INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES

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1 INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Local existence of strong solutions and weak-strong uniqueness for the compressible Navier-Stokes system on moving domains Ondřej Kreml Šárka Nečasová Tomasz Piasecki Preprint No PRAHA 7

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3 Local existence of strong solutions and weak-strong uniqueness for the compressible Navier-Stokes system on moving domains Ondřej Kreml Šárka Nečasová Tomasz Piasecki 3 Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 5, 5 67 Praha Czech Republic Institute of Applied Mathematics and Mechanics, University of Warsaw Banacha, -97 Warszawa, Poland Abstract We consider the compressible Navier-Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier-Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak-strong uniqueness principle for slip boundary conditions which remained so far open question. Keywords: compressible Navier-Stokes equations, time-dependent domain, strong solution, local existence, weakstrong uniqueness. MSC: 35Q3, 76N Introduction We consider a barotropic flow of a compressible viscous fluid in the absence of external forces. Such flow is described by the isentropic compressible Navier-Stokes system t ϱ + div x (ϱu) =, (.) t (ϱu) + div x (ϱu u) + x p(ϱ) = div x S( x u), (.) The work of O.K. was partially supported by the project 7AMB6PL6 of Czech Ministry of Education, Youth and Sports, in the framework of RVO: The work of Š.N. was partially supported by the project 7AMB6PL6 of Czech Ministry of Education, Youth and Sports(last section of paper). The rest of paper was supported by the Czech Science Foundation project GA7-747 in the framework of RVO: The work of T.P was partially supported by the Polish NCN Harmonia project UMO-4/4/M/ST/8 and his stay in Institute of Mathematics by project 7AMB6PL6.

4 where ϱ is the density of the fluid and u denotes the velocity. We assume that the stress tensor S is determined by the standard Newton rheological law ( S( x u) = µ x u + t xu ) 3 div xui + ηdiv x ui (.3) with µ > and η being constants. The pressure p(ϱ) is a given sufficiently smooth function of the density and we introduce the pressure potential as ϱ p(z) H(ϱ) = ϱ dz. (.4) z We are interested in proving local existence of strong solutions to the system of equations (.)-(.) on a moving domain = t with prescribed movement of the boundary. More precisely, the boundary of the domain t is assumed to be described by means of a given velocity field V(t, x), where t and x R 3. Assuming V is regular and solving the associated system of differential equations we set d ( ) dt X(t, x) = V t, X(t, x), t >, X(, x) = x (.5) τ = X (τ, ), where R 3 is a given domain. Moreover we denote Γ τ = τ and Q τ = {t} t =: (, τ) t. t (,τ) System (.)-(.) is supplied with the Navier slip boundary condition stated as the combination of the impermeability relation (u V) n Γτ = for any τ, (.6) where n(t, x) denotes the unit outer normal vector to the boundary Γ t, and equation ([Sn] tan + κ [u V] tan ) Γτ =, (.7) where κ represents a friction coefficient. In particular the choice κ = yields the complete slip boundary condition and in the limit κ + we recover the no-slip boundary condition (7.). Finally, the system of equations (.)-(.) is supplemented by the initial conditions ϱ(, ) = ϱ, u(, ) = u in. (.8) Global weak solutions to the compressible barotropic Navier Stokes system on a fixed domain were proved to exist in a pioneering work by Lions []. This theory was later extended by Feireisl and collaborators ([], [6], [7], [8]) to cover larger class of pressure laws. The existence theory in the case of moving domains was developed in [] for no-slip boundary conditions using the so-called Brinkman penalization and in [] for slip boundary conditions. Recently these results have been generalized to a complete system with thermal effects in [8] and [9]. There is also quite large amount of literature available concerning existence of strong solutions for the system (.) (.) (or even for more complex systems involving also heat conductivity of the fluid) on a fixed domain, such solutions are proved to exist either locally in time or globally provided the initial data are sufficiently close to a rest state, let us mention among others [], [3], [3], [3] in the framework of Hilbert spaces and [4],[5],[6] in the L p

5 setting. All these results are proved under the no-slip boundary conditions. The case of slip boundary condition on fixed domain was considered by Zaj aczkowski [34] and Hoff [7]. Free boundary problems for the system (.)-(.) has been investigated by Zaj aczkowski et al. ([35],[36]) where global existence of strong solutions in L setting is shown under assumption that the domain is close to a ball. Recently, Shibata and Murata [9] showed global well posedness using entirely different approach in the L p L q maximal regularity setting. For moving domains with given motion of the boundary local-in-time existence results of strong solutions (incompressible case) can be found in the L p setting by Hieber and al. [3]. Moreover, in the case of fluid-structure interaction we can mention work of Hieber for incompressible and also compressible case [5, 6]. The concept of relative entropies has been successfully used in the context of partial differential equations (see among others Carillo et al. [], Masmoudi [], Saint-Raymond [8], Wang and Jiang [33]). Germain [4] introduced a notion of solution to the system (.) (.) based on a relative entropy inequality with respect to a hypothetical strong solution. Similar idea was adapted by Feireisl et al. [3] who defined a suitable weak solution to the barotropic Navier-Stokes system based on a general relative entropy inequality with respect to any sufficiently smooth pair of functions. In [9] the authors used relative entropy inequality to prove the weak-strong uniqueness property. Doboszczak [5] proved both the relative entropy inequality as well as the weak-strong uniqueness property in the case of moving domain and no-slip boundary condition. In this paper we first prove the local existence of strong solutions to the system (.) (.). The general approach consist in nowadays classical method of splitting the problem into continuity and momentum equations and investigation of linearized problems. Here we adapt a mixed approach. The continuity exuation is solved directly on moving domain. For this purpose we extend the known existence theory for the linear transport equation based on the method of characteristics. Analogous result on a fixed domain has been applied for example in [34], however without proof which turns out to be quite involved and here we present it for the sake of completeness. To treat the linear momentum equation we apply the Lagrangian coordinates. The main difficulty in the resulting system on a fixed domain is in nonhomogeneous boundary conditions which, in spite of smallness of time, require careful treatment with appropriate extension of the boundary data. Having completed the proof of the local existence we show that weak solutions to the system (.) (.) on moving domains proved to exist in [] possess also an energy inequality (Proposition 6.). Using this energy inequality we prove the relative entropy (energy) inequality (Proposition 6.) and finally the weak-strong uniqueness property (Theorem.). We are now in a position to formulate the main results of our paper. The first one gives the local existence of strong solutions (a precise definition of function spaces defined on the moving domain is given in Section.). Theorem. Let R 3 be a bounded domain of class C. Assume that p(ϱ) is a C function of the density and V C 3 ((, T ) R 3 ). Assume further that u H 3 ( ), ϱ H ( ) and there exists positive constants c, c such that < c ϱ c. Then there exists (sufficiently small) T > and a unique solution (u, ϱ) to the system (.)-(.3) with boundary conditions (.6)-(.7) (or (7.)) and initial condition (.8) such that u L (, T, H ( t )) L (, T, H 3 ( t )), u t L (, T, H ( t )) L (, T, H ( t )), ϱ L (, T, H ( t )), ϱ t L (, T, H ( t )). The second main result concerns the weak-strong uniqueness principle. Theorem. let V C ([, T ]; C 3 c (R 3 )) be given. Assume that the pressure p C[, ) C (, ) satisfies p() =, p (ϱ) > for any ϱ >, p (ϱ) lim ϱ ϱ γ = p > for a certain γ > 3/. Let (ϱ, u) be a weak solution to the compressible Navier-Stokes system (.)-(.8) constructed in Theorem.. Let ( ϱ, ũ) be a strong solution to the same problem satisfying < inf ϱ sup ϱ < (.9) Q T Q T 3

