Existence and stability of periodic planar standing waves in phase-transitional elasticity with strain-gradient effects
aa r X i v : . [ m a t h . A P ] O c t Existence and stability of periodic planar standing waves inphase-transitional elasticity with strain-gradient effects
Jinghua Yao ∗ Abstract
Extending investigations of Antman & Malek-Madani, Schecter & Shearer, Slemrod,Barker & Lewicka & Zumbrun, and others, we investigate phase-transitional elasticitymodels of strain-gradient effect. We prove the existence of non-constant planar periodicstanding waves in these models with strain-gradient effects by variational methods andphase-plane analysis, for deformations of arbitrary dimension and general, physical,viscosity and strain-gradient terms. Previous investigations considered one-dimensionalphenomenological models with artificial viscosity/strain gradient effect, for which theexistence reduces to a standard (scalar) nonlinear oscillator. For our variational analysis,we require that the mean vector of the unknowns over one period be in the elliptic regionwith respect to the corresponding pure inviscid elastic model. For our (1-D) phase-planeanalysis, we have no such restriction, obtaining essentially complete information on theexistence of non-constant periodic waves and bounding homoclinic/heteroclinic waves.Our variational framework has implications also for time-evolutionary stability, throughthe link between the action functional for the traveling-wave ODE and the relativemechanical energy for the time-evolutionary system. Finally, we show that spectralimplies modulational nonlinear stability by using a change of variables introduced byKotschote to transform our system to a strictly parabolic system to which general resultsof Johnson–Zumbrun apply. Previous such results were confined to one-dimensionaldeformations in models with artificial viscosity–strain-gradient coefficients.
Keywords : elasticity, strain-gradient effect, periodic wave, Hamiltonian system. : 35Q74, 49A10, 49A22
Elasticity is the typical property of elastic media. Since they have a wide range of appli-cations, the mathematical study of elasticity has been an important topic (see [AB, AM,BLeZ, D, FP, K, RZ, S1, S2, S3, SS, Z1, Z2], etc., and references therein). However, up tonow, to the best of our knowledge, the study of phase-transitional elasticity has been carriedout only for phenomenological models [S1, S2, S3, SS, Z2] for one-dimensional shear flow, ∗ Department of Mathematics, Indiana University Bloomington, IN 47405, USA. Email: [email protected] INTRODUCTION τ > τ imposessignificant challenges in the mathematical analysis (see the discussions in [Ba, AN]).Besides the Hamitlonian structure of the standing wave equations, we prove that forthe general physical model, there exist non-constant periodic waves no matter whether theunknowns are scalar or not (see Theorem 6.13) under assumptions on the mean vector andperiodic of the wave. For some specific phase-transitional models, we given explicit condi-tions under which the non-constant oscillatory waves exist (see Section 7). In particular,for the one dimensional models, we use phase-plane analysis to get detailed information onthe wave phenomena (existence of periodic, homoclinic, heteroclinic waves, see Section 8).We address also the issue of nonlinear stability. Specifically, using a coordinate trans-formation introduced by Kotschote [K], we show that the class of systems studied here areeffectively strictly parabolic, in the sense that they may be transformed to an enlarged ELASTICITY MODELS WITH STRAIN-GRADIENT EFFECTS
In this section, we will follow the presentations of [AN, Ba, BLeZ, NPT]. Let Ω be thereference configuration which models an elastic body with constant temperature and den-sity. A typical point in Ω will be denoted by X . We use ξ : Ω × R + → R to denotethe deformation(i.e., the deformed position of the material point X ). Consequently, thedeformation gradient is given by F := ∇ X ξ , which we regard as an element in R × .Adopting the notations above, the equations of isothermal elasticity with strain-gradienteffect are given through the following balance of linear momentum(2.1) ξ tt − ∇ X · (cid:16) DW ( ∇ ξ ) + Z ( ∇ ξ, ∇ ξ t ) − E ( ∇ ξ ) (cid:17) = 0 . We make the following physical constraint on the deformation gradient (see [Ba, BLeZ]and [AN, NPT] for the physical background), prohibiting local self-impingement of thematerial:(2.2) det
F > . ELASTICITY MODELS WITH STRAIN-GRADIENT EFFECTS ∇ X · stands for the divergence of an approximate field. As in [D, NPT],for a matrix-valued vector field, we use the convention that the divergence is taken row-wise.In what follows, we shall also use the matrix norm | F | = ( tr ( F T F )) / , which is inducedby the inner product: F : F := tr ( F T F ) . In view of the second law of thermodynamics (see [Ba, PB]), the Piola-Kirchhoff stresstensor DW : R × → R × is expressed as the derivative of an elastic energy density W : R × → R + . Throughout the paper, we assume as in [AN, Ba, NPT] the elastic energydensity function W is frame-indifference. Let SO (3) be the group of proper rotations in R .Then the frame-indifference assumption can be formulated as(2.3) W ( RF ) = W ( F ) , ∀ F ∈ R × , ∀ R ∈ SO (3) . Also, the material consistency (to avoid interpenetration of matter, (2.2), [AN, Ba])requires the following important assumption:(2.4) W ( F ) → + ∞ as det F → . We emphasize that viscous stress tensor Z : R × × R × → R × depends on both thedeformation gradient F and the velocity gradient Q = F t = ∇ ξ t = ∇ v , where τ = ξ t . Fromphysical point of view, the stress tensor Z should also be compatible with principles ofcontinuum mechanics (balance of angular momentum, frame invariance, and the Claussius-Duhem inequality etc). For the related mathematical descriptions and corresponding stressforms see [AN, Ba, BLeZ] and references therein.The strain-gradient effect E is given by E ( ∇ ξ ) = ∇ X · D Ψ( ∇ ξ ) = h X i =1 ∂∂X i (cid:16) ∂∂ ( ∂ ij ζ k ) Ψ( ∇ ξ ) (cid:17)i j,k :1 ... for some convex density Ψ : R × × → R , compatible with frame indifference.The corresponding inviscid part of system (2.1)(2.5) ξ tt − ∇ X · (cid:16) DW ( ∇ ξ ) (cid:17) = 0can be written as(2.6) ( F, τ ) t + X i =1 ∂ X i (cid:16) ˜ G i ( F, τ ) (cid:17) = 0 . Above, (
F, τ ) : Ω → R represents conserved quantities, while ˜ G i : R → R are given by − ˜ G i ( F, τ ) = τ e i ⊕ τ e i ⊕ τ e i ⊕ h ∂∂F ki W ( F ) i k =1 , i = 1 ... e i denotes the i -th coordinate vector in R . EQUATIONS AND SPECIFIC MODELS ∇ X · (cid:16) E ( ∇ ξ ) (cid:17) = ∇ X · {∇ X · D Ψ( ∇ ξ ) } . In view of the orders of differentiation and convexity of Ψ, we may assume thatΨ ≥
0; Ψ(0) = 0; D Ψ(0) = 0; δId ≤ D Ψ( · ) ≤ M Id where δ, M are two positive real numbers and Id is an element in the space L ( R × × ; R × × ).The mapping relations (ignoring physical constraints) areΨ : R × × → R + ; D Ψ : R × × → R × × ; D Ψ : R × × → L ( R × × ; R × × )When the operator ∇ X · reduces to the operator ∂ x where x is a one dimension variable,(2.7) takes the form ∂ x { ∂ x D Ψ( ∂ x ξ ) } . If we identify ξ x as τ , then ∂ x ξ = τ x and (2 .
7) becomes ∂ x { ∂ x D Ψ( τ x ) } = ∂ x { D Ψ( τ x ) τ xx } . Note that D Ψ : R → L ( R ; R ) when ∇ X · reduces to ∂ x . So we assume that D Ψ( · ) as matrix function satisfy the assumption δId ≤ D Ψ( · ) ≤ M Id as operators.
In this paper, we focus on the interesting subclass of planar solutions, which are solutions inthe full 3D space that depend only on a single coordinate direction; that is, we investigatedeformations ξ given by ξ ( X ) = X + U ( z ) , X = ( x, y, z ) , U = ( U , U , U ) ∈ R . Corresponding to the above deformation or displacement ξ , the deformation gradient withrespect to X (3.1) F = U ,z U ,z U ,z = τ τ τ . We shall denote V = ( τ, u ) = ( τ , τ , τ , u , u , u ), where τ = U ,z , τ = U ,z , τ = 1 + U ,z and u = U ,t , u = U ,t , u = U ,t with the physical constraint τ >
0, corresponding todet
F > V .Writing W ( τ ) = W (cid:16) τ τ τ (cid:17) , we see that for all F as in (2.1) there holds ∇ X · ( DW ( F )) = ( D τ W ( τ )) z . EQUATIONS AND SPECIFIC MODELS DW ( F )).In this paper, we first study the problems (traveling wave ODEs, Hamiltonian ODEs,existence of standing waves) for general elastic potential energy and give a rather generalabstract existence result. Then we study local models by specifying the related terms insystem (2.1) as follows:
1. Elastic potential W.
As described in [FP], we shall study the phase-transitionalelastic potential: W ( F ) = | F T F − C − | · | F T F − C + | , which is a potential for anisotropic material with material frame indifference property. Westudy models involving this phase-transitional elasticity potential(See Appendix A for re-lated computations). It is important to notice that the elastic potential here does not satisfythe asymptotic behavior when det F → + . Especially this is an irrelevant assumption forshear models. Hence the related models are local models for the real physics.
