cid:68) B t Z i p , ˚ , Z j p , ˚ (cid:69) L À ε ´ } p } l } q } l } q } l À ερ { } q } l Combining estimates (5.43)-(5.44) with (5.42) and applying Young’s inequality yield R ` p Θ R B t } q } l ´ Cερ { } q } l ´ Cερ { } q } l ď (cid:104) B t Q, Q (cid:105) L , which when substituted into (5.41) implies(5.45) R ` p Θ R B t } q } l ` (cid:104) Π L p Q, Q (cid:105) L ď Cερ { } q } l ` Cερ { } q } l ´ (cid:104) B t w, Q (cid:105) L ´ (cid:104) B t Φ p , Q (cid:105) L ´ (cid:10) Π N p v K q , Q (cid:11) L ´ (cid:104) Π L p w, Q (cid:105) L ´ (cid:104) Π F p Φ p q , Q (cid:105) L . We bound the terms in (5.45) one by one. Since w P Z K˚ we may apply the expansion(5.27) of B t Q to deduce(5.46) | (cid:104) B t w, Q (cid:105) L | “ | (cid:104) w, B t Q (cid:105) L | “ ˇˇˇˇˇ ÿ j P Σ (cid:68) w, q j B t Z j p , ˚ (cid:69) L ˇˇˇˇˇ . From the estimate (5.13), H¨older’s inequality, and the l - l estimate (2.25) weobtain(5.47) | (cid:104) B t w, Q (cid:105) L | À ε ´ } p } l } w } L } q } l À ερ { } w } L } q } l . To bound the term involving B t Φ p we recall the expansion (5.7) for which theleading order term resides in the meandering slow space Z ˚ and hence is largelyorthogonal to Q by (4.56). In addition, since φ “ m ψ is perpendicular to ψ in L p R q , the projection the right-hand side of (5.7) onto Q is zero at leading order.With these observations, we deduce from Lemma 4.2 and the a priori bound of } p } l that(5.48) ˇˇ (cid:104) B t Φ p , Q (cid:105) L ˇˇ À ε { } ˆ p } V } p } l } q } l À ´ ε { | ln ε |} ˆ p } V ` ε { } ˆ p } V ¯ } q } l . For the inner product with the nonlinear term, Lemma 4.7 implies that } Q } L „} q } l so H¨older’s inequality yields(5.49) ˇˇ (cid:10) Π N p v K q , Q (cid:11) L ˇˇ À } N p v K q} L } q } l . For the L p w term, Lemma 4.7 and a priori estimates yield(5.50) ˇˇ (cid:104) Π L p w, Q (cid:105) L ˇˇ À ´ ε ` ε { } ˆ p } V ¯ } w } L } q } l ` ε } q } l À ερ { } w } L } q } l ` ε } q } l . The estimate the term involving the residual, F p Φ p q , is deferred to (5.55) of Lemma5.4. As a consequence of (5.47)–(5.50) and (5.55), the estimate (5.45) reduces to(5.51) R ` p Θ R B t } q } l ` (cid:104) Π L p Q, Q (cid:105) L À ερ { } q } l ` ερ { } q } l ` ´ } N p v K q} L ` ε { | ln ε |} ˆ p } V ` ε { } ˆ p } V ¯ } q } l ` ερ { } w } L } q } l . Since the system is pearling stable, Lemma 4.6 implies the existence of C ą ε for which, (cid:104) Π L p Q, Q (cid:105) L “ q T M ˚ p , q q ě Cε } q } l . The a priori estimates imply that | p | is small and hence p R ` p Θ q{p R q isbounded away from zero. Applying Young’s inequality to the right-hand side yields(5.52) B t } q } l ` Cε } q } l À ερ { } q } l ` ε ´ } N p v K q} L ` ε | ln ε | } ˆ p } V ` ε } ˆ p } V ` ε } w } L . It remains to bound } q } l . Taking the inner product of equation (5.40) with ř j P Σ q j Z j p , ˚ and estimating terms as above we obtain(5.53) } q } l À ´ (cid:42) Π L p Q, ÿ j P Σ q j Z j p , ˚ (cid:43) L ` ε p} q } l ` } w } L q ` } N p v K q} L ` ε | ln ε | } ˆ p } V ` ε } ˆ p } V . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW From the definition (4.65) of Q and the estimate (4.62) on M ˚ we rewrite the terminvolving Π L p Q as(5.54) ˇˇˇˇˇˇ (cid:42) Π L p Q, ÿ j P Σ q j Z j p , ˚ (cid:43) L ˇˇˇˇˇˇ “ ˇˇˇˇˇ ÿ i,j P Σ q i q j M ˚ ij p , q ˇˇˇˇˇ À } q } l } q } l . This establishes the l -estimate of q , and the estimate on B t } q } l follows from(5.52). (cid:3) The proof of Lemma 5.3 required the following estimate on the projection of theresidual to the pearling space.
Lemma 5.4.
Assuming (5.4) , the projection of the residual to pearling space sat-isfies the estimate (5.55) ˇˇ (cid:104) Π F p Φ p q , Q (cid:105) L ˇˇ À ε { } ˆ p } V } q } l ` ε { } ˆ p } V } q } l . Proof.
Subtracting off the far-field value of the residual and using the definition ofΠ , we have(5.56) (cid:104) Π F p Φ p q , Q (cid:105) L “ (cid:104) F p Φ p q ´ F m , Q (cid:105) L ´ | Ω | ż Ω p F p Φ p q ´ F m q d x ż Ω Q d x. From Lemma 3.5 and the estimate (4.60), the second term on the right-hand sideis estimated by(5.57) 1 | Ω | ż Ω p F p Φ p q ´ F m q d x ż Ω Q d x “ O ´ ε { } p } V } q } l ¯ . We use the expansion of F p Φ p q given in Lemma 3.2 to estimate the first term onthe right-hand side of (5.56). Examining the L -inner product of F and Q , sinceF “ p σ ˚ ´ σ q κ p f p z p q with f odd, the leading order vanishes since ψ has evenparity in z p . Integrating out z p we then deduce from Lemma 4.2 that(5.58) | (cid:104) F , Q (cid:105) L | “ ε { ˇˇˇˇˇ p σ ˚ ´ σ q ÿ j P Σ ż I p h p γ p q q j ˜Θ j d˜ s p ˇˇˇˇˇ À ε { | σ ˚ ´ σ |} ˆ p } V } q } l . Using the form of F for Lemma 3.2 we rewrite(5.59) (cid:104) F ´ F , Q (cid:105) L “ (cid:10) φ ∆ s p κ p , Q (cid:11) L ´ (cid:10) f p z p , γ p q ´ f , Q (cid:11) L . Similarly as the derivation of (5.58), we bound(5.60) ˇˇˇ (cid:10) f p z p , γ p q ´ f , Q (cid:11) L ˇˇˇ À ε { } q } l } ˆ p } V . For the curvature term ∆ s p κ p , since φ is perpendicular to ψ in L p R (cid:96) q , theleading order vanishes yielding(5.61) (cid:10) φ ∆ s p κ p , Q (cid:11) L “ ε { ÿ j P Σ q j ż I p ´ h p γ p q ∆ s p κ p ˜Θ j ` h p γ p q ∆ s p κ p ε ˜Θ j ¯ d˜ s p . Using the decomposition (2.51) for h k p k “ , q , we bound the difference | h k p γ p q ´ h k p γ p , q| by } ˆ p } V ,(5.62) ˇˇ (cid:10) φ ∆ s p κ p , Q (cid:11) L ˇˇ À ε { } ˆ p } V } ˆ p } V } q } l ` ε { ÿ j P Σ | q j | ˇˇˇˇˇż I p h ` γ p , ˘ ∆ s p κ p ˜Θ j ` h ` γ p , ˘ ∆ s p κ p ε ˜Θ j d˜ s p ˇˇˇˇˇ . From the expansion of κ p in Lemma 2.7, the estimate (4.17), the uniform bound on εβ p ,j for j P Σ, and the orthogonality of t ˜Θ j u in L p d˜ s p q with Σ and Σ disjoint,we deduce the first integral item on the right hand side can be bounded by(5.63) p} ˆ p } V } ˆ p } V ` } ˆ p } V } ˆ p } V q} q } l . Applying the V k bounds of ˆ p in (5.5), hence (5.62) and (5.63) yield(5.64) ˇˇ (cid:10) φ ∆ s p κ p , Q (cid:11) L ˇˇ À ε { } ˆ p } V } q } l . Combining estimates (5.60) and (5.64) with (5.59) implies(5.65) | (cid:104) F ´ F , Q (cid:105) L | À ε { } ˆ p } V } q } l ` ε { } ˆ p } V } q } l . In a similar manner we have ˇˇˇ (cid:68) F ´ F ` e ´ (cid:96)ν { ε F e , Q (cid:69) L ˇˇˇ À ε { } ˆ p } V } q } l , which combined with the estimates on F given by (5.58) and F from (5.65) yields(5.66) ˇˇ (cid:104) F p Φ p q ´ F m , Q (cid:105) L ˇˇ À ε { | σ ˚ ´ σ |} ˆ p } V } q } l ` ε { } ˆ p } V } q } l ` ε { } ˆ p } V } q } l . Combining estimates (5.66) and (5.57) with (5.56) completes the Lemma. (cid:3)
Estimates on the v K nonlinearity. The results of Lemmas 5.2, 5.3 and 5.10,incorporate L -bounds of the nonlinear term N p v K q . These are established in thefollowing Lemma. Lemma 5.5. If } v K } L p Ω q is bounded independent of ε , then (5.67) } N p v K q} L À ε ´ ´ ρ ´ (cid:104) L p w, w (cid:105) L ` } q p t q} l ¯ , Moreover, if v K “ w ` Q as in (5.10) then it admits upper bound } v K } L À ε ´ ´ ρ ´ (cid:104) L p w, w (cid:105) { L ` } q p t q} l ¯ . Proof.