6 with q > max{3; x ϱ L (, T ; L q ( t )), xũ L (, T, L q ( t )) (.) 6γ 5γ 6 }, and emanating from the same initial data. Then ϱ = ϱ, u = ũ in Q T. (.) Remark. Notice that the strong solution constructed in Theorem. satisfies the assumptions of Theorem. since by the imbedding theorem we have (.) for q 6 and 6γ 5γ 6 < 6 for γ > 3. The paper is organized as follows. In Section we discuss the proper definition of function spaces on moving domains, recall the existence theorem for weak solutions from [] and introduce the iterative scheme used in the proof of Theorem.. In Sections 3 and 4 we present the existence theory for the linear continuity and momentum equations respectively in appropriate function spaces. In Section 5 we show the convergence of the iterative scheme which completes the proof of Theorem.. Section 6 is dedicated to the proof of Theorem.. Finally, in Section 7 we present concluding remarks regarding different boundary conditions, regularity of V etc. Preliminaries. Function spaces To begin with we introduce the function spaces L p (, T, X( t )). For fixed T > we assume that there exists R > such that for all t [, T ] it holds t B R (), where B R () denotes the ball in R 3 of radius R centered at the origin. Then we define L p (, T, L q ( t )) := {u L p (, T, L q (B R ())), u(t, ) = in B R () \ t for a.e. t (, T )} (.) with the norm for p < and u L p (,T,L q ( t)) := ( T u(t) p L q ( t) dt ) p u L (,T,L q ( t)) := ess sup t (,T ) u(t) Lq ( t). Similarly we define spaces L p (, T, W l,q ( t )). Let l N and α be a multi-index. Then with the norm L p (, T, W l,q ( t )) := {u L p (, T, L q ( t )), α u L p (, T, L q ( t )) α l} u L p (,T,W l,q ( := t)) α u L p (,T,L q (. t)) α l The spaces of continuous functions in time with values in Lebesgue or Sobolev spaces in space variable C([, T ], W l,q ( t )) are defined similarly as in (.) using the large ball B R (). We also recall that we denote as usual H k := W k,. If no confusion may arise, we often drop the time interval and also the spatial domain to denote L p (W l,q ) := L p (, T, W l,q ( t )). Finally we introduce a compact notation for the regularity class of the velocity in Theorem.. For a function f defined on (, T ) t we define f X (T ) = f L (,T,H ( t)) L (,T ;H 3 ( t)) + f t L (,T,H ( t)) L (,T,H ( t)) (.) 4

7 and for a function f defined of (, T ) where is a fixed domain f Y(T ) = f L (,T,H ()) L (,T ;H 3 ()) + f t L (,T,H ()) L (,T,H ()). (.3) and obviously we denote by X (T ) and Y(T ) spaces for which above norms are finite.. Weak solutions For weak solutions it is enough to assume the initial condition in a more general form ϱ(, ) = ϱ, (ϱu)(, ) = (ϱu) in. (.4) We define weak solutions to the compressible Navier-Stokes system with slip boundary conditions on moving domains as follows. Definition. We say that the couple (ϱ, u) is a weak solution of problem (.)-(.) with boundary conditions (.6)-(.7) and initial conditions (.4) if ϱ L (, T ; L γ (R 3 )), ϱ a.e. in Q T. u, x u L (Q T ), (u V) n(τ, ) Γτ = for a.a. τ [, T ]. The continuity equation (.) is satisfied in the whole R 3 provided the density is extended by zero outside t, i.e. τ ϱϕ(τ, ) dx ϱ ϕ(, ) dx = (ϱ t ϕ + ϱu x ϕ) dxdt (.5) τ t for any τ [, T ] and any test function ϕ C c ([, T ] R 3 ). Moreover, equation (.) is also satisfied in the sense of renormalized solutions introduced by DiPerna and Lions [4]: τ b(ϱ)ϕ(τ, ) dx b(ϱ )ϕ(, ) dx = (b(ϱ) t ϕ + b(ϱ)u x ϕ + (b(ϱ) b (ϱ)ϱ) div x uϕ) dxdt (.6) τ t for any τ [, T ], any ϕ C c ([, T ] R 3 ), and any b C [, ), b() =, b (r) = for large r. The momentum equation (.) is replaced by the family of integral identities ϱu ϕ(τ, ) dx τ (ϱu) ϕ(, ) dx (.7) = τ for any τ [, T ] and any test function ϕ C c t (ϱu t ϕ + ϱ[u u] : x ϕ + p(ϱ)div x ϕ S( x u) : x ϕ) dxdt ([, T ] R 3 ) satisfying The existence of weak solutions to the problem (.)-(.8) was proved in []. ϕ n Γτ = for any τ [, T ]. (.8) 5

8 Theorem. Let R 3 be a bounded domain of class C +ν. Assume V and p(ϱ) satisfy the assumptions of Theorem.. Let the initial data (.4) satisfy ϱ L γ (R 3 ), ϱ, ϱ, ϱ R3 \ =, (ϱu) = a.a. on the set {ϱ = }, (ϱu) dx <. ϱ Then the problem (.)-(.7),(.4) admits a weak solution on any time interval (, T ) in the sense specified through Definition.. We also notice that the existence theorem for the problem with no-slip boundary condition was proved in []..3 Iterative scheme Theorem. will be proved with the method of successive approximations. At each step we will solve the linear system solving coupled linear continuity and momentum equations. Here we adapt nowadays classical approach ([3], [34]). The linear continuity equation is solved in a moving domain which is possible since the characteristics are well defined due to boundary condition (.6). Then in order to solve the momentum equation we use Lagrangian transformation determined by the velocity field V. It should be noticed that such approach is admissible since the transformation depends only on V and not on the solution, therefore is independent on the step of iteration. Then the crucial difficulty is in showing appropriate estimates that will give convergence of the iterative scheme. We restrict our presentation to the proof of the result for slip boundary condions. With our method this case is more complicated as it requires treatment of boundary terms which otherwise vanish. In view of the above considerations we define our iterative scheme as follows. We set ρ (t, X(t, x)) := ρ (x) and u (t, X(t, x)) := u (x) for all t (, T ). Assume we already have (ϱ n, u n ). We define the next step of approximation (ϱ n+, u n+ ) as follows.. We solve for ϱ n+ the linear continuity equation with the initial condition ϱ n+ (, x) = ϱ (x) in.. We solve for u n+ the linear momentum equation with boundary conditions t ϱ n+ + u n x ϱ n+ + ϱ n+ div x u n = in Q T (.9) ϱ n+ t u n+ µ x u n+ ( µ 3 + η) xdiv x u n+ = (.) ϱ n+ u n x u n x p(ϱ n+ ) =: F(ϱ n+, u n ) for all t (, T ) and initial condition u n+ (, x) = u (x) in. 3 Linear continuity equation in Q T (u n+ V) n Γt = (.) [S( x u n+ )n] tan + κ[u n+ V] tan Γt = (.) In order to have the iterative scheme well defined we have to solve in particular the linear continuity equation ϱ t + v x ϱ + ϱdiv x v = in Q T (3.) with (v V) n Γt =. As pointed out earlier, we treat the linear continuity equation (3.) directly in the moving domain and we do not use here any change of variables. The following result gives existence of solution to (3.). 6

