Compressible fluids interacting with plates -- regularity and weak-strong uniqueness
CCompressible fluids interacting with plates - regularity andweak-strong uniqueness
Srđan Trifunović ∗ Abstract
In this paper, we study a nonlinear interaction problem between compressible viscous fluids andplates. For this problem, we introduce relative entropy and relative energy inequality for the finiteenergy weak solutions (FEWS). First, we prove that for all FEWS, the relative energy inequality issatisfied and that the structure displacement enjoys improved regularity by utilizing the dissipationeffects of the fluid onto the structure. Then, we show that all FEWS enjoy the weak-strong uniquenessproperty, thus extending the classical result for compressible Navier-Stokes system to this fluid-structure interaction problem.
Keywords and phrases: fluid-structure interaction, compressible viscous fluid, elastic plate, weak-strong uniqueness
AMS Mathematical Subject classification (2020):
The well-posedness theory for the nonlinear interaction problems between fluids and thin elastic struc-tures (plates or shells) has seen quite a development in the last two decades. Such problems arise fromvarious physical phenomena and they are quite challenging for various reasons. For example, the fluiddomain changes in time and it depends on the displacement of the elastic structure - an unknown inthe system, it is a mixed-type coupled physical system, usually there are both hyperbolic and parabolicequations in the system etc. Majority of such results use the incompressible Navier-Stokes system tomodel the fluid. We mention [5, 14, 17, 24, 32, 33, 34, 37] for the existence of weak solutions, and[2, 18, 19, 25] for the existence of strong solutions. On the other hand, for the compressible Navier-Stokes system, there are only a handful of results which appeared in the last few years. First such resultwas due to Breit and Schwarzacher in [4], where they proved the existence of a weak solution for a shellinteracting with a compressible viscous fluid. Later, Mitra proved the existence of a unique local regularsolution in 2D, where the beam is viscoelastic in [29]. Maity and Takahashi obtained a strong solution in L p − L q framework for the interaction of heat-conducting fluid with a viscoelastic plate, and later Maity,Roy and Takahashi obtained a unique local regular solution in 3D [27], where the structure is modeledby a wave equation and doesn’t have any viscosity. Trifunović and Wang obtained the existence of aweak solution in 3D in [38], where the structure is modeled by a nonlinear thermoelastic (or just elastic)plate, by constructing a time-continuous splitting approximation scheme that decouples the fluid andthe structure. ∗ Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovića 4, 21000 Novi Sad, Serbia,email: [email protected] a r X i v : . [ m a t h . A P ] J a n hile weak solutions are in general global, or in the case of fluid-structure interaction exist up to acollision or self-collision of elastic structures, they are very often not unique and have a low regularitythat comes from the energy of the problem. On the other hand, strong (or more regular) solutions areusually unique and posses higher regularity, but are often local in time. Thus, a natural question arises- when do weak and strong solutions coincide if eminating from the same initial data, or in other words,when do weak solutions enjoy so called weak-strong uniqueness property? Note that this only gives apartial answer to the uniqueness problem, as it usually doesn’t hold globally in time and requires a moreregular initial data in order to ensure the existence of a strong solution in the first place. Some systems,like the incompressible Navier-Stokes equations, actually enjoy a better type of uniqueness properties.In particular, finite energy weak solutions in 2D are unique, while in 3D, a different type of a weakeruniquneness, so called Prodi-Serrin, is proven to hold. This means that a weak solution that has theintegrability u ∈ L at L bx , for a ∈ (2 , ∞ ) , b ∈ (3 , ∞ ) such that a + b = 1, is unique in the class of weaksolutions. On the other hand, the compressible Navier-Stokes system is not as fortunate, as only theresults concerning the weak-strong uniqueness are available so far (see [12, 11, 15]), at least without anyfurther simplifications.The uniqueness of fluid-structure interaction problems has been studied mostly for the interaction ofrigid bodies and incompressible viscous fluids. The following results are mentioned: the uniqueness in2D case with no-slip condition [16], the uniqueness and the energy identity in 2D with slip condition [3],the weak-strong uniqueness with slip condition [6] and the Serri-Prodin uniqueness with slip or no-slip[31]. The weak-strong uniqueness for compressible viscous fluid on moving domains, surrounding a rigidbody and filling a rigid body were studied in [26, 22, 13], respectively. In the mentioned results, whencomparing two solutions, the positions of the rigid bodies (and consequently the fluid domains) are notnecessarily the same as the rigid bodies move trough the fluid cavity. The usual approach to deal withthis issue is to transform coordinates of both solutions to be expressed on a common fixed domain andthen compare them there. One should note that to be even able to do this in the first place, the twosolutions need to have the same rigid body and the fixed part of the fluid domain. This also means thatif one of the two solutions is more regular, then the fluid cavity and the body for both of the solutionswill need to have the same (higher) regularity. Thus, the boundaries and their regularities don’t playany essential role in the analysis. However, when one deals with the interaction of fluids and elasticplates (or shells), the boundaries are not the same in general and there can also be a mismatch of theirregularities, if one of the solutions is more regular. Uniqueness of such problems was first studied byGuidoboni, Guidorzi, and Padula in [20], where a 1D beam interacts with a 2D incompressible fluid. Forthis problem, they proved the continuous time dependence of solutions on initial data, and consequentlythe uniqueness of such weak solutions. Very recently, Schwarzacher and Sroczinski in [35] studied theinteraction of incompressible viscous fluids and elastic plates, in 2D or 3D. Denoting the fluid velocitiesas u i and the plate displacements as w i , they proved that two finite energy weak solutions ( u , w )and ( u , w ) coincide if ∂ t u ∈ L t W − , x in 2D, or u ∈ L pt W ,qx and ∂ t u ∈ L t W − ,rx in 3D, for any p > , q >
3. The authors of this paper argue that the condition on ∂ t u was forgotten in [20] and it needsto be additionally imposed. This is because when ( u , w ) is compared with ( u , w ), it is mapped ontoa domain corresponding to the latter and this mapping (constructed in [20, section 6] and also used in[35]) is chosen to preserve the divergence-free condition in order to avoid the appearance of the pressure,but it doesn’t ensure that ∂ t u is in the right dual space of divergence-free functions corresponding tothe domain of the solution ( u , w ), so this is why this condition needs to be imposed additionally.In this paper, we study the finite energy weak solutions (FEWS) to the interaction problem ofcompressible viscous fluids and elastic or viscoelastic plates. The notions of relative entropy and relativeenergy inequality are introduced. First, it is proved that all FEWS satisfy the relative energy inequality.Then, by utilizing the dissipation effects of the fluid onto the structure, we prove that the additionalregularity ∆ w ∈ L t H sx can be obtained, for s = s ( γ, d, α ) >
0, where w is the displacement of the plate,2 is the adiabatic constant, d = 2 , α is the viscoelasticity constant.This is done by following the ideas from [32], where similar (but stronger) estimates were obtained forthe interaction of a incompressible viscous fluid and a nonlinear Koiter shell. Finally, we prove thatall FEWS enjoy the weak-strong uniqueness property. The proof uses the strategy from [12] (see also[11, 15]), and relies on using the relative entropy, which measures the distance between two solutions,and the relative energy inequality, which if tested by sufficiently regular solution of the same problem,gives the desired result by means of the Gronwall lemma. Since the fluid domains of two solutions arenot the same in general, the coordinates of both solutions are transformed to the ones of a common fixeddomain. The time and space derivatives are then replaced by the corresponding transformed derivativeswhich depend on the domain transformation, and thus on the displacement of the plate. This drasticallyincreases the difficulty of the analysis. To close these estimates, additional integrability is needed - ∇ w ∈ L t L ∞ x , which is always ensured by the mentioned regularity ∆ w ∈ L t H sx and the imbedding ofSobolev spaces (for any s > Here we deal with a compressible viscous fluid interacting with an elastic or viscoelastic plate, wherethe dimensions of the fluid and the plate are d and d −
1, respectively, for d = 2 ,
