Stabilization of a rigid body moving in a compressible viscous fluid
aa r X i v : . [ m a t h . A P ] O c t STABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLEVISCOUS FLUID
ARNAB ROY AND TAK ´EO TAKAHASHI
Universit´e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
Abstract.
We consider the stabilizability of a fluid-structure interaction system where the fluid isviscous and compressible and the structure is a rigid ball. The feedback control of the system acts onthe ball and corresponds to a force that would be produced by a spring and a damper connecting thecenter of the ball to a fixed point h . We prove the global-in-time existence of strong solutions for thecorresponding system under a smallness condition on the initial velocities and on the distance betweenthe initial position of the center of the ball and h . Then, we show with our feedback law, that thefluid and the structure velocities go to 0 and that the center of the ball goes to h as t → ∞ . Keywords.
Fluid-structure interaction, compressible Navier-Stokes system, global solutions, stabi-lization
AMS subject classifications.
Contents
1. Introduction and main result 1Notation 52. Local in time existence of solutions 62.1. Lagrangian change of variables 62.2. Analysis of a linear problem 82.3. Estimates of the nonlinear terms 102.4. Proof of Theorem 2.1 143. Global in time existence of solutions 153.1. A priori estimates 153.2. Proof of Theorem 1.1 224. Proof of Theorem 1.2 24References 271.
Introduction and main result
Let Ω ⊂ R be a bounded domain with C boundary occupied by a fluid and a rigid body. Wedenote by B ( t ) ⊂ Ω, the domain of the rigid body and we assume it is an open ball of radius 1 and ofcenter h ( t ), where t ∈ R + is the time variable. We suppose that the fluid domain F ( t ) = Ω \ B ( t ) isconnected. Date : October 17, 2019.
The fluid is modeled by the compressible Navier-Stokes system whereas the motion of the rigid bodyis governed by the balance equations for linear and angular momentum. We also assume the no-slipboundary conditions. The equations of motion of fluid-structure are: ∂ρ∂t + div( ρu ) = 0 t > , x ∈ F ( t ) , (1.1) ρ (cid:18) ∂u∂t + ( u · ∇ ) u (cid:19) − div σ ( u, p ) = 0 t > , x ∈ F ( t ) , (1.2) mℓ ′ = − Z ∂ B ( t ) σ ( u, p ) N d
Γ + w t > , (1.3) J ω ′ = − Z ∂ B ( t ) ( x − h ( t )) × σ ( u, p ) N d Γ t > , (1.4) h ′ = ℓ t > , (1.5) u ( t, x ) = 0 t > , x ∈ ∂ Ω , (1.6) u ( t, x ) = ℓ ( t ) + ω ( t ) × ( x − h ( t )) t > , x ∈ ∂ B ( t ) , (1.7) ρ (0 , · ) = ρ , u (0 , · ) = u in F (0) , (1.8) h (0) = h , ℓ (0) = ℓ , ω (0) = ω . (1.9)In the above equations, ρ = ρ ( t, x ) and u = u ( t, x ) represent respectively the density and the ve-locity of the fluid and the pressure of the fluid is denoted by p . We assume that the flow is in thebarotropic regime and we focus on the isentropic case where the relation between p and ρ is given bythe constitutive law: p = aρ γ , with a > γ > . The Cauchy stress tensor is defined as: σ ( u, p ) = 2 µ D ( u ) + λ div u I − p I , where D ( u ) = (cid:0) ∇ u + ∇ u ⊤ (cid:1) denotes the symmetric part of the velocity gradient ( ∇ u ⊤ is the transposeof the matrix ∇ u ) and λ, µ are the viscosity coefficients satisfying µ > , λ + µ > . Here ℓ and ω are the linear and angular velocities of the rigid body, N ( t, x ) is the unit normal to ∂ B ( t )at the point x ∈ ∂ B ( t ), directed to the interior of the ball and m , J are the mass and the moment ofinertia of the rigid ball respectively. The formulae for m and J are m = 43 πρ B , J = 2 m I , where ρ B > w (in (1.3)) is our control that we take as a feedback control: w ( t ) = k p ( t )( h − h ( t )) − k d ℓ ( t ) , (1.10)where k d > k p ( t ) > t →∞ h ( t ) = h , TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 3 whereas the velocities of the fluid and of the rigid ball go to 0:lim t →∞ u ( t ) = 0 , lim t →∞ ℓ ( t ) = 0 , lim t →∞ ω ( t ) = 0 . In literature, this type of control is known as Proportional-Derivative (PD) controller generated by aspring and a damper. The spring-damper is connected from the center of the ball to the fixed anchorpoint h and it is attracting the ball towards the point h .In order to give the precise statement of stabilization (Theorem 1.2), we first need a global in timeexistence result for (1.1)–(1.10) with (1.10). Such a result in the case without control is given in [1]by adapting a method introduced in [13].Here we will prove again this existence result, with the same approach but with a special attentionto the estimates on h ( t ) and with some modifications in the proof of [1] due to the feedback law (1.10).In order to state our result we introduce ρ the mean-value of ρ : ρ = 1 |F (0) | Z F (0) ρ ( x ) dx. (1.11)Note that, from equation (1.1) and Reynold’s Transport Theorem, we obtain Z F (0) ρ ( x ) dx = Z F ( t ) ρ ( t, x ) dx. For 0 T < T ∞ , we introduce the following space: b S T ,T = n ( ρ, u, ℓ, ω ) | ρ ∈ L ( T , T ; H ( F ( t ))) ∩ BC ([ T , T ]; H ( F ( t ))) ∩ H ( T , T ; H ( F ( t )) ∩ BC ([ T , T ]; H ( F ( t ))) ∩ H ( T , T ; L ( F ( t ))) ,u ∈ L ( T , T ; H ( F ( t ))) ∩ BC ([ T , T ]; H ( F ( t ))) ∩ H ( T , T ; H ( F ( t )) ∩ BC ([ T , T ]; H ( F ( t ))) ∩ H ( T , T ; L ( F ( t ))) ,ℓ ∈ H ( T , T ) , ω ∈ H ( T , T ) o . (1.12)Here BC k are the functions of class C k bounded with bounded derivatives. We set k ( ρ, u, ℓ, ω ) k b S T ,T = k ρ − ρ k L ∞ ( T ,T ; H ( F ( t ))) + k ρ − ρ k H ( T ,T ; H ( F ( t ))) + k ρ − ρ k W , ∞ ( T ,T ; H ( F ( t ))) + k ρ − ρ k H ( T ,T ; L ( F ( t ))) + k u k L ( T ,T ; H ( F ( t ))) + k u k L ∞ ( T ,T ; H ( F ( t ))) + k u k H ( T ,T ; H ( F ( t ))) + k u k W , ∞ ( T ,T ; H ( F ( t ))) + k u k H ( T ,T ; L ( F ( t ))) + k ℓ k H ( T ,T ) + k ℓ k W , ∞ ( T ,T ) + k ω k H ( T ,T ) + k ω k W , ∞ ( T ,T ) , (1.13)and for T > k ( ρ , u , ℓ , ω ) k b S T,T = k ρ − ρ k H ( F ( T )) + k u k H ( F ( T )) + | ℓ | + | ω | . Since we are working with regular solutions of (1.1)–(1.10), we need to introduce the following com-patibility conditions at initial time: u ( y ) = ℓ + ω × ( y − h ) for y ∈ ∂ B (0) , u = 0 on ∂ Ω , (1.14) − ρ div σ ( u , p ) = 0 on ∂ Ω , (1.15) STABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID − (cid:16) ω × ( ω × ( y − h )) (cid:17) − ρ div σ ( u , p )( y )= 1 m Z ∂ B (0) σ ( u , p ) n d Γ − k d ℓ + J − Z ∂ B (0) ( x − h ) × σ ( u , p ) n d Γ x × ( y − h )for y ∈ ∂ B (0) , (1.16)where p = aρ γ . Finally, we introduce the following notationΩ := { x ∈ Ω ; dist( x, ∂ Ω) > } . Our hypotheses on k p and k d are the following ones: k p ∈ C ( R + , [0 , , k p (0) = 0 , k p > , ∞ ) , k p ≡ T I , ∞ ) , k ′ p < k d T I (1.17)for some T I > Theorem 1.1.
Assume that Ω is non empty and connected. Let h ∈ Ω and ρ > . Assume w isgiven by the feedback law (1.10) with ( k p , k d ) satisfying (1.17) . There exists δ > such that for any h ∈ Ω , ρ ∈ H ( F (0)) , ρ > , u ∈ H ( F (0)) , ℓ , ω ∈ R , (1.18) satisfying the compatibility conditions (1.14) – (1.16) with k ( ρ , u , ℓ , ω ) k b S , + | h − h | δ, (1.19) the system (1.1) – (1.10) admits a unique strong solution ( ρ, u, ℓ, ω ) ∈ b S , ∞ , h ∈ L ∞ (0 , ∞ ) . Moreover,there exist C, η > such that k ( ρ, u, ℓ, ω ) k b S , ∞ + k p k p ( h − h ) k L ∞ (0 , ∞ ) C (cid:16) k ( ρ , u , ℓ , ω ) k b S , + | h − h | (cid:17) , (1.20)dist( h ( t ) , ∂ Ω) > η ( t > . (1.21)We are now in a position to state our stabilization result. Theorem 1.2.