9 Proposition 3. Assume ϱ H ( ), v L (, T, H ( t )) L (, T, H 3 ( t )). Then for T > sufficiently small there exists a unique solution ϱ to the linear transport equation (3.) such that Moreover the following estimates hold and ϱ C([, T ], H ( t )), t ϱ C([, T ], H ( t )). (3.) ϱ L (H ) C ϱ H φ( T v L (H 3 )) =: D (3.3) t ϱ L (H ) C T ϱ H φ( T v L (H 3 )) v L (H 3 ) (3.4) where φ( ) is an increasing, positive Lipschitz function. Moreover, if ϱ κ > then ϱ C(κ, T, v) >. Remark. In what follows we will denote by φ( ) an increasing, positive, sufficiently smooth function which may vary from line to line. In order to obtain the desired form of estimates with the factor T we use the solution formula provided by the method of characteristics. We have where ϱ(t, X(t, z)) = ϱ (z)exp ( X(t, z) = z + t t div x v(s, X(s, z))ds ), (3.5) v(s, X(s, z))ds. (3.6) Note that the mapping X(t, z) is different from the mapping X(t, z) introduced in (.5), since one is related to the velocity field v while the other to the velocity field V. Before we proceed with the proof of Proposition 3. we first need some properties of the mapping X and its derivatives. Lemma 3. Let v L (, T, H 3 ( t )). Let X(t, z) be defined by (3.6), i.e. for fixed t (, T ) we have X(t, ) : t. Then there exists sufficiently small T > such that for all t (, T ) there exists an inverse mapping z(t, ) : t, i.e. z(t, X(t, y)) = y for all y and X(t, z(t, y)) = y for all y t. Moreover it holds and x z(t, x) I L (Q τ ) E(τ) (3.7) zx(t, z) L (,T,L 4 ( )) φ( v L (,T,H 3 ( t))), (3.8) where I denotes the identity matrix, E(t) is a nonnegative function such that E(t) as t + and φ is an increasing positive function. Proof: [Lemma 3.] We have hence z X satisfies a system of ODE z X(t, z) = I + t x v(s, X(s, z)) z X(s, z)ds, (3.9) t z X = x v(t, X(t, z)) z X. 7

10 Multiplying the component of the above equation corresponding to zj X i by zj X i p zj X i and integrating over we obtain d dt zx p L p ( C ) xv H ( t) z X p L p (, ) therefore by Gronwall inequality ( ) T z X L (,T,L p ( )) Cexp v H 3 ( t)ds φ( v L (H 3 )) M. p <. (3.) Moreover, tracking the dependence of the above estimate on p we can justify the limit passage p to conclude that (3.) holds also for p =. In order to show the bound for the second gradient we differentiate (3.9) w.r.t. z and t obtaining t zx(t, z) xv(t, X(t, z))( z X(t, z)) + x v(t, X(t, z)) zx(t, z). Note that we don t really care about the precise structure of the right hand side, in order to obtain estimates, it is enough for us to know that we have term involving second gradient of v multiplied twice by first gradient of X and second gradient of X multiplied by first gradient of v. We proceed similarly as before. Multiplying the component corresponding to z iz j X k by z iz j X k z iz j X k and integrating over we obtain d dt zx(t, ) 4 L C 4 xv(t, X(t, )) z X(t, ) zx(t, ) 3 dx + C x v(t, X(t, )) zx(t, ) 4 dx C xv(t) L6 ( t) z X(t) L 4 ( ) zx(t) 3 L 4 ( ) + xv(t) H ( t) zx(t) 4 L 4 ( ), which by Sobolev embedding and Gronwall inequality implies (3.8). Now we are ready to finish the proof of (3.7). Recall that from (3.) we have z X(t, z) L ((,T ) ) M (3.) for some M > which depends on the L (H 3 ) norm of v. Therefore from (3.9) we have t z X(t, z) I = x v(s, X(s, z)) z X(s, z)ds M xv L (,τ,l ( CM t)) τ v L (,τ,h 3 ( t)) (3.) The expression on the right hand side of (3.) will be small for small times, which yields the invertibility of z X and also the bound (3.7). Now we proceed with the proof of Proposition 3.. Proof: [Proposition 3.] The solution formula (3.5) immediately gives the last statement of the lemma. Moreover, denoting ϱ(t, z) = ϱ(t, X(t, z)) we obtain from (3.5) ϱ(t, ) L ( t) C ϱ(t, ) L ( ) C ϱ L exp ( and consequently t div x vds ) L C ϱ L exp ( t div x v L ds ) (3.3) ϱ L (,T,L ( t)) C ϱ L (,T,L ( )) C ϱ L exp ( div x v L (,T,L )) ϱ L φ( T v L (,T,H 3 )). (3.4) 8

11 Differentiating (3.5) w.r.t z we obtain x ϱ(t, X(t, z)) z X(t, z) (3.5) t ( t ) = exp( div x v(s, X(s, z)) ds) z ϱ (z) ϱ (z) x div x v(s, X(s, z)) z X(s, z) ds and thus at least for small times x ϱ ( z X) ( z ϱ exp( t div x v) + ϱ exp( t div x v) t x div x v z X ds ). (3.6) Now, the first term in (3.6) can be estimated as before (see (3.3) and (3.4)) and with the second term we proceed as follows ( z X) ϱ exp( t div x v) t ( z X) L ( ) ϱ L ( ) exp( ( z X) L ( ) ϱ H ( )exp( x div x v z X ds L ( ) (3.7) t t div x v) L ( ) div x v L ( t)) t t ( x div x v L ( t) z X L ( )) ds ( x div x v L ( t) z X L ( )) ds Taking supremum over t (, T ), using the fact that both ( z X) L ( ) and z X L ( ) are bounded in time for sufficiently small T and using Hölder inequality similarly as in (3.4) to obtain the factor T we arrive at x ϱ L (,T,L ( t)) C ϱ H ( )φ( T v L (,T,H 3 ( t))). (3.8) In order to estimate the second derivatives of ϱ we differentiate (3.5) one more time with respect to z. Again we are not particularly interested in the precise structure of the resulting equation, for the purpose of the estimate it is enough to recognize all kinds of terms appearing on both sides of the resulting equation. We have xϱ( z X) + x ϱ zx zϱ exp( + ϱ exp( ϱ exp( t t t div x v) z ϱ exp( t div x v) t ( t div x v) ( x div x v z X) ds) ϱ exp( div x v) t ( xdiv x v( z X) ) ds. ( x div x v z X) ds t div x v) t ( x div x v zx) ds Since ( z X) is bounded for small times, our goal is to estimate the L (L ) norm of the right hand side of (3.9) and also of x ϱ zx. Let us start with the latter. Due to Lemma 3. we have the L (L 4 ) bound for zx and for x ϱ we use the formula (3.6). We show only the estimates in the space variable, the time variable is handled always the same way as in (3.4) or (3.8). We have zx( z X) exp( t div x v) z ϱ L zx L 4 C z X L 4 ( z X) L ( z X) L exp( t exp( div x v) t L div x v L ) ϱ H (3.9) z ϱ L 4 (3.) 9