3. The vertical platedisplacement is described by a scalar function w : Γ → R , where Γ ⊂ R is a connected bounded domainwith a Lipschitz boundary. The fluid fills the domain defined asΩ w ( t ) := { ( X, z ) : X ∈ Γ , − < z < w ( t, X ) } , t ∈ [0 , T ] . We will denote the graph of w as Γ w ( t ) = { ( X, z ) : X ∈ Γ , z = w ( t, X ) } , while the entire rigid part ofthe boundary ∂ Ω w ( t ) will be denoted as Σ := (Γ × {− } ) ∪ ∂ Γ × {− , } . Finally, by a slight abuse ofnotation, for a time interval I and a time dependent set B ( t ), we will write I × B ( t ) := [ t ∈ I { t } × B ( t ) , and the time-space cyllinders corresponding to our system will be denoted as Q wT := (0 , T ) × Ω w ( t ) , Γ wT := (0 , T ) × Γ w ( t ) , Γ T := (0 , T ) × Γ . The governing equations for our coupled fluid-structure interaction problem read:
The (visco)elastic plate equation on Γ T : ∂ t w + ∆ w − α∂ t ∆ w = − S w f fl · e ; (1) The compressible Navier-Stokes equations on Q wT : ∂ t ( ρ u ) + ∇ · ( ρ u ⊗ u ) = −∇ p ( ρ ) + S ( ∇ u ) , (2) ∂ t ρ + ∇ · ( ρ u ) = 0; (3) The fluid-structure coupling (kinematic and dynamic, resp.) on Γ T : ∂ t w ( t, X ) e = u ( t, X, w ( t, X )) , (4) f fl ( t, X ) = (cid:2) ( − p ( ρ ) I + ∇ · S ( ∇ u )) ν w (cid:3) ( t, X, w ( t, X )); (5)3 he boundary conditions (clamped and no-slip, resp.) : w ( t, x ) = ∂ ν w ( t, x ) = 0 , on (0 , T ) × ∂ Γ , (6) u = 0 , on (0 , T ) × Σ; (7)
The initial data: ρ (0 , · ) = ρ , ( ρ u )(0 , · ) = ( ρ u ) , w (0 , · ) = w , ∂ t w (0 , · ) = v . (8)Here, α ≥ S w ( t, X ) is the Jacobian of the transformation from theEulerian to the Lagrangian coordinates of the plate S w ( t, X ) = p |∇ w | , e = (0 , ,
1) (or e = (0 ,
1) if d = 2), ν w is the unit outward normal vector on Γ w , p is the pressuregiven by p ( ρ ) = ρ γ , where γ >
1, if d = 2 and α ≥ γ > , if d = 3, and α = 0, γ > , if d = 3 and α > S ( ∇ u ) is the Newton stress tensor given by S ( ∇ u ) := µ ∇ u + ( µ + λ )( ∇ · u ) I, with µ > λ + µ >
0, and ν is the normal vector on ∂ Γ. First, to introduce the weak formulation to the problem (1) − (8), the following spaces that come fromthe energy inequality (12) of our problem are defined:the structure displacement space W S (0 , T ) := W , ∞ (0 , T ; L (Γ)) ∩ L ∞ (0 , T ; H (Γ)) ∩ αH (Γ T ) , the space for the fluid density W D (0 , T ) := L ∞ (0 , T ; L γ (Ω w ( t ))) . the fluid velocity space W F (0 , T ) := L ∞ (0 , T ; L (Ω w ( t ))) ∩ L (0 , T ; H (Ω w ( t ))) , and the coupled fluid-structure space W F S (0 , T ) = { ( u , w ) ∈ W F (0 , T ) × W S (0 , T ) : γ | Γ w ( t ) u = ∂ t w e and γ | Σ u = 0 for a.a. t ∈ (0 , T ) } . Here, γ | Γ w ( t ) is the Lagrangian trace operator on Γ w ( t ), which is a continuous linear operator from H (Ω w ( t )) to H s (Γ), for any s < (see [30]).The weak solution of the problem (1) − (8) is defined as: The lower bounds for γ are imposed to ensure the existence of a weak solution (see Remark 2 . See [38] for the derivation of this weak form for smooth solutions. efinition 2.1. We say that ( ρ, u , w ) ∈ W D (0 , T ) × W F S (0 , T ) is a weak solution of the problem (1) - (8) if:1. The initial data ρ , ( ρ u ) , w , v , ∈ L γ (Ω w ) × L γγ +1 (Ω w ) × H (Γ) × L (Γ) and is assumed to satisfythe following compatibility conditions: ρ > , on { ( ρ u ) > } , ( ρ u ) ρ ∈ L (Ω w ) ,∂ ν w = w = 0 , on ∂ Γ ,w > − , on Γ . (9) ρ ∈ C w (0 , T ; L γ (Ω w ( t ))) and the continuity equation Z T Z R (cid:0) ρ∂ t ϕ + ρ u · ∇ ϕ (cid:1) = Z T ddt Z R ρϕ, (10) holds for all ϕ ∈ C ∞ ([0 , T ] × R ) .3. ρ u ∈ C w (0 , T ; L γγ +1 (Ω w ( t ))) and the coupled momentum equation Z Q wT ρ u · ∂ t q + Z Q wT ( ρ u ⊗ u ) : ∇ q + Z Q wT ρ γ ( ∇ · q ) − Z Q wT S ( ∇ u ) : ∇ q + Z Γ T ∂ t w∂ t ψ − Z Γ T ∆ w ∆ ψ − α Z Γ T ∂ t ∇ w · ∇ ψ = Z T ddt Z Ω w ( t ) ρ u · q + Z T ddt Z Γ ∂ t wψ, (11) holds for all q ∈ C ∞ ([0 , T ] × Ω w ( t )) and ψ ∈ C ∞ ([0 , T ] × Γ) such that q | Σ = 0 and q ( t, X, w ( t, X )) = ψ ( t, X ) e on Γ T .If, in addition ( ρ, u , w ) satisfies the following energy inequality Z Ω w ( t ) ( ρ | u | )( t ) + 1 γ − Z Ω w ( t ) ρ γ ( t ) + Z t Z Ω w ( t ) S ( ∇ u ) : ∇ u + 12 Z Γ | ∂ t w ( t ) | + 12 Z Γ | ∆ w ( t ) | + α Z t Z Γ | ∂ t ∇ w | ≤ Z Ω w ( ρ u ) ρ + 1 γ − Z Ω w ρ γ + 12 Z Γ | v | + 12 Z Γ | ∆ w | , (12) then we call ( ρ, u , w ) a finite energy weak solution . Theorem 2.1. ([38,
Theorem 2.1 ]) Let α = 0 and γ > or α > and γ > . Then thereexists a finite energy weak solution ( ρ, u , w ) ∈ W D (0 , T ) × W F S (0 , T ) in the sense of the Definition . . The lifespan T > is either any T < T ∗ , where T ∗ is the moment when the collision occurs min X ∈ Γ w ( T ∗ , X ) + 1 = 0 , or T = ∞ if no collision occurs. Remark 2.1. (1) In theorem above, the structure is modeled by a nonlinear thermoelastic plate model,but the same result holds for the linear elastic plate model given in (1) . In section 5, the nonlinearelastic/thermoelastic plate models are considered.(2) In [4], a weak solution in the sense very similar to the one in Definition . was constructed, for α = 0 and γ > . There however, the geometry is different - the elastic structure is a shell whichdeforms in the normal direction of the reference domain boundary.(3) One of the main difficulties in studying the existence of such weak solutions is the weak L conver-gence of the pressure. Since the structure regularity that comes from the energy is not strong enough Here, the density ρ is extended by 0 to R . o ensure that the fluid cavity is Lipschitz (and it also changes in time), the standard approach basedon Bogovskii operator fails in this framework. The alternative here consists in proving the additionalintegrability of the pressure inside the domain in the standard way and that the L norm of the pressuredoesn’t concentrate near the boundary, thus giving the equiintegrability sufficient to pass to the weak limit.The latter proof is very involved and it is based on constructing a test function which sort of representsthe distance with respect to the boundary. As a consequence, this construction in [4] and [38] variesgreatly due to different geometries, as the latter has corners at ∂ Γ × {− } and ∂ Γ × { } . If α = 0 , inorder to close the estimates in the mentioned proof, one needs the condition γ > / to compensate forthe low integrability of the structure velocity ∂ t w ∈ L t L px , for any p < , which comes from the traceregularity. If however α > , then one can use the regularity ∂ t w ∈ L t H x , so γ > / suffices. Thismeans that the same results as in [4, 38] hold for α > and γ > / , even though they weren’t explicitlyconsidered. Here, we refer to known results concerning strong and regular solutions to the problem (1) − (8):Reference, constants Regularity of initial data Regularity of solutions[29, Theorem 1.1 ] ρ ◦ A w ∈ H ρ ◦ A w ∈ C t H x ∩ C t H x d = 2 u ◦ A w ∈ H u ◦ A w ∈ L t H x ∩ C t H x ∩ H t H x ∩ C t H x ∩ H t L x γ ≥ w = 0 w ∈ C t H x ∩ H t H x ∩ C t H x ∩ H t H x α > v ∈ H ∩ C t H x ∩ H t L x [27, Theorem 1.1 ] ρ ◦ A w ∈ H ρ ◦ A w ∈ H t H x ∩ W , ∞ t H x d = 2 , u ◦ A w ∈ H u ◦ A w ∈ L t H x ∩ H t H x ∩ H t L x γ ≥ w ∈ H w ∈ L ∞ t H x ∩ W , ∞ t H x ∩ H t H x ∩ H t L x α ≥ v ∈ H [28, Theorem 1.1 ] ρ ◦ A w ∈ W ,q ρ ◦ A w ∈ W pt L qx ∩ L pt W qx d = 2 , u ◦ A w ∈ B − /p ) q,p u ◦ A w ∈ L pt W ,qx ∩ W ,pt L qx γ ≥ w ∈ B − /p ) q,p w ∈ L pt W ,qx ∩ W ,pt L qx α > v ∈ B − /p ) q,p p ∈ (1 , ∞ ) , q ∈ ( d, ∞ ) , p + q = − d Table 1:
List of strong and regular solutions to (1) − (8). The regularity of ρ, u is given in ALE coordinates, by means ofthe mapping A w from (21), and the compatibility conditions for the initial data are omitted as they are quite natural andcan be found in the respective references. Remark 2.2. (1) In the above table, B − /p ) q,p stands for Besov spaces (see [1]).(2) In [27, 29], the plate domain Γ is periodic, i.e. Γ = (
R/L Z ) × ( R/L Z ) and Γ = (
R/L Z ) ,respectively, for some L , L > , and ρ, u , w are periodic in horizontal coordinates as well. However, in[28], the reference fluid domain is smooth and contains a flat part which is deformable and constitutes aviscoelastic plate.(3) In [28], the fluid conducts heat and it is governed by the Navier-Stokes-Fourier system, but the sameresult holds for the barotropic case used in this paper, as it was pointed out in [28, Remark 1.3(5)].(4) In [27], the structure is actually governed by a wave equation. Nevertheless, the same result can beobtained for the plate equation, both for α = 0 and α > , as these cases have more regularity. However,in [28, 29], α > and the same results cannot be obtained in the same way if α = 0 . This is becauseviscoelasticity ensures maximal regularity [28] and the analiticity [29] properties, which play a key rolein the respective approaches. .3 Relative entropy, relative energy inequality and suitable weak solutions Relative energy (or entropy) inequality in the context of compressible Navier-Stokes system was firstintroduced by Germain in [15], and it was used as a tool in proving the weak-strong uniqueness propertyfor a sub-class of finite energy weak solutions which have have a L pt L qx -integrable gradient of density (forcertain p, q > ρ , u , w ) be a finite energy weak solution in the sense of Definition 2 .