With the notations and assumptions of Theorem 1.1, the solution ( ρ, u, h, ℓ, ω ) of (1.1) - (1.10) satisfies lim t →∞ k ρ ( t, · ) − ρ k H ( F ( t )) = 0 , lim t →∞ k u ( t, · ) k H ( F ( t )) = 0 , (1.22)lim t →∞ h ( t ) = h , lim t →∞ ℓ ( t ) = 0 , lim t →∞ ω ( t ) = 0 . (1.23)During the last two decades, there has been a considerable interest in fluid-structure interactionproblems involving moving interfaces. Broadly speaking, these types of models can be classified intotwo types: either the structure is moving inside the fluid or the structure is located at the boundaryof the fluid domain. Since in this article we are interested in studying the motion of body inside thecompressible fluid domain, below we mention related works from the literature concerning this caseonly.The global-in-time existence (up to contact) of weak solutions for compressible viscous flow (for γ >
2) in a bounded domain of R interacting with a finite number of rigid bodies has been studiedby Desjardins and Esteban [6]. In [9], Feireisl established the global existence result (for γ > / TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 5 boundary. Regarding strong solutions, the existence and uniqueness of global solutions for small initialdata have been achieved in [1] in the Hilbert space framework by Boulakia and Guerrero as long asno collisions occur. Their work is based on a method proposed in [13] for a viscous compressiblefluid (without structure). In a L p - L q setting, the authors in [12] proved the existence and uniquenessof local-in-time strong solutions for the system composed by rigid bodies immersed into a viscouscompressible fluid and in [11], the authors establish the global in time existence up to contact.Let us mention some works related to the large time behavior of fluid-structure interaction system.In [17], the authors analyze the fluid-structure model in one space dimension where the fluid is governedby the viscous Burgers equation and the solid mass is moving by the difference of pressure at bothsides of it. They obtain that the asymptotic profile of the fluid is a self-similar solution of the Burgersequation and the point mass enjoys the parabolic trajectory as t → ∞ . An extension of this work inseveral space dimensions is obtained in [14] for the heat equations in interaction with a rigid body.Their result is that as t → ∞ , the fluid solution behaves as the fundamental solution of the heatequation and the ball goes to infinity in bidimensional case whereas the ball remains in a boundeddomain in three dimension. Regarding the long-time behavior of a moving particle inside a Navier-Stokes fluid, the authors in [10] consider in particular the case of a ball falling over an horizontalplane and show that the velocity of the fluid goes to zero and the particle reaches the bottom of thecontainer asymptotically in time. In [7], the authors analyze the case of a rigid disk immersed into atwo-dimensional Navier-Stokes equations filling the exterior of the structure domain. They restrict tothe case of a solid and a fluid with the same density and for the linear case.Finally, let us mention two works using a control supported on the rigid body: [5] in the 1d case fora Burgers-particle system and [16] in the 3d case for a rigid ball moving into a viscous incompressiblefluid. The main difference between this study and the two previous references come from the fact thatin our case we need to deal with stronger solutions than in the incompressible case. In particular, toavoid compatibility conditions at t = 0 that involve the feedback control w , we take here k p dependingon time with k p (0) = 0.The plan of the paper is the following. In Section 2, we establish the local-in-time existence of solu-tions for the system (1.1)–(1.10). We then obtain a priori estimates in Section 3 to prove Theorem 1.1.Finally Section 4 is devoted to the asymptotic analysis of the solutions in order to prove Theorem 1.2. Notation.
For any a ∈ R , we set b B ( a ) = { x ∈ R | | x − a | < } , b F ( a ) = Ω \ b B ( a ) . In particular, B ( t ) = b B ( h ( t )) , F ( t ) = b F ( h ( t )) . In this article, to shorten the notation, we write H m and L instead of H m ( F (0)) and L ( F (0)).Assume X is Banach space. We need to consider a particular norm for H m (0 , T ; X ) if m ∈ N ∗ andif T ∈ R ∗ + . k f k H m ∞ (0 ,T ; X ) = k f k H m (0 ,T ; X ) + k f k W m − , ∞ (0 ,T ; X ) . (1.24)Using the Sobolev embedding, this norm is equivalent to the usual one, but the corresponding constantsdepend on T and that is the reason why we introduce such a notation.Assume X and X are Banach spaces. We also introduce the following spaces H m (0 , T ; X , X ) = L (0 , T ; X ) ∩ H m (0 , T ; X ) ( m > . In the case T ∈ R ∗ + , we also need to introduce the following norm for the above space: k f k H ∞ (0 ,T ; H ,L ) = k f k L (0 ,T ; H ) + k f k L ∞ (0 ,T ; H ) + k f k H (0 ,T ; L ) , (1.25) STABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID k f k H ∞ (0 ,T ; H ,L ) = k f k L (0 ,T ; H ) + k f k L ∞ (0 ,T ; H ) + k f k H (0 ,T ; H ) + k f k W , ∞ (0 ,T ; H ) + k f k H (0 ,T ; L ) . (1.26)Using interpolation results, we see again that the corresponding norm is equivalent to H (0 , T ; H )but the corresponding constants depend on T .2. Local in time existence of solutions
In order to prove Theorem 1.1, we first prove the existence and uniqueness of strong solutions ofsystem (1.1)-(1.10) for small times. More precisely, we show in this section the following result:
Theorem 2.1.
Let h ∈ Ω and ρ > . Assume w is given by the feedback law (1.10) with k d ∈ R and k p ∈ H loc ([0 , ∞ )) . There exist δ , C ∗ , T ∗ > such that for any h ∈ Ω , ρ ∈ H , u ∈ H , ℓ , ω ∈ R , (2.1) satisfying the compatibility conditions (1.14) – (1.16) with k ( ρ , u , ℓ , ω ) k b S , + | h − h | δ , (2.2) the system (1.1) - (1.9) admits a unique strong solution ( ρ, u, ℓ, ω ) ∈ b S ,T ∗ , h ∈ L ∞ (0 , T ∗ ) and k ( ρ, u, ℓ, ω ) k b S ,T ∗ + k h − h k L ∞ (0 ,T ∗ ) C ∗ (cid:16) k ( ρ , u , ℓ , ω ) k b S , + | h − h | (cid:17) . (2.3)2.1. Lagrangian change of variables.
Firstly, we use a Lagrangian change of variables to rewritethe system (1.1)–(1.10) in a fixed spatial domain: let introduce the flow X ( t, · ) : F (0) → F ( t ) definedby ∂X∂t ( t, y ) = u ( t, X ( t, y )) ,X (0 , y ) = y. Due to the boundary conditions, we have X ( t, y ) = ( h ( t ) + Q ( t )( y − h ) if y ∈ ∂ B (0) ,y if y ∈ ∂ Ω , where Q ( t ) ∈ SO (3) is the rotation matrix associated to the angular velocity ω : Q ′ = A ( ω ) Q, Q (0) = I . For any ω ∈ R , A ( ω ) is the skew-symmetric matrix: A ( ω ) = − ω ω ω − ω − ω ω . If u is regular enough, X is well-defined and X ( t, · ) is a C -diffeomorphism from F (0) onto F ( t ) for all t ∈ (0 , T ). We denote by Y ( t, · ) the inverse of X ( t, · ) and we consider the following change of variables e u ( t, y ) = Q ( t ) ⊤ u ( t, X ( t, y )) , e ρ ( t, y ) = ρ ( t, X ( t, y )) − ρ, (2.4) e h ( t ) = h ( t ) − h , e ℓ ( t ) = Q ( t ) ⊤ ℓ ( t ) , e ω ( t ) = Q ( t ) ⊤ ω ( t ) . (2.5)Note that now we have X ( t, y ) = y + t Z Q ( s ) e u ( s, y ) ds, ∀ y ∈ F (0) . (2.6) TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 7
Under the change of variables (2.4)-(2.5), the system (1.1)-(1.9) is transformed as follows: ∂ e ρ∂t + ρ div e u = F ( e ρ, e u, e ℓ, e ω, Q ) in (0 , T ) × F (0) , (2.7) ∂ e u∂t − µρ ∆ e u − λ + µρ ∇ (div e u ) = F ( e ρ, e u, e ℓ, e ω, Q ) in (0 , T ) × F (0) , (2.8) m e ℓ ′ = F ( e ρ, e u, e h, e ℓ, e ω, Q ) in (0 , T ) , (2.9) J e ω ′ = F ( e ρ, e u, e ℓ, e ω, Q ) in (0 , T ) , (2.10) e h ′ = Q e ℓ, Q ′ = Q A ( e ω ) in (0 , T ) , (2.11) e u = e ℓ + e ω × ( y − h ) on (0 , T ) × ∂ B (0) , (2.12) e u = 0 in (0 , T ) × ∂ Ω , (2.13) e ρ (0 , · ) = ρ ( · ) − ρ, e u (0 , · ) = u ( · ) , in F (0) , (2.14) e h (0) = h − h , e ℓ (0) = ℓ , e ω (0) = ω , Q (0) = I . (2.15)In the above equations, F , F , F , F are defined in the following way: F ( e ρ, e u, e ℓ, e ω, Q ) = − ( e ρ + ρ ) ∇ e u : h (( ∇ Y ( X )) Q ) ⊤ − I i − ( e ρ + ρ − ρ ) div e u, (2.16)for i = 1 , , F ) i ( e ρ, e u, e ℓ, e ω, Q ) = − ( e ω × e u ) i + µ e ρ + ρ X p,l,m ∂ e u i ∂y m ∂y l (cid:18) ∂Y m ∂x p ( X ) ∂Y l ∂x p ( X ) − δ mp δ lp (cid:19) + µ e ρ + ρ X p,l ∂ e u i ∂y l ∂ Y l ∂x p ( X ) + µ ∆ e u i (cid:18) ρ − ( e ρ + ρ ) ρ ( e ρ + ρ ) (cid:19) + λ + µ e ρ + ρ X p,l ∂ e u p ∂y l ∂ Y l ∂x p ∂x i ( X )+ λ + µ e ρ + ρ X p,l,m ∂ e u p ∂y m ∂y l (cid:18) ∂Y m ∂x p ( X ) − δ mp (cid:19) ∂Y l ∂x i ( X ) + λ + µ e ρ + ρ X p,l ∂ e u p ∂y p ∂y l (cid:18) ∂Y l ∂x i ( X ) − δ li (cid:19) + ( λ + µ )[ ∇ (div e u )] i (cid:18) ρ − ( e ρ + ρ ) ρ ( e ρ + ρ ) (cid:19) + aγ ( e ρ + ρ ) γ − X j,l Q ji ∂ e ρ∂y l ∂Y l ∂x j ( X ) , (2.17) F ( e ρ, e u, e h, e ℓ, e ω, Q ) = − m ( e ω × e ℓ ) − Z ∂ B (0) h µ (cid:16) Q ∇ e u ( ∇ Y ( X )) + ( Q ∇ e u ( ∇ Y ( X ))) ⊤ (cid:17) + λ ( Q ∇ e u ( ∇ Y ( X )) : I ) − a ( ρ + e ρ ) γ i n d Γ − k p Q ⊤ e h − k d e ℓ, (2.18) F ( e ρ, e u, e ℓ, e ω, Q ) = − Z ∂ B (0) ( y − h ) × h µ (cid:16) Q ∇ e u ( ∇ Y ( X )) + ( Q ∇ e u ( ∇ Y ( X ))) ⊤ (cid:17) + λ ( Q ∇ e u ( ∇ Y ( X )) : Id) − a ( ρ + e ρ ) γ i n d Γ . (2.19)Here n ( y ) = Q ( t ) ⊤ N ( t, x ) is the unit normal to ∂ B (0) at the point y ∈ ∂ B (0), directed to the interiorof the ball. STABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID
Analysis of a linear problem.