12 and zx( z X) exp( z X L 4 C z X L 4 t ( z X) L t div x v)ϱ ( x div x v z X) ( z X) L exp( t exp( div x v) t L L ϱ L div x v L ) ϱ H t t ( x v L 4 z X L ) ( v H 3 z X L ). Now let us treat the right hand side of (3.9). The first term can be handled similarly as above in (3.4). For the second term we have t t t zϱ exp( div x v) ( x div x v z X) z ϱ L 4 exp( t div x v) ( x v L 4 z X L ) (3.) L C ϱ H exp( t div x v L ) L t The third term on the right hand side of (3.9) is estimated as t ( t ϱ exp( div x v) ( x div x v z X) ds) L t ( ϱ L exp( t div x v) ( x v L 4 z X L ) C ϱ H exp( t L ( t div x v L ) ( v H 3 z X L ) The fourth term is estimated similarly as in (3.). We have t t ϱ exp( div x v) ( x div x v zx) L t ϱ L exp( t div x v) ( x v L 4 C ϱ H exp( t L t div x v L ) ( v H 3 ( v H 3 z X L ) ) z X L 4) zx L 4) The last term on the right hand side of (3.9) is the only one involving third derivatives of the velocity field v. We proceed as follows t t ϱ exp( div x v) ( xdiv x v( z X) ) (3.5) L t ϱ L exp( t div x v) ( 3 x v L z X L ) C ϱ H exp( t L t div x v L ) ( v H 3 z X L ) ) (3.) (3.3) (3.4)

13 As we mentioned above, handling the time variable is the same as in the estimates for ϱ and x ϱ. Since we have the L (H 3 ) bound for v and under the time integrals there is always only first power of the H 3 norm of v we always gain the factor T and finally end up with xϱ L (,T,L ( t)) C ϱ H ( )φ( T v L (,T,H 3 ( t))). (3.6) Combining (3.3), (3.8) and (3.6) we obtain (3.3). With the estimates of ϱ in L (H ) at hand we can use the equation (3.) to obtain estimates for the time derivative of ρ. We have t ϱ = v x ϱ ϱdiv x v. (3.7) Estimating the right hand side as t ϱ L (,T,L ( v t)) xϱ L (,T,L ( + ϱdiv t)) xv L (,T,L ( t)) (3.8) C ( ) T v L (,T,H ( t)) xϱ L (,T,L ( + ϱ t)) L (,T,H ( div t)) xv L (,T,L ( t)) C T ϱ H ( )φ( T v L (,T,H 3 ( t))) v L (,T,H ( t)). Next we apply the spatial gradient to (3.7) to obtain t x ϱ = v x x ϱ x ϱ x v x ϱdiv x v ϱ x div x v (3.9) and we estimate the L (L ) norm of the right hand side. Starting first with the norms in the x variable we have v xϱ L ( t) v L xϱ L C v H ϱ H (3.3) so altogether we conclude x ϱ x v L ( t) xϱ L 4 x v L 4 C v H ϱ H (3.3) ϱ x v L ( t) ϱ L x v L C v H ϱ H, (3.3) t x ϱ L (,T,L ( t)) C T v L (H ) ϱ L (H ) (3.33) 4 Linear momentum equation In this section we treat the linear momentum equation C T ϱ H ( )φ( T v L (,T,H 3 ( t))) v L (,T,H ( t)). ϱ t u µ x u ( µ 3 + η) xdiv x u = F in Q T, (4.) (u V) n Γτ =, ([Sn] tan + κ [u V] tan ) Γτ =. The next proposition gives existence of solutions to the linear momentum equation on moving domain and we deliberately skip emphasizing the domains of all function spaces in this proposition in order to shorten the notation. However we recall here that by L p (B) we mean in this lemma the function space L p (, T, B( t )). We recall that the space X (T ) was defined in (.).

14 Proposition 4. Let T > be sufficiently small. Let V, p(ϱ) satisfy the assuptions of Theorem.. Assume ϱ L (H ), ϱ t L (H ), F L (H ), F t L (L ), F() H ( ), u H 3 ( ) and let V C ((, T ) R 3 ). Then there exist a unique solution u to the system (4.) such that u X (T ) and the following estimate holds u X (T ) φ( ϱ L (H ), ϱ t L (H )) (4.) ( F L (H ) + F t L (L ) + F() H + u H 3 + V L (W, ) + V t L (H ) + V tt L (L )) ), where φ is a positive increasing function of its arguments. This time it is more convenient to convert the problem to fixed spatial domain. For this purpose we introduce the Lagrangian coordinates determined by V. As this is an important and independent step we present it in a separate subsection. 4. Lagrangian coordinates and linearization We start with rewriting the problem (4.) defined on Q T to a problem defined on a fixed spatial domain (, T ) using time dependent change of coordinates. To this end we use the formula (.5). We set ϱ(t, y) := ϱ(t, X(t, y)), ũ(t, y) := u(t, X(t, y)) (4.3) and we denote the components of the vector X as X = (X, X, X 3 ). To proceed we also need the inverse mapping to X(t, y) which we denote by Y(t, x), thus it holds for all t and all x t and again we denote the components of Y as Y = (Y, Y, Y 3 ). Differentiating (4.4) with respect to time we obtain X(t, Y(t, x)) = x (4.4) X t + X y Y t = (4.5) and thus Y t = V xy. (4.6) Using (4.6) we thus transform the time derivative of u i as follows u i t = ũ i t + yũ i Y t = ũ i t yũ i (V x Y). (4.7) The spatial derivatives transform just by multiplying by the Jacobian of the change of coordinates. Therefore the i-th component of the equation (4.) rewritten in terms of ϱ, ũ defined on fixed domain reads ϱ [ ũ i t ũ i y j V k Y j x k ] µ ũ i y k y l Y l x p Y k x p (4.8) µ ũ i x Y k ( µ y k 3 + η) ũ p Y l Y k ( µ y k y l x p x i 3 + η) ũ p Y k = y k x i x F i, p