2. Viscous stress tensor Z . We use the following tress tensor which is compatiblewith the principles of continuum mechanics (see [BLeZ]) Z ( F, Q ) = 2(det F ) sym ( QF − ) F − ,T . We note that the related Cauchy stress tensor T = 2(det F ) − Z F T = 2 sym ( QF − ) is theLagrangian version of the stress tensor 2 sym ∇ v written in the Eulerian coordinates. Forincompressible fluids 2 div ( sym ∇ v ), giving the usual parabolic viscous regularization of thefluid dynamics evolutionary system.
3. The Strain-gradient term E . For the strain-gradient effect we will chooseΨ( P ) = | P | , so that E ( ∇ ξ ) = ∇ X · ∇ ξ = △ X F, which is an extension of the 1Dcase of [S1]. We see that it is the strain-gradient term that makes the models have abun-dant wave phenomena. The system.
As a convention, we shall use x ∈ R as the space variable instead of z .So we have the following system(3.2) ( τ t − u x = 0; u t + σ ( τ ) x = ( b ( τ ) u x ) x − ( d ( τ x ) τ xx ) x . with(3.3) σ := − D τ W ( τ ), d ( · ) := D Ψ( · ) = Id and b ( τ ) = τ − .We are interested in the existence of periodic traveling waves of the above system, whichinvolves third order term because of the strain-gradient effect. See also Appendix 9.1.1 forsome structure properties of the system. TRAVELING WAVE ODE SYSTEM We seek traveling wave solution of the system (3 . τ ( x, t ) , u ( x, t )) := ( τ ( x − st ) , u ( x − st )),where s ∈ R is the wave speed. Let us denote in the following ′ as differentiation with respectto x − st . For convenience, we still use x to represent x − st (Indeed, we will show a bitlater that in fact s = 0 is necessary for the existence of periodic or homoclinic waves; seeequation (5.6)). With further investigation in mind, we write the related equations for thegeneral class of elastic models with strain-gradient effects. Now from system (3 . ( − sτ ′ − u ′ = 0; − su ′ + σ ( τ ) ′ = ( b ( τ ) u ′ ) ′ − ( d ( τ ′ ) τ ′′ ) ′ . Plugging the first equation into the second in the above system, we obtain the followingsecond-order ODE in τ :(4.2) s τ ′ + σ ( τ ) ′ = − ( b ( τ ) sτ ′ ) ′ − ( d ( τ ′ ) τ ′′ ) ′ . In view of d ( · ) = D Ψ( · ), we readily see:(4.3) s τ ′ + σ ( τ ) ′ = − ( b ( τ ) sτ ′ ) ′ − ( D Ψ( τ ′ ) τ ′′ ) ′ . Choosing a specific space point, say x , we integrate once to get:(4.4) s τ + σ ( τ ) + q = − sb ( τ ) τ ′ − D Ψ( τ ′ ) ′ Here q is an integral constant vector. Relating this with the elastic potential function W ,we have(4.5) − DW ( τ ) + s τ + q = − sb ( τ ) τ ′ − D Ψ( τ ′ ) ′ Note carefully that the integral constant vector is given by(4.6) q = { DW ( τ ) − s τ − sb ( τ ) τ ′ − D Ψ( τ ′ ) ′ } (cid:12)(cid:12)(cid:12) x = x . Defining G ( P ) := h P, D Ψ( P ) i − Ψ( P ), we see that dGdP = h P, D Ψ i . Here P ∈ R n andΨ : R n → R (for our purpose n = 1 , , G : R n → R a different scalar potential typefunction. Now we are ready to state a structural property about the traveling wave ODEsystem (4.5). Proposition 5.1.
When s = 0 , the system (4.5) is a Hamiltonian system with factor (cid:16) D Ψ( τ ′ ) (cid:17) − , preserving the Hamiltonian integral H ( τ, τ ′ ) = − W ( τ ) + qτ + G ( τ ′ ) ≡ constant . HAMILTONIAN STRUCTURE Here W can be taken in particular as the phase-transitional elastic potential (see AppendixA and also p. 36 of [BLeZ]), with ( a , a , a ) = ( τ , τ , τ ) ) W ( τ ) = (cid:16) τ + τ + ε ) + ( | τ | − − ε ) (cid:17) − ε τ , and Ψ( p ) as any convex function with Ψ(0) = 0 and d Ψ(0) = 0 , simplest case Ψ( P ) = | P | / . (Note: in this simple case G ( P ) = Ψ( P ) .)Proof. When s = 0, the traveling wave ODE (4.5) becomes:(5.1) − dW ( τ ) + q = − D Ψ( τ ′ ) ′ and the constant q = { DW ( τ ) − D Ψ( τ ′ ) ′ } (cid:12)(cid:12)(cid:12) x = x . Noticing the positive-definiteness of D Ψ( · ),we may write the ODE as a first order system by regarding τ, τ ′ as independent variables:(5.2) τ ′ = τ ′ = [ D Ψ( τ ′ )] − D Ψ( τ ′ ) τ ′ ; τ ′′ = − [ D Ψ( τ ′ )] − ( − DW ( τ ) + q )Now, consider the energy surface given by:(5.3) H ( τ, τ ′ ) := − W ( τ ) + qτ + G ( τ ′ ) . We see that(5.4) ∂∂τ ′ H ( τ, τ ′ ) = dG ( τ ′ ) dτ ′ = D Ψ( τ ′ ) τ ′ ; ∂∂τ H ( τ, τ ′ ) = − DW ( τ ) + q. Comparing (5.2), (5.4), we see that the traveling wave ODE is a Hamiltonian systemwith factor γ := [ D Ψ( τ ′ )] − . Thus, (4.5) preserves the Hamiltonian H . We can see thisalso by the explicit computation, writing ζ = x − st : ddζ H ( τ, τ ′ ) = ∂∂τ H ( τ, τ ′ ) τ ′ + ∂∂τ ′ H ( τ, τ ′ ) τ ′′ = γ ∂∂τ H ( τ, τ ′ ) ∂∂τ ′ H ( τ, τ ′ ) + γ ∂∂τ ′ H ( τ, τ ′ ) {− ∂∂τ H ( τ, τ ′ ) } = 0 . From the above structural information, we easily get a necessary condition for the exis-tence of periodic or homoclinic waves, extending results of [OZ] in a one-dimensional modelcase.
HAMILTONIAN STRUCTURE Theorem 5.2.
For (4.5) with s ≷ , there holds dH/dζ ≶ , where (5.5) H ( τ, τ ′ ) := − W ( τ ) + s | τ | + qτ + G ( τ ′ ) , so that no homoclinic or periodic orbits can occur unless s = 0 .Proof. Considering the evolution of ddζ H ( τ, τ ′ ) along the flow of traveling wave ODE system(4.5), we have ddζ H ( τ, τ ′ ) = ∂∂τ H ( τ, τ ′ ) τ ′ + ∂∂τ ′ H ( τ, τ ′ ) τ ′′ = h− D τ W ( τ ) + q + s τ, τ ′ i + h DG ( τ ′ ) , τ ′′ i = h− D τ W ( τ ) + q + s τ, τ ′ i + h D Ψ( τ ′ ) τ ′ , τ ′′ i = h− D τ W ( τ ) + q + s τ, τ ′ i + h D Ψ( τ ′ ) τ ′′ , τ ′ i = h− D τ W ( τ ) + q + s τ + D Ψ( τ ′ ) ′ , τ ′ i = h− sb ( τ ) τ ′ , τ ′ i . The conclusion thus follows from the positive definiteness of b ( τ ). The Hamiltonian system.