1. For any ρ , u ∈ C ∞ ([0 , T ] × Ω w ( t )) and w ∈ C ∞ ([0 , T ] × Γ) such that u | Σ = 0 and u ( t, X, w ( t, X )) = ∂ t w ( t, X ) e onΓ T , we define the relative entropy with respect to ( ρ , u , w ) as E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) ( t ) = 12 Z Ω w ( t ) ( ρ | u − u | )( t ) + 1 γ − Z Ω w ( t ) (cid:0) ρ γ − γρ γ − ( ρ − ρ ) − ρ γ (cid:1) ( t )+ 12 Z Γ | ∂ t w − ∂ t w | ( t ) + 12 Z Γ | ∆ w − ∆ w | ( t ) , (13)and the relative energy inequality with respect to ( ρ , u , w ) as E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) ( t ) + Z t Z Ω w ( t ) S ( ∇ u − ∇ u ) : ( ∇ u − ∇ u ) + α Z t Z Γ | ∂ t ∇ w − ∂ t ∇ w | ≤ E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) (0) + Z t R (cid:16) ρ , u , w , ρ , u , w (cid:17) , (14)where the remainder term reads R (cid:16) ρ , u , w , ρ , u , w (cid:17) := Z Ω w ( t ) S ( ∇ u ) : ( ∇ u − ∇ u ) + Z Ω w ( t ) ρ ( ∂ t u + u · ∇ u ) · ( u − u )+ γγ − Z Ω w ( t ) h ( ρ u − ρ u ) · ∇ ( ρ γ − ) + ( ρ − ρ ) ∂ t ( ρ γ − ) i + Z Ω w ( t ) ( ρ γ − ρ γ )( ∇ · u )+ Z Γ w ( t ) ( ∂ t w − ∂ t w ) ρ γ ν w · e − Z Γ ( ∂ t w − ∂ t w ) ∂ t w − Z Γ ( ∂ t w − ∂ t w )∆ w + α Z Γ ∂ t ∆ w ( ∂ t w − ∂ t w ) . (15)It is straightforward to see that by choosing ρ = 0 , u = 0 , w = 0, the relative energy inequality reducesto the energy inequality (12). Also, if one chooses w = w = 0, then relative entropy and relative energyinequality take the form of the ones defined in [12]. First, the following characterization of finite energy weak solutions is given:7 heorem 2.2. ( Main result I ) Let ( ρ, u , w ) be a finite energy weak solution in the sense ofDefinition (2.1) . Then, one has the following:(1) ( ρ, u , w ) satisfies the relative energy inequality (14) ;(2) ∆ w ∈ L (0 , T ; H s (Γ)) , where2D: α = 0 , γ > and < s < min (cid:8) − γ , (cid:9) ; α > , γ > and < s < min (cid:8) − γ , (cid:9) ;3D: α = 0 , γ > and < s < min (cid:8) − γ , (cid:9) ; α > : γ > and < s < min (cid:8) − γ , (cid:9) .Consequently, ∇ w ∈ L (0 , T ; L ∞ (Γ)) in all the above cases. The second result of this paper states that all finite energy weak solutions enjoy the weak-stronguniqueness property in the following sense:
Theorem 2.3. ( Main result II ) Let ( ρ , u , w ) be a finite energy weak solution in the sense ofDefinition . and let ( ρ , u , w ) be a strong solution of the problem (1) - (8) such that ρ ∈ L (0 , T ; W ,q (Ω w ( t ))) ∩ H (0 , T ; L q (Ω w ( t ))) , u ∈ L (0 , T ; W ,q (Ω w ( t ))) ∩ H (0 , T ; L q (Ω w ( t ))) ,w ∈ L (0 , T ; H (Γ)) ∩ H (0 , T ; L (Γ)) , and < inf (0 ,T ) × Ω w ( t ) ρ ≤ sup (0 ,T ) × Ω w ( t ) ρ < ∞ . If ρ (0 , · ) = ρ (0 , · ) , u (0 , · ) = u (0 , · ) , w (0 , · ) = w (0 , · ) , v (0 , · ) = v (0 , · ) , and2D: α ≥ , γ > and q > ;3D: α = 0 , γ ∈ (cid:0) , (cid:3) , ∇ u ∈ L ∞ ((0 , T ) × Ω w ( t )) and q > γ γ − ; α = 0 , γ > and q > ; α > , γ > and q > max { , γ γ − } ,then ( ρ , u , w ) ≡ ( ρ , u , w ) , a.e. in (0 , T ) × Ω w ( t ) . Remark 2.3.