15 where we have used Einstein summation convention. This can be rewritten as a usual linearized momentum equation with a transport term on the left hand side and a right hand side containing terms which are either small for small times or of lower order: where ϱ ũ t µ y(ũ) ( µ 3 + η) ydiv y ũ = F + ϱv y ũ + R( ϱ, ũ) =: F( ϱ, ũ). (4.9) R( ϱ, ũ) = ϱ ũ ( Yj V k y j x k δ jk + ( µ 3 + η) ũ p y k y l ) + µ ũ y k y l ( Yl x p x Y k δ lp e k ( ) Yl Y k δ lp δ kp x p x p ) + µ ũ y k x Y k + ( µ 3 + η) ũ p y k x Y k x p, where e j is the j-th unit vector. The boundary conditions are transformed in a nontrivial way as well. In particular, the condition (4.) transforms as (4.) (ũ V)(t, y) n(y) = (ũ V)(t, y) (n(y) n(x(t, y))) + (V(t, X(t, y)) V(t, y)) n(x(t, y)) =: d(ũ, V)(t, y) (4.) for y. Note however, that for small times, the expression d(ũ, V) will be small due to the fact that the mapping X is close to identity and its regularity is given by the regularity of V. Similarly the boundary condition (4.) 3 will also be transformed, however since it contains differentiation, the resulting expression is more complicated. We denote by τ, τ tangent vectors to. A lengthy yet straightforward computation yields for y the following: µ( y ũ + T y ũ)(t, y)n(y) τ k (y) + κ(ũ V)(t, y) τ k (y) = (4.) = µ ( y ũ(t, y)(i x Y) + ((I T x Y) T y ũ(t, y))t ) n(x(t, y)) τ k (X(t, y)) + µ( y ũ + T y ũ)(t, y)[(n(y) n(x(t, y))) τ k (X(t, y)) + n(t, y) (τ k (y) τ k (X(t, y)))] + κ(ũ V)(t, y) (τ k (y) τ k (X(t, y))) + κ(v(t, X(t, y)) V(t, y)) τ k (X(t, y)) =: B(ũ, V)(t, y). Again we emphasize that despite the rather complicated structure of B(ũ, V) it is easy to observe, that this expression will be small for small times as a consequence of a fact that X is close to identity for small times. The right hand side of (4.9) and boundary conditions (4.), (4.) contains the solution and variable coefficients dependent of the change of variables. However, all these quantities will remain small for small times in appropriate norms. Therefore what is important is the structure of the left hand side and in particular we will be able to solve the system (4.9), (4.), (4.) once we have solved the following linear problem on a fixed domain (, T ). ϱ u t µ y(u) ( µ 3 + η) ydiv y u = F, (4.3) (u V) n Γ = d, ( [S( y u)n] tan + κ [u V] tan ) Γ = B. For simplicity we denote the unknown of this system as u instead of ũ. This system is solved in the next subsection. 3

16 4. Solution of the linear momentum equation on a fixed domain In this subsection we work with a system of differential equations stated on a domain (, T ). We again skip emphasizing the domains of all function spaces in this subsection in order to shorten the notation and we note here that by L p (B) we mean here the function space L p (, T, B( )). We also skip the index y in differential operators. Lemma 4. Assume that B and d admits an extension to given by u b n Γ = d, ([ S( u b )n ] tan + κ [ u b] ) Γ = B (4.4) tan such that u b Y(t). Let p, V satisfy the assumptions of Theorem.. Assume further that ϱ L (H ), ϱ t L (H ), F L (H ), F t L (L ), u H 3 ( ) and V C ((, T ) R 3 ). Then there exists a unique solution u to the problem (4.3) such that u Y(T ) φ( ϱ L (H ), ϱ t L (H )) ( F L (H ) L (L ) + F t L (L ) + u b Y(T ) (4.5) + u H 3 + V L (W, ) + V t L (H ) + V tt L (L )) ), where φ denotes a positive increasing function of its arguments. The result is based on Lemmas.-.4 from [34] where the same linear system is considered, however with d = B = V =. Therefore we present here some details to show how we treat the inhomogeneous boundary data. For simplicity of notation we set for the rest of this subsection :=. We start with removing the inhomogeneity from the boundary data. For this purpose we take the extension u b defined by (4.4). Taking ũ = u u b we obtain ϱ t ũ µ (ũ) ( µ 3 + η) divũ = F ϱ t u b, (4.6) (ũ V) n Γ =, ([S( ũ)n] tan + κ [ũ V] tan ) Γ =, where F = F + µ (u b ) + ( µ 3 + η) divub. (4.7) We have to solve the above system, for simplicity we denote ũ again by u. We can also assume the friction coefficient κ =, positive friction yields only additional lower order terms which are easy to treat. First we can write the weak formulation of (4.6) and using u V as a test function (we have to test by a function with vanishing normal component at the boundary) we obtain the energy estimate u L (L ) + u L (L ) C[ F, V, u b t L (L )]. (4.8) Next we multiply (4.6) by t (u V)+εAu, where ε is sufficiently small and Au = µ u ( µ 3 +η) div u. Integrating over we get ϱ u t dx+ (u V) t Audx+ε Au dx = ε ϱu t Audx+ ϱu t V t dx+ ( F ϱ t u b ) (u t +εau)dx (4.9) 4

17 Integrating by parts the second term on the left hand side we obtain (u V) t Audx = (u V) t div S( u)dx = S( u) : (u t V t )dx = S( u) : u t dx S( u) : V t dx, (u V) t S( u)nds = where in the last step we decomposed S into tangential and normal part and used the boundary conditions. Notice that we have S( u) : u t dx C(µ, η) d dt u L. Moreover, the elliptic theory yields u L C Au L. Now we apply the Hölder and Young inequalities to most of the terms on the right hand side of (4.9) and use the fact that ϱ is bounded from below by a positive constant. The first term on the right hand side can be absorbed by the left hand side for sufficiently small ε and so we get u t L + d dt u L + ε u L C( V t H + F L + ϱub t L ) + C u L, The energy estimate (4.8) gives a bound on the last term of the right hand side. Therefore applying the Gronwall inequality we obtain u t L (L ) + u L (L ) + u L (L ) φ( ϱ L ( (,T )))[ V t L (H ) + F L (L ) + u b Y(T ) ]. (4.) Next we take the time derivative of (4.6), multiply by (u V) t and integrate. We get d ϱ u t dx + Au t (u V) t dx = ϱ t u t dx + ϱu tt V t dx + ϱ t u t V t dx dt + (F t ϱ t u b t) (u V) t dx ϱu b tt (u V) t dx. (4.) As before we integrate by parts the second term on the left hand side. Using the fact that (u V) t n = and the identity [S( u t )n] tan = t [S( u)n] tan = we obtain Au t (u V) t dx = S( u t ) : u t dx S( u t ) : V t dx. The condition (u V) t n = implies the Poincaré inequality for (u V) t which yields u t L C( u t L + V t H ). Now we examine the right hand side of (4.). For the first term we have by Poincaré inequality ϱ t u t dx ϱ t L 3 u t L ( u t L + V t H ) δ( u t L + V t H ) + C(δ) ϱ t L 3 ϱ u t dx, where δ is a sufficiently small number coming from application of Young inequality (we keep this notation in what follows). The remaining terms are estimated directly and we get from (4.) d ϱ u t dx + u t L {δ dt C u t L + C(δ ) ϱ u t dx + V t L 6( ϱ t L + u t 3 L ) } + C(δ )φ( ϱ L )( F t L + ϱ tu b t L + V t H ) + δ u tt L ϱu b tt (u V) t dx. (4.) 5

