Asymptotic Stability of a Compressible Oseen-Structure Interaction via a Pointwise Resolvent Criterion
AAsymptotic Stability of a Compressible Oseen-Structure Interactionvia a Pointwise Resolvent Criterion
Pelin G. Geredeli ∗ Department of MathematicsIowa State University, Ames-IA, USAFebruary 9, 2021
Abstract
In this study, we consider a linearized compressible flow structure interaction PDE modelfor which the interaction interface is under the effect of material derivative term. While thelinearization takes place around a constant pressure and density components in structure equa-tion, the flow linearization is taken with respect to a non-zero, fixed, variable ambient vectorfield. This process produces extra “convective derivative” and “material derivative” terms whichcauses the coupled system to be nondissipative.We analyze the long time dynamics in the sense of asymptotic (strong) stability in an invari-ant subspace (one dimensional less) of the entire state space where the continuous semigroupis “ uniformly bounded ”. For this, we appeal to the pointwise resolvent condition introducedin [18] which avoids many technical complexity and provides a very clean, short and easy-to-follow proof.
Key terms:
Flow-structure interaction, compressible flows, stability, resolvent, uniformlybounded semigroup, material derivative
The mathematical analysis of fluid structure interaction (FSI) problems constitutes a broadarea of research with applications in aeroelasticity, biomechanics and fluid dynamics [3–14, 19–22,29, 30, 38]. Such interactive dynamics between flow/fluid and a plate (or shell) are mathematicallyrealized by coupled PDE systems with compressible flow and elastic plate components. The analysisof these PDE systems is considered from many points of view [7, 8, 10, 21, 29, 30].In this work, we consider a linearized flow-structure PDE model with respect to some referencestate which results in the appearance of an arbitrary spatial flow field. In contrast to the incom-pressible case, having a compressible flow component presents a great many difficulties due to theincrease in the number of unknown variables. By the nature and physics of compressible flows,density can change by pressure forces and a new set of governing equations are necessarily derived ∗ email address: [email protected]. a r X i v : . [ m a t h . A P ] F e b long with the equations for the conservation of mass and momentum. These equations should bevalid for the flows (compressible) whose range of Mach number is M achN umber = M = velocitylocal speed of sound > . . The cases
M < . . < M < . . < M < .
2) or supersonic(1 . < M < . C − semigroup generated by thesolution. This asymptotic decay for solutions of the compressible flow-structure PDE model willbe stated within the context of the associated semigroup formulation and “frequency domain”approach. The FSI Geometry
Let the flow domain
O ⊂ R with Lipschitz boundary ∂ O . We assume that ∂ O = S ∪ Ω, with S ∩ Ω = ∅ , and the (structure) domain Ω ⊂ R is a flat portion of ∂ O with C − boundary. Inparticular, ∂ O has the following specific configuration:Ω ⊂ { x = ( x , x , } and surface S ⊂ { x = ( x , x , x ) : x ≤ } . (1)Additionally, the flow domain O should be curvilinear polyhedral domain which satisfies thefollowing conditions: • Each corner of the boundary ∂ O -if any- is diffeomorphic to a convex cone, • Each point on an edge of the boundary ∂ O is diffeomorphic to a wedge with opening < π. We note that these additional conditions on the flow domain O are necessary for the applicationof some elliptic regularity results for solutions of second order boundary value problems on cornerdomains [24, 27]. We denote the unit outward normal vector to ∂ O by n ( x ) where n | Ω = [0 , , ∂ Ω by ν ( x ). Some examples of geometries can be seen inFigure 1. Linearization and the PDE Model.
In what follows we provide some information about the linearization process and the PDE de-scription of the compressible flow-structure interaction system under consideration. Firstly, we notethat since the linear flow problem here is already of great technical complexity and mathematicalchallenge we assume that the pressure is a linear function of the density; p ( x, t ) = Cρ ( x, t ) , as istypically seen in the compressible flow literature, and it is chosen as a primary variable to solve. Forfurther and detailed explanations of the physical background concerning the relationship betweenpressure and density, the reader is referred to [10, 30].2 S O Figure 1: Polyhedral Flow-Structure GeometriesThe linearization takes place around an equilibrium point of the form { p ∗ , U , (cid:37) ∗ } where thepressure and density components p ∗ , (cid:37) ∗ are assumed to be scalars (for simplicity, assume p ∗ = (cid:37) ∗ =1), and a generally non-zero, fixed, ambient vector field U : O → R U ( x , x , x ) = [ U ( x , x , x ) , U ( x , x , x ) , U ( x , x , x )] . At this point, we emphasize that flow linearization is taken with respect to some inhomogeneouscompressible Navier-Stokes system; thus, U does not need to be divergence free, generally.Now, with respect to the above linearization, the small perturbations give the following physicalequations by generalizing the forcing functions:( ∂ t + U · ∇ ) p + div( u ) + (div U ) p = f ( x ) in O × R + , ( ∂ t + U · ∇ ) u − ν ∆ u − ( ν + λ ) ∇ div u + ∇ p + ∇ U · u + ( U · ∇ U ) p = F ( x ) in O × R + . (For further discussion, see also [10, 21].) When we delete some of the non-critical lower order andthe benign inhomogeneous terms in the above equations, this linearization gives rise to the followingsystem of equations, in solution variables u ( x , x , x , t ) (flow velocity), p ( x , x , x , t ) (pressure), w ( x , x , t ) (elastic plate displacement) and w ( x , x , t ) (elastic plate velocity): p t + U · ∇ p + div u +div( U ) p = 0 in O × (0 , ∞ ) u t + U · ∇ u + u · ∇ U − div σ ( u ) + ηu + ∇ p = 0 in O × (0 , ∞ )( σ ( u ) n − p n ) · τ = 0 on ∂ O × (0 , ∞ ) u · n = 0 on S × (0 , ∞ ) u · n = w + U · ∇ w on Ω × (0 , ∞ ) (2) w t − w − U · ∇ w = 0 on Ω × (0 , ∞ ) w t + ∆ w + [2 ν∂ x ( u ) + λ div( u ) − p ] Ω = 0 on Ω × (0 , ∞ ) w = ∂w ∂ν = 0 on ∂ Ω × (0 , ∞ ) (3)[ p (0) , u (0) , w (0) , w (0)] = [ p, u, w , w ] ∈ H . (4)3ere, H is given as follows: H = { [ p , u , w , w ] ∈ H : (cid:90) O p d O + (cid:90) Ω w d Ω = 0 } , (5)where H ≡ L ( O ) × L ( O ) × H (Ω) × L (Ω) (6)is the associated finite energy (Hilbert) space, topologized