In section 5, theorems . and . are extended to the case where the linear plate equation (1) is replaced by a elastic nonlinear plate equation or a thermoelastic nonlinear plate system. . The following proof will be carried in two parts. 8 .1 Part 1: relative energy inequality - Theorem . We start by choosing ( q , ψ ) = ( u , ∂ t w ) in the coupled momentum equation (11) on (0 , t ) to obtain Z Ω w ( t ) ( ρ u · u )( t ) + Z Γ ( ∂ t w ∂ t w )( t )= Z Ω w (0) ( ρ u · u )(0) + Z Γ ( ∂ t w ∂ t w )(0) + Z Q w t ρ u · ∂ t u + Z Q w t ( ρ u ⊗ u ) : ∇ u + Z Q w t ρ γ ( ∇ · u ) − Z Q w t S ( ∇ u ) : ∇ u | {z } = R Qw t S ( ∇ u ): ∇ u + Z Γ t ∂ t w ∂ t w − Z Γ t ∆ w ∆ ∂ t w − α Z Γ t ∂ t ∇ w · ∂ t ∇ w . (16)Next, in the continuity equation (10) on (0 , t ), choose ϕ = | u | to obtain12 Z Ω w ( t ) ( ρ | u | )( t ) = 12 Z Ω w (0) ( ρ | u | )(0) + Z Q w t ρ ∂ t u · u + Z Q w t ρ ( u · ∇ u ) · u , (17)and then choose ϕ = γγ − ρ γ − γγ − Z Ω w ( t ) ( ρ ρ γ − )( t )= γγ − Z Ω w (0) ( ρ ρ γ − )(0) + γγ − Z Q w t ρ ∂ t ( ρ γ − ) + γγ − Z Q w t ρ u · ∇ ( ρ γ − ) . (18)Noticing that, γγ − Z Q w t ρ u · ∇ ( ρ γ − ) = Z Q w t u · ∇ ( ρ γ ) = − Z Q w t ρ γ ( ∇ · u ) + Z Γ w t ρ γ ∂ t w ν w · e , by the divergence theorem, and Z t ddt Z Ω w ( t ) ρ γ = γγ − Z Q w t ρ ∂ t ( ρ γ − ) + Z Γ w t ρ γ ∂ t w ν · e , by the Raynolds transport theorem, one obtains Z Ω w ( t ) ρ γ ( t ) = Z Ω w (0) ρ γ (0) + Z Γ w t ρ γ ( ∂ t w − ∂ t w ) ν w · e + γγ − Z Q w t h ρ U · ∇ ( ρ γ − ) + ρ ∂ t ( ρ γ − ) i + Z Q w t ρ γ ( ∇ · u ) . (19)Finally, expressing the last three terms on the RHS of (16) as Z Γ t ∆ w ∂ t ∆ w ± (cid:16) Z Γ | ∆ w | (cid:17)(cid:12)(cid:12)(cid:12) t = (cid:16) Z Γ ∆ w ∆ w (cid:17)(cid:12)(cid:12)(cid:12) t + Z Γ t ( ∂ t w − ∂ t w )∆ w + (cid:16) Z Γ | ∆ w | (cid:17)(cid:12)(cid:12)(cid:12) t , − Z Γ t ∂ t w∂ t w ± (cid:16) Z Γ | ∂ t w | (cid:17)(cid:12)(cid:12)(cid:12) t = Z Γ t ( ∂ t w − ∂ t w ) ∂ t w + (cid:16) Z Γ | ∂ t w | (cid:17)(cid:12)(cid:12)(cid:12) t , − Z Γ t ∂ t ∇ w · ∂ t ∇ w ± Z Γ t | ∂ t ∇ w | = Z Γ ∂ t ∆ w ( ∂ t w − ∂ t w ) + Z Γ t | ∂ t ∇ w | , we sum (12) − (16) + (17) − (18) − (19), which then finally gives us (14). Note that the coupled momentum equation (11) and the continuity equation (10) hold on (0 , t ), for any t ∈ (0 , T ], dueto the weak continuity in time of ρ and ρ u and ∂ t w . .2 Part 2: the regularity estimates - Theorem . We start with introducing fractional Sobolev and Nikolskii functional spaces, respectively (see [1] and[36] for more details):
Definition 3.1.
For α ∈ (0 , , q ∈ (1 , ∞ ) , we say that g ∈ W α,q (Γ) if its norm satisfies || g || q W α,q (Γ) := (cid:16) Z Γ Z Γ | g ( x ) − g ( y ) | q | x − y | n + αq dxdy (cid:17) q + (cid:16) Z Γ | g ( x ) | q dx (cid:17) q < ∞ , and that g ∈ N α,q (Γ) if its norm satisfies || g || q N α,q (Γ) := sup i ∈{ ,...,d } sup h> (cid:16) Z Γ h (cid:12)(cid:12)(cid:12) g ( x + h e i ) − g ( x ) h α (cid:12)(cid:12)(cid:12) q dx (cid:17) q + (cid:16) Z Γ | g ( x ) | q dx (cid:17) q < ∞ , where e i its the i -th unit vector and Γ h = { x ∈ Γ : dist ( x, ∂ Γ) > h } . Given any < α < β < and g ∈ N β,q (Γ) , the following imbedding is true || g || L nqn − αq (Γ) ≤ C || g || W α,q (Γ) ≤ C || g || N β,q (Γ) ≤ C || g || W β,q (Γ) , for αq < n, or || g || C α − nq (Γ) ≤ C || g || W α,q (Γ) ≤ C || g || N β,q (Γ) ≤ C || g || W α,q (Γ) , for αq > n. Finally, for s, h > and a function q : Γ → R , we introduce the quotient D sh,e ( q )( y ) := q ( y + h e ) − q ( y ) h s , for any (unit) vector e ∈ R . Since the direction of the unit vector is not relevant, we will omit it and just write D sh ( q )( y ) := D sh,e ( q )( y ) instead, for any unit vector e . The following notation will be useful in the upcoming analysis:
Definition 3.2.
For a given < b ≤ ∞ and a domain E , denote by L b − ( E ) := T p
0. Also, we willonly prove the estimate for the 3D case, since the estimates in the 2D case can be obtained more easily.
Remark 3.1.
The following proof is the adaptation to the compressible case of [32, Theorem 1.2],where similar estimates were obtained in the context of incompressible viscous fluid interacting with aquasilinear elastic Koiter shell. R w be an extension operator defined as R w : f ( t, X ) f ( t, X ) z + 1 w ( t, X ) + 1 e . Fix h > q , ψ ) = ( R w [ D s − h D sh w ] , D s − h D sh w ) toobtain: || D sh ∆ w || L (Γ T ) = − Z Q wT ( ρ u ⊗ u ) : ∇ R w (cid:2) D s − h D sh [ w ] (cid:3) + Z Q wT ρ u · ∂ t R w (cid:2) D s − h D sh [ w ] (cid:3) + Z Q wT ρ γ (cid:0) ∇ · R w (cid:2) D s − h D sh [ w ] (cid:3)(cid:1) − µ Z Q wT ∇ u : ∇ R w (cid:2) D s − h D sh [ w ] (cid:3) − ( µ + λ ) Z Q wT ( ∇ · u )( ∇ · R w (cid:2) D s − h D sh [ w ] (cid:3) )+ || D sh ∂ t w || L (Γ T ) − α Z Γ T ∂ t ∇ w · ∇ D s − h D sh [ w ] − Z T ddt Z Ω ρ u · R w (cid:2) D s − h D sh [ w ] (cid:3) ) − Z T ddt Z Γ D s − h ∂ t wD sh w, (20)where we used the following commutative property − Z Γ f D s − h D sh g = Z Γ D sh f D sh g, f, g ∈ L (Γ) . The proof will follow once we show that right-hand side of (20) can be bounded by a constant independentof h . We start with Z Q wT ( ρ u ⊗ u ) : ∇ R w (cid:2) D s − h D sh [ w ] (cid:3) ≤ C || ρ || L ∞ (0 ,T ; L γ (Ω w ( t ))) || u || L (0 ,T ; L − (Ω w ( t ))) ||∇ R w (cid:2) D s − h D sh [ w ] (cid:3) || L ∞ (0 ,T ; L p (Ω w ( t ))) ≤ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D s − h D sh [ ∇ w ] z + 1 w + 1 + D s − h D sh [ w ] ∇ (cid:16) z + 1 w + 1 (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ (0 ,T ; L p (Ω w ( t ))) ≤ C (cid:0) || D s − h D sh [ ∇ w ] || L ∞ (0 ,T ; L p (Γ)) + || D s − h D sh [ w ] || L ∞ (0 ,T ; L p − (Γ)) (cid:1) ≤ C (cid:0) ||∇ w || L ∞ (0 ,T ; W s,p (Γ)) + || w || L ∞ (0 ,T ; W s,p − (Γ)) (cid:1) ≤ C || ∆ w || L ∞ (0 ,T ; L (Γ)) ≤ C, for any 0 < s < min n − γ , o and 42 − − s ) = 1 s ≥ p > max n , γ γ − o , by Hölder’s inequality and the imbedding of Sobolev spaces. Next, Z Q wT ρ u · ∂ t R w (cid:2) D s − h D sh [ w ] (cid:3) ≤ || ρ u || L (0 ,T ; L ( 6 γ γ − − (Ω w ( t ))) || ∂ t R w (cid:2) D s − h D sh [ w ] (cid:3) || L (0 ,T ; L p (Ω w ( t ))) ≤ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D s − h D sh [ ∂ t w ] z + 1 w + 1 − D s − h D sh [ w ] z + 1( w + 1) ∂ t w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (0 ,T ; L p (Ω w ( t ))) ≤ C (cid:0) || D s − h D sh [ ∂ t w ] || L (0 ,T ; L p (Γ)) + || D s − h D sh [ w ] ∂ t w || L (0 ,T ; L p (Γ)) (cid:1) ≤ C, for α = 0 and 0 < s < min n − γ , o and 41 + 4 s ≥ p > max n , γ γ − o , or for α > < s < min n − γ , o and 42 − − s ) = 1 s ≥ p > max n , γ γ − o . ∇ · R w (cid:2) D s − h D sh [ w ] (cid:3) = D s − h D sh [ w ] 1 w + 1 , so one can bound Z Q wT ρ γ (cid:0) ∇ · R w (cid:2) D s − h D sh [ w ] (cid:3)(cid:1) ≤ C || ρ || L ∞ (0 ,T ; L γ (Ω w ( t ))) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D s − h D sh [ w ] 1 w + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (0 ,T ; L ∞ (Γ)) ≤ C || D s − h D sh [ w ] || L (0 ,T ; H a (Γ)) ≤ C || w || L (0 ,T ; H s + a (Γ)) ≤ C, for any 0 < s, a < a + 2 s <
1. The remaining terms can be bounded in a similar fashion asthe previous ones, the proof is finished. . When working with free boundary problems, it is common to transform the coordinate system in orderto define the problem on a fixed domain. Also, in this proof, we will be working with two solutions whichare in general defined on different domains, so transforming them onto a common domain will allow usto compare them.