18 The first term on the right hand side can be absorbed by the left hand side and the second term will be treated with the Gronwall inequality. For the L norm of u t we use (4.). Integrating by parts the last term we get Therefore integrating (4.) in time we obtain T T ϱu b tt (u V) t dxdt = ϱu b t (u V) tt dxdt. u t L (,T ;L ) + u t L (,T ;L ) φ( ϱ L (H ), ϱ t L (H ))[ V t L (H ) + φ ( V t ) L 4 (L ) F t L (L )] + ε( u tt L (L ) + V tt L (L )) + C(ε) u b tϱ L (L ) =: Φ. (4.3) Next, computing Au from (4.6) we get a bound on Au L (,T ;L ) which due to (4.3) and ellipticity of A yields u L (L ) F L (L ) + ϱu b t L (L ) + ϱ L ((,T ) )Φ. (4.4) In order to show the bound on u L (H 3 ) we take the spatial gradient of (4.6) which yields Au = F ϱ (u t + u b t) (u t + u b t) ϱ. (4.5) We compute the L (, T ; L ) norm of the right hand side. From the Hölder inequality we get (u t + u b t) ϱ L (L ) + ϱ (u t + u b t) L (L ) φ( ϱ L (H )) u t + u b t L (H ). Therefore we obtain a bound on 3 u L (L ) which, together with previously obtained estimates on lower derivatives gives a bound on u L (H 3 ) assuming we can estimate u tt L (L ). Hence, in order to close the estimates we need to show a bound for u tt L (L ) which appears in the term Φ in (4.3). Taking the time derivative of (4.6) we obtain the elliptic problem for u t : divs( u t ) = F t ϱ t u t ϱu tt ϱ t u b t ϱu b tt, [S( u t )n] tan =, (u V) t n =, (4.6) recall that we have set κ =. For this problem the classical elliptic theory yields Integrating (4.7) with respect to time we get u t H C ( F t L + ϱ t (u t + u b t) L + ϱu tt L + ϱu b tt L ). (4.7) u t L (H ) C ( F t L (L ) + u b tt L (L ) + u tt L (L ) + ϱ t L (H )( u t L (H ) + u b t L (H )) ). (4.8) Now we multiply (4.6) by (u V) tt and integrate over : ϱ u tt dx divs( u t ) (u V) tt dx = ( F t ϱu b tt) (u V) tt dx+ Again, from the boundary condition (4.6) we have divs( u t ) (u V) tt dx = 6 ϱ t (u t +u b t) (V u) tt +ϱu tt V tt dx (4.9) S( u t ) : (u V) tt dx (4.3)

19 and for the second term on the right hand side of (4.9) we have ϱ t (u + u b ) t (V u) tt dx δ( u tt L + V tt L ) + C(δ)( ϱ t L 4 u t L 4). Treating similarly the other terms on the right hand side of (4.9) we obtain ϱ u tt L + d dt u t L δ u tt L + C(δ) ( F t L + ub tt L + V tt L + ϱ t H ( u t H + ub t H )). (4.3) Integrating this inequality in time we get u tt L (L ) + u t L (H ) φ( ϱ t L (H ) ) ( F t L (L ) + u b tt L (L ) + V tt L (L ) + u t () H ). (4.3) Remark 4. For κ > we obtain in (4.3) additional boundary term (u V) tt κ[(u V) t ] tan ds which contains u tt. However we can integrate this term in time T and the first term has good sign and second is given. d dt κ u V t dsdt = κ (u V) t (T ) ds κ (u V) t () ds Finally we comment the term u t () appearing on the right hand side of (4.3). To this end we differentiate (4.6) with respect to space and multiply by xi v t and formally take the resulting equation in t = (rigorously we show it for a smooth approximation and pass to the limit with the estimate) obtaining Combining (4.8),(4.3) and (4.33) we get u t () H C ( F() H + u() H 3). (4.33) u tt L (L )+ u t L (H )+ u t L (H ) φ( ϱ t L (H ) ) ( F t L (L ) + F() H + u b tt L (L ) + V tt L (L ) + u t () H ), which together with previous estimates and definition of F gives (4.5). Now, since (4.6) is a linear parabolic problem, the existence of solution follow from the estimates we have shown and classical theory of parabolic equations, see for example [3]. 4.3 Proof of Proposition 4. Let us denote We start with the estimate for the extension of the boundary data. A(t) = u Y(t) (4.34) 7

20 Lemma 4. Let V satisfy the assumptions of Theorem.. Then there exists an extension u b (u, V) defined by (4.4) of the boundary data d(u, V), B(u, V) given by (4.) and (4.) satisfying the estimate where E(t) is continuous and E() =. u b Y(T ) E(T )[ + (u V) Y(T ) ], (4.35) The proof consist in defining the extension in a special way to ensure that, roughly speaking, the regularity of u b is the same as the regularity of u. We derive an explicit formula for u b and using the assumed regularity of V and u together with smallness of time we show the estimate (4.35). As the proof it is quite technical, we show it in the Appendix. Next, we need the following estimate for the right hand side of the momentum equation in Lagrangian coordinates: Lemma 4.3 For R(ϱ, u) defined by (4.) we have for any ε > ϱv x u + R(ϱ, u) L (H ) + t [ϱv x u + R(ϱ, u)] L (L ) C[(ε + tc(ε) + E(t))A(t)], (4.36) where E(t) is small for small times. Proof: The first two terms on the RHS of (4.36) contain derivatives w.r.t. x and now we need estimates in Lagrangian coordinates y. However we can easily observe that ϱv x u = ϱv y u + R (ϱ, u) where R (ϱ, u) L (H ) + t R (ϱ, u) L (L ) E(t)A(t) with E(t) small for small times, therefore we can work with these terms directly. Let us treat the term ϱv u. From the interpolation inequality we have for any ε > Therefore using again (3.3) we get t u L (L 4 ) ε u L (H 3 ) + C(ε)t u L (H ). ϱ V u t ϱ L 8 V L 8 u L 4 C[ε u L (H 3 ) + tc(ε) u L (L ) ]. Next, recalling the definition (4.) of R(, ) we see that it contains terms with derivatives of u of order up to multiplied by quantities which are small for small times, therefore R(ϱ, u) L (H ) E(t) u L (H 3 ), where E(t) is small for small times. Let us estimate the time derivative. We have t ϱ V u t ϱ L (L ) V L (L ) t u t L (H ) Ct[A(t)] and treating similarly the other terms coming from the chain rule we obtain t [ϱv u] L (L ) C[εA(t) + C(ε)tA(t)]. 8

21 Finally, the structure of R implies t R(ϱ, u) L (L ) E(t)[ u L (H 3 ) + u t L (H )] E(t)A(t) with E(t) as above. This estimate completes the proof of (4.36). Now for clarity let us denote again functions defined in Lagrangian coordinates by ũ, ϱ. Combining (4.35) and (4.36) we see that the right hand side of the system (4.9),(4.),(4.) satisfies F( ϱ, ũ) L (H ) + F t ( ϱ, ũ) L (L ) + u b (ũ, V) Y(T ) F L (H ) + F t L (L ) + E(T ) ( ) A(t) + V L (H 3 ) + V t L (H ) + V tt L (L ) (4.37) where u b (ũ, V) is the extension of the boundary data d(ũ, V), B(ũ, V) given by Lemma 4. and E(t) is small for small times. Therefore from (4.5) we obtain the estimate (4.). Moreover, the right hand side of (4.9), (4.), (4.) is linear w.r.t. ũ, therefore the existence of a unique solution follows from the Banach fixed point theorem. Therefore, as Lagrangian transformation is a diffeomorphism for small times, u(t, x) = ũ(t, Y (y, x)) is a solution of (4.). 5 Proof of Theorem. 5. Boundedness of the sequence of approximations In this section we use the estimates for the linear problems to show that the sequence (ϱ n, u n ) defined by the iterative scheme described in Section.3 is bounded in the space where we are looking for the solution. Let us denote A n (t) = u n X (T ). (5.) Since our estimate for the linear momentum equation holds on fixed domain, we rewrite (.) in Lagrangian coordinates ϱ n+ t u n+ µ y u n+ ( µ 3 + η) ydiv y u n+ = R n, (u n+ V) n Γ = d(u n, V), ( [S( x u n+ )n] τ k + κ[u n+ V] τ k) Γ = B(u n, V), (5.) where R n = ϱ n+ (V y u n+ u n x u n ) x p(ϱ n+ ) + R(ϱ n+, u n+ ) (5.3) and R(, ) is defined in (4.). The following lemma gives the estimate L (H ) norm of the right hand side of (5.) and L (L ) norm of its time derivative. Lemma 5. For R(ϱ n+, u n+ ) and R n defined by (4.9) and (5.3) respectively we have for any ε > R n L (H ) + t R n L (L ) φ( ta n (t))[(ε + tc(ε) + E(t))A n (t) + E(t)A n+ (t)], (5.4) where E(t) is small for small times. 9