From the definition (5.3) of the nonlinear term N p v K q , with F given by (1.2)and L p given by (4.1), some rearrangements lead to the equalityN p v K q “ ´ ´ W p u q ´ W ¯´ ε ∆ v K ´ W p u q ` W ¯ ´ p ε ∆ ´ W ` εη q ´ W p u q´ W ´ W v K ¯ ´ ´ W p u q ´ W ´ W v K ¯´ ε ∆Φ p ´ W p Φ p q ¯ , where W , W , W are evaluated at Φ p if not otherwise specified and u “ Φ p ` v K .The function u is uniformly bounded in L since v K is by assumption and ε k ∇ k Φ p P EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW L is uniformly bounded for any integer k . We deduce that the nonlinear term Nsatisfies the pointwise bounds | N p v K q| À } W } C c ´ ε | ∇ v K | ` ε | ∆ v K || v K | ` | v K | ¯ , which yields the L estimate } N p v K q} L À ε } v K } L ` } v K } L ε } ∆ v K } L ` } v K } L . In two space dimensions the Gargliardo-Nirenberg inequalities imply(5.68) } ∇ v K } L À } ∇ v K } L } v K } L and } v K } L À } v K } { L } v K } { H , and the L -estimate of N p v K q reduces to(5.69) } N p v K q} L ď C } v K } L ` ε } ∇ v K } L ` } v K } L ˘ ď C } v K } { L ε } ∆ v K } { L ` C } v K } { L } v K } { H ď Cε ´ ´ } v K } L ` ε } ∆ v K } L ¯ . From the decomposition v K “ w ` Q , we have } v K } L À } w } L ` } q p t q} l , ε } ∆ v K } L À ε } ∆ w } L ` } q p t q} l , where we used that the local coordinate expression (2.10) of ε ∆ implies it is auniformly bounded operator on Z ˚ in L and hence on Q . The estimate (5.67)follows from the coercivity Lemma 4.9. Applying the estimate (5.68) leads to } v K } L ď ε ´ p} w } L ` } q } l q { p ε } ∆ w } L ` } q } l q { À ε ´ ´ ρ ´ (cid:104) L p w, w (cid:105) { L ` } q } l ¯ . The proof is complete. (cid:3)
Control of the meandering parameters p p t q . The evolution of the mean-dering parameters p describes the dynamics of the perturbed interface Γ p . Theyare determined by the condition v K K Z ˚ imposed in (5.1), which culminates inLemma 5.10. A key role in the analysis is played by the implicit dependence of theperturbed interface on p , seen in Definition 2.3 and justified in Lemma 2.5. Thisdependence improves the apporoximate orthogonality of the space Z ˚ and inducesthe N ˆ N matrix whose p k, j q -th component given by(5.70) T kj : “ (cid:28) B Φ p B p j , Z k p , ˚ (cid:29) L , for k, j P Σ . and which approximates the first fundamental form of the manifold induced bygraph of Φ p over the space Z ˚ . The matrix appears at leading order in the evolutionand captures the tangential motion of structure on the interface as the interfacestretches or contracts. Introducing the canonical unit basis t P k u k P Σ of R N , from(5.6) and (5.70) we have(5.71) (cid:104) T p , P k (cid:105) “ (cid:10) B t Φ p , Z k p , ˚ (cid:11) L for k P Σ . For clarity we extract the dominant off-diagonal terms in T that describe the re-distribution of structure on the evolving interface. These are incorporated in the p N ´ q ˆ N matrix U with p l, k q -th component(5.72) U lk : “ R R ` p Θ ż I p n ¨ n p ˜ s p ˜Θ l ˜Θ k d˜ s p , l “ , , ¨ ¨ ¨ , N ´ , k P Σ . For the following Lemma it is convenient to introduce(5.73) p “ p p , p , ¨ ¨ ¨ , p N ´ q “ N ´ ÿ k “ p k P k , and the constants m , m (5.74) m “ ż R L ´ p zφ q d z ; m “ ż R | zφ | d z. Lemma 5.6.
Under the a priori assumptions (5.4) , there exists a unit vector e “p e k q k P Σ for which (5.71) admits the expansion (5.75) ´ ε { µ m (cid:104) T p , P k (cid:105) “ p k ` p Θ R ` p Θ ` p k t k ě u ´ ˆ p T U P k ˘ ` O ´ } ˆ p } V } p } l ¯ e k ` O ´ p ε ` } ˆ p } V q} p } l ¯ e k , where m ą is defined in (1.9) and µ “ µ p σ, p q satisfies (5.76) µ p σ, p q : “ m m ` ε p σm ` η d m q R R ` p Θ “ ` O p| ln ε | ´ q . Proof.
We first establish the existence of uniformly l -bounded matrix E P R N ˆ N such that(5.77) T kj “ ´ m ε { µ p σ, p q ˜ δ kj ` R ` p Θ N ´ ÿ l “ p l ż I ˜Θ l ˜Θ j ˜Θ k d s p ´ B p j p ln | Γ p |q ˆ p T U P k ` O p} ˆ p } V q E jk t j ě u ¸ ` O p ε { q E kj . We first examine the case j P Σ zt , u . Applying (3.48), Lemma 4.2, and theorthogonality (4.46) we obtain(5.78) T kj “ ´ ε { m ż R p φ ` εφ q φ d z ż I n ¨ n p ´ ˜Θ j ´ B p j p ln | Γ p |q ˜ s p ¯ p ¯ ˜Θ k | γ p | d s p ` O p ε { q E kj . From the definition (3.5) of φ “ φ p σ q and (5.74) it is straightforward to derive(5.79) ż R p φ ` εφ q φ d z “ m ` ε p σm ` η d m q . To calculate the s p -integral over I p we first employ the identity (2.29) to deducethat(5.80) ˜Θ j n p ¨ n | γ p | “ γ p ¨ γ “ R ` p Θ R ˜Θ j ` R N ´ ÿ l “ p l ˜Θ l ˜Θ j , EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW and hence integrating(5.81) ż I ˜Θ j ˜Θ k n p ¨ n | γ p | d s p “ R ` p Θ R δ kj ` R N ´ ÿ l “ p l ż I ˜Θ l ˜Θ j ˜Θ k d s p . The result (5.77) follows for the case j P Σ zt , u from the definition (5.72) of U .The case j “ , n with the Cartesian unit vector E j P R in (5.80), yielding thes identity E j ¨ n p ? πR | γ p | “ ˜ R ` p Θ R ` R N ´ ÿ l “ p l ˜Θ l ¸ Θ j ` R N ´ ÿ l “ p l ˜Θ l Θ j , j “ , . Here Θ j appears instead of ˜Θ j , and we estimate the difference as | Θ j ´ ˜Θ j | À } ˆ p } V for j “ , j , the definition of ˜ s p , andthe approximation of | Γ p | given by (2.35). The claim (5.77) for the case j “ , O p} ˆ p } V q E jk .The triple ˜Θ integral on the right-hand side of (5.77) is approximated in Lemma2.11. We deduce(5.82) T kj “ ´ m ε { µ p σ, p q ˆ δ kj ` R ` p Θ p k Θ δ j t k ě u ´ B p j p ln | Γ p |q ˆ p T U P k ` O p} ˆ p } V q E jk t j ě u ˙ ` O p ε { q E jk . The result (5.75) follows from (5.82) and (2.36). (cid:3)
Corollary 5.7.
There exists a unit vector e “ p e k q for which U “ p U lk q defined in (5.72) satisfies (5.83) | ˆ p T U P k | À } ˆ p } V e k . Proof.
If we introduce(5.84) s “ πR | Γ p | ˜ s p P I “ r , πR s . then by the definition of ˜Θ i in (2.21) U lk in (5.72) can be rewritten as(5.85) U lk “ R R ` p Θ | Γ p | πR ż I n ¨ n p s Θ l Θ k d s Applying the approximation of | Γ p | in (2.35) and decomposing n p into the sum-mation of n and n p ´ n , we have(5.86) U lk “ ´ ` O p} ˆ p } V q ¯ ˆż I s Θ l Θ k d s ` ż I n ¨ p n p ´ n q s Θ l Θ k d s ˙ . The estimate (5.83) then follows from Lemma 2.9 . (cid:3)
The projection of the residual to the meandering approximate slow space Z ˚ drives the dynamics of meandering parameters p . We approximate this projectionin the following Lemma. Lemma 5.8.
