Low stratification of a heat-conducting fluid in time-dependent domain
Ondřej Kreml, Václav Mácha, Šárka Nečasová, Aneta Wróblewska-Kamińska
aa r X i v : . [ m a t h . A P ] D ec Low stratification of a heat-conducting fluid intime-dependent domain
Ondˇrej Kreml ∗ V´aclav M´acha ∗ ˇS´arka Neˇcasov´a ∗ Aneta Wr´oblewska-Kami´nska † December 18, 2018 Institute of Mathematics of the Academy of Sciences of the Czech RepublicˇZitn´a 25, 115 67 Praha 1, Czech Republic Institute of Mathematics, Polish Academy of Sciences´Sniadeckich 8, 00-656 Warszawa, Poland
Abstract
We study the low Mach number limit of the full Navier-Stokes-Fourier system in the caseof low stratification with ill-prepared initial data for the problem stated on moving domainwith prescribed motion of the boundary. Similarly as in the case of a fixed domain we recoveras a limit the Oberback-Boussinesq system, however we identify one additional term in thetemperature equation of the limit system which is related to the motion of the domain andwhich is not present in the case of a fixed domain. One of the main ingredients in the proofare the properties of the Helmholtz decomposition on moving domains and the dependenceof eigenvalues and eigenspaces of the Neumann Laplace operator on time.
The mathematical theory of singular limits of systems of equations describing fluid motiongoes back to the seminal work of Klainerman and Majda [23]. The motivation for a study ofsuch type of limits follows from the generality of corresponding equations. More precisely, ina case of the full Navier-Stokes-Fourier system the equations describe a spectrum of possiblemotions e.g. sound waves or models of gaseous stars in astrophysics. Such type of studyallows us to eliminate unimportant or unwanted modes of motion, as a consequence of scalingand asymptotic analysis. The aim of the asymptotic analysis of various physical systems isto derive a simplified system of equations which can be solved numerically or analytically seee.g. Zeytounian [27].The goal of the mathematical analysis of low Mach number limits is to fill up the gapbetween the compressible fluids and their ”idealized” incompressible models. There are twoways how to introduce the Mach number into the system, from the physical point of viewdifferent but from the mathematical point of view completely equivalent. The first approachconsiders a varying equation of state as well as the transport coefficients see work of Ebin[8], Schochet [25]. The second way is to evaluate qualitatively the incompressibility using ∗ The works of O.K., V.M. and ˇS.N. were supported by project GA16-03230S and by RVO 67985840. Part ofwork was done during stay of O.K. at Imperial College London which was supported by the grant Iuventus Plus0871/IP3/2016/74. † The work of A.W.-K. is partially supported by a Newton Fellowship of the Royal Society and by the grant Iuven-tus Plus 0871/IP3/2016/74 of Ministry of Sciences and Higher Education RP. Her stay at Institute of Mathematicsof Academy of Sciences, Prague was supported by 7AMB16PL060. he dimensional analysis. We rewrite our system in the dimensionless form by scaling eachvariable by its characteristic value, see Klein [11].The mathematical analysis of singular limits in the frame of strong solutions can be referredto works of Gallager [22], Schochet [25], Danchin [4], Hoff [10]. The seminal works of Lions[24] and the extension by Feireisl et al. [21] on the existence of global weak solutions in thebarotropic case gave a new possibility of rigorous study of the singular limits in the frameof weak solutions see work