22 Proof: Arguing as in the proof of (4.36) we can replace the derivatives w.r.t. x in the definition of R n with derivatives w.r.t. y. Since the pressure p is a C function of the density we have and therefore p(ϱ) = p (ϱ) ϱ + p (ϱ) ϱ ϱ( ϱ + ϱ), p(ϱ) L (H ) ϱ L (L ) t ϱ L (H ) ( + ϱ L (H ) ) φ( t u L (H 3 ))t ϱ L (L ), (5.5) where in the last passage we applied (3.3). Next, similarly to the proof of (4.36) we get and ϱ n+ (u n u n V u n+ ) L (H ) φ( t u L (H 3 ))[ε u L (H 3 ) + tc(ε) u L (L )]. (5.6) R(ϱ n+, u n+ ) L (H ) E(t) u n+ L (H 3 ), where E(t) is small for small times. Let us estimate the time derivative. For the pressure we have therefore from (3.3) and (3.4) we obtain t p(ϱ) ϱ( ϱ t + ϱ t ϱ), t p(ϱ) L (L ) φ( ta n (t))[εa n (t) + C(ε)tA n (t)]. (5.7) The remaining terms are again estimated like in the proof of (4.36) which leads to and t [ϱ n (u n u n V u n+ )] L (L ) φ( ta n (t))[εa n (t) + C(ε)tA n (t)]. (5.8) t R(ϱ n+, u n+ ) L (H ) E(t)[ u n+ L (H 3 ) + u n+,t L (H )] E(t)A n+ (t) (5.9) with E(t) as above. This estimate completes the proof of (5.4). With above lemmas we are ready to show the key estimate for the sequence of approximations Proposition 5. Let A n (t) be defined in (5.). Then there exists M > sufficiently large and T > such that A n (t) M for t T. (5.) Proof: The appropriate choice of the extension of the boundary data given by Lemma 4. ensures Therefore the estimate (4.) applied to (5.) yields u b tt(u n, V) Y(T ) E(T )A n (t). A n+ (t) φ( ta n (t)) [ (ε + tc(ε) + E(t))A n (t) + u H 3 + ϱ H + V Y(T ) for some increasing positive function φ( ). We conclude that there exists M = M(u, V) and T > such that (5.) holds.

23 5. Convergence of the sequence of approximations. Let us denote Subtracting (.) for n + and n we get supplied with boundary conditions Subtracting (.9) we obtain w n+ = u n+ u n, σ n+ = ϱ n+ ϱ n. ϱ n t w n+ µ w n+ ( µ 3 + η) div w n+ = = σ n+ t u n+ + σ n+ (u n u n ) + ϱ n+ (u n w n + w n u n ) (5.) + p (ϱ n ) σ n + ϱ n σ n p (ϱ n )(sϱ n + ( s)ϱ n )ds w n+ n Γt =, [S( w n+ )n] tan + κ[w n+ ] tan Γt =. (5.) t σ n+ + u n σ n+ + σ n+ div u n = ϱ n div σ n σ n ϱ n, σ n+ (, ) =. (5.3) In order to apply our estimates for the linear momentum equation we rewrite (5.) on a fixed domain using the transformation (.5). We obtain with boundary conditions ϱ n t w n+ µ y w n+ ( µ 3 + η) div yw n+ = ϱ n V y w n+ + R(ϱ n, w n+ ) + σ n+ t u n+ + σ n+ (u n x u n ) + ϱ n+ (u n x w n + w n x u n ) (5.4) + p (ϱ n ) x σ n + ϱ n σ n p (ϱ n )(sϱ n + ( s)ϱ n )ds =: R n (w n+ n)(y) Γ = w n+ (n(y) n(x(t, y))) =: d (w n+ ), (5.5) = µ ( y w n+ (t, y)(i x Y) + ((I T x Y) T y w n+ (t, y)) T ) n(x(t, y)) τ k (X(t, y)) + µ( y w n+ + T y w n+ )(t, y)[(n(y) n(x(t, y))) τ k (X(t, y)) + n(t, y) (τ k (y) τ k (X(t, y)))] + κw n+ (t, y) (τ k (y) τ k (X(t, y)))+ =: B (w n+ )(t, y). The left hand side of the system (5.4) has exactly the structure of (4.3). As we will not be able to close the estimate for w n in the regularity we have for the sequence u n (see Remark 5. below) we show the convergence in a weaker space. Let us denote B n (t) = w n L (H ) + w n L (H ) + w nt L (L ). (5.6) Repeating the proof of (4.) we obtain in particular the following (in fact classical) parabolic estimate B n (t) C[ R n L (L ) + B (w n+ ) L (H / ( )) + d (w n+ ) L (H 3/ ( ))]. (5.7) The structure of the boundary data clearly implies b(w n+ ) L (H / ( )) + d(w n+ ) L (H 3/ ( )) E(t) w n+ L (H ). (5.8) The following lemma gives the estimate of the right hand side of (5.4).

24 Lemma 5. Let R n be defined in (5.4). Then R n L (L ) C[ σ n+ L (L ) + σ n L (L ) + (ε + tc(ε) + E(t))( σ n L (L ) + B n (t))], (5.9) where E(t) is small for small times. Proof: Arguing as in the proofs of Lemma 5. and Proposition 5. we get R(ϱ n, w n+ ) L (H ) E(t) w n+ L (H 3 ). (5.) For the remaining terms, we can follow the proof of Lemma 5.. As we have explained there, we can replace the derivatives w.r.t x with derivatives w.r.t y. We have t Next, consider the term σ n+ u n u n σ n+ L (L ) u n L (H ) C σ n+ L (L ) [ε u n L (H 3 ) + tc(ε) u n L (L ) ]. ϱ n V y w n+ L (L ) ϱ n L (H ) V L (H ) Similarly and t t u n L (5.) w n+ L C[ε w n+ L (H ) + tc(ε) w n+ L (L )]. (5.) ϱ n+ (u n w n + w n u n ) L (L ) C[ε w L (H ) + tc(ε) w n+ L (L )] (5.3) σ n+ t u n+ L (L ) σ n+ L (L ) t u n+ L (H ), (5.4) Finally, under our regularity assumptions on the pressure we have and p (ϱ n ) σ n L (L ) C σ n L (L ) t p (ϱ n ) L C σ n L (H ) [ε ϱ n L (H ) + tc(ε) ϱ L (L ) ] (5.5) ϱ n σ n p (ϱ n )(sϱ n + ( s)ϱ n )ds L (L ) C σ n L (L ) ϱ n L. (5.6) Combining all above estimates and Proposition 5. we get (5.9). In order to show the convergence we need to estimate the norms of σ n on the right hand side of (5.9). For this purpose we investigate the equation (5.3). Again, as in the proof of Proposition 3., we stay in a moving domain and use the transformation Denoting X(t, z) = z + equation (5.3) rewrites as a nonhomogeneous ODE t σ n+ (t, z) = σ n+ (t, X(t, z)), u n (X(s, z))ds. (5.7) t σ n+ + σ n+ div x u n = (ϱ n div w n + w n ϱ n ), σ n+ (, ) =. (5.8)