In this section, we want to study the existence and regularityof the solution of the following linear system: ∂ e ρ∂t + ρ div e u = f in (0 , T ) × F (0) , (2.20) ∂ e u∂t − µρ ∆ e u − λ + µρ ∇ (div e u ) = f in (0 , T ) × F (0) , (2.21) m e ℓ ′ = f in (0 , T ) , (2.22) J e ω ′ = f in (0 , T ) , (2.23) e u = e ℓ + e ω × ( y − h ) on (0 , T ) × ∂ B (0) , (2.24) e u = 0 on (0 , T ) × ∂ Ω , (2.25) e u (0 , · ) = u ( · ) in F (0) , (2.26) e ρ (0 , · ) = e ρ in F (0) , (2.27) e ℓ (0) = ℓ , e ω (0) = ω . (2.28)We introduce the following set for T > S T = n ( e ρ, e u, e ℓ, e ω ) | e ρ ∈ H (0 , T ; H ) ∩ C ([0 , T ]; H ) ∩ H (0 , T ; L ) , e u ∈ H (0 , T ; H , L ) , e ℓ ∈ H (0 , T ) , e ω ∈ H (0 , T ) , e u = 0 on ∂ Ω , e u = e ℓ + e ω × ( y − h ) on ∂ B (0) , e ρ (0) = e ρ , e u (0) = u , e ℓ (0) = ℓ , e ω (0) = ω o , (2.29)equipped with the norm k ( e ρ, e u, e ℓ, e ω ) k S T := k e ρ k H ∞ (0 ,T ; H ) + k e ρ k W , ∞ (0 ,T ; H ) + k e ρ k H (0 ,T ; L ) + k e u k H ∞ (0 ,T ; H ,L ) + k e ℓ k H ∞ (0 ,T ) + k e ω k H ∞ (0 ,T ) . We recall that the norms k · k H ∞ (0 ,T ; H ) , k · k H ∞ (0 ,T ) are defined in (1.24) and k · k H ∞ (0 ,T ; H ,L ) isdefined in (1.26). The space S T is similar to b S T ,T defined by (1.12) except that here F ( t ) is replacedby F (0) and we add the boundary and initial conditions.Since ρ >
0, there exists δ > ρ > ρ > . In that case, the system (2.20)–(2.28) is well-posed:
Proposition 2.2.
Let us assume ρ > , (2.2) with δ as above and ( e ρ , u , ℓ , ω ) ∈ H × H × R × R , f ∈ L (0 , T ; H ) ∩ C ([0 , T ]; H ) ∩ H (0 , T ; L ) ,f ∈ H (0 , T ; H , L ) , f ∈ H (0 , T ) , f ∈ H (0 , T ) with u = ℓ + ω × ( y − h ) for y ∈ ∂ B (0) , u = 0 on ∂ Ω , (2.30) f (0) + µρ ∆ u + λ + µρ ∇ (div u ) = 0 on ∂ Ω , (2.31) f (0) + µρ ∆ u + λ + µρ ∇ (div u ) = m − f (0) + J − f (0) × ( y − h ) for y ∈ ∂ B (0) . (2.32) TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 9
Then the system (2.20) – (2.28) admits a unique solution ( e ρ, e u, e ℓ, e ω ) ∈ S T . Moreover, there exists C L > (nondecreasing with respect to T ) such that k ( e ρ, e u, e ℓ, e ω ) k S T C L (cid:16) k f k L (0 ,T ; H ) + k f k L ∞ (0 ,T ; H ) + k f k H (0 ,T ; L ) + k f k H ∞ (0 ,T ; L ,H ) + k f k H ∞ (0 ,T ) + k f k H ∞ (0 ,T ) + k e ρ k H + k u k H + | ℓ | + | ω | (cid:17) . (2.33) Proof.
We solve (2.20)-(2.28) like a cascade system: first, (2.22)-(2.23) admits a unique solution ( e ℓ, e ω )with k e ℓ k H ∞ (0 ,T ) + k e ω k H ∞ (0 ,T ) C (cid:16) k f k H ∞ (0 ,T ) + k f k H ∞ (0 ,T ) + | ℓ | + | ω | (cid:17) . (2.34)Next, we solve equation (2.21) with the boundary and initial conditions (2.24)-(2.26). First weconsider a lifting operator R , such that for any a, b ∈ R , R ( a, b ) ∈ C ∞ ( R ) satisfies R ( a, b ) = ( a + b × ( y − h ) on ∂ B (0) , ∂ Ω . Then e v = e u − R ( e ℓ, e ω ) satisfies ∂ e v∂t − µρ ∆ e v − λ + µρ ∇ (div e v ) = F = f + µρ ∆ R ( e ℓ, e ω ) + λ + µρ ∇ (cid:16) div R ( e ℓ, e ω ) (cid:17) − R ( e ℓ ′ , e ω ′ ) , e v = 0 on (0 , T ) × ∂ F (0) , e v (0 , · ) = e v = u − R ( ℓ , ω ) in F (0) . By using a standard Galerkin method (see [8, Chapter 7, Theorem 1, p.354]) and by using the regularityresult of Lam´e operator (see, for instance, [4, Theorem 6.3-6, p.296]), under the condition that ∂ F (0)is of class C , we can show the following result: if F ∈ L (0 , T ; H ) ∩ H (0 , T ; L ) , e v ∈ H ∩ H , with the condition F (0 , · ) + µρ ∆ e v + λ + µρ ∇ (div e v ) = 0 on ∂ F (0) , (2.35)then there exists a unique solution e v ∈ H (0 , T ; H , L ) with the estimate k e v k H ∞ (0 ,T ; H ,L ) C (cid:16) k F k H ∞ (0 ,T ; L ,H ) + k e v (0) k H (cid:17) . We note that condition (2.35) is equivalent to (2.31) and (2.32). We can use the relation e u = e v + R ( e ℓ, e ω )and the above estimate of e v to deduce the following estimate of e u : k e u k H ∞ (0 ,T ; H ,L ) C (cid:16) k f k H ∞ (0 ,T ; L ,H ) + k f k H ∞ (0 ,T ) + k f k H ∞ (0 ,T ) + k u k H + | ℓ | + | ω | (cid:17) . (2.36)Now, with the help of equation (2.20) satisfied by e ρ , we obtain k e ρ k H ∞ (0 ,T ; H ) + k e ρ k W , ∞ (0 ,T ; H ) + k e ρ k H (0 ,T ; L ) C (cid:16) k f k L (0 ,T ; H ) + k f k L ∞ (0 ,T ; H ) + k f k H (0 ,T ; L ) + k e u k L (0 ,T ; H ) + k e u k L ∞ (0 ,T ; H ) + k e u k H (0 ,T ; H ) + k e ρ k H (cid:17) . (2.37)Thus, we have proved the existence of solution in appropriate space for the system (2.20)-(2.28).Thanks to (2.34), (2.36) and (2.37), we have also obtained our required estimate (2.33). (cid:3) Estimates of the nonlinear terms.
For
T >
R >
0, we define the following subset of S T : S T,R = n ( e ρ, e u, e ℓ, e ω ) ∈ S T | k ( e ρ, e u, e ℓ, e ω ) k S T R o . (2.38)In what follows, R is fixed and the constants that appear can depend on R .Assume ( e ρ, e u, e ℓ, e ω ) ∈ S T,R . Then there exists a unique solution ( e h, Q ) ∈ H (0 , T ) of the followingequations e h ′ = Q e ℓ in (0 , T ) ,Q ′ = Q A ( e ω ) in (0 , T ) ,Q (0) = I , e h (0) = h − h , (2.39)and we can then define X by (2.6). From (2.38), there exists C = C ( R ) > k Q k H (0 ,T ) C, k Q − I k L ∞ (0 ,T ) CT, k e h k L ∞ (0 ,T ) | h − h | + CT / . (2.40)In particular, taking δ small enough in (2.2), there exists T = T ( R, δ , dist( h , ∂ Ω)) > c > b B ( e h ( t ) + h ) , ∂ Ω) > c > ∀ t ∈ [0 , T ] . (2.41)From now on, we assume T T and the constants may depend on T .Combining (2.6) and (2.38), we also deduce k∇ X − I k L ∞ (0 ,T ; H ) CT / . (2.42)In particular, using the embedding H ( F (0)) ֒ → W , ∞ ( F (0)) and (2.41), there exists T T suchthat X : F (0) → b F ( e h ( t ) + h ) is invertible and its inverse is denoted by Y .In the same spirit, using the initial condition on e ρ (see (2.29)), we have k e ρ + ρ − ρ k L ∞ (0 ,T ; H ) T / R. (2.43)Using the embedding H ( F (0)) ֒ → L ∞ ( F (0)) and (2.2) with δ small enough, there exists T T such that ρ e ρ + ρ ρ . (2.44)In particular, combining this with (2.38), for any α ∈ R , k ( e ρ + ρ ) α k L ∞ (0 ,T ; H ) C, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ∂ B (0) ( e ρ + ρ ) γ nd Γ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H (0 ,T ) CT / . (2.45)From the above construction and assuming T T , we can define the terms F , F , F , F by(2.16)-(2.19). To estimate these terms, we first give some estimates of X and Y : Lemma 2.3.