by the standard inner product:( y , y ) H = ( p , p ) L ( O ) + ( u , u ) L ( O ) + (∆ w , ∆ w ) L (Ω) + ( v , v ) L (Ω) (7)for any y i = ( p i , u i , w i , v i ) ∈ H , i = 1 , . It was shown in [7] and [29] that H ⊥ is the null space of the flow structure semigroup generatorclosely associated with (2)-(4). It will be shown below that solutions of (2)-(4), with initial datadrawn from H , decay asymptotically to the zero state. Also, the terms U · ∇ u + u · ∇ U constitutethe so-called Oseen (linear) approximation of the Navier-Stokes equations [42].The quantity η > τ in (2) is in the space T H / ( ∂ O ) of tangential vector fields of Sobolev index 1/2; thatis, τ ∈ T H / ( ∂ O ) = { v ∈ H ( ∂ O ) : v | ∂ O · n = 0 on ∂ O} . (8)(See e.g., p.846 of [17].) In addition, we take ambient field U ∈ V ∩ W where V = { v ∈ H ( O ) : v | ∂ O · n = 0 on ∂ O} , (9) W = { v ∈ H ( O ) : v ∈ L ∞ ( O ) , div ( v ) ∈ L ∞ ( O ) , and v | Ω ∈ C (Ω) } (10)(The vanishing of the boundary for ambient fields is a standard assumption in compressibleflow literature; see [23], [41], [32], [1].) Moreover, the stress and strain tensors in the flow PDEcomponent of (2)-(4) are defined respectively as σ ( µ ) = 2 ν(cid:15) ( µ ) + λ [ I · (cid:15) ( µ )] I ; (cid:15) ij ( µ ) = 12 (cid:18) ∂µ j ∂x i + ∂µ i ∂x j (cid:19) , 1 ≤ i, j ≤ , where Lam´e Coefficients λ ≥ ν > impermeability condition on Ω; namely, we assume that no fluidpasses through the elastic portion of the boundary during deflection [15, 28]. Also, note thatthe FSI problem under consideration has a material derivative term on the deflected interactionsurface which computes the time rate of change of any quantity such as temperature or velocity(and hence also acceleration) for a portion of a material in motion. For further details and thephysical explanation of the material derivative boundary conditions, see [10, 30].4 Previous Considerations
The long-time behavior, in particular the stability properties of incompressible/compressiblefluid/flow structure interaction (FSI) systems have been a popular topic treated by many authorsat different levels [3–7, 9, 13, 19, 20, 22, 29]. The PDE systems under consideration are generallyquite complicated, due to their unbounded hyperbolic-parabolic coupling mechanisms, both beingintrinsic to the underlying physics.In contrast to the large body of literature on incompressible FSI [3–6, 9, 11–14, 19, 20, 22, 38],existing work on compressible flows which interact with elastic solids is relatively limited due to theinherent mathematical challenges presented by the extra density (pressure) variable. In analyzingthese compressible FSI PDE models there are the following challenges: (i) Given that the flow-platevariables are coupled via boundary interfaces, the FSI geometry is inherently nonsmooth, [21, 23].Although such geometries are physically relevant, these domains also give rise to regularity issuesfor solutions; (ii) The linearization of the compressible FSI system under consideration takes placearound a rest state which includes a generally nonzero (not constant) ambient flow field U , andthe said pressure PDE will contain a “convective derivative” term U ∇ p which is strictly abovethe level of finite energy; (iii) The boundary conditions which couple flow and structure containnondissipative and unbounded terms which complicate the PDE analysis.A preliminary linearized compressible FSI model with U ≡ without material deriva-tive) was derived by I. Chueshov [21]. In this work, the author showed the wellposedness and theexistence of global attractors in the case that the structure equation has a von Karman nonlin-earity. However, in this pioneering work, the author noted that his methods (Galerkin approachand Lyapunov functionals) to show the wellposedness and long-term behavior of the correspondingsystem would not accommodate the case of interest; namely linearization about U (cid:54) = 0.The main reason for the difficulty is the need to control the “convective derivative” term U ∇ p which requires a decomposition of the fluid and pressure solution pair { u, p } with an eventual appealto some elliptic regularity results on nonsmooth domains. Subsequently, the suggested model wasanalyzed in [8] via a semigroup formulation and a wellposedness result with additional interior termsassociated to the U (cid:54) = 0 under a pure velocity matching condition at the interface was obtained.Then in [10], the authors re-visited the same problem after a careful derivation of fluid-structureinterface conditions written in terms of “material derivative” ( ∂t + U ·∇ ) w. This material derivativecomputes the time rate of change of any quantity such as temperature or velocity (and hence alsoacceleration) for a portion of a material in motion. However, since the material derivative term U ∇ w is unbounded and nondissipative, it adds an additional challenge to the analysis. In [10],this lack of boundedness and dissipativity were ultimately overcome by re-topologizing the finiteenergy space, and so the desired wellposedness result was obtained by an appropriate semigroupformulation. At this point, we should note that in both papers [8, 10], the obtained semigroup isunfortunately NOT uniformly bounded (in time) which prevents us to seek long term behavior ofthe solution to these corresponding problems.With a view of looking into stability properties of the flow structure PDE models consideredin [8, 10], it appeared natural to ask: Is it possible to obtain a semigroup wellposedness andsubsequently a stability result, with the semigroup being bounded uniformly in time, at least insome (inherently invariant) subspace of the finite energy space? This was indeed a very importantdeparture point in order to analyze the stability of these FSI problems since there was not any5ong time behavior result in the literature for such classes of FSI systems. Motivated by thisquestion, an initial result of uniform stability result for the solution to linearized compressible FSImodel ( without material derivative ) in an appropriate subspace was shown in [7, 29] by using linearperturbation theory and also some novel multipliers. Having established the uniform decay result for the solution to the “ material free ” compressibleFSI system in [7,29], our next goal was to analyze the long time behavior of the system, when underthe effect of “material term” on the interaction interface. In order to embark on this stability work,we