Under the a priori assumptions (5.4) , there exists a unit vector e “p e k q k P Σ such that for any initial data u satisfying the conditions of Lemma 5.1,we have ż Ω Π F p Φ p q Z k p , ˚ d x “ ε { m ˆ C p p q ´ c R p ` O p} ˆ p } V q ˙ δ k ´ ε { m c k p k t k ě u ` O ´ ε | p ´ p ˚ |} ˆ p } V , ε } ˆ p } V ¯ e k , where the smooth functions C “ C p p q and c k “ c k p p , σ qp k ě q take the form C p p q : “ ´ m m c πR ˆ σ ˚ ´ σ p q ´ ε ´ ¯ B ż Ω v d x ˙ ` ε c πR „ p R ` p Θ q ´ α p σ ˚ q ` Cε p ` εC p p qp p ´ p ˚ q ` ε C p p q ,c k : “ β k ´ R p R ` p Θ q « c Θ p p ´ p ˚ q ` O p ε | p ´ p ˚ | q` ε ˆ ´ m m σ ˚ ` β k ´ p R ` p Θ q ` α p σ ˚ q ˙ ` O p ερ q ff . Here C k p p q for k “ , are smooth functions of p whose form is not material tosubsequent analysis.Proof. Adding and subtracting the far-field value of the residual and using thedefinition of Π , we write(5.87) (cid:10) Π F p Φ p q , Z k p , ˚ (cid:11) L “ I k ` J k , with(5.88) I k : “ ż Ω p F p Φ p q ´ F m q Z k p , ˚ d x, J k : “ | Ω | ż Ω p F p Φ p q ´ F m q d x ż Ω Z k p , ˚ d x. From the estimate (4.58) with k P Σ the item J k takes the form(5.89) J k “ C ż Ω p F p Φ p q ´ F m q d x ˜ ε { ż I p ´ h p γ p q ˜Θ k ` h p γ p q ε ˜Θ k ¯ d˜ s p ¸ . From decomposition (2.51) the functions h , h can be rewritten as h j p γ p q “ h j p γ p , q ` p h j p γ p q ´ h j p γ p , qq , and the integral of the leading order term h p γ p , q reduces to(5.90) ż I p h p γ p , q ˜Θ k d˜ s p “ h p γ p , q ż I p Θ k d˜ s p “ C p p q δ k where the constant C p p q depends only on p , although its value may change fromline to line. Since the leading order term h p γ p , q is constant in s p , its integrals donot contribute to the right-hand side of identity (5.89). Moreover, for j “ , ˇˇˇˇˇż I p ` h j p γ p q ´ h j p γ p , q ˘ ˜Θ k d˜ s p ˇˇˇˇˇ À } ˆ p } V e k . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Applying (5.90) and (5.91), the estimate (5.89) reduces to(5.92) J k “ ε { ´ C p p q δ k ` O } ˆ p } V q e k ¯ ż Ω p F p Φ p q ´ F m q d x, which, combined with Lemma 3.5, yields new functions C p p q , C p p q such that(5.93) J k “ ε { ´ C p p q ` C p p qp σ ´ σ ˚ q ¯ δ k ` O ´ ε { } ˆ p } V ¯ e k . To deal with the term I k , we use the expansion (3.11) of F to separate into threeterms(5.94) I k “ ż Ω p F p Φ p q ´ F m q Z k p , ˚ d x “ I k ` I k ` I k ` e ´ (cid:96)ν { ε ´ O p} ˆ p } V q ` C p p q ¯ , where I kj for j “ , , I k : “ ε ż Ω F p Φ p q Z k p , ˚ d x, I k : “ ε ż Ω F p Φ p q Z k p , ˚ d x, I k : “ ε ż Ω F ě p Φ p q Z k p , ˚ d x. We estimate I kj for j “ , ,
3. From the form (4.47) of Z k p , ˚ , its leading order term ψ “ φ { m has odd parity in z p . From Lemma 3.2 F also has odd parity withrespect to z p , Using parity we rewrite I k as(5.96) I k “ ε { m ż I p ż R (cid:96) F φ χ ˜Θ k d z p d˜ s p ` ε { ż I p ż R (cid:96) F ´ ϕ ,k ˜Θ k ` ϕ ,k ε ˜Θ k ¯ p ´ εz p κ p q d z p d˜ s p . Using the general form of F from (3.12) and of the projection of F onto φ from(3.13), and integrating over z p , we have(5.97) I k “ ε { m ´ m ` Ce ´ (cid:96)ν { ε ¯ p σ ˚ ´ σ q ż I p κ p ˜Θ k d˜ s p ` ε { p σ ˚ ´ σ q ż I p ´ h p γ p q ˜Θ k ` h p γ p q ε ˜Θ k ¯ d˜ s p . Applying the first identity from (2.74) , the first term on the right hand side of(5.97) can be simplified. In addition, the second term can be dealt with as in thereductions of (5.90), (5.91), the details are omitted. These reductions imply(5.98) I k “ ε { m ´ m ` Ce ´ (cid:96)ν { ε ¯ p σ ˚ ´ σ q ˆ ´ c πR δ k ´ p β k ´ q p k R p R ` p Θ q t k ě u ˙ ` O ´ ε | σ ˚ ´ σ |} ˆ p } V ¯ e k ` ε { p σ ˚ ´ σ q ´ C p p q δ k ` O p ε { } ˆ p } V q e k ¯ . To estimate I k, we apply the expansion (3.13) for the inner product of F and φ ,the general form (3.12) of F , and integrate in z p to obtain(5.99) I k “ ε { p m ` Ce ´ (cid:96)ν { ε q ż I p ˜ ´ ∆ s p κ p ´ κ p ` ακ p ¸ ˜Θ k d˜ s p ` ε { ż I p ´ ` h , ∆ s p κ p ` h , ˘ ˜Θ k ` ` h , ∆ s p κ p ` h , ˘ ε ˜Θ k ¯ d˜ s p . We first address the ε { terms on the right-hand side of (5.99). From the expansion(2.46) of κ p , arguments similar to those for (5.91) imply(5.100) ˇˇˇˇˇż I p h p γ p q ∆ s p κ p ˜Θ k d˜ s p ˇˇˇˇˇ À } ˆ p } V e k . Applying (5.90)-(5.91) and the estimate above, the ε { integral in (5.99) has thevalue(5.101) ε { C p p q δ k ` O ´ ε { } ˆ p } V ¯ . The ε { integral is approximated from Lemma 2.12. Combining these two approx-imations with the a priori estimate } ˆ p } V ď ε { , we obtain(5.102) I k “ ε { m « ˆ p R ` p Θ q ´ α ˙ c πR δ k ´ p k t k ě u ˜ p β k ´ qp β k ´ q R p R ` p Θ q ` α p β k ´ q R p R ` p Θ q ¸ff ` ε { C p p q δ k ` ε } ˆ p } V e k ` O ´ ε { p σ ˚ ´ σ q} ˆ p } V ¯ e k . Similar estimates applied to the I k integral yield(5.103) I k “ ε { C p p q ` O ´ ε { } ˆ p } V ¯ e k . Combining (5.98), (5.102) and (5.103) with (5.94) yields the expansion(5.104) I k “ ´ ε { ˆ m m p σ ˚ ´ σ q ` εm ˆ α ´ p R ` p Θ q ˙˙ c πR δ k ` ε { p σ ˚ ´ σ q C p p q δ k ` ε { C p p q δ k ´ ε { m c k p k t k ě u ` O ´ ε p σ ˚ ´ σ q} ˆ p } V , ε } ˆ p } V ¯ e k , where the constant c k is given by(5.105) c k p p , σ q “ m m p σ ˚ ´ σ qp β k ´ q R p R ` p Θ q ` ε ˜ β k ´ p R ` p Θ q ` α ¸ β k ´ R p R ` p Θ q . To obtain the form of c k in Lemma 5.8 we recall (3.46) and expand σ ˚ “ σ ˚ ` εσ ˚ě ,so that(5.106) σ ˚ ´ σ “ ´ εσ ˚ě ` c m m Θ p p ´ p ˚ q ` O ´ ε | p ´ p ˚ | , } ˆ p } V ¯ , EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Using the a priori assumption } ˆ p } V À ε { ρ { , to simplify (5.106), we may reducethe right-hand side of (5.105) to(5.107) c k “ β k ´ R p R ` p Θ q « c Θ p p ´ p ˚ q ` O p ε | p ´ p ˚ | q` ε ˆ ´ m m σ ˚ ` β k ´ p R ` p Θ q ` α p σ q ˙ ` O p ερ q ff , where α “ α p σ q defined in (3.14 is a smooth function of σ . From the a prioriassumptions (5.5) the identity (5.106) has the asymptotic limit(5.108) σ ˚ ´ σ “ O p| ln ε | ´ q , which combined with (5.107) yields the desired form of c k . Finally, applying Corol-lary 3.6 yields a smooth function C “ C p p q for which(5.109) ´ m m c πR p σ ˚ ´ σ q “ ´ c m R p ´ m m c πR ˆ σ ˚ ´ σ p q ´ ε ´ ¯ B ż Ω v d x ˙ ` Cε p ` ε C p p q ` O p} ˆ p } V q . Combining (5.109) with (5.104) yields(5.110) I k “ « ´ ε { c m R p ´ ε { m m c πR ˆ σ ˚ ´ σ p q ´ ε ´ ¯ B ż Ω v d x ˙ ` ε { m c πR ˆ ˆ p R ` p Θ q ´ α ˙ ` ε { C p ` ε { p σ ´ σ ˚ q C p p q ` ε { C p p q` O ´ ε { } ˆ p } V ¯ ff δ k ´ ε { c k p k ` O ´ ε | σ ´ σ ˚ |} ˆ p } V q , ε } ˆ p } V ¯ e k . Substituting σ ´ σ ˚ above by (5.106), the main conclusion then follows from (5.87),(5.93) after absorbing all terms depending only on p , except for the leading orderlinear term c p , into C p p q . (cid:3) The zero’th meandering coordinate p “ p p t q controls the radius of the interfaceΓ p . By conservation of mass, its equilibrium value is determined by the initial data u “ Γ p σ q ` v and is given as the unique small solution of(5.111) c p { R “ C p p q . Where C p p q “ C p p q ` ε C p p q is defined in Lemma 5.8, with C p p q and C p p q given by(5.112) C p p q “ ´ m m c πR p σ ˚ ´ σ p qq , C p p q “ c πR ˆ p R ` p Θ q ´ α p σ ˚ q ` m ε ´ ¯ B m ż Ω v d x ˙ ` C p ` C p p qp p ´ p ˚ q ` εC p p q . The constraints on v from Lemma 5.1 imply that v has O p ε q mass. The function C p p q is smooth in p , and the existence and uniqueness of the a uniformly boundedsolution to (5.111) follow from a fixed point argument provided with ε small enough. Indeed, expanding p ˚ “ p ˚ , ` ε p ˚ , ` O p ε q , then a formal perturbation argumentdetermines(5.113)p ˚ , “ ´ m c Θ m p σ ˚ ´ σ p qq ;p ˚ , “ c Θ ˜ p R ` p ˚ , Θ q ´ α p σ ˚ q ` m ε ´ ¯ B m ż Ω v d x ¸ ` Cc Θ ` p ˚ , ˘ . The value of p ˚ can be determined, in principle, to any order of ε . For a giventotal mass, modulo translation there is a unique circular equilibrium profile with p σ, p , ˆ p q “ p σ ˚ , p ˚ , q . From Corollary 3.6, the equilibrium bulk density σ ˚ “ σ ˚ ` εσ ˚ ` O p ε q is related to the radial meander parameter p ˚ through(5.114) σ ˚ “ σ p q ` ε ´ ¯ B ż Ω v d x ´ c Θ m m p ˚ ` ε C p p ˚ q Therefore, with σ ˚ given in (1.8), we have(5.115) σ ˚ “ ´ c Θ m m p ˚ , “ m m ˜ ´ p R ` p ˚ , Θ q ` α p σ ˚ q ` C p p ˚ , q ¸ . With p ˚ given by (5.111), expanding C p p q near p ˚ gives(5.116) C p p q ´ c R p “ c p p ˚ ´ p q ` O p ε qp p ˚ ´ p q . This allows us to reformulate the projection of the residual Π F p Φ p q onto Z p , ˚ ,given in Lemma 5.8 as(5.117) ż Ω Π F p Φ p q Z p , ˚ d x “ ε { m p c { R ` O p ε qq p p ˚ ´ p q` O ´ ε } ˆ p } V , ε | σ ´ σ ˚ |} ˆ p } V , ε } ˆ p } V ¯ . Substituting (5.115) for σ ˚ and using the a priori assumption | p p t q| ď | ln ε | ´ ă ρ ,we rewrite c k as,(5.118) c k p p q “ c Θ R D kk p p ´ p ˚ q ` ε D kk R ` O p ερ D kk q , D kk : “ β k ´ . We recall that D is a diagonal matrix in R p N ´ qˆp N ´ q with diagonal entries D kk above, which induces a norm equivalent to V . Lemma 5.9.