Assume ( e ρ, e u, e ℓ, e ω ) ∈ S T,R . There exists a positive constant C depending only on R , F (0) such that, for all < T T , k∇ Y ( X ) − I k L ∞ (0 ,T ; H ) CT / , (2.46) (cid:13)(cid:13)(cid:13)(cid:13) ∂ Y l ∂x p ∂x i ( X ) (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂∂t ( ∇ Y ( X )) (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂∂t (cid:18) ∂ Y l ∂x p ∂x i ( X ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) C. (2.47) TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 11
Proof.
From (2.42) and the fact that L ∞ (0 , T ; H ) is an algebra, we deduce (2.46). This yields inparticular that k∇ Y ( X ) k L ∞ (0 ,T ; H ) C. (2.48)Writing ∂∂y m (cid:18) ∂Y l ∂x i ( X ) (cid:19) = X p ∂ Y l ∂x p ∂x i ( X ) ∂X p ∂y m and using (2.48), we deduce the estimate on ∂ Y l ∂x p ∂x i ( X ).From the expression (2.6), we have ∂∂t ( ∇ X ( t, · )) = Q ( t ) ∇ e u ( t, · ), and using ∂∂t ( ∇ Y ( X )) = −∇ Y ( X ) ∂∂t ( ∇ X ) ∇ Y ( X ) , we obtain the estimate of the second term in (2.47).Finally, we write ∂∂y m (cid:20) ∂∂t (cid:18) ∂Y l ∂x i ( X ) (cid:19)(cid:21) = X p ∂∂t (cid:18) ∂ Y l ∂x p ∂x i ( X ) (cid:19) ∂X p ∂y m + X p ∂ Y l ∂x p ∂x i ( X ) ∂∂t (cid:18) ∂X p ∂y m (cid:19) and from the previous estimate, we have (cid:13)(cid:13)(cid:13)(cid:13) ∂∂y m (cid:20) ∂∂t (cid:18) ∂Y l ∂x i ( X ) (cid:19)(cid:21)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X p ∂ Y l ∂x p ∂x i ( X ) ∂∂t (cid:18) ∂X p ∂y m (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) C. Thus, using (2.48), we deduce the estimate of the last term in (2.47). (cid:3)
Next we give some properties on F , F , F , F . Proposition 2.4.
There exist α > and a positive constant C depending on R , k p , k d , ρ and theother physical parameters, and on F (0) such that, for all < T T , for all ( e ρ, e u, e ℓ, e ω ) , ( e ρ , e u , e ℓ , e ω ) , ( e ρ , e u , e ℓ , e ω ) ∈ S T,R , k F ( e ρ, e u, e ℓ, e ω, Q ) k L (0 ,T ; H ) ∩ L ∞ (0 ,T ; H ) ∩ H (0 ,T ; L ) CT α , k F ( e ρ, e u, e ℓ, e ω, Q ) k H ∞ (0 ,T ; L ,H ) C (cid:16) T α + k ω × u k H + k aγρ γ − ∇ ρ k H (cid:17) , k F ( e ρ, e u, e h, e ℓ, e ω, Q ) k H ∞ (0 ,T ) C ( T α + | ω × ℓ | + | ℓ | + k ρ − ρ k H + k u k H ) , k F ( e ρ, e u, e ℓ, e ω, Q ) k H ∞ (0 ,T ) CT α , and k F ( e ρ , e u , e ℓ , e ω , Q ) − F ( e ρ , e u , e ℓ , e ω , Q ) k L (0 ,T ; H ) ∩ L ∞ (0 ,T ; H ) ∩ H (0 ,T ; L ) CT α k ( e ρ , e u , e ℓ , e ω ) − ( e ρ , e u , e ℓ , e ω ) k S T , k F ( e ρ , e u , e ℓ , e ω , Q ) − F ( e ρ , e u , e ℓ , e ω , Q ) k H ∞ (0 ,T ; L ,H ) CT α k ( e ρ , e u , e ℓ , e ω ) − ( e ρ , e u , e ℓ , e ω ) k S T , k F ( e ρ , e u , e h , e ℓ , e ω , Q ) − F ( e ρ , e u , e h , e ℓ , e ω , Q ) k H ∞ (0 ,T ) CT α k ( e ρ , e u , e ℓ , e ω ) − ( e ρ , e u , e ℓ , e ω ) k S T , k F ( e ρ , e u , e ℓ , e ω , Q ) − F ( e ρ , e u , e ℓ , e ω , Q ) k H ∞ (0 ,T ) CT α k ( e ρ , e u , e ℓ , e ω ) − ( e ρ , e u , e ℓ , e ω ) k S T . where Q, Q , Q , e h, e h , e h ∈ H (0 , T ) are given by (2.39) . Proof.
Using the definition (2.16) of F , (2.43), (2.29), (2.38), (2.46) we have the following estimates k F k L (0 ,T ; H ) C k ( e ρ + ρ ) k L ∞ (0 ,T ; H ) k∇ e u k L (0 ,T ; H ) k (( ∇ Y ) Q ) ⊤ − I k L ∞ (0 ,T ; H ) + C k ( e ρ + ρ − ρ ) k L ∞ (0 ,T ; H ) k div e u k L (0 ,T ; H ) CT α , (cid:13)(cid:13)(cid:13)(cid:13) ∂F ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; L ) C k ( e ρ + ρ ) k L ∞ (0 ,T ; H ) ((cid:13)(cid:13)(cid:13)(cid:13) ∇ ∂ e u∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; L ) k (( ∇ Y ) Q ) ⊤ − I k L ∞ (0 ,T ; H ) + k∇ e u k L (0 ,T ; H ) (cid:13)(cid:13)(cid:13)(cid:13) ∂∂t (( ∇ Y ( X )) Q ) ⊤ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) ) + CT / (cid:13)(cid:13)(cid:13)(cid:13) ∂ e ρ∂t (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) k∇ e u k L ∞ (0 ,T ; H ) (cid:13)(cid:13)(cid:13) (( ∇ Y ( X )) Q ) ⊤ (cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) + C k e ρ + ρ − ρ k L ∞ (0 ,T ; H ) (cid:13)(cid:13)(cid:13)(cid:13) div ∂ e u∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H ) CT α , (2.49) k F k L ∞ (0 ,T ; H ) C k e ρ + ρ k L ∞ (0 ,T ; H ) k∇ e u k L ∞ (0 ,T ; H ) k (( ∇ Y ) Q ) ⊤ − I k L ∞ (0 ,T ; H ) + C k e ρ + ρ − ρ k L ∞ (0 ,T ; H ) k div e u k L ∞ (0 ,T ; H ) CT / . Let us now estimate the L (0 , T ; H ) norm of F . Here we only estimate some terms in (2.17), theother terms can be estimated similarly. Using (2.45), (2.29), (2.38), (2.46), (2.47), (cid:13)(cid:13)(cid:13)(cid:13) e ρ + ρ ∂ e u i ∂y m ∂y l (cid:18) ∂Y m ∂x p ( X ) ∂Y l ∂x p ( X ) − δ mp δ lp (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H ) C (cid:13)(cid:13)(cid:13)(cid:13) e ρ + ρ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) (cid:13)(cid:13)(cid:13)(cid:13) ∂ e u i ∂y m ∂y l (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H ) (cid:13)(cid:13)(cid:13)(cid:13) ∂Y m ∂x p ( X ) ∂Y l ∂x p ( X ) − δ mp δ lp (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) CT α , (cid:13)(cid:13)(cid:13)(cid:13) e ρ + ρ ∂ e u i ∂y l ∂ Y l ∂x p ( X ) (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H ) CT / (cid:13)(cid:13)(cid:13)(cid:13) e ρ + ρ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) (cid:13)(cid:13)(cid:13)(cid:13) ∂ e u i ∂y l (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) (cid:13)(cid:13)(cid:13)(cid:13) ∂ Y l ∂x p ( X ) (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) CT α , (cid:13)(cid:13)(cid:13)(cid:13) ( e ρ + ρ ) γ − ∂ e ρ∂y l ∂Y l ∂x j ( X ) (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; H ) CT / k ( e ρ + ρ ) γ − k L ∞ (0 ,T ; H ) (cid:13)(cid:13)(cid:13)(cid:13) ∂ e ρ∂y l (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) (cid:13)(cid:13)(cid:13)(cid:13) ∂Y l ∂x j ( X ) (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ) CT α . For the estimate of the H (0 , T ; L ) norm of F , we also only give the estimates the L (0 , T ; L )norm of some terms of the time derivative F . Again, the other terms can be estimated similarly. TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 13
First, we write ∂∂t (cid:20) e ρ + ρ ∂ e u i ∂y m ∂y l (cid:18) ∂Y m ∂x p ( X ) ∂Y l ∂x p ( X ) − δ mp δ lp (cid:19)(cid:21) = − e ρ + ρ ) ∂ e ρ∂t ∂ e u i ∂y m ∂y l (cid:18) ∂Y m ∂x p ( X ) ∂Y l ∂x p ( X ) − δ mp δ lp (cid:19) + (cid:18) e ρ + ρ (cid:19) ∂ e u i ∂t∂y m ∂y l ∂Y m ∂x p ( X ) ∂Y l ∂x p ( X ) − δ mp δ lp ! + (cid:18) e ρ + ρ (cid:19) ∂ e u i ∂y m ∂y l ∂∂t (cid:18) ∂Y m ∂x p ( X ) ∂Y l ∂x p ( X ) (cid:19) ,∂∂t (cid:20) e ρ + ρ ∂ e u i ∂y l ∂ Y l ∂x p ( X ) (cid:21) = − e ρ + ρ ) ∂ e ρ∂t ∂ e u i ∂y l ∂ Y l ∂x p ( X ) + 1 e ρ + ρ ∂ e u i ∂t∂y l ∂ Y l ∂x p ( X )+ 1 e ρ + ρ ∂ e u i ∂y l ∂∂t (cid:18) ∂ Y l ∂x p ( X ) (cid:19) . Using (2.45), (2.29), (2.38), (2.46), (2.47), we deduce that the above terms is estimated in L (0 , T ; L )by CT α .Finally, to obtain the L ∞ (0 , T ; H ) estimate of the term F , we use the following inequality [15,Lemma 4.2]: sup t ∈ (0 ,T ) k F ( t ) k H C (cid:16) k F k L (0 ,T ; H ) + k F k H (0 ,T ; L ) + k F (0) k H (cid:17) , and since k F (0) k H k ω × u k H + k aγρ γ − ∇ ρ k H , we deduce the result for F .It remains to estimate F and F . We only consider F , the analysis for F is the same. From (2.18),we can see that the time derivative of F involves the following terms (and similar ones)( e ω × e ℓ ) ′ , ( k p Q ⊤ e h ) ′ , k d e ℓ ′ , Z ∂ B (0) Q ′ ∇ e u ∇ Y ( X ) + Q ∇ ∂ e u∂t ∇ Y ( X ) + Q ∇ e u ∂∂t ∇ Y ( X ) ! n d Γ − aγ Z ∂ B (0) ( ρ + e ρ ) γ − ∂ e ρ∂t n d Γ . Almost all the terms can be estimated in a direct way in L (0 , T ) by using (2.40), (2.45), (2.29), (2.38),(2.46). We have nevertheless to take care of Z ∂ B (0) Q ∇ ∂ e u∂t ∇ Y ( X ) n d Γ . For this term, we use standard interpolation result (see, for instance, [2, Lemma A.5]) to obtain (cid:13)(cid:13)(cid:13)(cid:13) ∇ ∂ e u∂t (cid:13)(cid:13)(cid:13)(cid:13) L / (0 ,T ; H / ) C (cid:13)(cid:13)(cid:13)(cid:13) ∇ ∂ e u∂t (cid:13)(cid:13)(cid:13)(cid:13) / L ∞ (0 ,T ; L ) (cid:13)(cid:13)(cid:13)(cid:13) ∇ ∂ e u∂t (cid:13)(cid:13)(cid:13)(cid:13) / L (0 ,T ; H ) , where C is independent of T . Using a trace result and (2.29), (2.38), we deduce an estimate of F ′ in L (0 , T ) of the form CT α . To end the estimate of F , we use that k F k L ∞ (0 ,T ) | F (0) | + T / k F k H (0 ,T ) . We have the following estimate: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ∂ B (0) ( ρ + e ρ (0)) γ n d Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ∂ B (0) ρ γ n d Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z ∂ B (0) ( ρ γ − ρ γ ) n d Γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C Z ∂ B (0) | ρ − ρ | d Γ . Thus, | F (0) | C ( | ω × ℓ | + | ℓ | + k ρ − ρ k H + k u k H ) . The estimates for the differences can be done in a similar way and we thus skip the correspondingproof. (cid:3)
Proof of Theorem 2.1.