From the above analysis, we see that necessarily s = 0,i.e., all traveling periodic waves are standing. The traveling wave ODE system reduces tothe following form with an integral constant q (5.6) − τ ′′ = − D τ W ( τ ) + q ; q = { D τ W ( τ ) − τ ′′ } (cid:12)(cid:12)(cid:12) x = x . If we take the Hamiltonian point of view, the corresponding Hamiltonian for the abovesystem is H ( τ, τ ′ ) = 12 | τ ′ ( x ) | + V ( τ, τ ′ ) , where V ( τ, τ ′ ) := q · τ ( x ) − W ( τ ( x )). The periodic solutions of the system are confined tothe surface H ( τ, τ ′ ) ≡ constant.In the following, we list the elastic potential and related information for the phase-transitional models we shall deal with in this paper for completeness and future study. Toget these models, we fix one or two directions of τ as zero, or, in the incompressible case, τ ≡ τ equation by a Lagrange multiplier corresponding to pressure). We refer thereader to [BLeZ], Section 3, for details of the derivations of these models. HAMILTONIAN STRUCTURE This model corresponds to setting τ = 1.(5.7) W ( τ ) = (cid:16) τ + 2( τ − ε ) + ( | τ | − ε ) (cid:17)(cid:16) τ + 2( τ + ε ) + ( | τ | − ε ) (cid:17) . Its gradient components are(5.8) D τ W ( τ ) = 8 τ ( | τ | + 1 − ε ) { | τ | + ε ) + ( | τ | − ε ) } ;(5.9) D τ W ( τ ) = 8 τ ( | τ | + 1 − ε ) { | τ | + ε ) + ( | τ | − ε ) } − τ ε . The Hessian components are(5.10) w := D τ τ W ( τ ) = 8( | τ | + 1 − ε + 2 τ ) { | τ | + ε ) + ( | τ | − ε ) } + 32 τ ( | τ | + 1 − ε ) ;(5.11) w = w := D τ τ W ( τ ) = 16 τ τ { | τ | + ε )+ ( | τ | − ε ) } + 32 τ τ ( | τ | + 1 − ε ) ;(5.12) w := D τ τ W ( τ ) = 8( | τ | +1 − ε +2 τ )[2( | τ | + ε )+( | τ | − ε ) ]+32[ τ ( | τ | +1 − ε ) − ε ] . τ ≡ ; τ ≡ . The elastic potential becomes(5.13) W ( τ ) = (cid:16) τ + 2 ε + ( τ − ε ) (cid:17) . The first order derivative is(5.14) D τ W ( τ ) = 8 τ ( τ + 1 − ε ) { τ + ε ) + ( τ − ε ) } . The second order derivative is(5.15) D τ τ W ( τ ) = 8(3 τ + 1 − ε ) { τ + ε ) + ( τ − ε ) } + 32 τ ( τ + 1 − ε ) . τ ≡ τ ≡ . Correspondingly, the elastic potential becomes(5.16) W ( τ ) = (cid:16) τ − ε ) + ( τ − ε ) (cid:17) × (cid:16) τ + ε ) + ( τ − ε ) (cid:17) The first order derivative is(5.17) D τ W ( τ ) = 8 τ ( τ + 1 − ε ) { τ + ε ) + ( τ − ε ) } − τ ε ;The second order derivative is(5.18) D τ τ W ( τ ) = 8(3 τ + 1 − ε ) { τ + ε ) + ( τ − ε ) } + 32 { τ ( τ + 1 − ε ) − ε } . HAMILTONIAN STRUCTURE Remark 5.3 (1D vs. 2D shear solutions) . Evidently, solutions of -D shear models I orII determine solutions of the full shear model obtained by adjoining τ ≡ or τ ≡ respectively. It is worth noting that the structure dW ( τ ) = τ f ( | τ | ) + c (0 , τ ) T for f a scalar-valued function and c a scalar constant yields that the only solutions τ ( x ) of (5.6) with η · τ ≡ c = constant for some constant vector η are those satisfying c f ( | τ | ) + cη τ ≡ constant , which gives by direct computation τ ≡ constant or else η · τ ≡ and η τ ≡ , inwhich case τ ≡ or τ ≡ . That is, the D systems derived here are the only solutionsof the D shear model that are not genuinely two-dimensional in the sense that they areconfined to a line in the τ -plane. In particular, if the mean of τ or τ over one period isnot zero, then we can be sure that the solution is genuinely two-dimensional. In this case τ = τ ≡ τ = τ . The potential and its derivatives are givenbelow. The elastic potential becomes(5.19) W ( τ ) = (cid:16) ε + ( τ − − ε ) (cid:17) The first order derivative is(5.20) D τ W ( τ ) = 8 τ ( τ − − ε ) { ε + ( τ − − ε ) } ;And the second order derivative is(5.21) w := D τ τ W ( τ ) = 8(3 τ − − ε ) { ε + ( τ − − ε ) } + 32 τ ( τ − − ε ) . First, we consider the case τ = ( τ , τ ) T ∈ R . The elastic potential W and derivatives areas follows.(5.22) W ( τ ) = (cid:16) τ − ε ) + ( | τ | − − ε ) (cid:17)(cid:16) τ + ε ) + ( | τ | − − ε ) (cid:17) ;The gradient components are(5.23) D τ W ( τ ) = 8 τ ( | τ | − ε ) { τ + ε ) + ( | τ | − − ε ) } − τ ε . (5.24) D τ W ( τ ) = 8 τ ( | τ | − − ε ) { τ + ε ) + ( | τ | − − ε ) } ;Similarly, we have the Hessian components(5.25) w := D τ τ W ( τ ) = 8( | τ | − ε + 2 τ )[2( τ + ε ) + ( | τ | − − ε ) ] + 32[ τ ( | τ | − ε ) − ε ] . HAMILTONIAN STRUCTURE w = w := D τ τ W ( τ ) = 16 τ τ { τ + ε )+( | τ | − − ε ) } +32 τ τ ( | τ | − ε )( | τ | − − ε )(5.27) w := D τ τ W ( τ ) = 8( | τ | − − ε + 2 τ ) { τ + ε ) + ( | τ | − − ε ) } + 32 τ ( | τ | − − ε ) Second, if we fix the τ direction and let τ := ( τ , τ ) T , we get another 2D compressiblemodel. We omit the details here as the form is obvious. In this case τ = ( τ , τ , τ ) T ∈ R . corresponding to the phase-transitional elastic potentialfunction W , we list the components of D W ( τ ) := ( w ij ) × .(5.28) w = 8( | τ | + 2 τ − ε ) { | τ | − τ + ε ) + ( | τ | − − ε ) } + 32 τ ( | τ | − ε ) (5.29) w = w = 16 τ τ { | τ | − τ + ε ) + ( | τ | − − ε ) } + 32 τ τ ( | τ | − ε ) (5.30) w = w = 16 τ τ { | τ | − τ + ε )+( | τ | − − ε ) } +32 τ τ ( | τ | − ε )( | τ | − − ε )(5.31) w = w = 16 τ τ { | τ | − τ + ε )+( | τ | − − ε ) } +32 τ τ ( | τ | − ε )( | τ | − − ε )(5.32) w = 8( | τ | + 2 τ − − ε ) { | τ | − τ + ε ) + ( | τ | − − ε ) } + 32 τ ( | τ | − − ε ) (5.33) w = 8( | τ | − ε + 2 τ )[2( τ + τ + ε ) + ( | τ | − − ε ) ] + 32[ τ ( | τ | − ε ) − ε ] . Remark 5.4.
Similarly as in Remark 5.3, we find that solutions of the D compressiblemodels are genuinely two-dimensional, in the sense that they are not confined to a line inthe τ -plane, unless they are solutions of the D model derived above: in particular, the meanover one period of ( τ , τ ) is zero. Likewise, solutions of the full D compressible model aregenuinely three-dimensional in the sense that they are not confined to a plane, unless theyare solutions of one of the D models derived above, in particular, the mean of τ or of τ over one period is zero. CALCULUS OF VARIATIONS. We note that, for the case of 1-D shear flow, the coefficients given by (3.3) become b, d ≡ constant, and the elastic potential W is of a generalized double-well form. Thus, we recoverfrom first principles the type of phenomenological model studied in [S1, S2, S3, SS, Z2],though with a slightly modified potential refining the quartic double-well potential assumedin the phenomenological models. The 2-D shear flow gives a natural extension to multi-dimensional deformations, which is also interesting from the pure Calculus of Variationspoint of view (see the following section), as a physically relevant example of a vectorial“real Ginzberg–Landau” problem of the type studied on abstract grounds by many authors.Finally, we note that the various compressible models give a different extension of thephenomenological models, to the case of “real” or nonconstant viscosity. In this section, we formulate the problem in the framework of Calculus of Variations andgive the proof of the existence result.
As a first step, we recall the notions of Sobolev spaces involving periodicity and introducethe space structure we are going to use (see [MW]). For fixed real number
T >
0, let C ∞ T be the space of infinitely differentiable T -periodic functions from R to R n (for our purpose n = 1 , , Lemma 6.1.
Let u, v ∈ L (0 , T ; R n ) . If the following holds: for every f ∈ C ∞ T , Z T ( u ( t ) , f ′ ( t )) dt = − Z T ( v ( t ) , f ( t )) dt, then Z T v ( s ) ds = 0 and there exists a constant vector c in R N such that u ( t ) = Z t v ( s ) ds + c a.e. on [0 , T ] . The function v := u ′ is called the weak derivative of u . Consequently, we have u ( t ) = Z t u ′ ( l ) dl + c, which implies the following: u (0) = u ( T ) = c ; CALCULUS OF VARIATIONS. u ( t ) = u ( s ) + Z ts u ′ ( l ) dl. Proof.
For the mean zero property, we could consider the specific test function f = e j . Forthe integral formulation, we could consider the use of Fubini Theorem and Fourier expansionof f to conclude ([MW]).Define the Hilbert space H T as usual (hence reflexive Banach space) with this innerproduct and corresponding norm: for u, v ∈ H T , h u, v i := Z T ( u, v ) + ( u ′ , v ′ ) ds ; k u k := Z T | u | + | u ′ | ds. Next, we collect some facts for later use.