of Desjardins and Grenier [5], Desjardinds, Grenier, Lions andMasmoudi [6].The mathematical theory of the full Navier-Stokes-Fourier system was studied by Feireisl.First he developed a concept of variational solution under the assumption that the pressurecan be decomposed into the elastic part and the thermal part. He introduced a definitionof the weak solution using the thermal energy inequality instead of the energy equation andcomplementing the system with the global total energy inequality [15, 16]. Later, he developeda new concept of a variational solution to the compressible Navier-Stokes-Fourier system basedon the principles of global energy conservation and nonnegative local entropy production. Thisconcept in particular allowed to develop important new results concerning the singular limitsof variational solutions to the full system, see Feireisl, Novotn´y [17]. Let us also mentionthat for the full system a low Mach number convergence on a short time interval within theframework of regular solutions was proved by Alazard [1].In real world applications there are many problems where the fluid interacts with a bound-ary of its container which is not fixed and moves either by a prescribed motion or the motion ofthe boundary is related to the motion of the fluid. The mathematical theory of such motionsthen becomes even more complex and additional difficulties arise. In this paper we study thefirst, and arguably the easier case, namely a problem in a moving domain whose motion isprescribed by a given velocity field V ( t, x ).In the barotropic case, the existence theory of global weak solution was proved by Feireislet al. [14, 19] for the Dirichlet and Navier type of boundary conditions, respectively. Moreover,in the framework of weak solutions the singular limit (low Mach number limit) in the case ofmoving domain was investigated by Feireisl et al. in [18, 20].Concerning the full Navier-Stokes-Fourier system, the global existence of weak/variationalsolutions was extended to the case of moving domain by Kreml et al. see [12, 13].The aim of this paper is to fill up the gap of theory of singular limits by examining the lowMach number limit for the full Navier-Stokes-Fourier system on moving domains. We considera low stratification with ill-prepared initial data. For a fixed domain the target system is theOberback-Boussinesq system and the convergence of sequence of variational solutions to theprimitive system to the weak solution of the target system was proved by Feireisl and Novotn´y[17]. Since the domain in our case is moving we can no longer assume that the potential ofthe driving force F ( x ) satisfies ´ Ω t F ( x ) d x = 0. This is the reason why in the limit we recoverthe Oberback-Boussinesq system with a new additional term.The paper is structured as follows. In Section 1, the variational formulation of the primitivesystem (scalled system) is introduced and the existence theorem is stated. Section 2 is devotedto the target system - limit system where the Oberback-Boussinesq system is recovered as alow Mach number limit of the full system which in particular differs from case of a fixeddomain. In Section 3 we state the uniform estimates and perform the limits in the continuityequation, entropy balance and momentum equation where the limit of the convective termremains unspecified.Finally, Section 4 is devoted to the study of the limit of the convective term. Firstly,we mention that in comparison with fixed domain the Helmholtz decomposition depends ontime. The main problem is a possible development of fast oscillations in the momenta ̺ ε u ε , ε → ̺ ǫ , ϑ ǫ satisfy a linear wave equation. We present the reduction to finite number of modesand as in the barotropic case on moving domain we deal with the fact that the eigenvaluesand eigenfunctions of the Neumann Laplace equation depend on time. .1 Primitive system Let us consider the full system on time dependent domain in low Mach number regime whichis given by the following ∂ t ̺ + div x ( ̺ u ) = 0 , (1.1) ∂ t ( ̺ u ) + div x ( ̺ u ⊗ u ) + 1 ε ∇ x p = div x S + 1 ε ̺ ∇ x F, (1.2) ∂ t ( ̺s ) + div x ( ̺s u ) + div x (cid:16) q ϑ (cid:17) = σ ε , (1.3)dd t ˆ (cid:18) ε ̺ | u | + ̺e − ε̺F (cid:19) d x = 0 . (1.4)The entropy production measure σ ε satisfies σ ε ≥ ϑ (cid:16) ε S : ∇ x u − q ϑ · ∇ x ϑ (cid:17) . (1.5)In particular the number ε > u char √ p char /̺ char ) tobe sufficiently small (the speed of sound dominates characteristic fluid velocity) and a Froudenumber to be equal √ ε which is related to the low stratification. We consider this systemof equations being mathematical formulations of the balance of mass, linear momentum,entropy and total energy respectively and to be satisfied on the space-time cylinder Q T = ∪ t ∈ (0 ,T ) { t } × Ω t describing a physical domain moving in time. Unknowns are the density ̺ : Q T [0 , ∞ ), the velocity u : Q T R and the temperature ϑ : Q T [0 , ∞ ). Thepotential of the external body force F = F ( x ) is assumed to be independent of time. Otherquantities appearing in these equations are functions of the unknowns, namely the stresstensor S , the internal energy e , the pressure p , the entropy s , and the entropy production rate σ ε . To be more precise, the time dependent domain Ω t is prescribed by movement of itsboundary on the time interval [0 , T ]. In order to describe this movement we consider a givenvelocity field V ( t, x ) for t ≥ x ∈ R which is smooth enough. Then the position of thedomain Ω t at time t > t X ( t, x ) = V (cid:16) t, X ( t, x ) (cid:17) , t > , X (0 , x ) = x, (1.6)and by a given bounded initial domain Ω ⊂ R asΩ τ = X ( τ, Ω ) , with Γ τ = ∂ Ω τ . (1.7)The system of equations (1.1)-(1.4) is complemented by the following boundary conditions.We assume that the boundary of the domain is impermeable, hence( u − V ) · n | Γ τ = 0 for any τ ≥ , (1.8)where n ( t, x ) denotes the unit outer normal vector to Γ t . We prescribe full slip boundarycondition for the velocity field u , meaning[ S n ] × n | Γ τ = 0 (1.9)and for the heat flux – the conservative boundary condition q · n | Γ τ = 0 . (1.10)Additionally, we assume that the moving domain does not change its volume ( | Ω τ | = | Ω | forany τ ≥ V such thatdiv x V ( τ, · ) = 0 for any τ ≥ . (1.11)Finally, the system (1.1)-(1.4) is supplemented with initial conditions ̺ , u , ϑ and wedenote e := e ( ̺ , ϑ ) and s := s ( ̺ , ϑ ) . In particular, we assume that the initial data areill-prepared and take the form ̺ ,ε = ̺ + ε̺ (1)0 ,ε , ϑ ,ε = ϑ + εϑ (1)0 ,ε , where ̺ > , ϑ > ˆ Ω ̺ (1)0 ,ε d x = 0 , ˆ Ω ϑ (1)0 ,ε d x = 0 for all ε > ̺ (1)0 ,ε , u ,ε , ϑ (1)0 ,ε are bounded measurable functions for all ε > . (1.14) .2 Hypotheses Motivated by [12, 17] we introduce the following set of assumptions, which allow to obtainthe existence of weak solutions.The stress tensor S is determined by the standard Newton rheological law S ( ϑ, ∇ x u ) = µ ( ϑ ) (cid:18) ∇ x u + ∇ tx u −