firstly needed to have a uniformly bounded (in time) semigroup, again in some (inherentlyinvariant) subspace of the finite energy space. This result was obtained in [30]. Now, in the presentwork, we provide a novel result on the asymptotic (strong) stability of compressible FSI modelunder said “ material derivative ” boundary conditions. This result ascertains that solution to theFSI model decays to zero state for all initial data taken from a subspace H which is “almost” theentire phase space H . This is, to the knowledge of the author , the first such result obtained with“ material derivative ” BC.By way of obtaining the aforesaid asymptotic (strong) stability result, we operate in the fre-quency domain which requires us to deal with some static equations, analogous to (2)-(3), andthe resolvent of the generator of the dynamical system. While we have already been aware of theneed to control the “convective derivative” term U ∇ p from our previous works, in this currentmanuscript the primary challenge is to have tight control of the material derivative term U ∇ w inthe coupling condition at the interface since it destroys the dissipative nature of the dynamics. Inthis regard, the main challenges associated with the analysis and the novelties are as follows: (i) Inner product adjustment for dissipativity:
In order to establish a long time behavior result forthe given FSI system, we expect that the energy of this system is decreasing. However, the presenceof the problematic convective and material derivative terms U ∇ p and U ∇ w adds a great challengeto the long term analysis since they can not be treated as a perturbation and moreover cause a lackof dissipativity property in the system. To deal with these issues and get the necessary dissipativityestimate we change the inner-product structure of the state space and we re-topologize the inher-ently invariant subspace with an inner product which is equivalent to the standard inner productdefined for the entire state space. In this construction, we make use of a multiplier which exploitsthe characterization of this invariant subspace and the Dirichlet map that extends boundary datadefined on the interaction interface to a harmonic function in the flow domain. See Theorem 1below and [30]. (ii) Use of pointwise resolvent criterion for stability:
Since the domain of the associated flow-structure generator is not compactly embedded into finite energy space H , (see (11)-(12) below)it would seem natural to appeal to well known strong stability result introduced by Arend-Batty[2]. However, this does not seem practicable for our model since the spectral analysis of thesemigroup generator will require a great amount of technicalities. For example, to analyze theresidual spectrum of the generator, we need to check that none of the points on the imaginary axis6re the eigenvalues of the adjoint operator of this generator. Unfortunately, it is not going to bestraightforward to get this result since the adjoint, itself, is given as the sum of three complicatedmatrix (see Lemma 3). Instead, we will use the very interesting pointwise resolvent criteriondeveloped by Chill and Tomilov [18, 40] for the stability of bounded semigroups which perfectlyworks for our model (See also [16] which gives a preliminary version of the resolvent criterion forstrong stability.) Hence we provide a very clean, short and easy-to-follow proof which does nottouch the –very complicated– adjoint operator and the challenges it might cause. Throughout, for a given domain D , the norm of corresponding space L ( D ) will be denoted as || · || D (or simply || · || when the context is clear). Inner products in L ( O ) or L ( O ) will be denotedby ( · , · ) O , whereas inner products L ( ∂ O ) will be written as (cid:104)· , ·(cid:105) ∂ O . We will also denote pertinentduality pairings as (cid:104)· , ·(cid:105) X × X (cid:48) for a given Hilbert space X . The space H s ( D ) will denote the Sobolevspace of order s , defined on a domain D ; H s ( D ) will denote the closure of C ∞ ( D ) in the H s ( D )-norm (cid:107) · (cid:107) H s ( D ) . We make use of the standard notation for the boundary trace of functions definedon O , which are sufficently smooth: i.e., for a scalar function φ ∈ H s ( O ), < s < , γ ( φ ) = φ (cid:12)(cid:12) ∂ O , which is a well-defined and surjective mapping on this range of s , owing to the Sobolev TraceTheorem on Lipschitz domains (see e.g., [39], or Theorem 3.38 of [37]). Also, C > H , define theoperators A = − U ·∇ ( · ) − div( · ) 0 0 −∇ ( · ) div σ ( · ) − ηI − U ·∇ ( · ) −∇ U · ( · ) 0 00 0 0 I [ · ] | Ω − [2 ν∂ x ( · ) + λ div( · )] Ω − ∆ ; (11)and B = − div( U )( · ) 0 0 00 0 0 00 0 U ·∇ ( · ) 00 0 0 0 (12)where D ( A + B ) ⊂ H is given by D ( A + B ) = { ( p , u , w , w ) ∈ L ( O ) × H ( O ) × H (Ω) × L (Ω) : properties ( A.i )–(
A.vi ) hold } : (A.i) U · ∇ p ∈ L ( O ) (A.ii) div σ ( u ) − ∇ p ∈ L ( O ) (So, [ σ ( u ) n − p n ] ∂ O ∈ H − ( ∂ O )) (A.iii) − ∆ w − [2 ν∂ x ( u ) + λ div( u )] Ω + p | Ω ∈ L (Ω) (by elliptic regularity theory w ∈ H (Ω))7 A.iv) ( σ ( u ) n − p n ) ⊥ T H / ( ∂ O ). That is, (cid:104) σ ( u ) n − p n , τ (cid:105) H − ( ∂ O ) × H ( ∂ O ) = 0 in D (cid:48) ( O ) for every τ ∈ T H / ( ∂ O ) (A.v) w + U · ∇ w ∈ H (Ω) (and so w ∈ H (Ω)) (A.vi) The flow velocity component u = f + (cid:101) f , where f ∈ V and (cid:101) f ∈ H ( O ) satisfies (cid:101) f = (cid:40) S ( w + U · ∇ w ) n on Ω(and so f | ∂ O ∈ T H / ( ∂ O )).Then the function Φ( t ) = [ p, u, w , w ] ∈ C ([0 , T ]; H ) that solves the problem (2)-(4) satisfies ddt Φ( t ) = ( A + B )Φ( t );Φ(0) = Φ (13)We should note that the semigroup generator ( A + B ) is H - invariant (see Lemma 5 of [30]).Moreover, in [30] we showed that the flow-structure PDE system (2)-(4), in the absence of the term u · ∇ U in the flow component, is associated with the generator of a C - contraction semigroup in H . Therefore, by a bounded perturbation argument, ( A + B ) generates a C - semigroup in H , which, however, might not be a contraction. Since the main goal of this manuscript is to show theasymptotic (strong) stability of the PDE system (2)-(4) for every initial data in the subspace H ,having a uniformly bounded semigroup will be our key point to embark on proving our main result,Theorem 4. For the sake of simplicity, we use the notation( A + B ) | H = ( A + B ) . In order to analyze the long time behavior of the solution to (2)-(4) in the reduced space H (definedin (5)), we have to guarantee that the dynamical system (2)-(4) is dissipative. But, unfortunately,the presence of the generally nonzero ambient vector field U and the material derivative termboundary conditions in the model cause the lack of dissipativity of the generator ( A + B ) (withrespect to the standard inner product given in (7).) Hence, to obtain a necessary dissipativityestimate for each solution variable, we re-topologize the phase space H with a new inner product tobe used in H and equivalent to the natural inner product given in (7). (See [30]) For the readersconvenience, we provide the details here as well:With the above notation let us take ϕ = [ p , u , w , w ] ∈ H , (cid:101) ϕ = [ (cid:101) p , (cid:101) u , (cid:101) w , (cid:101) w ] ∈ H . Thenthe new inner product is given as(( ϕ, (cid:101) ϕ )) H = ( p , (cid:101) p ) O + ( u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w ) , (cid:101) u − αD ( g · ∇ (cid:101) w ) e + ξ ∇ ψ ( (cid:101) p , (cid:101) w )) O The existence of an H ( O )-function (cid:101) f with such a boundary trace on Lipschitz domain O is assured; see e.g.,Theorem 3.33 of [37]. w , ∆ (cid:101) w ) Ω + ( w + h α · ∇ w + ξw , (cid:101) w + h α · ∇ (cid:101) w + ξ (cid:101) w ) Ω , (14)and in turn the norm (cid:107)| ϕ |(cid:107) H = (( ϕ, ϕ )) H = (cid:107) p (cid:107) O + (cid:107) u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w ) (cid:107) O + (cid:107) ∆ w (cid:107) + (cid:107) w + h α · ∇ w + ξw (cid:107) (15)for every ϕ = [ p , u , w , w ] ∈ H . Here, a) ξ = ( − C r U ) − (cid:113) ( − C r U ) − C ( C + C r U ) r U C + C r U ) , b) r U = (cid:107) U (cid:107) ∗ + (cid:107) U (cid:107) ∗ + (cid:107) U (cid:107) ∗ (16)where C , C > (cid:107) U (cid:107) ∗ which is given as (cid:107) U (cid:107) ∗ = (cid:107) U (cid:107) L ∞ ( O ) + (cid:107) div( U ) (cid:107) L ∞ ( O ) + (cid:107) U | Ω (cid:107) C (Ω) . (17)Also, (i) the function ψ = ψ ( f, χ ) ∈ H ( O ) is considered to solve the following BVP for data f ∈ L ( O ) and χ ∈ L (Ω) − ∆ ψ = f in O ∂ψ∂n = 0 on S ∂ψ∂n = χ on Ω (18)with the compatibility condition (cid:90) O f d O + (cid:90) Ω χd Ω = 0 . (19)We should note that by known elliptic regularity results for the Neumann problem on Lipschitzdomains–see e.g; [31]– we have (cid:107) ψ ( f, χ ) (cid:107) H ( O ) ≤ [ (cid:107) f (cid:107) O + (cid:107) χ (cid:107) ∂ O ] . (20) (ii) the map D ( · ) is the Dirichlet map that extends boundary data ϕ defined on Ω to a harmonicfunction in O satisfying: Dϕ = f ⇔ (cid:26) ∆ f = 0 in O f | ∂ O = ϕ | ext on ∂ O (21)where ϕ | ext = (cid:40) Sφ on ΩThen by, e.g., [37, Theorem 3.3.8], and Lax-Milgram Theorem, we deduce that D ∈ L (cid:0) H / (cid:15) (Ω); H ( O ) (cid:1) . (22) (iii) the vector field h α ( · ) is defined as h α ( · ) = U | Ω − αg, where g ( · ) is a C extension of the normalvector ν ( x ) (recall, with respect to Ω) and we specify the parameter α to be α = 2 (cid:107) U (cid:107) ∗ , (23)where (cid:107) U (cid:107) ∗ is as defined in (17). 9 Result I: Bounded Semigroup Wellposedness in H The bounded semigroup wellposedness is given as follows:
Theorem 1
With reference to problem (2)-(4), assume that U ∈ V ∩ W with (cid:107) U (cid:107) ∗ = (cid:107) U (cid:107) L ∞ ( O ) + (cid:107) div ( U ) (cid:107) L ∞ ( O ) + (cid:107) U | Ω (cid:107) C (Ω) sufficiently small. Then we have the following: (i) The operator ( A + B ) : D ( A + B ) ∩ H → H is maximal dissipative. In particular, it obeysthe following inequality for all φ = [ p , u , w , w ] ∈ D ( A + B ) ∩ H : Re (([ A + B ] ϕ, ϕ )) H ≤ (cid:18) −
14 + C δ r U (cid:19) (cid:104) ( σ ( u ) , (cid:15) ( u )) O + (cid:107) u (cid:107) O (cid:105) + (cid:18) −
12 + δC ∗ (cid:19) ξ (cid:104) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:105) , (24) where < δ < C ∗ , and r U is as given in (16). (ii) Consequently, the operator ( A + B ) : D ( A + B ) ∩ H → H generates a strongly continu-ous semigroup { e ( A + B ) t } t ≥ on H . Hence, for every initial data [ p, u, w , w ] ∈ H , the solution [ p ( t ) , u ( t ) , w ( t ) , w ( t )] of problem (2)-(4) is given continuously by p ( t ) u ( t ) w ( t ) w ( t ) = e ( A + B ) t puw w ∈ C ([0 , T ]; H ) . (25) Moreover, this semigroup is uniformly bounded in time with respect to the standard H -inner product.With respect to the special norm in (15), the semigroup { e ( A + B ) t } t ≥ is in fact a contraction. Proof.
Let A ≡ A + L U , (26)where L U = ∇ U · ( · ) 0 00 0 0 00 0 0 0 (27)with D ( A + B ) = D ( A + B ) . Therewith, it is established in [30] that A + B is maximal dissipative in H , for all (cid:107) U (cid:107) ∗ sufficiently small. In particular, we have for φ = [ p , u , w , w ] ∈ D ( A + B ) ∩ H :Re(([ A + B ] ϕ, ϕ )) H ≤ −
14 ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) , (28)where parameter ξ is as given in (16). (For the sake of completion, the proof of this estimate is10iven in the Appendix.) Given φ = [ p , u , w , w ] ∈ D ( A + B ) ∩ H , invoking (28) we have(([ A + B ] ϕ, ϕ )) H = (([ A + B ] ϕ, ϕ )) H − (( L U ϕ, ϕ )) H ≤ −
14 ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) − ( ∇ U · ( u ) , u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w )) O = −
14 ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) − (cid:90) ∂ O ( u · n ) U · [ u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w ))] d∂ O + (cid:90) O div ( u ) U · [ u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w ))] d O + (cid:90) O U · ( u · ∇ [ u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w ))]) d O (29)We estimate RHS implicitly using the regularity results of [25–27], which are valid under theassumptions made on the geometry. (I) The mapping in (21) satisfies (cid:107) D ( g · ∇ w ) (cid:107) H ( O ) ≤ C (cid:107) w (cid:107) H (Ω) (30)Thus, we have (I-a): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:90) ∂ O ( u · n ) U · D ( g · ∇ w ) e d∂ O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) (cid:107) w (cid:107) H (Ω) = (cid:113) (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) α (cid:112) (cid:107) U (cid:107) ∗ √ ξ (cid:112) ξ (cid:107) ∆ w (cid:107) Ω ≤ C δ (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) + δα (cid:107) U (cid:107) ∗ ξξ (cid:107) ∆ w (cid:107) ≤ C δ (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) + δC r U ξ ξ (cid:107) ∆ w (cid:107) ≤ C δ (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) + δCξ (cid:107) ∆ w (cid:107) . (31)At this point, we should emphasize that r U ξ will be bounded if (cid:107) U (cid:107) ∗ is small enough and so C > (cid:107) U (cid:107) ∗ . Also, we implicitly used the Sobolev Trace Theorem for the first inequality. (I-b): Again by (30), similarly (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α (cid:90) O U · (cid:16) u · ∇ D ( g · ∇ w ) e (cid:17) d O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C δ (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) + δCξ (cid:107) ∆ w (cid:107) (32)11 II)