Proposition 6.1. (Compact Sobolev Embedding property) H T ⊂⊂ C [0 , T ] compactly. Proposition 6.2. If u ∈ H T and (1 /T ) R T u ( t ) dt = 0 , then we have Wirtinger’s inequality Z T | u ( t ) | dt ≤ ( T / π ) Z T | u ′ ( t ) | dt and a Sobolev inequality | u | ∞ ≤ ( T / Z T | u ′ ( t ) | dt. The Compact Sobolev imbedding property will give us the required weak lower semi-continuity property for the nonlinear functionals. The Wirtinger’s inequality supplies usequivalent norms in related Sobolev spaces with mean zero property (see [MW] for completeproofs).
Now for a given real positive constant T , we consider problem (5.6) in H T ( − τ ′′ = − D τ W ( τ ) + q = − D τ ( W ( τ ) − q · τ ); τ (0) − τ ( T ) = 0; τ ′ (0) − τ ′ ( T ) = 0 . Let us first consider the cases and formulations without the physical restriction τ > τ := 1 T Z T τ ( x ) dx = m. CALCULUS OF VARIATIONS. m ∈ R n , n = 1 , , τ ′′ ( x ) = DW ( τ ) − q ; τ (0) = τ ( T ); τ ′ (0) = τ ′ ( T ); T R T τ ( x ) dx = m. If we seek periodic solutions, q can be determined by integrating the equations aboveover one period; that is, q = 1 T Z T DW ( τ ( x )) dx. Define v ( x ) = τ ( x ) − m . We see easily that T R T v ( x ) dx = 0 , and v ( x ) satisfies thesystem of equations:(6.2) v ′′ ( x ) = DW ( v + m ) − q ; v (0) = v ( T ); v ′ (0) = v ′ ( T ); T R T v ( x ) dx = 0 . For convenience, we rewrite the above system as(6.3) v ′′ ( x ) = DW ( v + m ) − DW ( m ) + DW ( m ) − q ; v (0) = v ( T ); v ′ (0) = v ′ ( T ); T R T v ( x ) dx = 0 . Define ˜ W ( v ) = W ( v + m ) − DW ( m ) · v and ˜ q = q − DW ( m ). We get the followingproblem(6.4) v ′′ ( x ) = D ˜ W ( v ) − ˜ q ; v (0) = v ( T ); v ′ (0) = v ′ ( T ); T R T v ( x ) dx = 0 . Here ˜ q is determined by integration: T R T D ˜ W ( v ) dx = ˜ q . Remark 6.3.
We require v > − m on [0 , T ] for models involving τ direction in view ofthe physical assumption (2 . . Define F ( v ) = W ( v + m ) − W ( m ) − DW ( m ) · v and introduce the functional(6.5) I ( v ) = Z T | v ′ | dx + Z T F ( v ) dx CALCULUS OF VARIATIONS. H T, := { v ∈ H T ; ¯ v = 1 T Z T v dx = 0 } . Proposition 6.4. is always a critical point of the functional I defined above on H T, .Proof. It is easy to verify that for φ ∈ H T, , there holds I ′ ( v ) φ = Z T v ′ · φ ′ + D ˜ W ( v ) · φ dx. Taking v = 0 and noticing that D ˜ W (0) = 0, we get the desired result. Remark 6.5.
For any Hamiltonian ODE, there exists such an equivalent variational, or“Lagrangian” formulation, according to the principle of least action; see [Lan]. By ourformulation, we make always a critical point and it corresponds to the constant solution.This geometric property supplies us nice way to exclude the possibility that the periodicsolution we find is constant, i.e., to help prove the periodic waves we find are oscillatory. Proposition 6.6.
Without physical restriction on τ , the critical point of I corresponds tothe solution of (5 . .Proof. This can be regarded as a simple consequence of Corollary 1.1 in [MW]. For com-pleteness, we write the details here. First, assume v solves(6.6) v ′′ ( x ) = D ˜ W ( v ) − ˜ q ; v (0) = v ( T ); v ′ (0) = v ′ ( T ); T R T v ( x ) dx = 0 . Multiplying the equation by φ ∈ H T, and integrating to get Z T v ′ φ ′ + D ˜ W ( v ) · φ dx = 0 , i.e v is a critical point of I .Next, we assume v is a critical point and φ ∈ H T . Then φ − ¯ φ ∈ H T, . Hence we have Z T v ′ · ( φ − ¯ φ ) ′ + Z T D ˜ W ( v ) · ( φ − ¯ φ ) dx = 0i.e. Z T v ′ · φ ′ + Z T D ˜ W ( v ) · φ − Z T D ˜ W ( v ) · ¯ φ dx = 0Noting that ¯ φ = T R T φ dx , we find that the left-hand side expression above is: Z T v ′ · φ ′ + Z T D ˜ W ( v ) · φ − Z T D ˜ W ( v ) · ( 1 T Z T φ dx ) dx = 0 CALCULUS OF VARIATIONS. T R T D ˜ W ( v ) dx = ˜ q , we get Z T v ′ · φ ′ + Z T ( D ˜ W ( v ) − ˜ q ) · φ dx = 0 , which implies v ′′ = D ˜ W ( v ) − ˜ q. Remark 6.7.
If we consider models involving the restriction v > − m , we need to considera variational problem with this constraint, which will make the admissible set not weaklyclosed. In view of the physical properties of the elastic potential function W (in particular, thepolynomial structure of the phase-transitional potential function), we can apply the directmethod of the calculus of variation to show existence. In order to deal with the integralconstant q , we may restrict the admissible sets (or choose proper function space) on whichwe consider the functional or use Lagrange multiplier to recover it by adding restrictionfunctional on the original space on which the functional is defined.In the following, we give some propositions of general nonlinear functionals. Thesepropositions and further materials can be found in [De, Ni, ZFC] and the references therein. Proposition 6.8.
Let X be a Banach space, I a real functional defined on X and U be asequentially weakly compact set in X . If I is weakly lower semi-continuous, then I attainsits minimum on U , i.e. there is x ∈ U , such that I ( x ) = inf x ∈ U I ( x ) .Proof. Let c := inf x ∈ U I ( x ). By definition of inf, there exists { x n } ⊂ U such that I ( x n ) → c .In view that U is sequentially weakly compact, { x n } admits a weakly convergent subse-quence, still denoted by { x n } . Denote x ∈ X the corresponding weak limit. Since U isweakly closed, we know x ∈ U . Noticing that weakly lower semi-continuity of I , we have c = lim n I ( x n ) ≥ I ( x ). By the definition of c , we in turn know I ( x ) = c > −∞ , whichcompletes the proof.It is well-known that a bounded weakly closed set in a reflexive Banach space is weaklycompact. In particular, a bounded closed convex set in reflexive Banach space is weaklycompact since weakly close and close in norm are equivalent for convex sets. Hence we havethe following corollaries: Corollary 6.9.
Let U be a bounded weakly closed set in a reflexive Banach space X and I be a weakly lower semi-continuous real functional on X . Then there exists x ∈ U suchthat I ( x ) = inf x ∈ U I . Definition 6.10.
A real functional I on a Banach space X is said to be coercive if lim | x | X → + ∞ I ( x ) = + ∞ . Corollary 6.11.
Any coercive weakly lower semi-continuous real functional I defined on areflexive Banach space X admits a global minimizer. CALCULUS OF VARIATIONS. In this part, we first give a general result for models with the physical assumption τ > v > − m . We will assume the following conditions on the potential W (A1) W ∈ C and W ( τ ) → + ∞ as τ → + . For τ ≤
0, define W ( τ ) = + ∞ ; (A2) There exist a positive constant C such that W ( τ ) ≥ Cτ for τ ∈ R n ( n = 1 , , (A3) There exists a constant vector m ∈ R := { m ∈ R ; m > } such that σ { D W ( m ) }∩ R − = ∅ . Here σ { D W ( m ) } is the spectrum set of D W ( m ).Assumption ( A
2) implies in particular that the potential is bounded from below. As-sumption ( A
3) amounts to saying that there is a point where the potential is concave. Fromthe physical point of view, this is quite reasonable.
Remark 6.12.
A simple kind of potential function is that for an isentropic polytropic gas,for which dW ( τ ) = cτ − γ , γ > . This yields W ( τ ) = c τ − γ , with < − γ < for γ in the typical range < γ < suggested by statistical mechanics [B], hence blowup as τ → at rate slower than c/τ . Indeed, a point charge model with inverse square lawyields in the continuum limit W ( τ ) ∼ τ − / for dimension , consistent with a monatomicgas law γ = 5 / . Thus, (A2) requires a near-range repulsion stronger than inverse square.Alternatively, one may assume not point charges but particles of finite radius, as is oftendone in the literature, in which case W ( τ ) = ∞ for τ ≤ α , α > , also satisfying (A2).However, in this case, a much simpler argument would suffice to yield τ ≥ α a.e. Theorem 6.13.