23 div x u I (cid:19) + η ( ϑ )div x u I , µ > , η ≥ . (1.15)We assume the viscosity coefficients µ and η are continuously differentiable functions of theabsolute temperature, namely µ, η ∈ C [0 , ∞ ) and satisfy0 < µ (1 + ϑ ) ≤ µ ( ϑ ) ≤ µ (1 + ϑ ) , sup ϑ ∈ [0 , ∞ ) | µ ′ ( ϑ ) | ≤ m, (1.16)0 ≤ η ( ϑ ) ≤ η (1 + ϑ ) . (1.17)The heat flux q satisfies the Fourier law for in the following form q = − κ ( ϑ ) ∇ x ϑ, (1.18)where the heat coefficient κ can be decomposed into two parts κ ( ϑ ) = κ M ( ϑ ) + κ R ( ϑ ) , where κ M , κ R ∈ C [0 , ∞ ) , (1.19)0 < κ R (1 + ϑ ) ≤ κ R ( ϑ ) ≤ κ R (1 + ϑ ) , (1.20)0 < κ M (1 + ϑ ) ≤ κ M ( ϑ ) ≤ κ M (1 + ϑ ) . (1.21)In the above formulas µ , µ , m , η , κ R , κ R , κ M , κ M are positive constants.The quantities p , e , and s are continuously differentiable functions for positive values of ̺ , ϑ and satisfy Gibbs’ equation ϑDs ( ̺, ϑ ) = De ( ̺, ϑ ) + p ( ̺, ϑ ) D (cid:18) ̺ (cid:19) for all ̺, ϑ > . (1.22)Further, we assume the following state equation for the pressure and the internal energy p ( ̺, ϑ ) = p M ( ̺, ϑ ) + p R ( ϑ ) , p R ( ϑ ) = a ϑ , a > , (1.23) e ( ̺, ϑ ) = e M ( ̺, ϑ ) + e R ( ̺, ϑ ) , ̺e R ( ̺, ϑ ) = aϑ , (1.24)and s ( ̺, ϑ ) = s M ( ̺, ϑ ) + s R ( ̺, ϑ ) , ̺s R ( ̺, ϑ ) = 43 aϑ . (1.25)According to the hypothesis of thermodynamic stability the molecular components satisfy ∂p M ∂̺ > ̺, ϑ > < ∂e M ∂ϑ ≤ c for all ̺, ϑ > . (1.26)Moreover, lim ϑ → + e M ( ̺, ϑ ) = e M ( ̺ ) > ̺ > , (1.27)and (cid:12)(cid:12)(cid:12)(cid:12) ̺ ∂e M ( ̺, ϑ ) ∂̺ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce M ( ̺, ϑ ) for all ̺, ϑ > . (1.28)We suppose that there is a function P satisfying P ∈ C [0 , ∞ ) , P (0) = 0 , P ′ (0) > , (1.29)and two positive constants 0 < Z < Z such that p M ( ̺, ϑ ) = ϑ P (cid:18) ̺ϑ (cid:19) whenever 0 < ̺ ≤ Zϑ , or, ̺ > Zϑ (1.30)and p M ( ̺, ϑ ) = 23 ̺e M ( ̺, ϑ ) for ̺ > Zϑ . (1.31) .3 Variational formulation of the primitive system We work with a variational formulation of the primitive system (1.1)-(1.4). Namely, theequation (1.1) is fulfilled in the sense of renormalized solutions introduced by DiPerna andLions [7]: ˆ T ˆ Ω t ̺B ( ̺ )( ∂ t ϕ + u · ∇ x ϕ ) d x d t = ˆ T ˆ Ω t b ( ̺ )div x u ϕ d x d t − ˆ Ω ̺ B ( ̺ ) ϕ (0) d x (1.32)for any ϕ ∈ C c ([0 , T ) × R ), and any b ∈ L ∞ ∩ C [0 , ∞ ) such that b (0) = 0 and B ( ̺ ) = B (1) + ´ ̺ b ( z ) z d z where we have ̺ ≥ , T ) × R .The momentum equation (1.2) is transferred to the following integral identity ˆ T ˆ Ω t (cid:18) ̺ u · ∂ t ϕ + ̺ [ u ⊗ u ] : ∇ x ϕ + 1 ε p ( ̺, ϑ )div x ϕ − S ( ϑ, ∇ x u ) : ∇ x ϕ + 1 ε ̺ ∇ x F · ϕ (cid:19) d x d t = − ˆ Ω ̺ u · ϕ (0 , · ) d x, (1.33)for any test function ϕ ∈ C c ( Q T ; R ) such that ϕ ( T, · ) = 0 and ϕ · n | Γ τ = 0 for any τ ∈ [0 , T ] . Moreover, u , ∇ x u ∈ L ( Q T ; R ) and ( u − V ) · n ( τ, · ) | Γ τ = 0 for a.a. τ ∈ [0 , T ] , (1.34)The entropy balance (1.3) is rewritten in the form of equation ˆ T ˆ Ω t ̺s ( ∂ t ϕ + u · ∇ x ϕ ) d x d t − ˆ T ˆ Ω t κ ( ϑ ) ∇ x ϑ · ∇ x ϕϑ d x d t + h σ ε ; ϕ i = − ˆ Ω ̺ s ϕ (0) d x (1.35)for all ϕ ∈ C ( Q T ) such that ϕ ( T, · ) = 0.Finally, the energy inequality has to cover the movement of the domain, hence we get ˆ Ω τ (cid:18) ε ̺ | u | + ̺e − ε̺F (cid:19) ( τ, · ) d x ≤ ˆ Ω (cid:18) ε ̺ u ) + ̺ e − ε̺ F − ε ̺ u · V (0) (cid:19) d x − ε ˆ τ ˆ Ω t (cid:18) ̺ ( u ⊗ u ) : ∇ x V − S : ∇ x V + ̺ u · ∂ t V + 1 ε ̺ ∇ x F · V (cid:19) d x d t + ε ˆ Ω τ ̺ u · V ( τ, · ) d x (1.36)for a.a. τ ∈ (0 , T ). Notice that the term containing div x V vanishes due to assumption (1.11)and (1.36) is an inequality, what differs this formulation from the one in [17] (see [12]).Let us remark that when writing f ∈ L ∞ (0 , T ; L q (Ω t )) for some q ∈ [1 , ∞ ) we mean thatthe mapping t → k f ( t, · ) k L q (Ω t ) is measurable and bounded function on time interval [0 , T ].We have Theorem 1.1.