As a linear operator from V k to V k ` , the diagonal matrix D isuniformly bounded away from zero and from above, i.e. (5.119) } D ˆ p } V k „ } ˆ p } V k ` . We may now derive the dynamics of the meandering parameters p . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Lemma 5.10.
Under the a priori assumptions (5.4) , for initial data satisfyingLemma 5.1, then the meandering parameters p “ p p , p , p , ˆ p q evolve according to (5.120) p “ ´ ε c R p p ´ p ˚ q ` d , p k “ O ´ ε | p ´ p ˚ |} ˆ p } V ¯ ` d k for k “ , , ˆ p “ ´ ε „ c Θ R p D ` I ` U T qp p ´ p ˚ q ` εR D ˆ p ` ˆ d. Here I is the p N ´ q by p N ´ q identity matrix, and U is defined in (5.72). Thevector d “ p d , d q with d “ p d , d , ˆ d q denotes error terms satisfying | d | À ε } ˆ p } V ` ε | ln ε | ´ | p ´ p ˚ | ` ´ ε { ` ε { p} q } l ` } w } L q ¯ } ˆ p } V ` ε { | ln ε |p} w } L ` } q } l ` } q } l q ` ε { } N p v K q} L , } d } l À ε | ln ε | ´ | p ´ p ˚ |} ˆ p } V ` ε | p ´ p ˚ | ` ´ ε ρ ` ε { p} q } l ` } w } L q ¯ } ˆ p } V ` ε { | ln ε | p} w } L ` } q } l ` } q } l q ` ε { } N p v K q} L . Moreover the flow p “ p p , p , p , ˆ p q admit upper bounds: (5.121) | p | À ε | p ´ p ˚ | ` | d | , } ˆ p } l À ε | p ´ p ˚ |} ˆ p } V ` ε } ˆ p } V ` } d } l , | p | ` | p | À ε | p ´ p ˚ |} ˆ p } V ` } d } l . Proof.
Projecting equation (5.2) onto Z k p , ˚ in L , using the definition (5.70) of T ,and the decomposition (5.10) of v K yields (cid:104) T p , P k (cid:105) l “ ´ (cid:10) Π F p Φ p q , Z k p , ˚ (cid:11) L ´ (cid:10) B t Q ` B t w, Z k p , ˚ (cid:11) L ´ (cid:10) Π L p p Q ` w q , Z k p , ˚ (cid:11) L ´ (cid:10) Π N p v K q , Z k p , ˚ (cid:11) L . Multiplying this identity by ´ ε { µ { m and using Lemmas 5.6, 5.8 and the identity(5.117) we obtain(5.122) p k ` p Θ R ` p Θ p p k t k ě u ´ ˆ p T U P k q “ ε ˆ c µ R ` O p ε q ˙ p p ˚ ´ p q δ k ´ ε µ c k p k t k ě u ` d k , in which the error terms d k ’s satisfy(5.123) d k “ ε { µ m „ (cid:10) B t Q ` B t w, Z k p , ˚ (cid:11) L ` (cid:10) Π L p p Q ` w q , Z k p , ˚ (cid:11) L ` (cid:10) Π N p v K q , Z k p , ˚ (cid:11) L ` O ´ ε } ˆ p } V ¯ δ k ` O p} ˆ p } V } p } l q e k ` O ´´ ε ` } ˆ p } V ¯ } p } l ¯ e k ` O ´ ε { | p ´ p ˚ |} ˆ p } V , ε { } ˆ p } V ¯ e k . For the case k “ µ in (5.76) allow us to write(5.124) ˆ ´ Θ R ` p Θ ˆ p T U P ˙ p “ ε c R p p ˚ ´ p q ` O ` ε | ln ε | ´ | p ´ p ˚ | ˘ ` d . Dividing by the coefficient of p and using Corollary 5.7 yields the ODE for p witha new error term(5.125) ˜ d “ p ` O p} ˆ p } V qq ` O p| ln ε | ´ | p ´ p ˚ |q ` d ˘ . For the case k ě to eliminate p from (5.122), c k in (5.118)and µ in (5.76), arriving at(5.126) p k “ ´ c ε p p ´ p ˚ q Θ R ˆ p U P k ´ ε ˆ c Θ R p p ´ p ˚ qp D kk ` q ` ε D kk R ˙ p k t k ě u ` ˜ d k , where the error terms with the aids of Corollary 5.7 and Lemma 5.9 can be writtenas(5.127) ˜ d k “ d k ` O ´ } ˆ p } V | d | , ε | ln ε | ´ | p ´ p ˚ |} ˆ p } V , ε ρ } ˆ p } V ¯ e k . In particular for k “ ,
2, (5.126) implies the bound(5.128) | p | ` | p | À ε | p ´ p ˚ |} ˆ p } V ` } ˜ d } l . In the remainder of the proof we omit the titles for d . To bound d and d in l , weestimate the dominant ε { terms on the right-hand side of (5.123). Using Lemma4.7 we have(5.129) ˇˇˇ (cid:10) Π L p p Q ` w q , Z k p , ˚ (cid:11) L ˇˇˇ À ε p ` } ˆ p } V q} q } l e k ` ε p ` } ˆ p } V q} w } L e k , while by H¨older’s inequality we obtain(5.130) ˇˇˇ (cid:10) Π N p v K q , Z k p , ˚ (cid:11) L ˇˇˇ À } N p v K q} L e k . Since w P Z K˚ , using (5.13) we have(5.131) ˇˇˇ (cid:10) B t w, Z k p , ˚ (cid:11) L ˇˇˇ “ ˇˇˇ ´ (cid:10) w, B t Z k p , ˚ (cid:11) L ˇˇˇ À ε ´ } p } l } w } L . To bound the projection of B t Q , we calculate(5.132) (cid:10) B t Q, Z k p , ˚ (cid:11) L “ ÿ j P Σ q j (cid:68) Z j p , ˚ , Z k p , ˚ (cid:69) L ` ÿ j P Σ q j (cid:68) B t Z j p , ˚ , Z k p , ˚ (cid:69) L , and employ the approximate orthonormality (4.56) of Z ˚ , the estimate (5.13), the l - l estimate from Lemma 2.4, and the a priori assumption } p } l À ε { | ln ε | ` ε } ˆ p } V to derive the bound(5.133) ˇˇˇ (cid:10) B t Q, Z k p , ˚ (cid:11) L ˇˇˇ À ε } ˆ p } V } q } l e k ` ε ´ } p } l } q } l e k À ε p} q } l ` } q } l q} ˆ p } V e k ` ε | ln ε |} q } l e k . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Combining the estimates (5.129)-(5.131), and (5.133) with (5.123),(5.125), (5.127)yields(5.134) | d k | À } ˆ p } V | d | e k ` ´ ε } ˆ p } V ` ε | ln ε | ´ | p ´ p ˚ | ¯ δ k ` ε ρ } ˆ p } V p ´ δ k q` } ˆ p } V } p } l e k ` ´ ε ` } ˆ p } V ¯ } p } l e k ` ε | ln ε | ´ | p ´ p ˚ |} ˆ p } V e k ` ε { } ˆ p } V e k ` ε { p} q } l ` } w } L q } ˆ p } V e k ` ε { | ln ε |p} q } l ` } q } l ` } w } L q e k ` ε { } N p v K q} L e k . To obtain the bounds of d and d in Lemma 5.10 it suffices to bound | p | and } p } l ,which in turn control } p } l ď | p | ` } p } l . From the ODE for p , we have(5.135) | p | À ε | p ´ p ˚ | ` | d | , and hence(5.136) } p } l ď } p } l ` | p | À } p } l ` ε | p ´ p ˚ | ` | d | . The matrix D is diagonal, and from Lemma 5.9, it is norm bounded from V k ` to V k . Using these norm bounds we have the estimates(5.137) } p } l À ε | p ´ p ˚ |} ˆ p } V ` ε } ˆ p } V ` } d } l À ε { | p ´ p ˚ | ` ε } ˆ p } V ` } d } l . We eliminate the p terms in (5.134) by substituting (5.136) with } p } boundedfrom (5.137) into the right-hand side of (5.134). Modulo the a priori assumption } ˆ p } V ď ε { , we arrive at the upper-triangular bounds(5.138) | d | À ε } ˆ p } V ` } ˆ p } V } d } l ` ε | ln ε | ´ | p ´ p ˚ | ` ε { p} q } l ` } w } L q} ˆ p } V ` ε } ˆ p } V ` ε { | ln ε |p} q } l ` } q } l ` } w } L q ` ε { } N p v K q} L , } d } l À ε | ln ε | ´ | p ´ p ˚ |} ˆ p } V ` p} ˆ p } V ` ε q| d | ` ε { p} q } l ` } w } L q} ˆ p } V ` ε ρ } ˆ p } V ` ε { | ln ε |p} q } l ` } q } l ` } w } L q ` ε { } N p v K q} L . Using the second estimate above to eliminating } d } l in the first estimate yieldsthe final upper bound for d . The final l -estimate for d follows from (5.138) andthe a priori assumptions (5.4). (cid:3) The system for ˆ p is weakly nonlinear, the sign of the matrix multiplying ˆ p onthe right-hand side of (5.120) determines the growth or decay of ˆ p . For the analysisof the long time convergence, the following Lemma establishes the definiteness ofthis matrix. Corollary 5.11.