Assume (A1), (A2) and (A3). If ( πT ) < λ ( m ) , then we have a physicalnonconstant periodic wave solution for the problem (3 . for which the mean over one periodof τ is m . Here − λ ( m ) is the smallest eigenvalue of D W ( m ) . In the following Lemmas of this section, we assume assumption (A1), (A2) and (A3)hold. Define two subsets of H T, by A := { v ∈ H T, ; v > − m } ; A := { v ∈ H T, ; v ≥ − m } . Remark 6.14.
The admissible set A is not weakly closed in H T, . Lemma 6.15.
Under assumptions (A1)-(A3), I is a coercive functional on H T, .Proof. By the definition of I , we just need to consider the part R T F ( v ) dx . By assumption(A2), we have CALCULUS OF VARIATIONS. Z T F ( v ) dx = Z T W ( v + m ) − W ( m ) − DW ( m ) · v dx = Z T W ( v + m ) − W ( m ) dx ≥ − W ( m ) T > −∞ . By the above lemma, we see that for sufficient large R the minimizers of I on A i arerestricted to the sets ¯ A i := A i ∩ B H T, [0 , R ] for i = 1 , B H T, [0 , R ] is the closed ballwith center 0 and radius R in H T, . Define S i := { v ∈ A i ; I ( v ) = inf ˜ v ∈A i I (˜ v ) } . Obviously,we have S i := { v ∈ ¯ A i ; I ( v ) = inf ˜ v ∈A i I (˜ v ) } . Lemma 6.16. ¯ A i is a weakly compact set in H T, .Proof. ¯ A is bounded by its definition. Since H T is reflexive, we know ¯ A is weakly se-quentially compact. Also, ¯ A is convex. Indeed, we can use the definition of convexity of aset to check this easily. An appeal to Sobolev embedding theorem yields that ¯ A is closedin norm topology of H T . For a convex set, closeness in norm topology and weak topologycoincides, hence we have that ¯ A is weakly closed. Putting this information together, wehave shown that ¯ A is weakly compact. Lemma 6.17. I is a weakly lower semi-continuous functional on H T, .Proof. Let v n → v weakly in H T, . By Sobolev imbedding, we have v n → v uniformlyin [0 , T ]. Hence we have R T F ( v n ) dx → R T F ( v ) dx . Because of the mean zero property, R T | v ′ | dx is of norm form, hence it is a weakly lower semi-continuous functional. Lemma 6.18. S = ∅ and v ≥ − m + ǫ for v ∈ S under the assumption of Theorem 6.12.Here ǫ is a positive constant.Proof. By Proposition 6.8, S = ∅ . Note that 0 ∈ A , I (0) = 0 and hence I ( v ) ≤ v ≥ − m + ǫ . Indeed, suppose there were x ∈ [0 , T ] such that v ( x ) = − m . Then by Sobolev embedding there would be a positive constant K suchthat | v ( x ) + m | = | ( v ( x ) + m ) − ( v ( x ) + m ) | ≤ K | x − x | / for x ∈ [0 , T ]. Byassumption ( A I ( v ) = R T (1 / | v ′ | dx + R T W ( v + m ) − W ( m ) dx ≥ R T CK | x − x | − dx − R T W ( m ) dx = + ∞ , a contradiction. Lemma 6.19. S = S under the assumption of Theorem 6.12. CALCULUS OF VARIATIONS. Proof.
Consider the second variation. An easy computation shows that for v, φ in H T, I ′′ ( v ) : ( φ ⊗ φ ) = Z T | φ ′ | dx + Z T D W ( v + m ) : ( φ ⊗ φ ) dx. To show 0 S , consider I ′′ (0) : ( φ ⊗ φ ) = Z T | φ ′ | dx + Z T D W ( m ) : ( φ ⊗ φ ) dx. Let ˜ φ ( x ) = η sin( πxT ) for 0 < η < m and v ∈ R be a unit eigenvector corresponding to − λ ( m ). We see that φ ( x ) := ˜ φ ( x ) v ∈ A . Since 0 is a critical point of I on H T, and I ′′ (0) : ( φv ⊗ φv ) = Z T η ( 2 πT ) (cos( 2 πxT )) dx − λ ( m ) Z T η (sin( 2 πxT )) dx = η T { ( 2 πT ) − λ ( m ) } < . Hence we see that 0 S and S = S is obvious. Proof of Theorem 5.11.
Combining Lemma 6.14-Lemma 6.18, we finish the proof of Therem6.12.
Remark 6.20.
The condition ( πT ) < λ ( m ) in Theorem 6.12 on the period T , is readilyseen by Fourier analysis to be the sharp criterion for stability of the constant solution τ ≡ m , u ≡ . Equivalently, it is the Hopf bifurcation condition as period is increased,marking the minimum period of bifurcating periodic waves. Thus, it is natural, and noreal restriction. On the other hand, there may well exist minimizers at whose mean mW is convex; this condition is sufficient but certainly not necessary. Likewise, there existsaddle-point solutions not detected by the direct approach. In the scalar case τ ∈ R , the condition that D W ( m ) have a negative eigenvalue is equiva-lent to convexity of the Hamiltonian H at the equilibrium ( m, τ ∈ R d , d >
1, if D W ( m ) < D W that becomes negative and not alleigenvalues, and so these methods cannot be directly applied.It is an interesting question to what extent such standard methods could be adaptedto the situation of a Hamiltonian potential (in our case − W ) with a single convex mode. EXISTENCE OF PERIODIC SOLUTIONS FOR SPECIFIC MODELS W at m . Finally, it would be interesting to find natural and readily verifiableconditions for existence of saddle-point solutions in this context. In this section, we focus on the existence of periodic waves for the incompressible models(i.e. models with τ ≡ D models I, II and the 2D incompressible model.First note that these models do not involve the τ direction. Hence we have no conditioncorresponding to ( A W has good growth rate when | τ | → + ∞ for τ = τ , τ or ( τ , τ ). This will makeour functionals coercive. Hence we have the following: Theorem 7.1.
For the incompressible models, there exist non-constant periodic standingwaves respectively if the mean m (either vector or scalar) satisfies ( A and ( πT ) < λ ( m ) .When m is scalar, assumption ( A means m lies in the elliptic region of the viscoelasticitysystem (2 . .Proof. It is easy to see the corresponding functionals are coercive, weakly lower semi-continuous functionals on the reflexive Banach spaces H T, . Hence Corollary 6.11 applies.The verification that the global minimizers respectively are not zero is entirely the same asin Lemma 6.18 by considering the second variation.Next, we specify the corresponding conditions in Theorem 7 . The condition is(7.1) ( 2 πT ) < − m + 1 − ε ) { m + ε ) + ( m − ε ) } − m ( m + 1 − ε ) . In particular, if m = 0, the condition reads(7.2) ( 2 πT ) < − − ε )(2 ε + ε ) . Condition (7 .
2) illustrate that our assumption is not a void assumption. Also, in the meanzero case, (7 .
2) holds only if ε >
1. Comparing this with the existence result by phase-planeanalysis (see in particular section 8.1), we see that these results match very well.
EXISTENCE OF PERIODIC SOLUTIONS FOR SPECIFIC MODELS The condition is(7.3) ( 2 πT ) < − m + 1 − ε ) { m + ε ) + ( m − ε ) } − { m ( m + 1 − ε ) − ε } . In particular, if m = 0, the condition reads(7.4) ( 2 πT ) < − − ε )(2 ε + ε ) + 32 ε = 8( ε + ε + 2 ε ) . Condition (7 .
4) implies in particular that for any ε >
0, we have long-periodic oscillatorywaves. Similarly, for any given
T >
0, we have oscillatory waves as long as ε >
In this case D W ( m ) is given by its components w := D τ τ W ( m ) = 8( | m | +1 − ε +2 m ) { | m | + ε )+( | m | − ε ) } +32 m ( | m | +1 − ε ) ; w = w := D τ τ W ( m ) = 16 m m { | m | + ε ) + ( | m | − ε ) } + 32 m m ( | m | + 1 − ε ) ; w := D τ τ W ( m ) = 8( | m | +1 − ε +2 m )[2( | m | + ε )+( | m | − ε ) ]+32[ m ( | m | +1 − ε ) − ε ];The corresponding condition is ( πT ) < λ ( m ). This is obviously a rather mild condition.To see this, we can consider in particular the mean m = ( m , T or (0 , m ) T . Then theresults on the two 1D incompressible models readily give the conclusion because we havediagonal matrices.Based on the analysis of these conditions, we have in particular ( m = 0 case): Theorem 7.2.
For the 2D shear model, 1D shear model I and II, we have the followingexistence result of periodic viscous traveling/standing waves:(1) Given any ε > ε ≥ , for any T satisfying T > T ( ε ) > , system (5 . hence (2 . has a nonconstant periodic solution with some appropriate integral constant q ; For the 1Dmodel II we have ε = 0 .(2) Given any T > , for any ε satisfying ε > ε ( T ) > , system (5 . hence (2 . has anonconstant periodic solution with some appropriate integral constant q . Remark 7.3.
From the above theorem, we see in particular indeed for the D shear model,we have infinitely many nontrivial periodic viscous traveling waves with appropriate corre-sponding q values. In particular, we have a sequence of waves with minimum positive period T → + ∞ .