Let Ω ⊂ R be a bounded domain of class C ν with some ν > , and let V ∈ C ([0 , T ]; C c ( R ; R )) be given. Assume that hypothesis (1.15) – (1.31) are satisfied, let F ∈ W , ∞ ( R ) and let ε > but sufficiently small s.t. ̺ ,ε ≥ , ϑ ,ε > .Then the problem (1.1) – (1.4) with boundary conditions (1.8) , (1.9) , (1.10) and initial con-ditions (1.12) – (1.14) , where the entropy production rate σ ε satisfies (1.5) , admits a variationalsolution on any finite time interval (0 , T ) . Namely, the trio ( ̺ ε , u ε , ϑ ε ) satisfies (1.32) – (1.36) .Moreover • ̺ ε ∈ L ∞ (0 , T ; L (Ω t )) , ̺ ε ≥ , ̺ ε ∈ L q ( Q T ) for certain q > , • u ε , ∇ x u ε ∈ L ( Q T ) , ̺ ε u ε ∈ L ∞ (0 , T ; L (Ω t )) , • ϑ ε > a.a. on Q T , ϑ ε ∈ L ∞ (0 , T ; L (Ω t )) , ϑ ε , ∇ x ϑ ε ∈ L ( Q T ) , and log ϑ ε , ∇ x log ϑ ε ∈ L ( Q T ) , • ̺ ε s ( ̺ ε , ϑ ε ) , ̺ ε s ( ̺ ε , ϑ ε ) u ε , q ( ϑ ε ) ϑ ε ∈ L ( Q T ) , roof. This theorem for ε = 1 and F = 0 has been recently proved in [12]. In order toaccommodate nonzero forcing F = 0 we do not need any additional technique in the proof,we just handle lower order terms ̺ ∇ x F in the momentum equation and ̺F in the energyinequality. In particular one easily observes that using the penalization technique as in [12],both these terms become equal to zero on the ”solid” part of the artificial domain B whenwe choose the initial density ̺ to be equal to zero on the solid part. The scaling by ε doesnot represent any additional difficulties in the proof of existence of variational solutions. First, we introduce G ( t ) := Ω t V ( t, x ) · ∇ x F ( x ) d x = 1 | Ω t | ˆ Ω t V ( t, x ) · ∇ x F ( x ) d x. (2.1)We claim that the following version of the Oberback-Boussinesq system is recovered as thelow Mach number limit div x U = 0 (2.2) ̺ ( ∂ t U + div x ( U ⊗ U )) + ∇ x Π − µ ( ϑ )∆ x U = r ∇ x F (2.3) ̺ c p ( ∂ t Θ + div x (Θ U )) − κ ( ϑ )∆ x Θ − α ̺ ϑ U · ∇ x F = − α ̺ ϑG (2.4) r + ̺ α Θ = 0 . (2.5)This system is considered on a time dependent domain Q T and supplemented with boundaryconditions( U ( τ, · ) − V ( τ, · )) · n | Γ τ = 0 , (( ∇ x U + ∇ tx U ) n ) × n | Γ t = 0 , ∇ x Θ · n | Γ t = 0 (2.6)and initial data U (0 , · ) = U , Θ = Θ(0 , · ) in Ω (2.7)Here α = ̺ ∂ ϑ p∂ ̺ p ( ̺, ϑ ) and c p = ∂ ϑ e ( ̺, ϑ ) + α ϑ̺ ∂ ϑ p ( ̺, ϑ ).Note that the difference with respect to classical Oberback-Boussinesq system is the pres-ence of additional forcing term in the equation for temperature variation Θ. As we will seelater, the presence of this term in the system is related to the fact that unlike in the case ofa fixed domain, it is no longer possible to assume ˆ Ω t F ( x ) d x = 0for all t ∈ [0 , T ). It is however interesting to notice, that G ≡ F and in that case one endsup with a usual Oberback-Boussinesq system.We also note that G ( t ) can be written as G ( t ) = Ω t div x ( V ( t, x ) F ( x )) d x = 1 | Ω t | ˆ Γ t F V · n d x and combined with the boundary condition (2.6) we have G ( t ) = Ω t div x ( U ( t, x ) F ( x )) d x = Ω t U ( t, x ) · ∇ x F ( x ) d x, yielding that (2.4) can be written as ̺ c p ( ∂ t Θ + div x (Θ U )) − κ ( ϑ )∆ x Θ = α ̺ ϑ (cid:18) div x ( F U ) − div x ( F U ) (cid:19) . Definition 2.1.