Suppose that | p ˚ ´ p | ! ε , then there exists µ ą , independentof ε , such that the coefficient matrix appearing in (5.120) satisfies Θ R p D ` I ` U T qp p ´ p ˚ q ` εR D ě µε D , in particular it is positive definite.Proof. The matrix D is uniformly invertible, independent of ε , and the matrix D ´ p D ` I ` U T q is uniformly bounded. It follows that under the conditions imposedon p ˚ ´ p guarantee that the D term is dominant. (cid:3) Proof of the main theorem.
The results of the previous sections providethe tools to establish the nonlinear stability of nearly circular bilayer equilibriumand recover the transient evolution of the interface in the curve lengthening regime.The proof takes three main steps:Step 1 We make a priori assumptions p A q in (5.140) on the variables w, q , p and apply energy estimates from Lemmas 5.2, 5.3 and 5.10 to closethese assumptions for w , q and p without restriction to the value of T . These are global time estimates.Step 2 In the regime t P r , T s with transient growth time T „ ε ´ | ln ε | { ,then p ´ p ˚ is not sufficiently small to render the p flow dis-sipative. We develop estimates on the meander parameters p “p p , p , p , ˆ p q P R N , showing that while p may grow by ε ´ { on the T transient, the radial and bulk parameters p and σ are sufficientlyclose to equilibrium values p ˚ and σ ˚ .Step 3 For t ą T , the p evolution is dissipative and we recover the conver-gence of the quasi-steady profile Φ p to its equilibrium value Φ p ˚ onthe O p ε ´ q time scale. Proof of Thoerem 1.1.
From the assumptions on the initial data, Lemma,5.1 im-plies the existence of an initial decomposition u “ Φ p p q ` v K “ Φ p p q ` Q p q p qq` w with Q P Z ˚ p p p qq and w P Z K˚ p p p qq satisfying,(5.139) | p p q| À ε { } v } L À ε , } p p q} V À ε { } v } H À ε, } w } L ` ε } w } H À ε { , } q p q} l À ε { . We establish the existence of a constant, θ P p , q independent of ε and T ą p A q $’’’’’&’’’’’% (cid:104) L p w, w (cid:105) L À ε ρ ´p ` θ q , } q } l À ε ρ ´p ` θ q , } p } l À ´ ε { | ln ε | ` ε } ˆ p } V ¯ , | p ˚ ´ p | ! ε { | ln ε | { , } ˆ p } V À ερ, ż t } ˆ p } V d τ À ε ´ , hold uniformly for all t P r , T s . The existence of a strictly positive T is assured by(5.139), relation (5.113) and assumption (1.30) on | σ p q ´ σ ˚ | . The V bound onˆ p affords uniform L bounds on the curvature κ p that prevent self-intersection ofthe associated curve Γ p . A key step in argument is to establish the existence of θ which closes the energy estimates for sufficiently small values of the spectral cut-off ρ , and for all ε P p , ε s . Assumption (5.140) implies that the a priori assumptions(5.4) hold. Step 1: Global estimates of w, q , p . From (5.106) and the p bound in ( A ) we havethe estimate | σ ´ σ ˚ | À | p ´ p ˚ | ` O p ε q À ε { | ln ε | . This bound guarantees that the equilibrium pearling assumption (1.21) implies thepearling stability condition (4.64) holds uniformly for all t P r , T s . In particular EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Lemma 5.3 applies. We restate the key estimates of Lemmas 5.2 and 5.3 as(5.141)dd t (cid:104) L p w, w (cid:105) L ` } L p w } L À } p } l ` ε | p ´ p ˚ | ` ε ρ ´ p} q } l ` } q } l q ` ε ` } N p v K q} L ; B t } q } l ` Cε } q } l À ε } w } L ` ε ´ } N p v K q} L ` ε | ln ε | } ˆ p } V ` ε } ˆ p } V , and recall from Lemma 2.4 that } ˆ p } V ` k À ε ´ k } ˆ p } V for k “ ,
2. Using ( A ) andbounding } w } L from the coercivity (4.73 we find that the } q } L bound in Lemma5.3 improves to(5.142) } q } l À } q } l ` } N p v K q} L ` ε , where the third term on the right-hand side comes from } ˆ p } V . Combining this withthe coercivity estimate (4.73), assumption ( A ), and the bound on } p } L from (5.5)we obtain(5.143) dd t (cid:104) L p w, w (cid:105) L ` Cρ (cid:104) L p w, w (cid:105) L À ε ` } N p v K q} L , where C ą ε and ρ , and the dominant power of ε term carriesover from the inhomogeneous term on the right-hand side of (5.141). From Lemma5.5 and ( A ) the nonlinear term N p v K q is asymptotically negligible(5.144) } N p v K q} L À ε ´ ´ ρ ´ (cid:104) L p w, w (cid:105) L ` } q } l ¯ À ε ρ ´p ` θ q ` ε ρ ´p ` θ q . The estimate (5.143) reduces to(5.145) dd t (cid:104) L p w, w (cid:105) L ` Cρ (cid:104) L p w, w (cid:105) L À ε . Integrating this estimate in time we obtain(5.146) (cid:104) L p w, w (cid:105) L ď (cid:10) L p p q w , w (cid:11) L e ´ Cρ t ` Cε ρ ´ , À ` } w } L ` ε } w } H ˘ ` ε ρ ´ À ε ρ ´ , where we used (5.139) to bound w . For any fixed value of θ ą ρ sufficiently small to close the a priori estimate ( A ) on w for all t P r , T s . Turning to the q estimate in (5.141) we use (5.146), convexity (4.73), ( A ) and(5.144) to bound the first three terms on the right-hand side,(5.147) B t } q } l ` Cε } q } l À ε ρ ´ ` ε } ˆ p } V , where the dominant ε -term arises from } w } L . We integrate this inequality andapply the initial estimates (5.139) to } q } l and ( A ) to bound the L pr , T s , V q norm of p to obtain,(5.148) } q } l ď e ´ Cεt } q p q} l ` Cε ρ ´ ` Cε ż t } ˆ p } V d τ À ε ρ ´ . Taking θ ą p A q on } q } l . The estimates(5.121) on p given in Lemma 5.10, can be combined to yield } p } l À ε | p ´ p ˚ | ` ε } ˆ p } V ` | d | ` } d } l , and using the estimates ( A ) and (5.144) we see the error terms | d | ` } d } l arelower order, and hence(5.149) } p } l À ε { | ln ε | ` ε } ˆ p } V . Since the a priori bounds only appeared as a lower order term in the derivation of(5.149), this estimate closes. In the Step 2 we close the a priori bounds on } ˆ p } V , } ˆ p } V in L pr , T sq and | p ˚ ´ p | over r , T s . Step 2: Transient growth estimates.