Using (20) and (30), we have as in (I-a), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) O div ( u ) U · [ u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w ))] d O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C δ ( (cid:107) U (cid:107) ∗ + (cid:107) U (cid:107) ∗ ) (cid:107) u (cid:107) H ( O ) + δCξ (cid:104) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:105) (33) (III) The mapping (18) satisfies | ψ ( p , w ) (cid:107) H ( O ) ≤ (cid:20) (cid:107) p | O + (cid:107) w | H
12 + (cid:15) (Ω) (cid:21) (34)Consequently, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ (cid:90) ∂ O ( u · n ) U · ∇ ψ ( p , w ) d∂ O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ (cid:90) O U · ( u · ∇ [ ∇ ψ ( p , w ))]) d O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cξ (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) [ (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) Ω ] ≤ C δ (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) + δCξ (cid:104) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:105) (35)Applying the estimates (31),(32), (33) and (35) to RHS of (29), we now haveRe(([ A + B ] ϕ, ϕ )) H ≤ −
14 ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) + C δ (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) + C δ (cid:107) U (cid:107) ∗ (cid:107) u (cid:107) H ( O ) + δCξ (cid:104) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:105) which yields the estimate (24), upon a use of Korn’s inequality and this gives the dissipativity of A + B, for U sufficiently small and 0 < δ < C ∗ . Moreover, it is shown in [30] that A + B ismaximal dissipative, then by a perturbation argument–see e.g., p. 211 of [33]– A + B is likewisemaximal dissipative. This concludes the proof of Theorem 1. This section is devoted to address the issue of asymptotic behavior of the solution whose existence-uniqueness is guaranteed by Theorem 1. In this regard, we show that the system given in (2)-(4) is strongly stable in H . That is, given any [ p, u, w , w ] ∈ H , the corresponding solution[ p ( t ) , u ( t ) , w ( t ) , w ( t )] ∈ C ([0 , T ]; H ) of (2)-(4) satisfieslim t →∞ (cid:107)| [ p ( t ) , u ( t ) , w ( t ) , w ( t )] |(cid:107) H = 0 . Our proof will be based on an ultimate appeal to the pointwise criterion [18, pp. 26, Theorem 8.4(i)]: 12 heorem 2
Let A be the generator of a bounded C -semigroup on a Hilbert space X . If thereexists a dense set M ⊂ X such that lim α → + √ αR ( α + iβ ; A ) x = 0 for every x ∈ M and every β ∈ R , then the semigroup is stable. There are reasons why we use here the resolvent criterion in [18], instead of the more wellknownapproaches in [36] and [2]. For one, the domain of the flow-structure generator A + B is notcompactly embedded into H , and so the classical stabilizability approach in [36] is not possible.In addition, the spectrum criterion for stability in [2] is not applicable here since it would entailthe elimination of all three parts of σ ( A + B ); particularly, the analysis of residual spectrum willbe of great technical complexity and mathematical challenge since it requires one to prove that iβ ( β (cid:54) = 0) is not an eigenvalue of the adjoint of the generator A + B . In order to justify our concernrelated to a residual spectrum analysis and to see how technical calculations the adjoint of thegenerator will require, we provide here the adjoint operator ( A + B ) ∗ (For a detailed proof, thereader is referred to [30, Lemma 13]): Lemma 3
With reference to problem (2)-(4), the adjoint operator ( A + B ) ∗ : D (( A + B ) ∗ ) ∩ H ⊂ H → H of the semigroup generator A + B is defined as ( A + B ) ∗ = A ∗ + B ∗ = U ·∇ ( · ) div ( · ) 0 0 ∇ ( · ) div σ ( · ) − ηI + U ·∇ ( · ) − ∇ U · ( · ) 0 00 0 0 − I − [ · ] Ω − [2 ν∂ x ( · ) + λ div ( · )] Ω ∆ + div ( U )( · ) 0 0 00 div ( U )( · ) 0 0˚ A − { div ([ U , U ]) + U ·∇ ) } ( · ) | Ω ˚ A − { div [ U , U ]+ U ·∇} [2 ν∂ x ( · ) + λ div ( · )] Ω + − div ( U )( · ) 0 0 00 0 0 00 0 − ˚ A − (cid:110) ( div [ U , U ]+ U ·∇ )∆ ( · ) (cid:111) + U ·∇ ( · ) + ∆ ˚ A − ∇ ∗ ( ∇· ( U ·∇ ( · ))) 00 0 0 0 = L + L + B ∗ (36) Here, ∇ ∗ ∈ L ( L (Ω) , [ H (Ω)] (cid:48) ) is the adjoint of the gradient operator ∇ ∈ L ( H (Ω) , L (Ω)) andthe domain of ( A + B ) ∗ | H is given as D (( A + B ) ∗ ) ∩H = { ( p , u , w , w ) ∈ L ( O ) × H ( O ) × H (Ω) × L (Ω) : properties ( A ∗ . i ) – ( A ∗ . vii ) hold } , where
13. ( A ∗ . i ) U · ∇ p ∈ L ( O )2. ( A ∗ . ii ) div σ ( u ) + ∇ p ∈ L ( O ) (So, [ σ ( u ) n + p n ] ∂ O ∈ H − ( ∂ O ))3. ( A ∗ . iii ) ∆ w − [2 ν∂ x ( u ) + λ div( u )] Ω − p | Ω ∈ L (Ω)4. ( A ∗ . iv ) ( σ ( u ) n + p n ) ⊥ T H / ( ∂ O ). That is, (cid:104) σ ( u ) n + p n , τ (cid:105) H − ( ∂ O ) × H ( ∂ O ) = 0 in D (cid:48) ( O ) for every τ ∈ T H / ( ∂ O )5. ( A ∗ . v ) The flow velocity component u = f + (cid:101) f , where f ∈ V and (cid:101) f ∈ H ( O ) satisfies (cid:101) f = (cid:40) Sw n on Ω(and so f | ∂ O ∈ T H / ( ∂ O ))6. ( A ∗ . vi ) [ − w + U ·∇ w + ∆ ˚ A − ∇ ∗ ( ∇· ( U ·∇ w ))] ∈ H (Ω) , (and so w ∈ H (Ω))7. ( A ∗ . vii ) (cid:82) O [ U · ∇ p +div ( u )] d O + (cid:82) Ω ˚ A − { (div[ U , U ] + U ·∇ )([ p + 2 ν∂ x ( u ) + λ div( u )] Ω ) } d Ω − (cid:82) Ω ˚ A − (cid:110) (div[ U , U ]+ U ·∇ )∆ w (cid:111) d Ω+ (cid:82) Ω [ U ·∇ w + ∆ ˚ A − ∇ ∗ ( ∇· ( U ·∇ w ))] d Ω= 0 . Looking at the expression for A ∗ + B ∗ in (36), it seems that showing this adjoint operator hasempty null space–necessary for ruling out residual spectrum of A + B –would be a difficult exercise.Now, our stability result is as follows: Theorem 4
The bounded C − semigroup (cid:8) e ( A + B ) t (cid:9) t ≥ given in Theorem 1 is strongly stable underthe condition that (cid:107) U (cid:107) ∗ = (cid:107) U (cid:107) L ∞ ( O ) + (cid:107) div ( U ) (cid:107) L ∞ ( O ) + (cid:107) U | Ω (cid:107) C (Ω) is small enough . That is, the solution φ ( t ) = [ p ( t ) , u ( t ) , w ( t ) , w ( t )] of the PDE system (2)-(4)tends asymptotically to the zero state for all initial data φ ∈ H . Proof.