We start by introducing the fixed domainΩ := Γ × ( − , , and the family of so called arbitrary Lagrangian-Eulerian (ALE) mappings A w ( t ) : Ω → Ω w ( t )( X, z ) ( X, ( z + 1) w ( t, X ) + z ) . (21)For an arbitrary (vector or scalar) function f defined on Ω w ( t ), corresponding to A w , we denote the pull-back , transformed gradient and transformed divergence of f on Ω, respectively, as f w := f ◦ A w , ∇ w f w := ( ∇ f ) w = ∇ f w ( ∇ A w ) − ◦ A w , ∇ w · f w := Tr( ∇ w f w ) . where ( ∇ A − w ) ◦ A w = − z + 1 w + 1 ∂ x w − z + 1 w + 1 ∂ y w w + 1 . The
Jacobian and the
ALE velocity of the mapping A w , respectively, read J := det ∇ A w = 1 + w, w := ddt A w = ( z + 1) ∂ t w e , and the time-space cylinder is denoted as Q T := (0 , T ) × Ω . i = 1 , f , denote ∇ i f := ∇ w i f , ∇ i · f := ∇ w i · f , A i := A w i , S i := S w i , ν i := ν w i , and the pull-backs r i := ρ w i i = ρ i ◦ A i , U i := u w i i = u i ◦ A i . Finally, the subscript for the Jacobian will be omitted J := J = 1 + w , since it will appear frequently. Let ( ρ , u , w ) be a finite energy weak solution in the sense of Definition 2 . r , U ∈ C ∞ ([0 , T ] × Ω) and w ∈ C ∞ (Γ T ) such that U | Σ = 0 and U | Γ ×{ } = ∂ t w e . For ( r , U , w ) = ( ρ ◦ A , u ◦ A , w ), the relative entropy on the fixed reference domain with respect to ( r , U , w ) will bealso denoted by E and it reads as E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( t ) = 12 Z Ω ( Jr | U − U | )( t ) + 1 γ − Z Ω J (cid:0) r γ − γr γ − ( r − r ) − r γ (cid:1) ( t )+ 12 Z Γ | ∂ t w − ∂ t w | ( t ) + 12 Z Γ | ∆ w − ∆ w | ( t ) . (22)By using the identity ddt ( q ◦ A i ) = ( ∂ t q ) ◦ A i + w i · ∇ i ( q ◦ A i ) , on Ω , (23)where q is a sufficiently regular function defined on Q w i T , and Z Γ w ( t ) ( ∂ t w − ∂ t w ) ρ γ Iν · e = Z Γ ( ∂ t w − ∂ t w ) r γ S Iν · e | {z } =1 , one can easily see that the triplet ( r , U , w ) satisfies the relative energy inequality on the fixedreference domain with respect to ( r , U , w ) of the form E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( t ) + Z Q t J | S ( ∇ U − ∇ U ) : ( ∇ U − ∇ U ) + α Z t Z Γ | ∂ t ∇ w − ∂ t ∇ w | ≤ E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) (0) + Z t R (cid:16) r , U , w, r , U , w (cid:17) (24) Here, the weak solution on the fixed reference domain is not defined explicitly since it will not be required in the proof,but this definition can be found in [38, section 2.3]. remainder term reads R (cid:16) r , U , w , r , U , w (cid:17) := Z Ω J S ( ∇ U ) : ( ∇ U − ∇ U ) + Z Ω Jr ( ∂ t U + U · ∇ U ) · ( U − U ) − Z Ω Jr (cid:2) w · ∇ U (cid:3) · ( U − U ) + γγ − h Z Ω J ( r U − r U ) · ∇ ( r γ − )+ J ( r − r ) (cid:0) ∂ t ( r γ − ) − w · ∇ ( r γ − ) (cid:1)i + Z Ω J ( r γ − r γ )( ∇ · U )+ Z Γ ( ∂ t w − ∂ t w ) r γ − Z Γ ( ∂ t w − ∂ t w ) ∂ t w − Z Γ ( ∂ t w − ∂ t w )∆ w + α Z Γ ∂ t ∆ w ( ∂ t w − ∂ t w ) . (25) Here, we compare (see figure 1) a finite energy weak solution ( ρ , u , w ) and a strong solution ( ρ , u , w )belonging to the regularity class specified in Theorem 2 .
3, which emanate from the same initial data.
Figure 1:
The weak and the strong solution on both physical and fixed reference domain coordinates.
The goal is to prove that E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( t ) ≤ C Z t h ( τ ) E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( τ ) dτ, h ∈ L (0 , T ) , for a.a. t ∈ (0 , T ), then the desired result will follow immediately by Gronwall’s lemma. Step 1: transforming the remainder term R (cid:16) r , U , w , r , U , w (cid:17) First, one can see that ( r , U , w ) satisfies the following transformed problem in the strong sense ∂ t w + ∆ w = − S (cid:2) ( S ( ∇ U ) − r γ I ) ν (cid:3) · e , on Γ T ,∂ t r − w · ∇ r + ∇ · ( r U ) = 0 , in Q T ,∂ t U − w · ∇ U + U · ∇ U = − r ∇ ( r γ ) + r ∇ · S ( ∇ U ) , in Q T , (26)14y using (23), and that r ∈ L (0 , T ; W ,q (Ω)) ∩ H (0 , T ; L q (Ω)) , U ∈ L (0 , T ; W ,q (Ω)) ∩ H (0 , T ; L q (Ω)) ,w ∈ L (0 , T ; H (Γ)) ∩ H (0 , T ; L (Γ)) , < inf Q T r ≤ sup Q T r < ∞ , with q = q ( d, α, γ ) being specified in Theorem 2 .