1D EXISTENCE BY PHASE-PLANE ANALYSIS. Remark 7.4.
Remarks 5.3 and 5.4 show that solutions of the specific D model of Section5 with m , m , and m nonzero are genuinely -dimensional in the sense that they are notconfined to a plane in τ -space, and that solutions of the various D models of Section 5are genuinely two-dimensional if the means of both components are nonzero. That is, wehave constructed by the variational approach solutions that are not obtainable by the planarphase-portrait analysis of the D case (see just below). On the other hand, a dimensionalcount reveals that, generically, the periodic solutions nearby a D solution are all D, andlikewise the periodic solutions nearby a D solution are all D. In this section, we discuss how to generate periodic waves for 1D models. In (5 . q = D τ W ( τ − ) − τ ′′− . Here ( τ − , τ ′′− ) is the vector evaluated at some specificspace value x . If there indeed exist periodic-T waves, q = T R T DW ( τ ( x )) dx .By the variational formulation and the usual bootstrap argument, we conclude thatthe periodic waves are classical solutions of the system (5 . . x , x ) in a period [0,T]such that τ ′ ( x ) = 0 and τ ′′ ( x ) = 0 , etc., if such periodic solution did exist for τ scalar.The reason is that τ ( x ) cannot be always monotone and convex in view of periodicity (thisapplies to all derivatives). Hence we can make the integration constant have the form q = D τ W ( τ ( x )) for convenience a priori. Then we can show existence, which in turn guar-antees the a priori assumption. Hence, we could assume q = D τ W ( τ − ) to show existence.We adopt this convention in the following analysis.The guiding principle is that the ODE systems are planar Hamiltonian systems. To getcomplete and clear pictures of the phase-portraits, we just need to specify the “potentialenergy” term V ( τ, τ − ) in the Hamiltonian H ( τ, τ ′ ). τ ≡ τ ≡ . In this section, we denote τ = τ . We use similar notation in other sections. Recall theelastic potential W ( τ ) = (cid:16) τ + 2 ε + ( τ − ε ) (cid:17) and its first and second order derivatives D τ W ( τ ) = 8 τ ( τ + 1 − ε ) { τ + ε ) + ( τ − ε ) } ; D τ τ W ( τ ) = 8(3 τ + 1 − ε ) { τ + ε ) + ( τ − ε ) } + 32 τ ( τ + 1 − ε ) . The traveling wave ODE and corresponding Hamiltonian system are(8.1) τ ′′ = W ′ ( τ ) − W ′ ( τ − ) . (8.2) ( τ ′ = τ ′ ; τ ′′ = W ′ ( τ ) − W ′ ( τ − ) .
1D EXISTENCE BY PHASE-PLANE ANALYSIS. | τ ′ | H ( τ, τ ′ ) − V ( τ ; τ − ) ≡ E − V ( τ, τ − ) . Here E are constants corresponding to energy level curves of H ( τ, τ ′ ) and V ( τ, τ − ) := qτ − W ( τ ).First, we determine the number of equilibria of the Hamiltonian system, hence focus onthe solution of W ′ ( τ ) = q .Note that W ′ ( τ ) is an odd function on the real line, hence we just need to study itsgraph on the interval (0 , ∞ ). In view of the expression of W ′ ( τ ), we need consider the cases:(1) 0 < ε ≤
1; (2) ε > < ε <
1, we have W ′′ ( τ ) > τ real, hence W ′ ( τ ) is strictly monotoneincreasing and W ′ (0) = 0; W ′ ( τ ) > , for τ > W ′ ( τ ) < , for τ < τ − , the solution of W ′ ( τ ) = W ′ ( τ − ) is τ − and unique.Similar analysis holds true for ε = 1. Considering the definition of V ( τ, τ − ), we have Proposition 8.1.
When < ε ≤ , V ( τ ; τ − ) has exactly one critical point τ − , which mustbe a global maximum. Remark 8.2.
In this case, our Hamiltonian system admits no periodic orbit for any τ − (orequivalently, for any q ). Next, consider the case ε >
1. In this case, we can see from the expression of W ′ ( τ )that W ′ ( τ ) has three distinct zeros: −√ ε −
1, 0, √ ε −
1. A qualitative graph of W ′ ( τ )is as follows: −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−400−300−200−1000100200300400 q * −q * τ w’( τ ) q τ τ τ l,r τ − τ
1D EXISTENCE BY PHASE-PLANE ANALYSIS. V ( τ , τ , − ) is as follows −2 −1 0 1 2−700−600−500−400−300−200−1000100200 V ( τ , τ , − ) τ τ τ γ τ Proposition 8.3.
The function W ′ ( τ ) has exactly two critical points.Proof. By symmetry, we do the following computations: Denote τ := X and ε := a > f ( X ) := (3 X + 1 − a )[2( X + a ) + ( X − a ) ] + 4 X ( X + 1 − a ) has exactly one zero when X > f (0) < f ( a − ) >
0, we know that f ( X ) has a root on (0 , a − ).Also note that f ( X ) > a − , ∞ ), hence we just need to show that f ( X ) admits aunique zero on (0 , a − ). Computing the derivative, we have f ′ ( X ) = 3[7 X + 10(1 − a ) X + a + 2 a + 2(1 − a ) ] . Denote ∆ = 100(1 − a ) − a + 2 a + 2(1 − a ) ] . If ∆ ≤
0, we know that f ′ ( X ) ≥
0, hence f ( X ) is monotone increasing, which impliesthat f ( X ) admits a unique zero;If ∆ >
0, we will have two positive roots for f ′ ( X ) = 0 and the smaller one is a − −√ ∆14 .However, we can show that a − −√ ∆14 ≥ a − , hence the function f ( X ) is monotone in-creasing on the interval (0 , a − ), which also implies the uniqueness of the zero.Now we have a clear picture on the potential W ′ ( τ ) (see the graph for W ′ ( τ )). Proposition 8.4.
When ε > , the function W ′ ( τ ) is an odd function with 3 zeros and 2critical points and goes to infinity when τ → + ∞ .
1D EXISTENCE BY PHASE-PLANE ANALYSIS. W ′ ( τ ) as q ∗ > − q ∗ , for convenience denoting Q = q ∗ . Then we have the following property: Proposition 8.5.
Assume ε > . When | q | > Q , the equation W ′ ( τ ) = q has exactly onesolution; When | q | = Q , the equation W ′ ( τ ) = q has exactly 2 solutions; When | q | < Q , theequation W ′ ( τ ) = q has exactly 3 solutions. As the solutions of W ′ ( τ ) = q correspond to the critical points of V ( τ ; τ − ), we have: Theorem 8.6.
For | q | ≥ Q , the Hamiltonian system admits no periodic orbit; For | q | < Q ,the Hamiltonian system admits a family of nontrivial periodic orbits. Further if q = 0 , theHamiltonian system also admits a heteroclinic orbit.Proof. For | q | ≥ Q , we know that V ( τ ; τ − ) has a global maximum and hence the Hamilto-nian admits no periodic orbit. For the case | q | < Q , we know that the potential V ( τ ; τ − )must have exactly 3 critical points with 2 local maxima and 1 local minimum. Also, weknow that V ( τ ; τ − ) has strictly lower energy at the local minimum than at the two localmaxima. Hence the existence of a family of periodic orbits follows. Further, when q = 0,the energies at the two local maxima of V ( τ ; τ − ) are the same, hence we get an heteroclinicorbit. Remark 8.7.
We may compare the two energy values of V ( τ ; τ − ) at the two local maxima.If they are equal (when q = 0 in particular), we have a heteroclinic orbit. In general, theyare not equal to each other, which yields a homoclinic orbit. τ ≡ τ ≡ . Recall the elastic potential W ( τ ) = (cid:16) τ − ε ) + ( τ − ε ) (cid:17) × (cid:16) τ + ε ) + ( τ − ε ) (cid:17) and the relevant derivatives D τ W ( τ ) = 8 τ ( τ + 1 − ε ) { τ + ε ) + ( τ − ε ) } − τ ε ; D τ τ W ( τ ) = 8(3 τ + 1 − ε ) { τ + ε ) + ( τ − ε ) } + 32 { τ ( τ + 1 − ε ) − ε } . As above, we list the Hamiltonian system and the potential V ( τ ; τ − ). The system is:(8.4) ( τ ′ = τ ′ ; τ ” = W ′ ( τ ) − W ′ ( τ − ) . The potential is V ( τ ; τ − ) = qτ − W ( τ ).
1D EXISTENCE BY PHASE-PLANE ANALYSIS. Remark 8.8.
There is a slight difference with the D shear model I in the function W ′ ( τ ) .Because of this difference, we do not need to restrict the positive number ε to get periodicorbits for the parameter q in proper range. The conclusions are completely the same when ≥ ε > as in D shear model I when ε > . We have the following:
Proposition 8.9.
When ≥ ε > , the behavior of the function W ′ ( τ ) is the same as thatof the function W ′ ( τ ) in the D shear model I when ε > . In fact, W ′′ ( τ ) is monotoneincreasing in this case for τ > . Remark 8.10.