The proof relies on the pointwise resolvent criterion given in Theorem 2, and hence toobtain the strong stability result it will be enough to show that the resolvent operator R ( a + ib ; [ A + B ]) = (( a + ib ) I − [ A + B ]) − obeys the limit estimate: lim a → + (cid:13)(cid:13)(cid:12)(cid:12) √ aR ( a + ib ; [ A + B ]) φ ∗ (cid:12)(cid:12)(cid:13)(cid:13) H = 0 (37)14here a > b ∈ R and given φ ∗ ∈ H . Here, we invoke the special inner product (( · , · )) H (definedin (14)) to get the necessary estimates but since the norms || · || H and || · || H are equivalent, weobtain the strong stability with respect to the standard inner product as well. Let φ = p u w w = R ( a + ib ; [ A + B ]) φ ∗ ∈ D ( A + B ) ∩ H , φ ∗ = p ∗ u ∗ w ∗ w ∗ ∈ H Then φ solves the following static PDE system: ( a + ib ) p + U · ∇ p + div u +div( U ) p = p ∗ in O ( a + ib ) u + U · ∇ u + ∇ U · u − div σ ( u ) + ηu + ∇ p = u ∗ in O ( σ ( u ) n − p n ) · τ = 0 on ∂ O u · n = 0 on Su · n = w + U · ∇ w on Ω (38) ( a + ib ) w − w − U · ∇ u = w ∗ ( a + ib ) w + ∆ w + [2 ν∂ x ( u ) + λ div( u ) − p ] Ω = w ∗ on Ω w = ∂w ∂ν = 0 on ∂ Ω (39)We invoke the dissipativity estimate (24) in Theorem 1 (i):( a + ib ) (cid:107)| φ |(cid:107) H − (([ A + B ]) φ, φ )) H = (( φ ∗ , φ )) H , or a (cid:107)| φ |(cid:107) H − Re (([ A + B ]) φ, φ )) H = Re (( φ ∗ , φ )) H which gives (cid:18) − C δ r U (cid:19) (cid:104) ( σ ( u ) , (cid:15) ( u )) O + (cid:107) u (cid:107) O (cid:105) + (cid:18) − δC ∗ (cid:19) ξ (cid:104) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:105) ≤ (cid:12)(cid:12) Re (( φ ∗ , φ )) H (cid:12)(cid:12) . (40)In turn, from the boundary condition w = u · n − U · ∇ w and (40), we get (cid:107) w (cid:107) Ω ≤ C ξ, U (cid:113)(cid:12)(cid:12) Re (( φ ∗ , φ )) H (cid:12)(cid:12) (41)where we have also used implicitly the Sobolev Trace Theorem. Now, combining (40) and (41),and using the equivalence of norms (cid:107)|·|(cid:107) H and (cid:107)·(cid:107) H on H , we have then (cid:107)| φ |(cid:107) H ≤ C ξ, U (cid:113)(cid:12)(cid:12) Re (( φ ∗ , φ )) H (cid:12)(cid:12) . Scaling the above inequality by √ a, followed by Young’s Inequality gives now √ a (cid:107)| φ |(cid:107) H ≤ √ aC ξ, U (cid:107)| φ ∗ |(cid:107) H (cid:107)| φ |(cid:107) H ≤ √ a C ξ, U (cid:107)| φ ∗ |(cid:107) H + √ a (cid:107)| φ |(cid:107) H . √ a (cid:107)| φ |(cid:107) H ≤ √ a C ξ, U (cid:107)| φ ∗ |(cid:107) H , and so lim a → + √ a (cid:107)| φ |(cid:107) H = 0 . This gives the desired limit estimate (37) and concludes the proof of Theorem 4.
In this paper, one of the key points used to analyze the stability properties of the PDE system(2)-(4) is the dissipativity estimate (24) in the inherently invariant subspace of the finite energyspace. To show this, we appeal to the dissipativity result given in [30] for an operator which isclosely associated our generator A + B . For the readers convenience, we also provide its proof. Lemma 5
Let A be the operator defined in (26). Then, with reference to problem (2)-(4), thesemigroup generator ( A + B ) : D ( A + B ) ∩ H ⊂ H → H is dissipative with respect to innerproduct (( · , · )) H for (cid:107) U (cid:107) ∗ = (cid:107) U (cid:107) L ∞ ( O ) + (cid:107) div ( U ) (cid:107) L ∞ ( O ) + (cid:107) U | Ω (cid:107) C (Ω) small enough. In particular,for ϕ = [ p , u , w , w ] ∈ D ( A + B ) ∩ H , Re (([ A + B ] ϕ, ϕ )) H ≤ − ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) , (42) where ξ is specified in (75). Proof.