We say that U , Θ is a weak solution to the Oberback-Boussinesq system (2.2) – (2.5) if the following holds: • div x U ( τ, · ) = 0 and ( U ( τ, · ) − V ( τ, · )) · n | Γ τ = 0 for a.a. τ ∈ (0 , T ) , The equation ˆ T ˆ Ω t ̺ U · ∂ t ϕ + ̺ ( U ⊗ U ) : ∇ x ϕ − µ ( ϑ ) (cid:0) ∇ x U + ∇ tx U (cid:1) : ∇ x ϕ = − ˆ T ˆ Ω t r ∇ x F · ϕ d x d t − ˆ Ω U · ϕ (0 , · ) d x holds for all ϕ ∈ C ∞ c ( Q T ) such that ϕ ( T, · ) = 0 , div x ϕ = 0 and ϕ · n | Γ t = 0 , • equation (2.4) is satisfied a.a. in Q T and ∇ x Θ · n | Γ t = 0 in a sense of traces for a.a. τ ∈ (0 , T ) , • Boussinesq relation (2.5) holds, • U , ∇ x U ∈ L ( Q T ) , ess sup t ∈ (0 ,T ) k U ( t, · ) k L (Ω t ) < c , • the mapping t
7→ k Θ( t, · ) k L q (Ω t ) belongs to W ,qloc ((0 , T ]) ∩ C ([0 , T ]) and the mapping t
7→ k Θ( t, · ) k W ,q (Ω t ) belongs to L qloc ((0 , T ]) for certain q > . Theorem 2.1.
Let Ω ⊂ R be a bounded domain of class C ν with some ν > , and let V ∈ C ([0 , T ]; C c ( R ; R )) be given and satisfy (1.11) . Assume that hypothesis (1.15) – (1.31) are satisfied, let F ∈ W , ∞ ( R ) and let ε > but sufficiently small s.t. ̺ ,ε ≥ , ϑ ,ε > .Let the trio ( ̺ ε , u ε , ϑ ε ) be a variational solution to the problem (1.1) – (1.4) with boundaryconditions (1.8) , (1.9) , (1.10) and initial conditions (1.12) – (1.14) on any finite time interval (0 , T ) , and where entropy production rate satisfies (1.5) .Then ess sup t ∈ (0 ,T ) k ̺ ε − ̺ k L q (Ω t ) → as ε → for certain q > , (2.8) and, for a suitable subsequence, u ε ⇀ U weakly in L ( Q T ) for ε → , (2.9) ∇ x u ε ⇀ ∇ x U weakly in L ( Q T ) for ε → , (2.10) ϑ ε − ϑε ⇀ Θ weakly in L q ( Q T ) for ε → with certain q > , (2.11) where the couple U and Θ is a weak solution according to Definition 2.1 to the Oberbeck-Boussinesq system (2.2) – (2.5) with boundary condition (2.6) and initial data (2.7) U = H [ u ] , Θ = ϑc p (cid:16) ∂ ̺ s ( ̺, ϑ ) ̺ (1)0 + ∂ ϑ s ( ̺, ϑ ) ϑ (1)0 + αF (cid:17) in Ω , (2.12) where u ,ε ∗ ⇀ u , ̺ (1)0 ,ε ∗ ⇀ ̺ (1)0 , ϑ (1)0 ,ε ∗ ⇀ ϑ (1)0 weakly ∗ in L ∞ (Ω ) for ε → . In the above H stand for Helmholtz projection onto the space of solenoidal functions onΩ . Before stating uniform estimates let us recall here some basic notations and results which weneed in proving our convergence results. We refer to [17].First of all we fix a smooth function χ ∈ C ∞ c ((0 , ∞ ) × (0 , ∞ )) such that 0 ≤ χ ≤ , χ = 1on the set O ess , where we define O ess = [ ̺/ , ̺ ] × [ ϑ/ , ϑ ] , O res = (0 , ∞ ) \ O ess . Namely, O ess is a neighborhood of the target density and temperature.Then, we introduce the decomposition on essential and residual part of a measurablefunction h as follows: we define the decomposition h = [ h ] ess + [ h ] res , with [ h ] ess := χ ( ̺ ε , ϑ ε ) h , [ h ] res = (1 − χ ( ̺ ε , ϑ ε )) h . ince div x V = 0, from the energy inequality (1.36) and the entropy balance (1.35) we get " ˆ Ω t ̺ | u | + 1 ε H ϑ ( ̺, ϑ ) − ( ̺ − ̺ ) ∂H ϑ ( ̺, ϑ ) ∂̺ − H ϑ ( ̺, ϑ ) ! − ̺ − ̺ε F d x t = τt =0 + ϑε σ ε (cid:2) ∪ s ∈ [0 ,t ] Ω s (cid:3) ≤ − ˆ τ ˆ Ω t ̺ u ⊗ u : ∇ x V − S : ∇ x V − ̺ u · ∂ t V d x d t + (cid:20) ˆ Ω t ̺ u · V ( t, · ) (cid:21) t = τt =0 − ˆ τ ˆ Ω t ̺ − ̺ε ∇ x F · V d x d t (3.1)where H ϑ ( ̺ ε , ϑ ε ) = ̺ ε ( e ( ̺ ε , ϑ ε ) − ϑs ( ̺ ε , ϑ ε ))(see [17, Chapter 2.2.3]) is a Helmholtz function.By (3.1) and with [17, Lemma 5.1] we obtain the following set of estimates. The detailscan be found in [17, Chapter 5]. Lemma 3.1.