3. Using this regularity, we test (24) with ( r , U , w )(by the density argument), and then transform the remainder term given in (25) by (26), as follows. The 2nd term on RHS of (25): Z Q t Jr ( ∂ t U + U · ∇ U )( U − U )= Z Q t Jr ( ∂ t U + U · ∇ U )( U − U ) + Z Q t Jr (cid:2) U · ( ∇ − ∇ ) U (cid:3) · ( U − U )+ Z Q t Jr (cid:2) ( U − U ) · ∇ U (cid:3) · ( U − U ) ± Z Q t Jr ( w · ∇ U )( U − U ) (26) = − Z Q t J r r ∇ ( r γ ) + Z Q t J r (cid:2) ∇ · S ( ∇ U ) (cid:3) · ( U − U )+ Z Q t Jr (cid:2) U · ( ∇ − ∇ ) U (cid:3) · ( U − U ) + Z Q t Jr (cid:2) ( U − U ) · ∇ U (cid:3) · ( U − U )+ Z Q t Jr ( w · ∇ U )( U − U ) . (27) The 2nd term on RHS of (27): Z Q t J r r (cid:2) ∇ · S ( ∇ U ) (cid:3) · ( U − U )= Z Q t J r ( r − r ) (cid:2) ∇ · S ( ∇ U ) (cid:3) · ( U − U ) + Z Q t ( J − J ) (cid:2) ∇ · S ( ∇ U ) (cid:3) · ( U − U )+ Z Q t J (cid:2) ∇ · S ( ∇ U ) (cid:3) · ( U − U ) , (28)where the last term can be transformed by partial integration on the physical domain Z Q t J ∇ · S ( ∇ U )( U − U ) = Z t Z Ω w ( t ) ∇ · S ( ∇ u )( u − U ◦ A − )= − Z t Z Ω w ( t ) S ( ∇ U ) : ( ∇ u − ∇ U ◦ A − ) + Z t Z Γ w ( t ) (cid:2) ( S ( ∇ u )) ν (cid:3) · e ( ∂ t w − ∂ t w )= − Z Q t J S ( ∇ U ) : ( ∇ U − ∇ U ) + Z Γ t S (cid:2) ( S ( ∇ U )) ν (cid:3) · e ( ∂ t w − ∂ t w ) . (29)15 he 4th term on RHS of (25) and the 1st term on RHS of (27): γγ − h Z Q t J ( r U − r U ) · ∇ ( r γ − ) + J ( r − r ) (cid:2) ∂ t ( r γ − ) − w · ∇ ( r γ − ) (cid:3)i − Z Q t J r r ∇ ( r γ ) | {z } = γγ − Jr ∇ ( r γ − ) = γγ − Z Q t J ( r − r ) h U · ∇ ( r γ − ) + ∂ t ( r γ − ) − w · ∇ ( r γ − ) i + Z Q t r ( ∇ − ∇ )( r γ − )( U − U ) ± γγ − Z Q t J ( r − r ) h U · ∇ ( r γ − ) − w · ∇ ( r γ − ) i = γγ − Z Q t J ( r − r ) h ∂ t ( r γ − ) + U · ∇ ( r γ − ) − w · ∇ ( r γ − ) | {z } = − ( γ − ∇ · U ) r γ − , by (26) i + γγ − Z Q t J ( r − r ) h U · ( ∇ − ∇ )( r γ − ) − ( w − w ) · ∇ ( r γ − ) − w · ( ∇ − ∇ )( r γ − ) i + Z Q t Jr ( ∇ − ∇ )( r γ − ) · ( U − U ) . (30) The 5th term on RHS of (25) and the 1st term on RHS of (30): Z Q t J ( r γ − r γ )( ∇ · U ) − γ Z Q t J ( r − r )( ∇ · U ) r γ − = Z Q t J h r γ − γr γ − ( r − r ) − r γ i ( ∇ · U ) + γ Z Q t J ( r − r ) r γ − ( ∇ − ∇ ) · U . (31) The plate equation (26) × ( ∂ t w − ∂ t w ): Z Γ t ( ∂ t w − ∂ t w ) ∂ t w + Z Γ t ( ∂ t w − ∂ t w )∆ w − α Z Γ t ∂ t ∆ w ( ∂ t w − ∂ t w ) − Z Γ t ( ∂ t w − ∂ t w ) r γ S Iν · e | {z } =1 + Z Γ t ( ∂ t w − ∂ t w ) S (cid:2) ( S ( ∇ U )) ν (cid:3) · e = 0 . (32)Now, the remainder term given in (25) is transformed by using the identities (27) , (28) , (29) , (30) and(31) (in that order), which then by (24) gives us: E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( t ) + Z Q t J | S ( ∇ U − ∇ U ) | ≤ Z Q t J S ( ∇ U ) : ( ∇ U − ∇ U ) − Z Q t J S ( ∇ U ) : ( ∇ U − ∇ U ) Z Q t J r ( r − r ) (cid:2) ∇ · S ( ∇ U ) (cid:3) · ( U − U ) + Z Q t ( J − J ) (cid:2) ∇ · S ( ∇ U ) (cid:3) · ( U − U )+ γγ − Z Q t J ( r − r ) h U · ( ∇ − ∇ )( r γ − ) − ( w − w ) · ∇ ( r γ − ) − w · ( ∇ − ∇ )( r γ − ) i + Z Q t Jr (cid:2) U · ( ∇ − ∇ ) U (cid:3) · ( U − U ) − Z Q t Jr (cid:2) ( U − U ) · ∇ U (cid:3) · ( U − U )+ Z Q t Jr ( ∇ − ∇ )( r γ − ) · ( U − U ) + γ Z Q t J ( r − r ) r γ − ( ∇ − ∇ ) · U + Z Q t J (cid:2) r γ − γr γ − ( r − r ) − r γ (cid:3) ( ∇ · U ) + Z Q t Jr h w · ∇ U − w · ∇ U i · ( U − U ) . (33)The aim is to bound the right-hand side by C ( δ ) Z t h E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | , (34)16or some h ∈ L (0 , T ) and a small δ . This will be done in step 3. Step 2: studying the difference ( r − r )Denote f ( x, y ) := x γ − γy γ − ( x − y ) − y γ . x, y ≥ . It is easy to see that for any
C > f ( x, C ) is convex in x and f ( C, C ) = ∂ x f ( C, C ) = 0. This meansthat for a fixed
C >
0, one can choose c = c ( C ) > f ( x, C ) ≥ c | x − C | ∧ γ , x ∈ h C , C i . Since 0 < c r = inf Q T r ≤ sup Q T r = C r , there is a constant c = c ( c r , C r , γ ) for which f ( x, r ) ≥ c | x − r | ∧ γ , for r ≤ x ≤ r , (35) f ( x, r ) ≥ c (1 + x ∧ γ ) , for x ∈ R +0 \ h r , r i . (36)Now, for 0 ≤ r ( t ) ≤ r ( t )2 , one has || r ( t ) − r ( t ) || L ∧ γ (cid:0)(cid:8) ≤ r ( t ) ≤ r t )2 (cid:9)(cid:1) ≤ Z(cid:8) ≤ r ( t ) ≤ r t )2 (cid:9) c ≤ Z(cid:8) ≤ r ( t ) ≤ r t )2 (cid:9) c (1 + r γ ( t )) ≤ C E (cid:16) ( r, U , w ) | ( r , U , w ) (cid:17) ( t ) , from (36), then for r ( t )2 ≤ r ( t ) ≤ r ( t ) || r ( t ) − r ( t ) || L ∧ γ (cid:0)(cid:8) r t )2 ≤ r ( t ) ≤ r ( t ) (cid:9)(cid:1) ≤ c || r ( t ) − r ( t ) || ∧ γL ∧ γ (cid:0)(cid:8) r t )2 ≤ r ( t ) ≤ r ( t ) (cid:9)(cid:1) ≤ c E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( t ) , from (35), and finally for r ( t ) ≥ r ( t ) || r ( t ) − r ( t ) || L ∧ γ ( { r ( t ) ≥ r ( t ) } ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) − r ( t ) r ( t ) (cid:17) r ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∧ γ ( { r ( t ) ≥ r ( t ) } ) ≤ c ( r ) || r ( t ) || L ∧ γ ( { r ( t ) ≥ r ( t ) } ) ≤ C (cid:16) Z { r ( t ) ≥ r ( t ) } r ∧ γ (cid:17) ∧ γ ≤ C E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ∧ γ ( t ) ≤ C E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( t ) , from (36), since E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( t ) ≤ C , so by combining previous three inequalities, oneobtains || r ( t ) − r ( t ) || L ∧ γ (Ω) ≤ C E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) ( t ) . (37) Step 3: closing the estimates