Proof.
We are going to establish the local in time existence of (2.7)-(2.19). In order to do this we usea fixed-point argument.Assume ρ > δ satisfying the smallness assumptions introduced in the above section and let usconsider ( ρ , u , h , ℓ , ω ) satisfying (2.1), (2.2). Recall that from (2.44), we have ρ ρ ρ C > ρ, δ and the geometry such that C (cid:16) k ω × u k H + k aγρ γ − ∇ ρ k H + | ω × ℓ | + | ℓ | + k ρ − ρ k H + k u k H (cid:17) C e δ (2.50)where C is the constant appearing in Proposition 2.4 and where we have set e δ = k ρ − ρ k H + k u k H + | h − h | + | ℓ | + | ω | δ . We now fix
R > R = 2 C L C e δ , (2.51)where C L is the continuity constant in estimate (2.33). We take T T , where T = T ( R ) is the timeobtained in the above section.Let us define the following mapping N : S T,R → S
T,R (2.52)( e ρ, e u, e ℓ, e ω ) ( b ρ, b u, b ℓ, b ω ) . (2.53)For ( e ρ, e u, e ℓ, e ω ) ∈ S T,R , we define X by (2.6), e h and Q by (2.39) and F , F , F , F by (2.16)-(2.19).Then ( b ρ, b u, b ℓ, b ω ) is the solution of ∂ b ρ∂t + ρ div b u = F ( e ρ, e u, e ℓ, e ω, Q ) in (0 , T ) × F (0) , (2.54) ∂ b u∂t − µρ ∆ b u − λ + µρ ∇ (div b u ) = F ( e ρ, e u, e ℓ, e ω, Q ) in (0 , T ) × F (0) , (2.55) m b ℓ ′ = F ( e ρ, e u, e h, e ℓ, e ω, Q ) in (0 , T ) , (2.56) J b ω ′ = F ( e ρ, e u, e ℓ, e ω, Q ) in (0 , T ) , (2.57) b u = b ℓ + b ω × ( y − h ) on (0 , T ) × ∂ B (0) , (2.58) b u = 0 in (0 , T ) × ∂ Ω . (2.59) b ρ (0 , · ) = ρ ( · ) − ρ, b u (0 , · ) = u ( · ) in F (0) , (2.60) b ℓ (0) = ℓ , b ω (0) = ω . (2.61) TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 15
In order to show that N is well-defined, we apply Proposition 2.2 to the above system. First we notethat (1.14)–(1.16) yield the compatibility conditions (2.30)–(2.32). More precisely, the first conditionis exactly condition (1.14). Using the expression of F in (2.17), we have h F ( e ρ, e u, e ℓ, e ω, Q ) i (0 , · ) = − ω × u + 1 ρ ∇ p , where p = aρ γ . Thus, (1.15) yields the second condition.On the other hand, using the expressions of F and F in (2.18) and (2.19), we have h F ( e ρ, e u, e ℓ, e ω, Q ) i (0 , · ) = − m ( ω × ℓ ) − Z ∂ B (0) σ ( u , p ) n d Γ − k d ℓ , h F ( e ρ, e u, e ℓ, e ω, Q ) i (0 , · ) = − Z ∂ B (0) ( y − h ) × σ ( u , p ) n d Γ . These expressions of F (0 , · ) and F (0 , · ) show that (1.16) gives the third condition (2.32). We thusdeduce from Proposition 2.2 the existence and uniqueness of ( b ρ, b u, b ℓ, b ω ) ∈ S T . Combining (2.33),Proposition 2.4, (2.50) and (2.51), we obtain k ( b ρ, b u, b ℓ, b ω ) k S T R CT α . In particular, taking T small enough, we deduce that N is well defined.Next we show that N is a contraction. Let ( e ρ , e u , e ℓ , e ω ), ( e ρ , e u , e ℓ , e ω ) ∈ S T,R . For j = 1 , , weset N ( e ρ j , e u j , e ℓ j , e ω j ) := ( b ρ j , b u j , b ℓ j , b ω j ) . Using Proposition 2.2 and Proposition 2.4, we obtain k ( b ρ , b u , b ℓ , b ω ) − ( b ρ , b u , b ℓ , b ω ) k S T CT α k ( e ρ , e u , e ℓ , e ω ) − ( e ρ , e u , e ℓ , e ω ) k S T . Thus N is a contraction in S T,R for T small enough.Finally, using (2.51) and (2.39), we deduce k ( e ρ, e u, e ℓ, e ω ) k S T + k h − h k L ∞ (0 ,T ) C e δ = C (cid:16) k ρ − ρ k H + k u k H + | h − h | + | ℓ | + | ω | (cid:17) that yields (2.3). (cid:3) Global in time existence of solutions
A priori estimates.
We have already established a local-in-time existence result in Theorem 2.1.In order to obtain the global in time existence of the solutions, we need an appropriate a prioriestimates. We recall that k · k b S ,T is introduced in (1.13). We also introduce the following notation toshorten the notation: for Z = L p or Z = W k,p , we set: W k, ∞ T ( Z ) = W k, ∞ (0 , T ; Z ( F ( t ))) , H kT ( Z ) = H k (0 , T ; Z ( F ( t ))) , for k = 1 , ,W , ∞ T ( Z ) = L ∞ T ( Z ) = L ∞ (0 , T ; Z ( F ( t ))) , H T ( Z ) = L T ( Z ) = L (0 , T ; Z ( F ( t ))) . The main tool to prove the global in time existence of the solutions is the following proposition:
Proposition 3.1.
Let h ∈ Ω and ρ > . Assume the feedback law (1.10) with ( k p , k d ) satisfying (1.17) . There exist ε , C > with ε δ such that if ( ρ, u, h, ℓ, ω ) is a solution of system (1.1) – (1.10) with k ( ρ, u, ℓ, ω ) k b S ,T ε , (3.1) then the following estimate holds: k ( ρ, u, ℓ, ω ) k b S ,T + k p k p ( h − h ) k L ∞ (0 ,T ) C (cid:16) k ( ρ , u , ℓ , ω ) k b S , + | h − h | (cid:17) . (3.2) Proof.