Given ϕ = [ p , u , w , w ] ∈ D ( A + B ) ∩ H , we have(([ A + B ] ϕ, ϕ )) H = ( − U ∇ p − div( u ) − div( U ) p , p ) O +( −∇ p + div σ ( u ) − ηu − U ∇ u , u − αD ( g · ∇ w ) e ) O +( −∇ p + div σ ( u ) − ηu − U ∇ u , ξ ∇ ψ ( p , w )) O − α ( D ( g · ∇ [ w + U ∇ w ]) e , u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w )) O + ξ ( ∇ ψ ( − U ∇ p − div( u ) − div( U ) p , w + U ∇ w ) , u − αD ( g · ∇ w ) e ) O + ξ ( ∇ ψ ( − U ∇ p − div( u ) − div( U ) p , w + U ∇ w ) , ∇ ψ ( p , w )) O +(∆ w , ∆ w ) Ω + (∆( U ∇ w ) , ∆ w ) Ω +( p | Ω − [2 ν∂ x ( u ) + λ div( u )] | Ω , w + h α · ∇ w + ξw ) Ω +( h α · ∇ [ w + U ∇ w ] , w + h α · ∇ w + ξw ) Ω − (∆ w , w + h α · ∇ w + ξw ) Ω + ξ ( w + U ∇ w , w + h α · ∇ w + ξw ) Ω . A + B ] ϕ, ϕ )) H = − ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O + 12 (cid:90) O div( U )[ | u | − | p | ] d O +2 i Im[( p , div( u )) O + (∆ w , ∆ w ) Ω ] − i Im[( U ∇ p , p ) O + ( U ∇ u , u ) O ]+ (cid:88) j =1 I j , (43)where above the I j are given by: I = ( ∇ p − div σ ( u ) + ηu + U ∇ u , αD ( g · ∇ w ) e ) O − α ( p | Ω − [2 ν∂ x ( u ) + λ div( u )] | Ω , g · ∇ w ) Ω , (44) I = ( −∇ p + div σ ( u ) − ηu − U ∇ u , ξ ∇ ψ ( p , w )) O − ξ (∆ w , w ) Ω + ( p | Ω − [2 ν∂ x ( u ) + λ div( u )] | Ω , ξw ) Ω , (45) I = − α ( D ( g · ∇ [ w + U ∇ w ]) e , u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w )) O , (46) I = ξ ( ∇ ψ ( − U ∇ p − div( u ) − div( U ) p , w + U ∇ w ) , u − αD ( g · ∇ w ) e ) O , (47) I = ξ ( ∇ ψ ( − U ∇ p − div( u ) − div( U ) p , w + U ∇ w ) , ∇ ψ ( p , w )) O , (48) I = (∆( U ∇ w ) , ∆ w ) Ω − (∆ w , h α · ∇ w ) Ω , (49) I = ( h α · ∇ [ w + U ∇ w ] , w ) Ω , (50) I = ( h α · ∇ [ w + U ∇ w ] , h α · ∇ w + ξw ) Ω + ξ ( w + U ∇ w , w + h α · ∇ w + ξw ) Ω . (51)where we also recall the definition h α = U | Ω − αg. In the course of estimating the terms (44)-(51)above, we will invoke the polynomial r ( a ) = a + a + a , and for the simplicity, we set r U = r ( (cid:107) U (cid:107) ∗ ) . We start with I ; integrating by parts, we have I = − α ( p , div[ D ( g · ∇ w ) e ]) O + α ( σ ( u ) , (cid:15) ( D ( g · ∇ w ) e ) O + αη ( u , D ( g · ∇ w ) e ) O + α ( U ∇ u , D ( g · ∇ w ) e ) O (52)Using the fact that Dirichlet map D ∈ L ( H + (cid:15) (Ω) , H ( O )), we have I ≤ r U C (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:111) (53)17e continue with I ; using the definition of the map ψ ( · , · ) in (18) and integrating by parts we get I = − ξ (cid:90) O | p | d O − ξ ( σ ( u ) , (cid:15) ( ∇ ψ ( p , w ))) O + ξ (cid:104) σ ( u ) n − p n , ( ∇ ψ ( p , w ) , n ) n (cid:105) ∂ O − η ( u , ξ ∇ ψ ( p , w )) O ( − U ∇ u , ξ ∇ ψ ( p , w )) O − (∆ w , ξw ) Ω +( p | Ω − [2 ν∂ x ( u ) + λ div( u )] | Ω , ξw ) Ω , whence we obtain I ≤ − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) + ξr U C (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:111) + ξC (cid:110) (cid:107) u (cid:107) H ( O ) [ (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) Ω ] (cid:111) . (54)For I : recalling the boundary condition( u ) | Ω = w + U ∇ w , making use of Lemma 6.1 of [10] and considering the assumptions made on the geometry, we have I ≤ αC (cid:107) g · ∇ ( u ) (cid:107) H − (Ω) (cid:107) u − αD ( g · ∇ w ) e + ξ ∇ ψ ( p , w ) (cid:107) O ≤ C (cid:104) r U (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) ∆ w (cid:107) (cid:111) + ξ (cid:110) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:111)(cid:105) (55)where we have also implicitly used the Sobolev Embedding Theorem. To continue with I : I = ξ ( ∇ ψ ( − U ∇ p − div( U ) p , , u − αD ( g · ∇ w ) e ) O + ξ ( ∇ ψ ( − div( u ) , u · n ) , u − αD ( g · ∇ w ) e ) O = I a + I b (56)Since U · n | ∂ O = , we have that ( U ∇ p +div( U ) p ) ∈ [ H ( O )] (cid:48) with (cid:107) U ∇ p + div( U ) p (cid:107) [ H ( O )] (cid:48) ≤ C (cid:107) U (cid:107) ∗ (cid:107) p (cid:107) O . (57)By Lax-Milgram Theorem, we then have I a ≤ Cξ (cid:107)∇ ψ ( − U ∇ p − div( U ) p , (cid:107) O (cid:107) u − αD ( g · ∇ w ) e (cid:107) O ≤ Cξr U (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:111) (58)and similarly I b ≤ Cξr U (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) ∆ w (cid:107) (cid:111) . (59)18ow, applying (58)-(59) to (56) gives I ≤ Cξr U (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:111) . (60)Estimating I : we proceed as before done for I and invoke (57), Lax Milgram Theorem and theestimate (20) to have I ≤ Cξ (cid:104) (cid:107) U (cid:107) ∗ (cid:110) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:111) + (cid:107) u (cid:107) H ( O ) (cid:105) (61)For I , in order to estimate the second term in (49), we follow the standard calculations used forthe flux multipliers and the commutator symbol given by[ P, Q ] f = P ( Qf ) − Q ( P f ) (62)for the differential operators P and Q . Hence, − (∆ w , h α · ∇ w ) Ω = ( ∇ ∆ w , ∇ ( h α · ∇ w )) Ω (63)= − (∆ w , ∆( h α · ∇ w )) Ω + (cid:90) ∂ Ω ( h α · ν ) | ∆ w | d∂ Ω , (64)where, in the first identity we have directly invoked the clamped plate boundary conditions, andin the