For the range ε > , numerics suggest that the behaviors are also the sameas we may show that the function W ′ ( τ ) has exactly three solutions and two critical points.We have a small problem to verify this by direct computation though we just need to showthat f ( X ) > evaluated at the larger root of f ′ ( X ) ( f ( X ) is defined similar as in D shearmodel I as the second derivatives of the two potentials differ with a constant ε . Evenwithout this, we still can conclude the existence of periodic orbits since the potential V ( τ ; τ − ) admits a minimum. Together with the existence obtained by variational argument, we knowthat there are still infinitely many nontrivial periodic waves for any ε > . For this model, we need to pay special attention to the physical restriction τ > τ = τ ≡
0; let τ = τ . The potential and its derivatives are given below.The elastic potential becomes W ( τ ) = (cid:16) ε + ( τ − − ε ) (cid:17) . Its first and second orderderivative are D τ W ( τ ) = 8 τ ( τ − − ε ) { ε + ( τ − − ε ) } and w := D τ τ W ( τ ) = 8(3 τ − − ε ) { ε + ( τ − − ε ) } + 32 τ ( τ − − ε ) . We write V = V ( τ, q, ε ) := qτ − W ( τ ) in this section to emphasize the analyticaldependence of V on the parameters q and ε (because V is a polynomial). As in previoussections, we see: Proposition 8.11. (1) W ′ ( τ ) = q always has one, two or three roots when | q | > Q , | q | = Q or | q | < Q for some positive Q . In the case that W ′ ( τ ) = q has 3 distinct roots, we denotethem from small to large by τ l , τ m and τ r .(2) W ′ ( τ ) has exactly two critical points. As before, in order to analyze the existence of periodic or homoclinic/heteroclinic waves,we just need to consider the potential energy V ( τ, q, ε ). Further, in order to have physicalwaves, we need necessarily that − Q < q <
0. In this situation, the two roots τ m and τ r of W ′ ( τ ) = q are positive. Noticing that τ = τ m is a local minimizer of V ( τ, q, ε ), there is aperiodic annulus around τ m . Hence we have the following proposition: TIME-EVOLUTIONARY STABILITY Proposition 8.12.
When − Q < q < , there always exists a periodic annulus. To show existence of physical homoclinic orbit, we just need to compare the values of V (0 , q, ε ) =: V (0) and V ( τ r , q, ε ) =: V ( r ). We have Proposition 8.13.
Let − Q < q < . If V (0) > V ( r ) , there is a physical homoclinic orbit;If V (0) ≤ V ( r ) , there is no physical homoclinic orbit. In particular, for the case q = 0, the 3 distinct roots of W ′ ( τ ) = q are easily seen to be τ l = −√ ε , τ m = 0 and τ r = − τ l . So V (0 , , ε ) = − (2 ε + (1 + ε ) ) < V ( τ r , , ε ) = − (2 ε ) . By continuity, we have the following conclusion: Proposition 8.14.
There exists a constant η > such that if − η < q ≤ , then there existsno physical homoclinic orbit.Proof. When q = 0, V (0 , , ε ) < V ( τ r , , ε ). Thus, by continuous dependence and Proposi-tions 8.11 and 8.13, we have the relation V (0) < V ( m ) holds when q < − Q < q < ε = 0 then proceed by a perturbation argument. When ε = 0,the corresponding elastic energy function and its derivatives are:˜ W ( τ ) = ( τ − ;˜ W ′ ( τ ) = 8 τ ( τ − ;˜ W ′′ ( τ ) = 8( τ − (7 τ − . Note that ˜ W ′′ ( τ ) = 0 has roots τ = ± , ± p / V ( τ, q, ε = 0) := qτ − ˜ W ( τ ), we need | q | < ˜ W ′ ( − p /
7) = 8( ) q to have a homoclinic wave. For physical ones, we need − ) q < q <
0. Consider thecase q → − ) q from the right, we see the largest root τ m of ˜ W ′ ( τ ) = q tends to p / V ( τ = 0 , q, ε = 0) → − ˜ W (0) = − V ( τ m , q, ε = 0) → − ) q − ˜ W ( p / < −
1. By Proposition 8.11 and Proposition 8.13,we have:
Proposition 8.15.
For the compressible 1D model, assume − Q < q < ( Q as in Propo-sition 8.11). Then, when ε > and q + Q are small, we have a physical homoclinic orbit. We conclude by discussing briefly the question of time-evolutionary stability of elastic trav-eling waves with strain-gradient effects.
TIME-EVOLUTIONARY STABILITY A very useful observation regarding the earlier phenomenological models τ t − u x = 0, u t + dW ( τ ) x = bu xx − dτ xxx , b, d > b , d , specifically, d < b /
4, the system can betransformed by the change of independent variable u → ˜ u := u − cτ x , c ( b − c ) = d to thefully parabolic system τ t − ˜ u x = cτ xx , ˜ u t + dW ( τ ) x = ( b − c )˜ u xx , thus allowing the treatment of nonlinear stability by standard parabolic techniques, takinginto account, for example, sectorial structure, parabolic smoothing, etc.Quite recently, this observation has been profoundly generalized by M. Kotschote [K],who showed that a somewhat different transformation in similar spirit may be used to con-vert elasticity or fluid-dynamical equations with strain-gradient (resp. capillarity) effects toquasilinear fully parabolic form, in complete generality , not only to the cases 4 d > b previ-ously uncovered for the phenomenological model, but to the entire class of physical modelsconsidered here. For further discussion/description of this transformation, see Appendix9.1.1.This reduces the question of nonlinear stability to a standard format already well studied.In particular, it follows that (except possibly in nongeneric boundary cases of neutrallystable spectrum) nonlinear stability is equivalent to spectral stability, appropriately defined.This follows for heteroclinic and homoclinic waves by the analysis of [HZ], and for periodicwaves by the analysis of [JZ]. For precise definitions of the notions of spectral stability, werefer the reader to those references; in the shock wave (heteroclinic or homoclinic) case, seealso the discussion of [BLeZ]. Spectral stability may be efficiently determined numericallyby Evans function techniques, as in for example [BHRZ, BLeZ, BHZ, BJNRZ1, BJNRZ2].We intend to carry out such a numerical study in a followup work [BYZ].
We now show how to apply the approach of Kotschote in our context and verify that wethereby obtain the structural properties needed to apply the general theory of [HZ]. Sowe shall in the following verify the structural properties of the elastic model with strain-gradient effect and related modified systems obtained by the apporach of Kotschote [K]. Tobe clear, we collect these related systems. The original system is(9.1) ( τ t − u x = 0 u t + σ ( τ ) x = ( b ( τ ) u x ) x − ( d ( τ x ) τ xx ) x . The analysis of [JZ] concerns modulational stability, or stability with respect to localized perturbationson the whole line; co-periodic stability may be treated by standard semigroup techniques [He]. Spectralanalyses of [OZ, BYZ, PSZ] suggest that modulational stability occurs rarely if ever for viscoelastic waves.
TIME-EVOLUTIONARY STABILITY σ = − D τ W ( τ ), d ( · ) = D Ψ( · ), d ( · ) = Id and(9.2) b ( τ ) = τ − Introducing the phase variable z := τ x , we may write (9 .
1) as a quasilinear second-ordersystem(9.3) τ t + z x − u x = τ xx z t = u xx u t + σ ( τ ) x = ( b ( τ ) u x ) x − ( d ( z ) z x ) x . Remark 9.1.