The key control mechanism with the flow is therelaxation of p to its equilibrium value p ˚ on the t “ O p ε ´ q time scale. Indeed wefix the transient growth time T : “ R c ε ´ | ln ε | { , where c is defined in (3.32),and close the remaining a priori bounds over the interval r , T s . We multiply the k “ p p ´ p ˚ q and apply Young’s inequality, deducing(5.150) B t p p ´ p ˚ q ` ε c R p p ´ p ˚ q À ε ´ d . Bounds on the error term d are established in Lemma 5.10. Applying estimates(5.142) on } q } l , (5.144) on } N p v K q} L , and the a priori assumptions in (5.140),yields the bound(5.151) | d | À ε | ln ε | ´ | p ´ p ˚ | ` ε ρ ` ε { } ˆ p } V , where the dominant inhomogeneous ε -term on the right-hand side arises from thebound on } ˆ p } V . Reporting this estimate to (5.150), integrating, and using the L pr , T sq bound on } ˆ p } V from (5.140) yields(5.152) | p ´ p ˚ | ď e ´ c ε tR | p p q ´ p ˚ | ` C ˆ ε ρ ` ε ż t } ˆ p } V d τ ˙ , ď e ´ c ε tR | p p q ´ p ˚ | ` Cε ρ . For t P r , T s with T as defined above, the bound on | p ´ p ˚ | decreases mono-tonically for initial values | p ´ p ˚ | " ρε , satisfying(5.153) | p p t q ´ p ˚ | ď e ´ c ε t R | p p q ´ p ˚ | ` ερ @ t P r , T s . This closes the a priori assumption on | p ´ p ˚ | in (5.140) on the interval r , T s .However this estimate is self-improving, as the same argument can be applied in-ductive on intervals r kT , p k ` q T s for values of k P Z ` such that p k ` q T ă T and values of T for which the L pr , T sq a priori estimate on } ˆ p } V holds uniformly.In particular we deduce that (5.153 holds for all t P r , T s and that(5.154) | p p t q ´ p ˚ | À ερ, @ t P r T , T s . To complete the transient growth estimates we address the bounds on ˆ p . Takethe dot product of the ˆ p equation (5.120) with D ˆ p , and recall Lemma 5.9. Theestimates of Lemma 5.10 yield the bound(5.155) B t } ˆ p } V ` Cε } ˆ p } V ď c µ Θ { R ε | p ´ p ˚ ||p D ` U T q ˆ p ¨ D ˆ p |`} ˆ d } l } ˆ p } V , where the constant C on the left-hand side takes the value C “ c µ Θ { R ą U term, and the estimate } D ˆ p } l ď } ˆ p } V we have a c ˚ ą |p D ` U T q ˆ p ¨ D ˆ p | ď c ˚ | ˆ p } V | ˆ p } V , EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW and hence(5.157) B t } ˆ p } V ` Cε } ˆ p } V ď C ˚ ε | p ´ p ˚ |} ˆ p } V } ˆ p } V ` } ˆ d } l } ˆ p } V , where we have introduced(5.158) C ˚ : “ c ˚ c µ Θ { R ą , To estimate the ˆ d term, we recall its form from Lemma 5.10, the estimates (5.144)on the nonlinearity, (5.142) on } q } l , and the a priori assumptions to drive thebound(5.159) } ˆ d } l ď } d } l À ε | ln ε | ´ | p ´ p ˚ |} ˆ p } V ` ε ρ } ˆ p } V ` ε { ρ ´p ` θ q , with the dominant inhomogeneous ε -term arises from the nonlinearity. Substitutingthis bound into (5.157) we absorb the ε ρ } ˆ p } V term into the positive term on theleft-hand side, obtaining(5.160) B t } ˆ p } V ` Cε } ˆ p } V ď ´ C ˚ ε | p ´ p ˚ |} ˆ p } V ` Cε { ρ ´p ` θ q ¯ } ˆ p } V , Using Young’s inequality to separate out and absorb the } ˆ p } V into the positiveterm on the left-hand side, and using a priori estimates to bound } ˆ p } V and thebound (5.153) on | p ´ p ˚ | for t P r , T s , we obtain(5.161) B t } ˆ p } V ` Cε } ˆ p } V ď C ˚ ε | p ´ p ˚ | } ˆ p } V ` Cε ρ ´ p ` θ q , ď C ˚ M ε | ln ε | e ´ c R ε t } ˆ p } V ` Cε ρ ´ p ` θ q , where in the last line we introduced the scaled modulus of the distance of the initialradius from equilibrium(5.162) M : “ | p p q ´ p ˚ | p ε | ln ε |q ´ . To bound the differential inequality (5.161), we introduce g p t q : “ c R ε | ln ε | e ´ c R ε t , and the indefinite integral(5.163) G p t q “ C ˚ M ż t g p τ q d τ “ C ˚ M | ln ε |p ´ e ´ c R ε t q . Multiplying the inequality (5.161) by e ´ G p τ q and integrating from τ “ τ “ t we obtain the bound(5.164) } ˆ p } V ` Cε ż t } ˆ p } V d τ ď e G p t q } ˆ p p q} V ` Cε ρ ´ p ` θ q ż t e G p t q´ G p τ q d τ. From the definition of G p t q , we have(5.165) e G p t q ď exp ! C ˚ M | ln ε | ´ ´ e ´ c R ε t ¯) ď ε ´ C ˚ M , while we may bound G p t q ´ G p τ q ď ε ´ C ˚ M in the last integral. For t P r , T s with T “ ε ´ | ln ε | { we deduce the bound(5.166) } ˆ p } V ` Cε ż t } ˆ p } V d τ ď ε ´ C ˚ M ´ } ˆ p p q} V ` Cε | ln ε | { ρ ´ p ` θ q ¯ . Imposing the condition(5.167) C ˚ M ă , and using the initial data estimate } ˆ p p q} V À ε from (5.139) we obtain(5.168) } ˆ p } V ` Cε ż t } ˆ p } V d τ ď ε ´ C ˚ M C ´ ε ` ε | ln ε | { ρ ´ p ` θ q ¯ ď ερ, @ t P r , T s , where the choice of ρ depends only upon the constant C , the value of ε and the sizeof 1 ´ C ˚ M ą . This closes the a priori bounds on ˆ p in the V and L pr , T s ; V q norms and establishes the a priori bounds on the transient scale. Step 3: Long-time scales t Á ε ´ . For t ą T the estimate (5.154) establishes thatthe driving term | p ´ p ˚ | À ερ, and hence from Corollary 5.11 the key diagonalelements of the evolution for ˆ p in (5.120) are uniformly positive for ρ ą T to and establish the exponential convergence to equilibriumwe derive decay estimates on ˆ p . The proof follows the derivation of the a prioriestimates on ˆ p in Step 2 , we outline the main points. We take the dot product ofthe ˆ p equation (5.120) with D ˆ p and of the p equation with p and add the result.Absorbing quadratic terms into the left-hand side, and estimating the error terms } ˆ d } l ď } d } l and d from Lemma 5.10 we obtain,(5.169) B t ´ } ˆ p } V ` | p ´ p ˚ | ¯ ` C ´ ε } ˆ p } V ` ε | p ´ p ˚ | ¯ À ε | ln ε | ` } w } L ` } q } l ` } q } l ˘ ` ε ´ } N p v K q} L . Applying the coercivity Lemma 4.9 to } w } L in terms of the L p bilinear form, the l estimate of q from Lemma 5.3 yield the estimate(5.170) B t ´ } ˆ p } V ` | p ´ p ˚ | ¯ ` C ´ ε } ˆ p } V ` ε | p ´ p ˚ | ¯ À ε | ln ε | ρ ´ } L p w } L ` ε | ln ε | } q } l ` ε ´ } N p v K q} L . We use the q and p estimates to replace the corresponding terms in the w and q bounds of Lemmas 5.2 (5.16) and 5.3 respectively, obtaining the simplified bounds(5.171) B t (cid:104) L p w, w (cid:105) L ` } L p w } L À ε ρ ´ } q } l ` ε | p ´ p ˚ | ` ε } ˆ p } V ` ε } ˆ p } V ` } N p v K q} L ; B t } q } l ` Cε } q } l À ε } w } L ` ε ´ } N p v K q} L ` ε | ln ε | } ˆ p } V ` ε } ˆ p } V ;Applying Young’s inequality and the coercivity Lemma 4.9 to the bound (5.67) onN p v K q we obtain(5.172) } N p v K q} L À ε ´ ´ ρ ´ (cid:104) L p w, w (cid:105) L ` } q } l ¯ À ε ρ ´ } L p w } L ` ε ρ ´ } q } l . We define the quantity(5.173) A p t q : “ ε ´ ρ (cid:104) L p w, w (cid:105) L ` ε ´ ρ ´ } q } l ` } ˆ p } V ` p p ´ p ˚ q , and combine the estimates (5.170), (5.171) and (5.172) and appropriate absorbterms into the positive entries on the left-hand side. We obtain the existence of aconstant ν ą A p t q ` νε A p t q ` νε } ˆ p } V ď . EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Applying variation of constants formula we obtain the long time estimates e νε t A p t q ` νε ż tT e νε τ } ˆ p } V p τ q d τ ď A p T q . From the a priori estimates valid at t “ T we obtain the exponential decay(5.174) A p t q ď e ´ νε t A p T q ď ερe ´ νε t , ε ż tT e νε τ } ˆ p } V p τ q d τ À ερ, which combined with estimates given in Step 1 and