second we have used the fact that w = ∂ ν w = 0 on ∂ Ω which yields that ∂∂ν ( h α · ∇ w ) = ( h α · ν ) ∂ w ∂ν = ( h α · ν )(∆ w (cid:12)(cid:12) ∂ Ω ) . (See [34] or [35, p.305]). Using the commutator bracket [ · , · ], we can rewrite the latter relation as − (∆ w , h α · ∇ w ) Ω = − (∆ w , [∆ , h α · ∇ ] w ) Ω − (∆ w , h α · ∇ (∆ w )) Ω + (cid:90) ∂ Ω ( h α · ν ) | ∆ w | d∂ Ω . With Green’s relations once more: − (∆ w , h α · ∇ w ) Ω = − (∆ w , [∆ , h α · ∇ ] w ) Ω − (cid:90) ∂ Ω ( h α · ν ) | ∆ w | d∂ Ω+ 12 (cid:90) Ω (cid:2) div( h α ) (cid:3) | ∆ w | d Ω − i Im(∆ w , h α · ∇ (∆ w )) Ω + (cid:90) ∂ Ω ( h α · ν ) | ∆ w | d∂ Ω . (65)Thus, − (∆ w , h α · ∇ w ) Ω = − (∆ w , [∆ , h α · ∇ ] w ) Ω + 12 (cid:90) ∂ Ω ( h α · ν ) | ∆ w | d∂ Ω+ 12 (cid:90) Ω (cid:2) div( h α ) (cid:3) | ∆ w | d Ω − i Im(∆ w , h α · ∇ (∆ w )) . (66)19ince h α = U (cid:12)(cid:12) Ω − αg , where g is an extension of ν ( x ), we will have then − Re(∆ w , h α ·∇ w ) Ω = 12 (cid:90) ∂ Ω ( U · ν − α ) | ∆ w | d∂ Ω+ 12 (cid:90) Ω div( h α ) | ∆ w | d Ω − Re(∆ w , [∆ , h α ·∇ ] w ) Ω (67)Since we can explicitly compute the commutator[∆ , h α · ∇ ] w =(∆ h )( ∂ x w ) + 2( ∂ x h )( ∂ x w ) + 2( ∂ x h )( ∂ x w ) + (∆ h )( ∂ x w )+ 2div( h α )( ∂ x ∂ x w ) , and (cid:12)(cid:12)(cid:12)(cid:12) [∆ , h α · ∇ ] w (cid:12)(cid:12)(cid:12)(cid:12) L (Ω) ≤ r U || ∆ w || L (Ω) . (68)combining (67)-(68) we eventually get − Re(∆ w , h α · ∇ w ) Ω ≤ (cid:90) ∂ Ω [ U · ν − α ] | ∆ w | d∂ Ω + Cr U (cid:107) ∆ w (cid:107) . (69)Moreover, for the first term of (49), we have(∆( U ∇ w ) , ∆ w ) Ω = ( U ∇ w ) , ∆ w ) Ω − ([ U ·∇ , ∆] w , ∆ w ) Ω = (cid:90) ∂ Ω ( U · ν ) | ∆ w | d∂ Ω − (cid:90) ∂ Ω div( U ) | ∆ w | d∂ Ω − ([ U ·∇ , ∆] w , ∆ w ) Ω − (cid:90) Ω ∆ w U ·∇ (∆ w ) d Ωwhere we also use the commutator expression in (62). This gives usRe(∆( U ∇ w ) , ∆ w ) Ω ≤ (cid:90) ∂ Ω ( U · ν ) | ∆ w | d∂ Ω + Cr U (cid:107) ∆ w (cid:107) . (70)Now applying (69)-(70) to (49), we obtainRe I ≤ (cid:90) ∂ Ω [ U · ν − α | ∆ w | d∂ Ω + Cr U (cid:107) ∆ w (cid:107) . (F)To estimate I : since w ∈ H (Ω) , we haveRe( h α · ∇ w , w ) Ω = − (cid:90) Ω div( h α ) | w | d Ω= − (cid:90) Ω div( h α ) | ( u ) − U ∇ w | d Ω20fter using the boundary condition in ( A . v ) . Applying the last relation to RHS of (50) and recallingthat h α = U | Ω − αg, we getRe I = Re( h α · ∇ w , w ) Ω + Re( h α · ∇ ( U ∇ w ) , ( u ) − U ∇ w ) O ≤ Cr U (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) ∆ w (cid:107) (cid:111) (71)where we also implicitly use Sobolev Trace Theorem. Lastly, for the term I , we proceed in amanner similar to that adopted for I and we have I = ( h α · ∇ ( u ) , h α · ∇ w + ξw ) Ω + ξ (( u ) , ( u ) − U · ∇ w + h α · ∇ w + ξw ) Ω ≤ C (cid:2) r U + ξ (cid:3) (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) ∆ w (cid:107) (cid:111) + Cξ (cid:104) (cid:107) u (cid:107) H ( O ) + r U (cid:110) (cid:107) u (cid:107) H ( O ) + (cid:107) ∆ w (cid:107) (cid:111)(cid:105) (72)Now, if we apply (53)-(72) to RHS of (43), we obtainRe(([ A + B ] ϕ, ϕ )) H ≤ − ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) + (cid:90) ∂ Ω [ U · ν − α | ∆ w | d∂ Ω+ C (cid:2) r U + ξr U + ξ + ξ (cid:3) (cid:107) u (cid:107) H ( O ) + C (cid:2) r U + ξr U + ξ + ξ r U (cid:3) (cid:110) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:111) + Cξ (cid:107) u (cid:107) H ( O ) {(cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) Ω } . (73)We recall now the value of α = 2 (cid:107) U (cid:107) ∗ to getRe(([ A + B ] ϕ, ϕ )) H ⊥ N ≤ − ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) + (cid:2) ( C + C r U ) ξ + C r U ξ + C r U (cid:3) (cid:110) (cid:107) p (cid:107) O + (cid:107) ∆ w (cid:107) (cid:111) + 12 (cid:110) ( σ ( u ) , (cid:15) ( u )) O + η (cid:107) u (cid:107) O (cid:111) + C (cid:2) r U + ξr U + ξ + ξ (cid:3) (cid:107) u (cid:107) H ( O ) (74)where the positive constants C , C and C are obtained with the application of Holder-Young andKorn’s inequalities and C depends on the constant in Korn’s inequality. We now specify ξ be azero of the equation ( C + C r U ) ξ + ( C r U −
12 ) ξ + C r U = 0 . Namely, ξ = − C r U C + C r U ) − (cid:113) ( − C r U ) − C ( C + C r U ) r U C + C r U ) (75)21here the radicand is nonnegative for (cid:107) U (cid:107) ∗ sufficiently small. Then (74) becomesRe(([ A + B ] ϕ, ϕ )) H ≤ − ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O − ξ (cid:107) p (cid:107) O − ξ (cid:107) ∆ w (cid:107) − ( σ ( u ) , (cid:15) ( u )) O − η (cid:107) u (cid:107) O C K (cid:2) r U + ξr U + ξ + ξ (cid:3) (cid:110) ( σ ( u ) , (cid:15) ( u )) O + η (cid:107) u (cid:107) O (cid:111) . With ξ as prescribed in (75), we now have the dissipativity estimate (42), for (cid:107) U (cid:107) ∗ small enough.(Here we also implicitly re-use Korn’s inequality and C K is the constant there). This concludes theproof of Lemma 5. The author would like to thank the National Science Foundation, and acknowledge her partialfunding from NSF Grant DMS-1907823.
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