Here, we will prove the right-hand side of inequality (33) can be controlled by (34), as follows.17 he 1st and the 2nd term on RHS of (33): Z Q t J S ( ∇ U ) : ( ∇ U − ∇ U ) − Z Q t J S ( ∇ U ) : ( ∇ U − ∇ U )= Z Q t ( J − J ) S ( ∇ U ) : ( ∇ U − ∇ U ) + Z Q t J S (( ∇ − ∇ ) U ) : ( ∇ U − ∇ U )+ Z Q t J S ( ∇ U ) : ( ∇ − ∇ )( U − U ) ≤ C Z t || w − w || L ∞ (Γ) ||∇ U || L p (Ω) ||∇ A || L ∞− (Ω) ||∇ U − ∇ U || L − (Ω) ||∇ A || L ∞− (Ω) + C Z t || A − A || W , ∞− (Ω) ||∇ U || L p (Ω) ||∇ A || L ∞− (Ω) ||∇ U − ∇ U || L − (Ω) ||∇ A || L ∞− (Ω) + C Z t ||∇ U || L p (Ω) ||∇ A || L ∞ (Ω) || A − A || W , ∞− (Ω) ||∇ U − ∇ U || L − (Ω) ≤ C Z t || ∆ w − ∆ w || L (Γ) || U − U || W , − (Ω) ≤ C ( δ ) Z t E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U | , for any p ∈ (2 , J, J ≤ C and ∇ U ∈ L ∞ (0 , T ; L p (Ω)), where we also used || U − U || W , − (Ω) ≤ C ||∇ A || L ∞− (Ω) Z Ω J | S ( ∇ U − ∇ U ) | , and || ( ∇ − ∇ ) f || L p − (Ω) ≤ C || A − A || W , ∞− (Ω) ||∇ f || L p (Ω) ≤ C || ∆ w − ∆ w || L (Γ) ||∇ f || L p (Ω) , for any f ∈ W ,p (Ω), by the imbedding of Sobolev spaces. The 3rd term on RHS of (33): Z Q t J r ( r − r ) (cid:2) ∇ · S ( ∇ U ) (cid:3) · ( U − U ) ≤ C Z t || r − r || L ∧ γ (Ω) ||∇ · S ( ∇ U ) || L q (Ω) | {z } := h ∈ L (0 ,T ) || U − U || L − (Ω) ≤ C ( δ ) Z t h || r − r || L ∧ γ (Ω) + δ Z Q t J | S ( ∇ U − ∇ U ) | ≤ C ( δ ) Z t h E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | , for q > max (cid:8) , γ γ − (cid:9) , by using (37) and the following inequalities ||∇ · S ( ∇ U ) || L q (Ω) ≤ C (cid:16) ||∇ U || L q (Ω) || ( ∇ w ) || L ∞ (Γ) + ||∇ U || L ∞ (Ω) || ∆ w || L ∞ (Γ) (cid:17) , || U − U || L − (Ω) ≤ C || U − U || W , − (Ω) ≤ C ||∇ A || L ∞− (Ω) Z Ω J | S ( ∇ U − ∇ U ) | . The 4th term on RHS of (33): Z Q t ( J − J ) ∇ · S ( ∇ U )( U − U ) ≤ Z t || w − w || L ∞ (Γ) h || U − U || L − (Ω) ≤ C ( δ ) Z t h E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | . he 5th term on RHS of (33) - case α = 0 , γ > / Z Q t J ( r − r ) h U · ( ∇ − ∇ )( r γ − ) − ( w − w ) · ∇ ( r γ − ) − w · ( ∇ − ∇ )( r γ − ) i ≤ C Z t || r − r || L ∧ γ (Ω) h || ∆ w − ∆ w || L (Γ) || U · ∇ ( r γ − ) || L q (Ω) | {z } := h ∈ L (0 ,T ) + || ∂ t w − ∂w || L − (Γ) ||∇ ( r γ − ) || L q (Ω) | {z } := h ∈ L (0 ,T ) + || ∆ w − ∆ w || L (Γ) || w · ∇ ( r γ − ) || L q (Ω) | {z } := h ∈ L (0 ,T ) i ≤ C ( δ ) Z t ( h + h + h ) || r − r || L ∧ γ (Ω) + C Z t || ∆ w − ∆ w || L (Γ) + δ Z t || ∂ t w − ∂w || H ( 12 ) − (Γ) ≤ C ( δ ) Z t ( h + h + h + 1) E (cid:16) ( r , U , w ) | ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | , for any q > max { , γ γ − } , where we used || ∂ t w − ∂w || L − (Γ) ≤ C || ∂ t w − ∂w || W ( 12 ) − , − (Γ) ≤ C || U − U || W , − (Ω) ≤ C Z Ω J | S ( ∇ U − ∇ U ) | , (38)by imbedding and trace inequality. The 5th term on RHS of (33) - case α > , γ > / Z Q t J ( r − r ) h U · ( ∇ − ∇ )( r γ − ) − ( w − w ) · ∇ ( r γ − ) − w · ( ∇ − ∇ )( r γ − ) i ≤ C Z t || r − r || L ∧ γ (Ω) h || ∆ w − ∆ w || L (Γ) || U · ∇ ( r γ − ) || L q (Ω) | {z } := h ∈ L (0 ,T ) + || ∂ t w − ∂w || L ∞− (Γ) ||∇ ( r γ − ) || L q (Ω) | {z } := h ∈ L (0 ,T ) + || ∆ w − ∆ w || L (Γ) || w · ∇ ( r γ − ) || L q (Ω) | {z } := h ∈ L (0 ,T ) i ≤ C ( δ ) Z t ( h + h + h + 1) E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δα Z t || ∂ t ∇ w − ∂ t ∇ w || L (Γ) , for any q > The 6th term on RHS of (33): Z Q t Jr (cid:2) U · ( ∇ − ∇ ) U (cid:3) · ( U − U ) ≤ C Z t || r || L γ (Ω) || U || L ∞ (Ω) || ∆ w − ∆ w || L (Γ) ||∇ U || L ∞ (Ω) | {z } = h ∈ L (0 ,T ) || U − U || L − (Ω) ≤ C ( δ ) Z t h E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | . The 7th term on RHS of (33): Z Q t Jr (cid:2) ( U − U ) · ∇ U (cid:3) · ( U − U ) ≤ Z t h ||∇ w || L ∞ (Γ) ||∇ U || L ∞ (Ω) | {z } := h ∈ L (0 ,T ) Z Ω Jr | U − U | i ≤ C Z t h E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) . since ∇ w ∈ L (0 , T ; L ∞ (Γ)), by Theorem 2 . he 8th term on RHS of (33): Z Q t Jr ( ∇ − ∇ )( r γ − ) · ( U − U ) ≤ C Z t || r || L γ (Ω) ||∇ r || L p (Ω) | {z } := h ∈ L (0 ,T ) || ∆ w − ∆ w || L (Γ) || U − U || L − (Ω) ≤ C ( δ ) Z t h E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | . for any p > γ γ − . The 9th term on RHS of (33): Z Q t J ( r − r ) r γ − ( ∇ − ∇ ) · U ≤ C Z t ||∇ U || L ∞ (Ω) | {z } := h ∈ L (0 ,T ) || r − r || L ∧ γ (Ω) || ∆ w − ∆ w || L (Γ) ≤ C Z t h E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) . The 10th term on RHS of (33): Z Q t J (cid:2) r γ − γr γ − ( r − r ) − r γ (cid:3) ( ∇ · U ) ≤ Z t h ||∇ w || L ∞ (Γ) ||∇ U || L ∞ (Ω) | {z } := h ∈ L (0 ,T ) Z Ω J (cid:0) r γ − γr γ − ( r − r ) − r γ (cid:1)i ≤ C Z t h E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) . The 11th term on RHS of (33) - case α = 0, γ ∈ ( , ∇ U ∈ L ∞ ( Q T ), one can bound Z Q t Jr h w · ∇ U − w · ∇ U i · ( U − U ) Z Q t J ( r − r ) h w · ∇ U − w · ∇ U i · ( U − U ) + Z Q t Jr h w · ∇ U − w · ∇ U i · ( U − U ) ≤ C Z t || r − r || L γ (Ω) ||∇ U || L ∞ (Ω) h || ∂ t w ∇ w || L − (Γ) + || ∂ t w ∇ w || L ∞ (Γ) i| {z } := h ∈ L (0 ,T ) || U − U || L − (Ω) + C Z t || r || L ∞ (Ω) ||∇ U || L ∞ (Ω) h ||∇ w || L ∞− (Γ) || ∂ t w − ∂ t w || L (Γ) + || ∂ t w || L ∞ (Γ) || ∆ w − ∆ w || L (Γ) i || U − U || L − (Ω) ≤ C ( δ ) Z t h h || r − r || L γ (Ω) + || ∂ t w − ∂ t w || L (Γ) + || ∆ w − ∆ w || L (Γ) i + δ Z Q t J | S ( ∇ U − ∇ U ) | ≤ C ( δ ) Z t ( h + 1) E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | . he 11th term on RHS of (33) - case α = 0, γ > Z Q t Jr h w · ∇ U − w · ∇ U i · ( U − U ) ≤ C Z t ||√ r || L γ (Ω) ||∇ U || L ∞ (Ω) | {z } := h ∈ L (0 ,T ) h ||∇ w || L ∞− (Γ) || ∂ t w − ∂ t w || L − (Γ) + || ∂ t w || L ∞ (Γ) || ∆ w − ∆ w || L (Γ) i ||√ r ( U − U ) || L (Ω) ≤ C Z t || ∆ w − ∆ w || L (Γ) + C ( δ ) Z t h Z Ω Jr | U − U | + δ Z Q t J | S ( ∇ U − ∇ U ) | ≤ C ( δ ) Z t ( h + 1) E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | , by (38). The 11th term on RHS of (33) - case α > γ > / Z Q t Jr h w · ∇ U − w · ∇ U i · ( U − U ) ≤ C Z t ||√ r || L (Ω) ||∇ U || L ∞ (Ω) | {z } = h ∈ L (0 ,T ) h ||∇ w || L ∞− (Γ) || ∂ t w − ∂ t w || L ∞− (Γ) + || ∂ t w || L ∞ (Γ) || ∆ w − ∆ w || L (Γ) i ||√ r ( U − U ) || L (Ω) ≤ C Z t || ∆ w − ∆ w || L (Γ) + C ( δ ) Z t h Z Ω Jr | U − U | + δα Z t || ∂ t ∇ w − ∂ t ∇ w || L (Γ) ≤ C Z t ( h + 1) E (cid:16) ( r , U , w ) (cid:12)(cid:12)(cid:12) ( r , U , w ) (cid:17) + δα Z t || ∂ t ∇ w − ∂ t ∇ w || L (Γ) Combining previous inequalities, the proof is finished.