The proof follows closely the idea of [1, Proposition 8]. We only repeat some parts of the proofto estimate ( h − h ). We define ρ ∗ ( t, x ) = ρ ( t, x ) − ρ and we rewrite (1.1)-(1.9) as follows ∂ρ ∗ ∂t + u · ∇ ρ ∗ + ρ div u = f ( ρ ∗ , u, h, ω ) t ∈ (0 , T ) , x ∈ F ( t ) ,∂u∂t − div σ ∗ ( u, ρ ∗ ) = f ( ρ ∗ , u, h, ω ) t ∈ (0 , T ) , x ∈ F ( t ) ,mℓ ′ = − Z ∂ B ( t ) σ ∗ ( u, ρ ∗ ) N d
Γ + k p ( h − h ( t )) − k d ℓ ( t ) + f ( ρ ∗ , u, h, ω ) t ∈ (0 , T ) ,J ω ′ = − Z ∂ B ( t ) ( x − h ) × σ ∗ ( u, ρ ∗ ) N d
Γ + f ( ρ ∗ , u, h, ω ) t ∈ (0 , T ) ,h ′ = ℓ t ∈ (0 , T ) ,u ( t, x ) = 0 , t ∈ (0 , T ) , x ∈ ∂ Ω ,u ( t, x ) = ℓ ( t ) + ω ( t ) × ( x − h ( t )) , t ∈ (0 , T ) , x ∈ ∂ B ( t ) ,ρ ∗ (0 , · ) = ρ − ρ, u (0 , · ) = u in F (0) ,h (0) = h , ℓ (0) = ℓ , ω (0) = ω , (3.3)In the above system (3.3) m = mρ , J = Jρ , k p = k p ρ , k d = k d ρ , µ = µρ , λ = λρ ,σ ∗ ( u, ρ ∗ ) = 2 µ D ( u ) + λ div u I − p ∗ ρ ∗ I , p ∗ = aγρ γ − , and f ( ρ ∗ , u, h, ω ) = − ρ ∗ div u,f ( ρ ∗ , u, h, ω ) = − ( u · ∇ ) u − (cid:18) ρ − ρ ∗ + ρ (cid:19) div (2 µ D ( u ) + λ div u I )+ (cid:0) p ∗ − aγ ( ρ ∗ + ρ ) γ − (cid:1) ∇ ρ ∗ ,f ( ρ ∗ , u, h, ω ) = − Z ∂ B ( t ) (cid:18) p ∗ ρ ∗ − a ( ρ ∗ + ρ ) γ ρ (cid:19) N d Γ ,f ( ρ ∗ , u, h, ω ) = − Z ∂ B ( t ) ( x − h ) × (cid:18)(cid:18) p ∗ ρ ∗ − a ( ρ ∗ + ρ ) γ ρ (cid:19) N (cid:19) d Γ . We take ε small enough in (3.1) so that ρ ∗ + ρ > ρ . After some calculations (that we skipped here), we obtain k f k L T ( H ) + k f k L ∞ T ( H ) + k f k H T ( L ) + k f k L T ( H ) + k f k L ∞ T ( H ) + k f k H T ( L ) + k f k H (0 ,T ) + k f k H (0 ,T ) C k ( ρ, u, ℓ, ω ) k b S ,T , TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 17 and (cid:13)(cid:13)(cid:13)(cid:13) ∂ρ ∗ ∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) H + | ℓ ′ (0) | + | ω ′ (0) | C (cid:16) k ρ − ρ k H + k u k H + | ℓ | + | h − h | + k f k L ∞ T ( L ) + k f k L ∞ T ( H ) + k f k L ∞ T + k f k L ∞ T (cid:17) . In particular, if we can show k ( ρ, u, ℓ, ω ) k b S ,T + k p k p ( h − h ) k L ∞ (0 ,T ) C (cid:16) k f k L T ( H ) + k f k L ∞ T ( H ) + k f k H T ( L ) + k f k L T ( H ) + k f k L ∞ T ( H ) + k f k H T ( L ) + k f k H (0 ,T ) + k f k H (0 ,T ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂ρ ∗ ∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) L + k ρ − ρ k H + (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) H + k u k H ( F (0)) + | h − h | + | ℓ ′ (0) | + | ℓ | + | ω ′ (0) | + | ω | + k ( ρ, u, ℓ, ω ) k b S ,T + k ( ρ, u, ℓ, ω ) k b S ,T (cid:17) , (3.4)then k ( ρ, u, ℓ, ω ) k b S ,T + k p k p ( h − h ) k L ∞ (0 ,T ) C (cid:16) k ( ρ, u, ℓ, ω ) k b S ,T + k ( ρ, u, ℓ, ω ) k b S ,T + k ρ − ρ k H + k u k H + | h − h | + | ℓ | + | ω | (cid:17) . The condition (3.1) with ε small enough combined with the above relation yields (3.2). The proof of(3.4) is done below. (cid:3) The proof of (3.4) (that is necessary to finish the proof of Proposition 3.1) is done in a precise wayin [1, Section 4] in the case k p = 0 and k d = 0. The presence of the corresponding terms only changesthe two lemmas on time regularity (Lemma 13 and Lemma 14 in [1]). Here we state these two lemmasin our case and give the idea of their proofs with a particular attention to the feedback term. Thenusing these two lemmas and the elliptic results [1, Section 4], we can deduce (3.4) and thus end theproof of Proposition 3.1. Lemma 3.2.
Let k=0,1. For every ε > , there exists a constant C > such that k ρ ∗ k W k, ∞ T ( L ) + k u k H kT ( H ) + k u k W k, ∞ T ( L ) + k ℓ k W k, ∞ (0 ,T ) + k ℓ k H k (0 ,T ) + k ω k W k, ∞ (0 ,T ) + k p k p ( h − h ) k L ∞ (0 ,T ) ε (cid:16) k ρ ∗ k H kT ( L ) + k ℓ k H k (0 ,T ) + k ω k H k (0 ,T ) (cid:17) + C (cid:16) k f k H kT ( L ) + k f k H kT ( L ) + k f k H k (0 ,T ) + k f k H k (0 ,T ) + k ρ − ρ k L + k u k L + | h − h | + | ℓ | + | ω | + (cid:13)(cid:13)(cid:13)(cid:13) ∂ρ ∗ ∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) L + | ℓ ′ (0) | + | ω ′ (0) | + k ( ρ, u, ℓ, ω ) k / b S ,T + k ( ρ, u, ℓ, ω ) k b S ,T (cid:17) . (3.5) Proof of Lemma 3.2.
Case k = 0 . We multiply equation (3.3) by p ∗ ρ ∗ /ρ , (3.3) by u , (3.3) by ℓ and(3.3) by ω : Z F ( t ) (cid:18) p ∗ ρ | ρ ∗ | + | u | (cid:19) dx + t Z Z F ( s ) (cid:0) µ | D ( u ) | + λ | div u | (cid:1) dx ds + m | ℓ | + J | ω | + k p t ) | h − h ( t ) | + k d t Z | ℓ | ds = t Z Z F ( s ) ( f p ∗ ρ ∗ /ρ + f · u ) dx ds + t Z ( f · ℓ + f · ω ) ds + t Z Z F ( s ) (cid:16) p ∗ ρ | ρ ∗ | div u + div (cid:18) | u | u (cid:19) (cid:17) dx ds + t Z k ′ p s ) | h − h ( s ) | ds + Z F (0) p ∗ ρ | ρ − ρ | dy + Z F (0) | u | dy + m | ℓ | + J | ω | . (3.6)Following standard calculation, we have t Z Z F ( s ) (cid:18) p ∗ ρ | ρ ∗ | div u + div (cid:18) | u | u (cid:19)(cid:19) dx ds C k ( ρ, u, ℓ, ω ) k b S ,T . (3.7)It only remains to estimate T Z k ′ p s ) | h − h ( s ) | ds T I Z k ′ p s ) | h − h ( s ) | ds k k ′ p k L ∞ (0 ,T ) T I | h − h | + T I Z s Z ℓ ( z ) dz ds ! . (3.8)By using H¨older’s inequality and (1.17), we obtain T Z k ′ p s ) | h − h ( s ) | ds C | h − h | + k d T Z | ℓ | ds. (3.9)Combining (3.6), (3.7), (3.9) and Young’s inequality, we deduce the result for k = 0. TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 19
Case k = 1 . By differentiating (3.3) with respect to t , we obtain: ∂∂t (cid:18) ∂ρ ∗ ∂t (cid:19) + ( u · ∇ ) ∂ρ ∗ ∂t + ρ div ∂u∂t = G t ∈ (0 , T ) , x ∈ F ( t ) ,∂∂t (cid:18) ∂u∂t (cid:19) − div σ ∗ (cid:18) ∂u∂t , ∂ρ ∗ ∂t (cid:19) = G t ∈ (0 , T ) , x ∈ F ( t ) ,mℓ ′′ = − Z ∂ B ( t ) σ ∗ (cid:18) ∂u∂t , ∂ρ ∗ ∂t (cid:19) N d
Γ + [ k p ( h − h ( t ))] ′ − k d ℓ ′ ( t ) + G t ∈ (0 , T ) ,J ω ′′ = − Z ∂ B ( t ) ( x − h ) × σ ∗ (cid:18) ∂u∂t , ∂ρ ∗ ∂t (cid:19) N d
Γ + G t ∈ (0 , T ) ,h ′ = ℓ t ∈ (0 , T ) ,∂u∂t ( t, x ) = 0 , t ∈ (0 , T ) , x ∈ ∂ Ω ,∂u∂t ( t, x ) = ℓ ′ ( t ) + ω ′ ( t ) × ( x − h ( t )) + G , t ∈ (0 , T ) , x ∈ ∂ B ( t ) . (3.10)where G = ∂f ∂t − (cid:18) ∂u∂t · ∇ (cid:19) ρ ∗ , G = ∂f ∂t , G = ∂f ∂t − Z ∂ B ( t ) ℓ · ∇ ( σ ∗ ( u, ρ ∗ ) N ) d Γ ,G = ∂f ∂t − Z ∂ B ( t ) ℓ · ∇ (( x − h ) × ( σ ∗ ( u, ρ ∗ )) N ) d Γ + Z ∂ B ( t ) ℓ × ( σ ∗ ( u, ρ ∗ )) N ) d Γ ,G = − ( ℓ · ∇ ) u. As in the first case, we multiply equation (3.10) by p ∗ ρ ∂ρ ∗ ∂t , equation (3.10) by ∂u∂t , equation (3.10) by ℓ ′ , and equation (3.10) by ω ′ . After some computations, we find Z F ( t ) p ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12) ∂ρ ∗ ∂t (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂t (cid:12)(cid:12)(cid:12)(cid:12) ! dx + t Z Z F ( s ) µ (cid:12)(cid:12)(cid:12)(cid:12) D (cid:18) ∂u∂t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + λ (cid:12)(cid:12)(cid:12)(cid:12) div ∂u∂t (cid:12)(cid:12)(cid:12)(cid:12) ! dx ds + m | ℓ ′ ( t ) | + J | ω ′ ( t ) | + k d t Z | ℓ ′ ( s ) | ds = t Z Z F ( s ) (cid:18) G p ∗ ρ ∂ρ ∗ ∂t + G · ∂u∂t (cid:19) dx ds + t Z ( G · ℓ ′ + G · ω ′ ) ds + t Z Z ∂ B ( s ) G · σ ∗ (cid:18) ∂u∂t , ∂ρ ∗ ∂t (cid:19) N d Γ ds + t Z Z F ( s ) p ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12) ∂ρ ∗ ∂t (cid:12)(cid:12)(cid:12)(cid:12) div u dx ds + t Z Z ∂ F ( s )
12 div (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂t (cid:12)(cid:12)(cid:12)(cid:12) u ! dx ds + t Z [ k p ( s )( h − h ( s ))] ′ · ℓ ′ ( s ) ds + Z F (0) p ∗ ρ (cid:12)(cid:12)(cid:12)(cid:12) ∂ρ ∗ ∂t (0) (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂t (0) (cid:12)(cid:12)(cid:12)(cid:12) ! dy + m | ℓ ′ (0) | + J | ω ′ (0) | . (3.11) We have the following estimates as in [1, Lemma 13]: k G k L T ( L ) + k G k L T ( L ) + k G k L (0 ,T ) + k G k L (0 ,T ) + k G k L T ( L ( ∂ B ( t )) + t Z Z F ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ρ ∗ ∂t (cid:12)(cid:12)(cid:12)(cid:12) div u + div (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂t (cid:12)(cid:12)(cid:12)(cid:12) u !! dx ds C (cid:13)(cid:13)(cid:13)(cid:13) ∂f ∂t (cid:13)(cid:13)(cid:13)(cid:13) L T ( L ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂f ∂t (cid:13)(cid:13)(cid:13)(cid:13) L T ( L ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂f ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂f ∂t (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ) + k ( ρ, u, ℓ, ω ) k b S ,T + k ( ρ, u, ℓ, ω ) k b S ,T (cid:17) (3.12)It only remains to estimate the term coming from the feedback: t Z [ k p ( s )( h − h ( s ))] ′ · ℓ ′ ( s ) ds = t Z k ′ p ( s )( h − h ( s )) · ℓ ′ ( s ) ds − t Z k p ( s ) ℓ ( s ) · ℓ ′ ( s ) ds = t Z k ′ p ( s )( h − h ( s )) · ℓ ′ ( s ) ds + t Z k ′ p s ) | ℓ ( s ) | ds − k p t ) | ℓ ( t ) | , and proceeding as in (3.8), we have the following estimates t Z [ k p ( s )( h − h ( s ))] ′ · ℓ ′ ( s ) ds C | h − h | + T Z | ℓ ( s ) | ds + k d T Z | ℓ ′ ( s ) | ds. (3.13)We can estimate k ℓ k L (0 ,T ) with (3.5) for k = 0 . With this remark and combining (3.11), inequality(3.13) and the above estimates we deduce (3.5) for k = 1 . (cid:3) Lemma 3.3.