Step 2 is sufficient to establishthe a priori assumptions up to T “ 8 , and the exponential decay estimates of ˆ p in(1.32). Furthermore, reexamining the estimates on w and q in light of the decayestimates (5.174) and (5.140) we deduce(5.175) } w } L ` ε } w } H ` } q } l À ε ρ ´ ^ ε ρ ´ e ´ νε t , @ t P r , and the second bound of (1.32) . Since u ´ Φ p “ w ` Q , these results establish theexponential decay of u ´ Φ p in (1.31). The evolution equations (1.25)-(1.27) forthe meander parameters p and associated remainder estimates follow from Lemma5.10 and bounds on the leading order terms of d from (5.140).Finally, the motion of the translation parameters is given by equation (1.26) for k “ ,
2. Integrating these we find(5.176) ż p| p | ` | p |q d t À ż ´ ε | p ´ p ˚ |} ˆ p } V ` ε } ˆ p } V ¯ d t ` ε ρ ´ ż ` } w } L ` ε } w } H ` } q } l ˘ d t. Applying (5.153) which holds on r , , in light of } ˆ p } V À ε { ^ } ˆ p } V , the firstintegral on the right hand side can be bounded by ż ε { e ´ c ε t R d t ` ε ˆż e ´ νε t d t ˙ { ˆż e νε t } ˆ p } V ˙ { À ε { ;Separating the time period to two parts, i.e. r ,
8q “ r , T ˚ s Y p T ˚ , with T ˚ “ ε ´ | ln ε | ν , and applying the bounds (5.175) wisely on different time interval, wearrive at ε ρ ´ ż ` } w } L ` ε } w } H ` } q } l ˘ d t À ε { ρ ´ | T ˚ | ` ε ρ ´ ż T ˚ e ´ νε t d t À ε { ρ ´ | ln ε | ` ε ρ ´ e ´ ν ε T ˚ ż T ˚ e ´ ν ε t d t À ε { ρ ´ | ln ε | . Combining the previous two estimates with (5.176) implies ż p| p | ` | p |q d t À ε { ρ ´ | ln ε | . We then deduce from the mean-value theorem that for any t , t P r , the trans-lational motion can be bounded by | p k p t q ´ p k p t q| À ż | p k | d t À ε { ρ ´ | ln ε | ď ε { | ln ε | . In particular p p , p q converge to a constant vector p p ˚ , p ˚ q which is the translateof the equilibrium solution. (cid:3) Discussion
We have shown that under the mass-preserving L -gradient flow of the FCH freeenergy, nearly circular bilayer interfaces evolve according to a finite dimensionalODE system, (1.25)-(1.27), generated in effect as a Galerkin expansion of the cur-vature flow (1.22) associated to the finite dimensional family t Γ p u of perturbedinterfaces. There are several directions in which the results could be improved orextended. It is not difficult to image that the ODE system is rigorously related tothe associated finite dimensional normal velocity system. It is less clear to whatextend the ODE system produces a flow that approximates the infinite dimensionalregularized curve lengthening (RCL) flow (1.15). Indeed, in many situations tak-ing increasing numbers of modes in a Galerkin approximation lead to convergenceof the ordinary differential system to the underlying partial differential equation.However here the number of modes is constrained by the the choice of ε and thevalue of the spectral cut-off ρ ą . Moreover the singular nature of the RCL flowmakes its flow increasingly difficult to control as ε Ñ ` . It is worthwhile to ask,given these constraints, to what extend and for what classes of initial interfaces theODE system approximates the RCL flow (1.15).Extensions to higher space dimensions and more complex geometries are alwaysdesirable, but there is a significant high-value target within three dimensions. ForΩ Ă R formal results, [4], have derived competition between spatially extendedcodimension one (bilayer) and codimension two (filament) structures. These havedifferent bulk density equilibrium, σ b for bilayers and σ f for filaments, that aregenerically not equal. The structure with the lower bulk density equilibrium absorbsmass from the bulk more efficiently, and can effectively consume the less efficientmorphology from a distance. In this sense the value of σ is slowly fed by thestructure with the higher equilibrium value. This produces a transient in which σ is not at equilibrium, but is also potentially not far from the equilibrium valuefor the growing morphology. This framework has the potential to produce longtransients that are within the reach of analytical tools.While the FCH free energy was motivated by small-angle X-ray scattering studiesof microemulsions, [32], [20], and [21], a rigorous derivation of the FCH free energyfrom more fundamental models would provide considerable insight into the scalingand choices possible for many of the parameters. Work of Choksi and Ren, [8]established the Ohta-Kawasaki free energy as a long-chain limit of a self-consistentmean field theory for diblock polymers. In particular their follow-up paper, [9],considered diblocks immersed within a homopolymer, deriving a continuum modelfor the a diblock-homopolymer blend in the long-chain limit. This approach seemsamenable to a short-chain limit, in which the homopolymer approximates a solventand the Florey-Huggins parameters for each component of the diblock can be ad-justed to mimic the hydrophilic-hydrophobic interactions of an amphiphilic diblockwith a solvent (homopolymer) phase. Such a model is evocative of amphiphilicblends, deriving a continuum reduction would clarify the relation between the FCHand these important statistical physics models. EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Acknowledgement
Both authors would like to thank Gurgen Hayrapetyan for sharing preliminaryresults on this problem for the weak FCH that arose out of his thesis.
Appendix
We use the implicit function theorem to establish the decomposition of the initialdata, completing Lemma 5.1.
Lemma 6.1.
Let X be a normed linear space, and E ˆ D be an open subset of X ˆ V r contains p , q . Let F : E ˆ D Ñ V r satisfy F p , q “ . Assume that F is continuous at p , q and D p F P L p V r , V r q exists in D , moreover } D p F } L p V r ˆ V r , V r q ` } D v D p F } L p V r ˆ X, V r q ď c, }r D p F p , qs ´ } L p V r , V r q ď ˜ c, Then for any v P X in a small neighborhood of containing the ball of radius L c ˜ c ,for which F p v, q also satisfies } F p v, q} V r ď c ˜ c , there exists p “ p p v q satisfying } p } V r ď L c ˜ c such that F p v, p p v qq “ . Moreover, } p p v q} V r ď c } F p v, q} V r . Proof.
Define(6.1) G v p p q “ : p ´ r D p F p , qs ´ F p v, p q . Observe that any point is a fixed point of G v if and only if it is a zero of F p v, ¨q .That is, for a given p , G v p p q “ p if and only if F p v, p q “
0. We first prove that G v p p q is a closed map on t p : } p } V r ď { c ˜ c u provided with } v } X ď { c ˜ c . In fact, } G v p p q} V r ď } G v p p q ´ G v p q} V r ` } G v p q} V r ď } p } V r ` }r D p F p , qs ´ } L p V r , V r q } F p v, q} V r ď } p } V r ` ˜ c } F p v, q} V r ă { c ˜ c. Let p and p be two elements in V r with } p i } V r ď { c ˜ c and v P X with } v } X ď { c ˜ c . } G v p p q ´ G v p p q} V r ď sup p “ λ p `p ´ λ q p } D p G v p p q} L p V r , V r q } p ´ p } V r . Here D p G v p p q “ r D p F p qs ´ p D p F p , q ´ D p F p v, p qq . It is then straightforward to check for } p } V r ă L c ˜ c and } v } X ď L c ˜ c that } D p G v p p q} L p V r , V r q ď . By contraction mapping principle, there exists p ˚ such that G v p p ˚ q “ p ˚ and } p ˚ } V r ď } G v p p ˚ q ´ G v p q} V r ` } G v p q} V r ď } p ˚ } V r ` }r D p F p , qs ´ } L p V r , V r q } F p v, q} V r . The proof is complete. (cid:3)
Define F i to be a real-valued functionals of v and parameters p , explicitly givenby F i p v ; p q “ ż Ω ´ Φ ` v p x q ´ Φ p ¯ Z i p , ˚ d x, for i P Σ p Γ p , ρ q . We apply Theorem 6.1 to F “ p F i q i P Σ by verifying its assump-tions and establish the following result. Lemma 6.2.
Let Γ be a circle associated with whiskered coordinates p s, z q . Let ď r ď and } v } L ` ε } v } H ď ε , then there exists p “ p p v q such that F p v ; p p v qq “ and } p p v q} V r À ε p ´ r q{ } v } L . Proof.