Here, we study the interaction between compressible viscous fluids and nonlinear plates. The goal is toprove that the main results of this paper can be extended to such interaction problems. We note thatthe existence of weak solutions for the following interaction problems were constructed in [38], while theexistence of strong solutions seem to be non-existent in the literature.
Here, the equation (1) is replaced by ∂ t w + ∆ w − α∂ t ∆ w + F ( w ) = − S w f fl · e , where F satisfies the following assumptions:(A1) Mapping F is locally Lipschitz from H − (cid:15) (Γ) into H − (Γ) for some (cid:15) >
0, i.e. ||F ( w ) − F ( w ) || H − (Γ) ≤ C R || w − w || H − (cid:15) (Γ) , (39)for a constant C R >
0, for any || w i || H (Γ) ≤ R ( i = 1 , F is locally Lipschitz from H (Γ) into H − a (Γ) for some 0 ≤ a < /
2, i.e. ||F ( w ) − F ( w ) || H − a (Γ) ≤ C R || w − w || H (Γ) , (40)for a constant C R >
0, for any || w i || H (Γ) ≤ R ( i = 1 , F ( w ) has a potential in H (Γ), i.e. there exists a Fréchet differentiable functional Π( w ) on H (Γ)such that Π ( w ) = F ( w ), and there are 0 < κ < / C ∗ ≥
0, such that the following inequalityholds, κ || ∆ w || L (Γ) + Π( w ) + C ∗ ≥ , for all w ∈ H (Γ) . (41)Moreover, Π is bounded on bounded sets in H (Γ). Remark 5.1. (1) This plate model is a generalization of Kirchhoff, von Karman and Berger plates (see[7, 8, 9]).(2) In [38], the condition ( A was not necessary for the existence proof and was thus removed. However,in the proof that follows, it is needed. Here, we give a sketch of the proofs of theorems 2 . . ρ , u , w )be a finite energy weak solution. In this case, the relative entropy stays the same, while in the relativeenergy inequality takes the form E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) ( t ) + Z t Z Ω w ( t ) S ( ∇ u − ∇ u ) : ( ∇ u − ∇ u ) + α Z t Z Γ | ∂ t ∇ w − ∂ t ∇ w | ≤ E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) (0) + Z t R (cid:16) ρ , u , w , ρ , u , w (cid:17) − Z t hF ( w ) , ∂ t w − ∂ t w i (42)It is easy to prove that that any finite energy weak solution satisfies this inequality. Next, to obtain theregularity estimates in Theorem 2 . − R Γ T hF ( w ) , D sh w i appears, which canbe controlled as − Z t hF ( w ) , D s − h D sh w i ≤ Z t h ||F ( w ) − F (0) || H − a (Γ) + ||F (0) || H − a (Γ) i || D s − h D sh w || H a (Γ) ≤ Z t C R ( || w || H (Γ) + 1) || w || H (Γ) ≤ C by ( A
2) and for any s < , where we have used the fact that the constant C R in (40) is uniform, since w , w are uniformly bounded in L ∞ (0 , T ; H (Γ)).Finally, to obtain the Theorem 2 . − R Γ T hF ( w ) −F ( w ) , ∂ t w − ∂ t w i on the right-hand side, which can be controlled as − Z t hF ( w ) − F ( w ) , ∂ t w − ∂ t w i ≤ Z t ||F ( w ) − F ( w ) || H − a (Γ) || ∂ t w − ∂ t w || H a (Γ) ≤ C Z t || w − w || H (Γ) || ∂ t w − ∂ t w || H a (Γ) ≤ C ( δ ) Z t E (cid:16)(cid:2) ( ρ , u , w ) | ( ρ , u , w ) (cid:3)(cid:17) + δ Z Q t J | S ( ∇ U − ∇ U ) | , by (38). If α >
0, then we can choose a ≤ − Z Γ T hF ( w ) − F ( w ) , ∂ t w − ∂ t w i ≤ Z t ||F ( w ) − F ( w ) || H − a (Γ) || ∂ t w − ∂ t w || H a (Γ) ≤ C Z t || w − w || H (Γ) || ∂ t w − ∂ t w || H (Γ) ≤ C ( δ ) Z t E (cid:16)(cid:2) ( ρ , u , w ) | ( ρ , u , w ) (cid:3)(cid:17) + αδ Z t || ∂ t w − ∂ t w || H (Γ) . .2 A thermoelastic semilinear plate Here, the equation (1) is replaced by a system ∂ t w + ∆ w − α∂ t ∆ w + F ( w ) = S w f fl · e ,∂ t θ − ∆ θ − ∂ t ∆ w = 0 , where θ : Γ → R is the temperature of the plate and F satisfies the assumptions ( A − ( A E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) ( t ) = 12 Z Ω w ( t ) ( ρ | u − u | )( t ) + 1 γ − Z Ω w ( t ) (cid:0) ρ γ − γρ γ − ( ρ − ρ ) − ρ γ (cid:1) ( t )+ 12 Z Γ | ∂ t w − ∂ t w | ( t ) + 12 Z Γ | ∆ w − ∆ w | ( t ) + 12 Z Γ | θ − θ | ( t )(43)while the relative energy inequality is replaced by E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) ( t ) + Z t Z Ω w ( t ) J S ( ∇ u − ∇ u ) : ( ∇ u − ∇ u )+ α Z t Z Γ | ∂ t ∇ w − ∂ t ∇ w | + Z t Z Γ |∇ θ − ∇ θ | ≤ E (cid:16)(cid:2) ( ρ , u , w ) | ( ρ , u , w ) (cid:3)(cid:17) (0) + Z t R (cid:16) ρ , u , w , ρ , u , w (cid:17) − Z t hF ( w ) , ∂ t w − ∂ t w i− Z t Z Γ ( θ − θ ) ∂ t θ − Z t Z Γ ( θ − θ )∆ θ . (44)The proofs of theorems 2 . . Here, the equation (1) is replaced by a system ∂ t w + ∆ w − α∂ t ∆ w + ∆(∆ w ) = S w f fl · e ,∂ t θ − ∆ θ − ∂ t ∆ w = 0 . Remark 5.2.
Such a thermoelastic plate model was first studied in [21] and later in [23]. This highorder nonlinearity comes from a thermoelastic plate model in which a nonlinear coupling between theelastic, magnetic and thermoelastic fields is considered.
In this case, the relative entropy takes the form E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) ( t ) = 12 Z Ω w ( t ) ( ρ | u − u | )( t ) + 1 γ − Z Ω w ( t ) (cid:0) ρ γ − γρ γ − ( ρ − ρ ) − ρ γ (cid:1) ( t )+ 12 Z Γ | ∂ t w − ∂ t w | ( t ) + 12 Z Γ | ∆ w − ∆ w | ( t ) + 12 Z Γ | θ − θ | ( t )+ Z Γ h (∆ w ) − (∆ w ) i (∆ w − ∆ w ) , (45) Note that in this case, there is an additional equation in the weak formulation. E (cid:16) ( ρ , u , w ) (cid:12)(cid:12)(cid:12) ( ρ , u , w ) (cid:17) ( t ) + Z t Z Ω w ( t ) J S ( ∇ u − ∇ u ) : ( ∇ u − ∇ u )+ α Z t Z Γ | ∂ t ∇ w − ∂ t ∇ w | + Z t Z Γ |∇ θ − ∇ θ | ≤ E (cid:16)(cid:2) ( ρ , u , w ) | ( ρ , u , w ) (cid:3)(cid:17) (0) + Z t R (cid:16) ρ , u , w , ρ , u , w (cid:17) − Z t Z Γ (∆ w ) (∆ w − ∆ w ) − Z t Z Γ ( θ − θ ) ∂ t θ − Z t Z Γ ( θ − θ )∆ θ . (46)In this case, in the proof of Theorem 2 .
3, when we choose ( q , ψ ) = ( R [ D s − h D sh w ] , D s − h D sh w ) in thecoupled momentum equation (11), the nonlinear term can be expressed as Z Γ T (∆ w ) D s − h D sh [∆ w ] = Z Γ T D sh (∆ w ) D sh ∆ w ≥ Z Γ T | D s h ∆ w | . This then gives us ∆ w ∈ L (0 , T ; W s , (Γ)) ∩ L (0 , T ; H s (Γ)), where s satisfies the same conditions as inTheorem 2 . Acknowledgments:
This research was partially carried out during the author’s PhD studies at ShanghaiJiao Tong University, where it was supported by the National Natural Science Foundation of China(NNSFC) under Grant No. 11631008.
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The author states that there is no conflict of interest.
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