Let k = 0 , . There exists a constant C > such that (cid:13)(cid:13)(cid:13)(cid:13) ∂ρ ∗ ∂t (cid:13)(cid:13)(cid:13)(cid:13) H kT ( L ) + k u k W k, ∞ T ( H ) + (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂t (cid:13)(cid:13)(cid:13)(cid:13) H kT ( L ) + k ℓ ′ k H k (0 ,T ) + k ω ′ k H k (0 ,T ) C (cid:16) k ρ ∗ k W k, ∞ T ( L ) + k u k H kT ( H ) + k ℓ k W k, ∞ (0 ,T ) + k ℓ k H k (0 ,T ) + k p k p ( h − h ) k L ∞ (0 ,T ) + k f k H kT ( L ) + k f k H kT ( L ) + k f k H k (0 ,T ) + k f k H k (0 ,T ) + k ρ − ρ k L + (cid:13)(cid:13)(cid:13)(cid:13) ∂ρ∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) L + k u k H + (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) H + | h − h | + | ℓ | + | ℓ ′ (0) | + | ω | + k ( ρ, u, ℓ, ω ) k / b S ,T + k ( ρ, u, ℓ, ω ) k b S ,T (cid:17) . (3.14) TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 21
Proof of Lemma 3.3.
Case k = 0. We multiply equation (3.3) by ∂ρ ∗ ∂t , (3.3) by ∂u∂t , (3.3) by ℓ ′ and(3.3) by ω ′ . After standard computations, we find t Z Z F ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ρ ∗ ∂t (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂t (cid:12)(cid:12)(cid:12)(cid:12) ! dx ds + Z F ( t ) (cid:0) µ | D ( u ) | + λ | div u | (cid:1) dx + t Z (cid:0) m | ℓ ′ | + J | ω ′ | (cid:1) ds + k d | ℓ ( t ) | − Z t k p ( h − h ) · ℓ ′ ds C t Z Z F ( s ) | div( (cid:0) µ | D ( u ) | + λ | div u | (cid:1) u ) | dx ds + t Z Z F ( s ) (cid:18) p ∗ ρ ∗ div ∂u∂t − ρ ∂ρ ∗ ∂t div u (cid:19) dx ds + t Z Z F ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ρ ∗ ∂t (cid:12)(cid:12)(cid:12)(cid:12) | ( u · ∇ ρ ∗ ) | dx ds + Z F (0) |∇ u | dx + k d | ℓ | + t Z Z F ( s ) (cid:0) | f | + | f | (cid:1) dx + | f | + | f | ds + t Z Z ∂ B ( s ) G · σ ∗ ( u, ρ ∗ ) N d Γ ds ! . (3.15)The terms in the right-hand side of (3.15) can be estimated as in Lemma 14 in [1]. We only estimate − Z t k p ( h − h ) · ℓ ′ ds = − k p ( t )( h − h ) · ℓ ( t ) + t Z (cid:16) k ′ p ( h − h ) · ℓ − k p | ℓ | (cid:17) ds (3.16)and thus (cid:12)(cid:12)(cid:12)(cid:12) − Z t k p ( h − h ) · ℓ ′ ds (cid:12)(cid:12)(cid:12)(cid:12) C | h − h | + k p | h − h | + | ℓ | + t Z | ℓ ( s ) | ds . Case k = 1. We multiply (3.10) by ∂ ρ ∗ ∂t , (3.10) by ∂ u∂t (3.10) by ℓ ′′ and (3.10) by ω ′′ . Followingthe proof of Lemma 14 in [1], we find t Z Z F ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ ρ ∗ ∂t (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ u∂t (cid:12)(cid:12)(cid:12)(cid:12) ! dx ds + Z F ( t ) µ (cid:12)(cid:12)(cid:12)(cid:12) D (cid:18) ∂u∂t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + λ (cid:12)(cid:12)(cid:12)(cid:12) div ∂u∂t (cid:12)(cid:12)(cid:12)(cid:12) ! dx + t Z (cid:18) m | ℓ ′′ | + J | ω ′′ | (cid:19) ds + 2 k d ( | ℓ ′ ( t ) | − | ℓ ′ (0) | ) − t Z k ′ p ( s )( h − h ( s )) · ℓ ′′ ( s ) ds + t Z k p ( s ) ℓ ( s ) · ℓ ′′ ( s ) ds C (cid:16) k ρ ∗ k W , ∞ T ( L ) + k u k H T ( H ) + k ℓ k W , ∞ (0 ,T ) + k ℓ k H (0 ,T ) + k f k H T ( L ) + k f k H T ( L ) + k f k H (0 ,T ) + k f k H (0 ,T ) + k ρ − ρ k L + (cid:13)(cid:13)(cid:13)(cid:13) ∂ρ∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) L + k u k H + (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂t (0 , · ) (cid:13)(cid:13)(cid:13)(cid:13) H + | ℓ | + | ℓ ′ (0) | + | ω | + k ( ρ, u, ℓ, ω ) k b S ,T + k ( ρ, u, ℓ, ω ) k b S ,T (cid:17) . (3.17)We estimate the additional term due to the feedback: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Z ( − k ′ p ( s )( h − h ( s )) · ℓ ′′ ( s ) + k p ( s ) ℓ ( s ) · ℓ ′′ ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C | h − h | + T Z | ℓ ( s ) | ds + m t Z | ℓ ′′ ( s ) | ds, (3.18)and this allows us to prove this case. (cid:3) Proof of Theorem 1.1.
Proof.