25 The solution for this equation is given by the following explicit formula t σ n+ (t, z) = exp[ div x u n (s, X(x, z))ds] (5.9) t { ( ) r } ϱ n div x w n + w n x ϱ n )(r, X(r, z) exp[ div x u n (τ, X(τ, z))dτ] dr. Remark 5. Notice that the above formula contains x ϱ n and this is the reason why we have to work with lower regularity. Namely, if we would like to have the regularity X (T ) then we would have to estimate L (L ) of the derivatives of the right hand side of (5.4), therefore x σ n which would require information about 3 xϱ n. Such situation when lack of sufficient information on the denstity requires to work in lower regularity is typical for the compressible Navier-Stokes system, see for example [34],[7]. The required estimate is provided in the following Lemma 5.3 Let σ n+ solve (5.3). Then σ n+ L (L ) (ε + tc(ε))b n (t), σ n+ L (L ) CB n (t). (5.3) Proof: The proof relies on the formula (5.9) and follows the proof of Proposition 3., therefore here we show only the main ideas. The solution formula (5.9) clearly implies σ n (t, z) L C C ϱ n L (H ) t t (ϱ n div w n )(s, ) L + (w n ϱ n )(s, ) L ds (5.3) w n H ds C t ε w n L + C(ε) w n L C[εB n (t) + tc(ε)b n (t)], and the regularity of the change of coordinates together with smallness of time yields the same estimate for σ n (t, x) which gives the first statement of (5.3). Let us denote e(t, z) = exp[ t div x u n (s, X(s, z))ds]. (5.3) Differentiating (5.9) w.r.t z we get t { ( ) } x σ n+ = x z z [e(t, z)] ϱ n div x w n + w n x ϱ n )(r, X(r, z) e(r, z) dr }{{} I [ t { ( ) } ] + e(t, z) ϱ n div x w n + w n x ϱ n )(r, X(r, z) e(r, z) dr. }{{} I Applying the estimate (3.7) to the change of coordinates (5.7) we obtain t t t x z I L C [ div x u n (s, X(s, z))ds] L div u n L 4 { ϱ n div w n + w n ϱ n }ds L 4, 3

26 where all the derivatives are already w.r.t. x. Therefore ( t ) t x z I L C div u n L 4ds { ϱ n div w n L 4 + w n ϱ n L 4}ds C[ε u n L (H 3 ) + C(ε) u n+ L (L )] ϱ n L ( (,T )) w n L (H ) C[ε + tc(ε)]b n (t). (5.33) With I we have t { r } I = e(t, z) ( z div x u n )e(r, z)( ϱ n div w n + w n ϱ n ) dr } {{ } I t + e(t, z) { e(r, z)( ϱn div w n ϱ n div w n + w n ϱ n + w n ϱ n ) } dr. } {{ } I The first part can be estimated exactly as I : x z I L C[ε + tc(ε)]b n (t). (5.34) However, in I we cannot get smallness in time because of the presence of second derivatives of w n, namely and therefore t ϱ n div w n L ϱ n L ( (,T )) w n L (L ) CB n (t), x z I L CB n (t). (5.35) Combining (5.33), (5.34) and (5.35) we obtain the second statement of (5.3). We are now in a position to prove the following Lemma which gives the Cauchy condition for the sequence B n (t), and therefore completes the proof of convergence of the sequence of approximations. Lemma 5.4 There exists T > and < K < such that Proof: Combinining (5.7), (5.9) and (5.3) we obtain B n (t) KB n (t) for t T. (5.36) B n (t) (ε + t(ε) + E(t))(B n (t) + B n (t)) which implies (5.36). Now we conclude the proof of Theorem. in a standard way. The sequence u n converges strongly in the space with topology given by the norm (5.6). On the other hand, the bound (5.) implies weak convergence up to a subsequence in X (T ). The estimates for the linear transport equations imply analogous convergences for the sequence of densities in appropriate spaces. Therefore the limit (u, ϱ) satisfies the regularity given in Theorem.. 4

27 6 Weak-strong uniqueness 6. Energy inequality In the remainder of the paper we assume that κ, η =, however all the results hold (with appropriate modifications of formulae) also with κ, η >. Our first result is a crucial observation allowing for proving the main theorem about weak-strong uniqueness. Proposition 6. Let the assumptions of Theorem. be satisfied. Then the problem (.)-(.7), (.4) admits a weak solution on any time interval (, T ) in the sense specified through Definition., satisfying moreover the energy inequality in the following form + τ τ ( ) ϱ u + H(ϱ) (τ, ) dx + ( ) (ϱu) + H(ϱ ) dx + ϱ t ( µ ( x u + t xu 3 div xui µ xu + t xu t 3 div xui dxdt (6.) (ϱu V)(τ, ) dx (ϱu) V(, ) dx τ ) ) : x V ϱu t V ϱu u : x V p(ϱ)div x V dxdt. τ Proof: We follow the same series of approximations and penalizations as it is introduced in [, Section 3] in the proof of Theorem.. The starting point is thus the modified energy inequality written on the fixed domain B which is a ball large enough such that t B for all t [, T ] and V = on B, see formula (3.) in [] B ( ϱ u + H(ϱ) + δ ) β ϱβ (τ, ) dx + τ B + ε τ Γ t (u V) n + τ ( + (µ ω x u + t xu ) B 3 div xui B ds x dt B µ ω x u + t xu 3 div xui ( ϱ,δ (ϱu),δ + H(ϱ,δ ) + ( ) (ϱu V)(τ, ) (ϱu),δ V(, ) dx δ ) β ϱβ,δ dxdt (6.) : x V ϱu t V ϱu u : x V p(ϱ)div x V δ ) β ϱβ div x V dxdt. Passing first with ε to zero, it is not difficult to observe that using the a priori estimates available, all the terms on the right hand side of (6.) converge to their counterparts. On the left hand side the last term is positive and thus can be omitted. Finally, using the convexity of S( x u) : x u we have T B T S ω ( x u) : x u dxdt lim inf S ω ( x u ε ) : x u ε dxdt. (6.3) ε B Next, passing with ω to zero, we first observe that all the terms which include the density can be rewritten as the integrals over t instead of integrals over B using the fundamental Lemma 4. in []. The viscosity term on the right hand side can be treated easily, in particular the integral over B \ t vanish due to the fact that µ ω on this set. On the left hand side we split the integral of the term with stress tensor into two parts, the integral over B \ t can be omitted since it is positive and on t we use the fact that µ ω = µ is constant and thus we can use again the convexity of S( x u) : x u to obtain similar inequality as (6.3). dx 5