Firstly, we compute ∇ p F at origin p v, p q “ p , q and show its inverseadmits an upper bound. Differentiating F i with respect to p j yields(6.2) B F i B p j p v ; p q “ ż R (cid:96) ż I ´ Φ ` v ´ Φ p ¯ B Z i p , ˚ B p j J d s d z ´ ż R (cid:96) ż I B Φ p B p j Z i p , ˚ J d s d z. Notice that when v “ p “ , the first integration is zero, and Φ p “ φ p z p q ` εφ p z p ; σ q` ε φ p s p , z p q` ε φ ě p z p q , the derivative of Φ p to p j is obtained throughderivative with respect to z p , s p , σ p p q by chain rule. Applying Lemma 2.10, B z p B p j ˇˇˇˇ p “ “ ´ ε Θ j p s q , B s p B p j ˇˇˇˇ p “ “ εz | γ |p ´ εzκ q Θ j ; and ψ “ ˆ φ “ φ m . In light of (5.77) with p “ , then we obtain B F i B p j p q “ ´ ż R (cid:96) ż I B Φ p B p j ˇˇˇˇ p “ Z i ˚ J d s d z “ m ε { µ p σ p q , q δ ij ` ε { I ij . Here error term I ij can be written as(6.3) I ij “ h p γ q (cid:104) Θ i , Θ j (cid:105) J ` h p γ q (cid:10) ε Θ i , Θ j (cid:11) J ` h p γ q (cid:10) Θ i , ε Θ j (cid:11) J ` h p γ q (cid:10) ε Θ i , ε Θ j (cid:11) J ` h p γ q (cid:104) , Θ i (cid:105) J (cid:104) , Θ j (cid:105) J . To prove the boundedness of the inverse operator p ∇ p F p , qq ´ , it is equivalentto show that ∇ p F p , q is away from zero as a linear operator from V r to V r . Inorder to achieve the goal, we estimate }p ∇ p F q v } V r : “ } v ¨ ∇ p F } V r ě m εµ } v } V r ´ ε { } I v } V r , which implies the conclusion if } I v } V r ď } v } V r . This holds by integrating by parts. As a summary, the above argument implies }r D p F p , qs ´ } L p V r , V r q À ε { . In addition, note that F i p q “ . In order to apply the implicit function theorem,it remains to check F “ p F i q i : V r ˆ H Ñ V r is a C -map. In fact, for r ď EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW bound } D p F } L p V r ˆ V r , V r q ď ÿ i,j,k P Σ β k β i β j ˆ B F k B p i B p j ˙ δ r ` N sup i,j P Σ ÿ k P Σ ˆ B F k B p i B p j ˙ δ r À ε ´ ` ε ´ } v } L ` ` ε } ∆ s h p γ p q} L ¯ δ r ` ε ´ ´ } v } L ` ` ε ´ } ˆ p } V ¯ δ r À ε ´ δ r ` ε ´ δ r . and } D v D p F } L p V r ˆ H , V r q ď sup } v } H r “ β rj ˇˇˇˇˇż Ω v B Z j p , ˚ B p i d x ˇˇˇˇˇ À ż Ω ˇˇˇˇˇ B Z j p , ˚ B p i ˇˇˇˇˇ d x À ε ´ . With ˜ c „ ε { and c „ ε ´ { for r “ c „ ε ´ { for r “
0, the implicit functiontheorem given in Lemma 6.1 holds, there exists p “ p p v q such that F i p v, p p v qq “ v in a neighbourhood of 0. Now we aim at giving an upper bound of } p } V r , in light of Lemma 6.2, we bound } F p v, q} V r “ ÿ i β ri ˇˇˇˇż Ω Z i ˚ v d x ˇˇˇˇ ď ε ´ r } v } L . The conclusion follows. (cid:3)
These tools permit the proof of Lemma 5.1.
Proof of Lemma 5.1.
Employing Lemma 6.2 we have for u “ Φ ` v with } v } H ď ε , there exists p p q “ p p , v q such that u “ Φ ` v “ Φ p p q ` v K , where v K P Z K˚ p ρ q . Moreover, v K is given by v K “ Φ ´ Φ p ` v , which furthermore implies for k “ , } v K } H k À sup λ Pr , s ÿ j ˆ›››› p j B Φ p B p j ›››› H k ˙ ˇˇˇˇ p “ λ p p q ` } v } H k À ε ´ k ´ { } p p q} l ` ε ´ { } p p q} V k ` } v } H k . Now we decompose v K into two parts: v K “ Q ` w , in which w is perpendicular to the pearling eigen-modes t Z j p , ˚ u . And then Q isgiven by Q “ ÿ j P Σ q j p q Z j p p q , ˚ with q j p q “ (cid:68) v K , Z j p p q , ˚ (cid:69) L } Z j p p q , ˚ } L “ (cid:68) v K , Z j p p q , ˚ (cid:69) L . By the definition of q “ p q ,j q j P Σ , we have } q p q} l ď } v K } L , } q p q} l ď } v K } L . Finally we estimate w , indeed, } w } H k ď } Q } H k ` } v K } H k ď ε ´ k } v K } L ` } v K } H k . The proof is complete. (cid:3)
References Nicholas D. Alikakos, Peter W. Bates, Xinfu Chen (1994).
Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model.
Arch. Rational Mech. Anal., , 165-205.2.
Nicholas D.Alikakos, Giorgio Fusco (2003).
Ostwald ripening for dilute systems underquasistationary dynamics.
Commun. Math. Phys., , 429-479.3.
Nicholas D. Alikakos, Giorgio Fusco, Georgia Karali (2004).
Ostwald ripening in twodimensions—the rigorous derivation of the equations from the Mullins–Sekerka dynamics , J.Differential Equations, , 1-49.4.
Andrew Christlieb, Noa Kraitzman, Keith Promislow (2018).
Competition and complex-ity in amphiphilic polymer morphology. submitted.5.
Amy Novick Cohen (2008).
Chapter 4: The Cahn–Hilliard Equation , Handbook of Differen-tial Equations: Evolutionary Equations. , 201-228.6. Peter Canham (1970).
The minimum energy of bending as a possible explanation of thebiconcave shape of the human red blood cell , J. Theoretical. Biology, (1), 61-81.7. Xingfu Chen (1993).
The Hele Shaw problem and area-preserving curve shortening motion.
Arch. Rat. Mech. Anal., , 117-151.8.
Rustum Choksi, Xiaofeng Ren (2003).
On the derivation of a density functional theory formicrophase separation of diblock copolymers , J. Stat. Phys., , 151-176.9.
Rustum Choksi, Xiaofeng Ren
Diblock copolymer/homopolymer blends: Derivation of adensity functional theory , Physica D, , 100-119.10.
Arjen Doelman, Gurgen Hayrapetyan, Keith Promislow, Brian Wetton (2014).
Mean-der and Pearling of Single-Curvature Bilayer Interfaces in the Functionalized Cahn-HilliardEquation.
SIAM Journal on Mathematical Analysis, (6), 3640-3677.11. Qiang Du, Chun Liu, R. Ryham, Xiaoqiang Wang (2005).
Modeling the spotaneous curva-ture effects in static cell membrane deformations by a phase field formulation.
Comm. Pureand Appl. Anal., (3), 537-548.12. Qiang Du, Chun Liu, Rolf Ryham, and Xiaoqiang Wang. A phase field formulation of theWillmore problem.
Nonlinearity (3) (2005): 1249.13. Qiang Du, Chun Liu, Xiaoqiang Wang (2006).
Simulating the defomation of vesicle mem-branes under elastic bending energy in three dimensions.
J. Comp. Phys., (2), 757-777.14.
Shibin Dai, Keith Promislow (2013, May).
Geometric evolution of bilayers under the func-tionalized Cahn-Hilliard equation.
In Proc. R. Soc. A (Vol. , No. 2153, p. 20120505). TheRoyal Society.15.
Shibin Dai, Keith Promislow (2015).
Competitive geometric evolution of amphiphilic inter-faces.
SIAM Journal on Mathematical Analysis, (1), 347-380.16. Shibin Dai, Keith Promislow (2019)
Geometrically Constrained Minimizers in Dispersionsof Strongly Hydrophobic Amphiphilic Polymer, submitted.17.
L. C. Evans, J. Spruck (1989)
Motion of level sets by mean curvature. I.
J. Diff. Geom., ,635-681.18. Nir Gavish, Gurgen Hayraphetyan, Keith Promislow, Li Yang (2011)
Curvature drivenflow of bilayer surfaces.
Physica D Nonlinear Phenomena, (7), 675-693.19.
Nir Gavish, Jaylan Jones, Zhengfu Xu, Andrew Chriestlieb, Keith Promislow (2012).
Variational models of network formation and ion transport: applications to perflurousufonateionomer membranes.
Polymers, (4), 630-655.20.
G. Gommper, M. Schick (1990).
Correlation between structural and interfacial properties ofamphiphilic systems.
Phys. Rev. Lett., G. Gommper, J. Goos (1994).
Fluctuating interfaces in microemulsions and sponge phases.
Phys. Rev. E, Helfrich, Wolfgang (1973).
Elastic properties of lipid bilayers: theory and possible exper-iments.
Zeitschrift f¨ur Naturforschung C, (11-12), 693-703.23. Gerhard Huisken (1984).
Flow by mean curvature of convex surfaces into spheres.
J. Diff.Geom, (1), 237-266.24. Gurgen Hayrapetyan, Keith Promislow (2015).
Spectra of functionalized operators aris-ing from hypersurfaces.
Zeitschrift f¨ur angewandte Mathematik und Physik, 66(3), 631-662.
EGULARIZED CURVE LENGTHENING FROM THE STRONG FCH FLOW Todd Kapitula, Keith Promislow (2013).
Spectral and dynamical stability of nonlinearwaves (Vol. ). New York: Springer.26.
I.M. Lifschitz, V.V. Slyosov (1961).
The kinetics of precipitation from supersaturated solidsolutions , J. Phys. Chem. Solids, , 35-50.27. Luciano Modica (1987).
The gradient theory of phase transitions and the minimal interfacecriterion.
Archive for Rational Mechanics and Analysis, (2), 123-142.28. Noa Kraitzman, Keith Promislow (2018).
Pearling Bifurcations in the strong Functional-ized Cahn-Hilliard Free Energy.
SIAM Math. Analysis , 3395?3426.29. R.L.Pego (1989).
Front migration in the nonlinear Cahn-Hilliard equation , Prod. R. Soc.Lond. A, , 261-278.30.
Keith Promislow, Qiliang Wu (2015),
Existence of pearled patterns in the planar function-alized Cahn–Hilliard equation.
Journal of Differential Equations, (7): 3298-3343.31.
Matthias R¨oger, Reiner Sch¨atzle (2006).
On a modified conjecture of De Giorgi.
Mathe-matische Zeitschrift, (4), 675-714.32.
M. Tuebner, R. Strey (1987).
Origin of scattering peaks in microemulsions , J. Chem. Phys. , 3195-3200.33. C. Wagner (1961).
Theorie der Alterung von Niedershlagen dursch Umlosen , Z. Electrochem, , 581-594. Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.
E-mail address : [email protected] Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.
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