We combine Theorem 2.1 and Proposition 3.1 to establish our result. Note that we can take δ small enough in Theorem 2.1 so that (2.2) yields h ∈ Ω and ρ > . Since h ∈ Ω , there exists η > h , ∂ Ω) > η. We can assume that δ η where δ is the constant in (2.2).Let us fix δ = min δ , ε C ∗ , ε C (cid:18) C ∗ √ k p ( T ∗ ) (cid:19) , η p k p ( T ∗ ) C , (3.19) TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 23 where the constants δ , C ∗ are appeared in Theorem 2.1, ε , C are introduced in Proposition 3.1.Since ( ρ , u , h , ℓ , ω ) satisfies (1.18)-(1.19) and δ δ , we can apply Theorem 2.1 to obtain theexistence of solution of system (1.1)–(1.10) in (0 , T ∗ ) and k ( ρ, u, ℓ, ω ) k b S ,T ∗ + k h − h k L ∞ (0 ,T ∗ ) C ∗ (cid:16) k ( ρ , u , ℓ , ω ) k b S , + | h − h | (cid:17) . In particular, from (1.19) and (3.19), k ( ρ, u, ℓ, ω ) k b S ,T ∗ + k h − h k L ∞ (0 ,T ∗ ) C ∗ δ ε δ . (3.20)Thus dist( h ( t ) , ∂ Ω) > η for t ∈ [0 , T ∗ ] and Proposition 3.1 gives k ( ρ, u, ℓ, ω ) k b S ,T ∗ + k p k p ( h − h ) k L ∞ (0 ,T ∗ ) C (cid:16) k ( ρ , u , ℓ , ω ) k b S , + | h − h | (cid:17) . (3.21)Using that ( ρ, u, h, ℓ, ω ) is solution of (1.1)–(1.10), one can check that( ρ ( T ∗ , · ) , u ( T ∗ , · ) , h ( T ∗ ) , ℓ ( T ∗ ) , ω ( T ∗ ))satisfies the compatibility conditions (1.14)-(1.16) and, from (3.20), we have k ( ρ ( T ∗ , · ) , u ( T ∗ , · ) , ℓ ( T ∗ ) , ω ( T ∗ )) k b S T ∗ ,T ∗ + | h − h ( T ∗ ) | δ . We can thus apply again Theorem 2.1 to extend our solution on ( T ∗ , T ∗ ) and using (3.21), we find k ( ρ, u, ℓ, ω ) k b S T ∗ , T ∗ + k p k p ( h − h ) k L ∞ ( T ∗ , T ∗ ) C ∗ (cid:16) k ( ρ ( T ∗ , · ) , u ( T ∗ , · ) , ℓ ( T ∗ ) , ω ( T ∗ )) k b S T ∗ ,T ∗ + | h − h ( T ∗ ) | (cid:17) C ∗ C p k p ( T ∗ ) (cid:16) k ( ρ , u , ℓ , ω ) k b S , + | h − h | (cid:17) . (3.22)Thus, combining (3.21) and (3.22), and using (3.19), we obtain k ( ρ, u, ℓ, ω ) k b S , T ∗ + k p k p ( h − h ) k L ∞ (0 , T ∗ ) C C ∗ p k p ( T ∗ ) ! (cid:16) k ( ρ , u , ℓ , ω ) k b S , + | h − h | (cid:17) C C ∗ p k p ( T ∗ ) ! δ ε . Applying Proposition 3.1, we deduce k ( ρ, u, ℓ, ω ) k b S , T ∗ + k p k p ( h − h ) k L ∞ (0 , T ∗ ) C (cid:16) k ( ρ , u , ℓ , ω ) k b S , + | h − h | (cid:17) . (3.23)In particular dist( h ( t ) , ∂ Ω) > η for t ∈ [ T ∗ , T ∗ ]. Moreover, from (3.22) and (3.19), k ( ρ (2 T ∗ , · ) , u (2 T ∗ , · ) , ℓ (2 T ∗ ) , ω (2 T ∗ )) k b S T ∗ , T ∗ + | h − h (2 T ∗ ) | C C ∗ p k p ( T ∗ ) δ ε δ . Then, we repeat the argument on [ jT ∗ , ( j + 1) T ∗ ], j ∈ N ∗ and we use that k p is non-decreasing toconclude the proof. (cid:3) Proof of Theorem ρ − ρ ∈ H (0 , ∞ ; H ( F ( t ))) , u ∈ H (0 , ∞ ; H ( F ( t )) , ℓ, ω ∈ H (0 , ∞ )so that ([3, Corollary 8.9, p.214]),lim t →∞ k ρ ( t, . ) − ρ k H ( F ( t )) = 0 , lim t →∞ k u ( t, . ) k H ( F ( t )) = 0 , lim t →∞ ℓ ( t ) = 0 , lim t →∞ ω ( t ) = 0 . (4.1)In the rest of the section, we show lim t →∞ h ( t ) = h that completes the proof of Theorem 1.2. Inorder to do this, we need the notion of weak solutions for the problem (1.1)-(1.9). First, we extend ρ and u in R by the formula ρ = ρ in F ( t ) , m π = ρ B in B ( t ) , R \ Ω . u = u in F ( t ) ,ℓ ( t ) + ω ( t ) × ( x − h ( t )) = u B in B ( t ) , R \ Ω . Then we consider the following notion of weak solutions (see [9]).
Definition 4.1.
A triplet ( ρ, u, h ) is a weak solution to (1.1) - (1.9) on (0 , T ) if ρ > , ρ ∈ L ∞ (0 , T ; L γ (Ω)) ∩ C ([0 , T ]; L (Ω)) , u ∈ L (0 , T ; H (Ω)) ,u = ℓ ( t ) + ω ( t ) × ( x − h ( t )) in B ( t ) , h ′ = ℓ, T Z Z R (cid:20) ρ ∂φ∂t + ( ρu ) · ∇ φ (cid:21) dx dt = 0 , T Z Z R (cid:20) b ( ρ ) ∂φ∂t + ( b ( ρ ) u ) · ∇ φ + (cid:0) b ( ρ ) − b ′ ( ρ ) ρ (cid:1) div u φ (cid:21) dx dt = 0 , for any φ ∈ C ∞ c ((0 , T ) × R ) and for any b ∈ C ( R ) such that b ′ ( z ) = 0 for z large enough; T Z Z R (cid:20) ( ρu ) · ∂φ∂t + ( ρu ⊗ u ) : D ( φ ) + aρ γ div φ (cid:21) dx dt = T Z Z R (2 µ D ( u ) + λ div u I ) : D ( φ ) dx dt + T Z w · ℓ φ dt, (4.2) for any φ ∈ C ∞ c ((0 , T ) × Ω) , with φ ( t, y ) = ℓ φ ( t ) + ω φ ( t ) × ( y − h ( t )) in a neighborhood of B ( t ) ; fora.e. t ∈ [0 , T ] , the following energy inequality holds: Z Ω (cid:18) ρ ( t, x )2 | u ( t, x ) | + aγ − ρ γ ( t, x ) (cid:19) dx + t Z Z Ω (cid:0) µ | D ( u ) | + λ | div u | (cid:1) dx dt C Z { ρ (0) > } (cid:18) | q ( x ) | ρ (0 , x ) + aγ − ρ γ (0 , x ) (cid:19) dx + t Z w · ℓ dt ; TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 25 and ρ (0 , · ) = ρ , ( ρu )(0 , · ) = q, h (0) = h . We now state a result on the weak compactness of the set of weak solutions to the problem (1.1)-(1.9)obtained in [9, Theorem 9.1].
Theorem 4.2.
Let ( ρ n , u n , h n ) be a sequence of weak solutions to (1.1) - (1.9) on (0 , T ) × Ω with theinitial condition ( ρ ,n , u ,n , h ,n ) and forcing term w n for each n > . Assume that { w n } is a sequenceof bounded and measurable functions such that w n → w weakly * in L ∞ (0 , T ) , along with ρ n → ρ in L γ ( R ) , (4.3) ρ n u n = q n → q in L ( R ) , (4.4) where ρ , q satisfy the following compatibility conditions q = 0 a.e. on the set { x ∈ Ω | ρ = 0 } , | q | ρ ∈ L (Ω) . (4.5) Moreover, let Z { ρ n > } (cid:18) | q n | ρ n + aγ − ρ γn (0) (cid:19) dx → Z { ρ > } (cid:18) | q | ρ + aγ − ρ γ (cid:19) dx (4.6) and h n → h . (4.7) Then there is a subsequence such that ρ n → ρ in C ([0 , T ]; L ( R )) ,u n → u weakly in L (0 , T ; H (Ω)) ,h n → h uniformly in (0 , T ) . where ( ρ, u, h ) is a weak solution of the problem (1.1) - (1.9) on (0 , T ) × Ω with the initial conditions ( ρ , q, h ) . With the help of above result, we can now prove Theorem 1.2.
Proof of Theorem . From (1.21), there exist h ∗ ∈ Ω and { t n } ⊂ R ∗ + such that t n → ∞ , lim n →∞ h ( t n ) = h ∗ . Define ρ ∗ = b F ( h ∗ ) ρ + b B ( h ∗ ) ρ B . Writing ρ ( t n , · ) − ρ ∗ = [ ρ ( t n ) − ρ ] F ( t n ) + ρ [ F ( t n ) − b F ( h ∗ ) ] + ρ B [ B ( t n ) − b B ( h ∗ ) ] , and using (4.1), we deduce ρ ( t n , · ) t n →∞ −−−−→ ρ ∗ in L γ ( R ) ,ρ ( t n , · ) u ( t n , · ) t n →∞ −−−−→ L ( R ) ,ρ ( t n , · ) | u ( t n , · ) | t n →∞ −−−−→ L ( R ) . We set ρ n = ρ ( t n ) , u n = u ( t n ) , h n = h ( t n ) , that satisfy (4.3), (4.4), (4.5), (4.6) and (4.7) with { ρ n > } = { ρ > } = Ω. We also define ρ n ( t, x ) = ρ ( t + t n , x ) , u n ( t, x ) = u ( t + t n , x ) , h n ( t ) = h ( t + t n ) , ℓ n ( t ) = ℓ ( t + t n ) , that is a weak solution to (1.1)-(1.9) in the sense of Definition 4.1 (since it is a strong solution) withinitial conditions ( ρ n , u n , h n ) and with w n ( t ) = k p ( t )( h − h n ( t )) − k d ℓ n ( t ) . From Theorem 1.1, we have that w n ⇀ b w weakly * in L ∞ (0 , T ) . Thus, we can apply Theorem 4.2 and we deduce that up to a subsequence for
T > ρ n → b ρ in C ([0 , T ]; L ( R )) ,u n → b u weakly in L (0 , T ; H (Ω)) ,h n → b h in L ∞ (0 , T ) , (4.8)with ( b ρ, b u, b h ) is a weak solution of (1.1)-(1.9) such that b ρ (0 , · ) = ρ ∗ , ( b ρ b u )(0 , · ) = 0 , b h (0) = h ∗ , and with b w ( t ) = k p ( t )( h − b h ( t )) − k d b ℓ ( t ) . Moreover up to a subsequence, T Z k D ( u n ( t, · )) k L (Ω) dt = t n + T Z t n k D ( u ( t, · )) k L (Ω) dt n →∞ −−−→ . The above limit and (4.8) yield D b u = 0 in (0 , T ) × Ω . Thus, we deduce that b u = 0 in (0 , T ) × Ω. In particular, we have b h ′ ( t ) = 0, ∀ t ∈ (0 , T ). This gives, b h = h ∗ in (0 , T ) . Consequently, (4.2) gives T Z Z R a ( b ρ ) γ div φ dx dt = T Z k p ( h − h ∗ ) · ℓ φ dt, for all φ ∈ C ∞ c ((0 , T ) × Ω) , with φ ( t, y ) = ℓ φ ( t ) + ω φ ( t ) × ( y − h ( t )) in a neighborhood of B ( t ). Thenwe take div φ = 0 , φ ( t, · ) = ( h − h ∗ ) ζ ( t ) in B ( t ) , with ζ ∈ C ∞ c ((0 , T )) , so that T Z | h − h ∗ | k p ( t ) ζ ( t ) dt = 0 , ∀ ζ ∈ C ∞ c ((0 , T )) . Since, k p = 0, h ∗ = h . (cid:3) TABILIZATION OF A RIGID BODY MOVING IN A COMPRESSIBLE VISCOUS FLUID 27
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