This transformation, introduced in [K], is similar in spirit to but more gen-eral than the one introduced by Slemrod [S1, S2, S3] and used in [OZ] for an artificialviscosity/capillarity model. We can slightly modify the above system in the second equation. Then we have thefollowing system(9.4) τ t + z x − u x = τ xx z t + z x = u xx + τ xx u t + σ ( τ ) x = ( b ( τ ) u x ) x − ( d ( z ) z x ) x . If we write the above system (9.3) and (9.4) in matrix form U t + f ( U ) x = ( B ( U ) U x ) x using the variable U := τzu , the corresponding matrix B becomes: I I − d ( z ) b ( τ ) ; I I I − d ( z ) b ( τ ) . Also, the corresponding matrix Df ( U ) for the two systems are: I − I Dσ ( τ ) 0 ; A transformation ( τ, u ) → ( τ, u − cτ x ) reducing the model to a parabolic system of the same size. TIME-EVOLUTIONARY STABILITY I − I I Dσ ( τ ) 0 . Proposition 9.2. (Strict parabolicity) Systems (9 . and (9 . are both strictly parabolicsystems in the sense that the spectrum of B have positive real parts.Proof. Comparing the two matrices B above, we know they have the same spectrum. Weprove this proposition for τ, z, u ∈ R . The lower dimension cases becomes easier and thecomputations are totally the same . Pick one of the B ′ s , say I I − d ( z ) b ( τ ) . To compute the spectrum of B , consider the characteristic polynomial det( λI − B ) = 0,which is det ( λ − I λI − I + d ( z ) λ − b ( τ ) = 0 . Doing Laplace expansion and elementary column transformation , we getdet { ( λ − I } det { ( λ I − λb ( τ ) + d ( z )) } = ( λ − ( λ − λτ + 1) ( λ − λτ + 1) = 0 . From the first factor of the above degree 9 polynomial, we get three equal root 1 whichhas positive real parts . The other 6 roots also have positive real parts noticing that τ > Proposition 9.3. (Nonzero characteristic speeds) The corresponding first order systems of(C.4) has nonzero characteristic speed at τ where the matrix D W ( τ ) are strictly positivedefinite.Proof. Again, we prove this for τ, z, u ∈ R . To prove the corresponding first order systemis non-characteristic, we consider the spectrum of the matrix I − I I Dσ ( τ ) 0 . We getthe following system for the characteristic speed:det( λI − Df ( U )) = λI − I I ( λ − I − Dσ ( τ ) 0 λI = 0 . By direct computation, we know:(9.5) det( λI − Df ( U )) = det (cid:16) ( λ − I (cid:17) det (cid:16) − Dσ ( τ ) − λ I (cid:17) = 0 . TIME-EVOLUTIONARY STABILITY (cid:16) − Dσ ( τ ) − λ I (cid:17) = det (cid:16) D W ( τ ) − λ I (cid:17) = 0. Hence the propositionfollows. Proposition 9.4. (Same spectrum) For system (9 . and its first order system, the matrix Df ( U ) and B − Df ( U ) have the same spectrum.Proof. We prove this for the variables τ, z, u ∈ R . It is easy to verify that B − = I − b ( τ ) b ( τ ) − I − I I . Since Df ( U ) = I − I I Dσ ( τ ) 0 , we immediately get B − Df ( U ) = I − I − Dσ ( τ ) 0 b ( τ )0 I . Considering the corresponding eigenvalue problem, we have:det (cid:16) λI − B − Df ( U ) (cid:17) = det λI − I I Dσ ( τ ) λI − b ( τ )0 ( λ − I = 0 . Doing Laplace expansion and performing basic transformation, we get:det (cid:16) λI − I (cid:17) det (cid:16) λ I + Dσ ( τ ) (cid:17) = det (cid:16) λI − I (cid:17) det (cid:16) λ I − D W ( τ ) (cid:17) = 0 , which implies the conclusion by noticing (9 . More fundamentally, perhaps, there is a relation between variational stability of periodicwaves and their time-evolutionary stability as solutions of (2.1). In particular, the energyfunctional that we minimized in constructing periodic solutions is essentially the self-samefunctional that defines the mechanical energy of the system, a Lyapunov functional thatdecreases with the flow of (2.1); as we show below in Section 9.2.1. This gives a stronglink between the two notions of stability. Indeed, it can be used to directly show that theperiodic waves constructed as minimizers of the associated variational problem are time-evolutionarily stable with respect to co-periodic perturbations (Remark 9.6).
TIME-EVOLUTIONARY STABILITY necessary and sufficient for time-evolutionary stability (co-periodic stability, in the case ofperiodic waves, and variational stability constrained by a prescribed mean). Moreover, thesearguments yield at the same time the curious fact that unstable spectra of the linearizedoperator about the wave must, if it exists, be real . These properties give additional insight,and additional avenues by which time-evolutionary stability may be studied.
In particular, this shows that the waves we have constructed are the (co-periodically) stable ones.
However, these are not necessarily the only stable waves, as we did not con-struct all minimizers of the variational problem, but only those with mean satisfying anonconvexity condition. Just recently (in particular, after the completion of the analysis ofthis paper), there has been introduced in [PSZ] a different, more direct argument showingequivalence of variational and time-evolutionary stability, which yields at the same timeconcise conditions for variational stability. These yield in particular that the sharp condi-tion for stability is not the condition of nonconvexity of W at m , defined as the mean overone period of τ , but rather the “averaged” condition of nonconvexity of the Jacobian withrespect to m of the mean over one period of DW ( τ ). See [PSZ] for further details. In this section, we discuss further the relation between variational stability and time-evolutionary stability with respect to co-periodic perturbations of periodic waves. It iswell-known that the physical foundation of calculus of variation is the principle of leastaction (also known as Hamilton’s principle, Maupertuis’ principle, see [Lan, MW, PB] andthe references therein). Hence the functional in calculus of variation represents some sortof “energy” accordingly.To be self-contained, we recall the system (3.2) (which is (9.6) below now)(9.6) ( τ t − u x = 0; u t + σ ( τ ) x = ( b ( τ ) u x ) x − ( d ( τ x ) τ xx ) x Here the functions b, d are the same as before. From [BLeZ] we know the following isthe associated entropy in the sense of hyperbolic system of conservation laws:(9.7) η ( τ, u ) = u W ( τ ) + Ψ( τ x ) . Consider the following mechanical energy for a given positive period T (9.8) E = Z T η ( τ, u ) dx = Z T u W ( τ ) + Ψ( τ x ) dx TIME-EVOLUTIONARY STABILITY E decreases along the flow for periodic boundarycondition, with dissipation as follows ddt E ( τ, u ) = Z T (cid:0) uu t + DW ( τ ) τ t + D Ψ( τ x ) τ xt (cid:1) dx = Z T − u x b ( τ ) u x + u x d ( τ x ) τ xx + u x σ ( τ ) + DW ( τ ) u x + D Ψ( τ x ) u xx ) dx = − Z T u x b ( τ ) u x dx ≤ . Hence we see immediately that periodic traveling wave solutions must have u ≡ constant vector,from which we then find easily the speed s = 0 in view of the relation u t = − su x = 0 and τ t − u x = 0.Next, we adopt the periodic Sobolev space framework to discuss the relation betweenthe least action functional (6 .
5) and the relative mechanical energy (9.9) E ( τ, u ; < τ >, < u > ) := E ( τ, u ) − E ( < τ >, < u > ) − D E ( < τ >, < u > ) · ( τ, u ) , where < τ > := T R T τ ( x ) dx as in physics literature and similarly for u .After a brief computation, we get that the relative mechanical energy (“relative entropy”in the sense of hyperbolic systems of conservation law) is given by E ( τ, u ; < τ >, < u > ) = Z T Ψ( τ x ) + W ( τ ) − DW ( < τ > ) · τ − W ( < τ > ) dx + Z T | u | − | < u > | − < u > · u dx. From the system (3.2) we know that the structure is preserved under the transformation u → u + c where c is an arbitrary constant vector. Hence without loss of generality, we canlet u ≡
0. Hence we get the following expression by further choosing < u > = m as before E ( τ,
0; 0 , m ) = Z T Ψ( τ x ) + W ( τ ) − DW ( m ) · τ − W ( m ) dx = Z T | τ x | + W ( τ ) − DW ( m ) · τ − W ( m ) dx Defining the translated variable v ( x ) = τ ( x ) − m , we get the relation between the leastaction functional and the relative entropy E ( v,
0; 0 , m ) = I ( v ) + constant , where the constant is given by T ( DW ( m ) · m ). APPENDIX: PHASE-TRANSITIONAL ELASTICITY Remark 9.5.
The relative entropy in the sense of hyperbolic system of conservation law isa rather common construction, meant to be stationary about the reference configuration (inour case < τ > = m and < u > = 0 ). Remark 9.6.
The discussion above also sheds some light on the relation between time-evolutionary properties and variational structure of the Hamiltonian structure of our prob-lem (See [Z2] for further discussions). By the method of [GSSI, GSSII], we see that themean-constraint minimizers we constructed are necessarily stable in the time-evolutionarysense with respect to co-periodic perturbation if they are stable in the variational sense.
A Appendix: Phase-transitional elasticity
In this appendix, we collect some computations for the phase-transitional elasticity ([BLeZ],[FP]and references therein). W ( F ) := (cid:12)(cid:12)(cid:12) F T F − C − (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) F T F − C + (cid:12)(cid:12) , where C ± = ( F T F ) ± := ± ε ± ε ε , F ± = ± ε . Evidently, W is minimized among planar deformations at the two values A ± = ± ε . Indeed, we have then W ( F ) = (cid:16) τ + 2( τ − ε ) + ( | τ | − − ε ) (cid:17)(cid:16) τ + 2( τ + ε ) + ( | τ | − − ε ) (cid:17) , where, as in [BLeZ], A = τ τ τ and F = τ τ τ , so that(A.1) F T F = τ τ τ τ | τ | and(A.2) W ( F ) = (cid:12)(cid:12)(cid:12) τ τ + ετ τ + ε | τ | − − ε (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) τ τ − ετ τ − ε | τ | − − ε (cid:12)(cid:12)(cid:12) For the convenience of the reader, we note that the the vector ( a , a , a ) in [BLeZ] corresponds to( τ , τ , τ ) here. EFERENCES F T F − C ± ) = τ ∗ ∗∗ ( τ ∓ ε ) ∗∗ ∗ τ + ( τ ∓ ε ) + ( | τ | − − ε ) . Acknowledgement.
The author is grateful to his thesis advisor, Professor Kevin Zumbrun, for the illuminatingdiscussions, precious guidance and encouragement during this work. He would like also tothank Blake Barker for helpful discussions.
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