Let assumptions of the Theorem 1.1 be satisfied. Let { ( ̺ ε , u ε , ϑ ε ) } ε> be asequence of weak solutions obtained in Theorem 1.1. Then the following estimates hold ess sup t ∈ (0 ,T ) ˆ Ω t [1( t )] res d x ≤ ε c, (3.2)ess sup t ∈ (0 ,T ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) ̺ ε − ̺ε (cid:21) ess ( t ) (cid:13)(cid:13)(cid:13)(cid:13) L (Ω t ) ≤ c, (3.3)ess sup t ∈ (0 ,T ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) ϑ ε − ϑε (cid:21) ess ( t ) (cid:13)(cid:13)(cid:13)(cid:13) L (Ω t ) ≤ c, (3.4)ess sup t ∈ (0 ,T ) ˆ Ω t (cid:18) [ ̺ ε ] res + [ ϑ ε ] res (cid:19) ( t ) d x ≤ ε c, (3.5)ess sup t ∈ (0 ,T ) k√ ̺ ε u ε k L (Ω t ) ≤ c, (3.6) σ ε [ Q T ] ≤ ε c, (3.7) ˆ T k u ε ( t ) k W , (Ω t ) d t ≤ c, (3.8) ˆ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) ϑ ε − ϑε (cid:19) ( t ) (cid:13)(cid:13)(cid:13)(cid:13) W , (Ω t ) d t ≤ c, (3.9) ˆ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) log( ϑ ε ) − log( ϑ ) ε (cid:19) ( t ) (cid:13)(cid:13)(cid:13)(cid:13) W , (Ω t ) d t ≤ c, (3.10) ˆ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) ̺ ε s ( ̺ ε , ϑ ε ) ε (cid:21) res ( t ) (cid:13)(cid:13)(cid:13)(cid:13) qL q (Ω t ) d t ≤ c, (3.11) ˆ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) ̺ ε s ( ̺ ε , ϑ ε ) ε (cid:21) res u ε ( t ) (cid:13)(cid:13)(cid:13)(cid:13) qL q (Ω t ) d t ≤ c, (3.12) ˆ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) κ ( ϑ ε ) ϑ ε (cid:21) res (cid:18) ∇ x ϑ ε ε (cid:19) ( t ) (cid:13)(cid:13)(cid:13)(cid:13) qL q (Ω t ) d t ≤ c. (3.13)Let us introduce the following notation ̺ (1) ε = ̺ ε − ̺ε , ϑ (1) ε = ϑ ε − ϑε . We assume that ̺ ε is extended to the whole space R by a constant ̺ . Similarly, weextend also the velocity and the temperature to the whole space by a standard extension E t : W , (Ω t ) W , ( R ) which is uniformly bounded with respect to t ∈ [0 , T ] (we refer to E t in the notation so from now onit holds that ϑ ε = E t ϑ ε and u ε = E t u ε .As a consequence of the above Lemma 3.1 we obtain the following convergences (for detailssee [17, Chapter 5.3]) ̺ ε − ̺ → L ∞ (0 , T ; L / ( R )) , (3.14) (cid:18) t → ˆ Ω t ( ̺ ε − ̺ ) ϕ d x (cid:19) → C ([0 , T ]) for all ϕ ∈ L r ′ (Ω t ) with r ∈ [1 , / , (3.15) ϑ ε − ϑ → L ∞ (0 , T ; L ( R )) , (3.16) ̺ (1) ε ∗ ⇀ ̺ (1) weakly ∗ in L ∞ (0 , T ; L / ( R )) , (3.17) ϑ (1) ε ⇀ ϑ (1) weakly in L ( Q T ) , (3.18) ∇ x ϑ (1) ε ⇀ ∇ x ϑ (1) weakly in L ( Q T ) , (3.19) u ε ⇀ U weakly in L (0 , T ; L ( R )) . (3.20) u ε ⇀ U weakly in L (0 , T ; W , ( R )) , (3.21) (cid:20) ϑ ε − ϑε (cid:21) ess ∗ ⇀ ϑ (1) weakly ∗ in L ∞ (0 , T ; L ( R )) , (3.22) (cid:20) ̺ ε s ( ̺ ε , ϑ ε ) ε (cid:21) res → L q ( Q T ) for certain q > , (3.23) h ̺ ε ε i res → L ∞ (0 , T ; L ( R )) , (3.24)[ ̺ ε ] ess [ s ( ̺ ε , ϑ ε )] ess − s ( ̺, ϑ ) ε ∗ ⇀ ̺ (cid:16) ∂ ̺ s ( ̺, ϑ ) ̺ (1) + ∂ ̺ s ( ̺, ϑ ) ϑ (1) (cid:17) weakly ∗ in L ∞ (0 , T ; L ( R )) , (3.25)[ ̺ ε ] ess [ s ( ̺ ε , ϑ ε )] ess − s ( ̺, ϑ ) ε u ε ⇀ ̺ (cid:16) ∂ ̺ s ( ̺, ϑ ) ̺ (1) + ∂ ϑ s ( ̺, ϑ ) ϑ (1) (cid:17) U weakly in L (0 , T ; L ( R )) , (3.26) (cid:20) κ ( ϑ ε ) ϑ (cid:21) ess ∇ x (cid:18) ϑ ε − ϑε (cid:19) ⇀ κ ( ϑ ) ϑ ∇ x ϑ (1) weakly in L (0 , T ; L ( R )) , (3.27) (cid:20) κ ( ϑ ε ) ϑ ε (cid:21) res ∇ x (cid:18) ϑ ε ε (cid:19) → L q ( Q T ) for certain q > , (3.28) ̺ ε u ε ⊗ u ε ⇀ ̺ U ⊗ U weakly in L (0 , T ; L ( R )) , (3.29) S ε ⇀ µ ( ϑ )( ∇ x U + ∇ tx U ) weakly in L q ( Q T ) for certain q > . (3.30) We recall the Reynolds transport theorem:
Theorem 3.1.
Let a general function f = f ( t, x ) belong to C (( t , t ); W , ∞ (Ω t )) and let V ∈ C ( R + × R ) . Then for each t ∈ ( t , t ) there exists a finite derivative dd t ˆ Ω t f ( t, x ) d x = ˆ Ω t ( ∂ t f ( t, x ) + div x ( f V )( t, x )) d x. We proceed with the limit in the continuum equation. By (3.14) and (3.21) we obtain theboundary condition U · n = V · n on Γ τ (3.31)in the sense of traces. Moreover, passing with ε → b ( ̺ ) ≡ B ( ̺ ) ≡ ̺ ˆ T ˆ Ω t ( ∂ t ϕ + U · ∇ x ϕ ) d x d t = − ̺ ˆ Ω ϕ (0 , · ) d x for all ϕ ∈ C c ([0 , T ) × R ). We use the transport Theorem 3.1 to conclude thatdiv x U = 0 a.e. in Q T . ndeed, ˆ T ˆ Ω t ( ∂ t ϕ + U · ∇ x ϕ ) d x d t = ˆ T ˆ Ω t ( ∂ t ϕ − div x U ϕ ) d x d t + ˆ T ˆ Γ t U · n ϕ d S d t = ˆ T ˆ Ω t ( ∂ t ϕ − div x U ϕ ) d x d t + ˆ T ˆ Γ t V · n ϕ d S d t = ˆ T ˆ Ω t ( ∂ t ϕ − div x U ϕ + div x ( V ϕ )) d x d t = − ˆ T ˆ Ω t div x U ϕ d x d t + ˆ T (cid:18) ddt ˆ Ω t ϕ d x (cid:19) d t = − ˆ T ˆ Ω t div x U ϕ d x d t − ˆ Ω ϕ (0 , · ) d x and thus ´ T ´ Ω t div x U ϕ d x d t = 0 for all ϕ ∈ C c ([0 , T ) × R ). Similarly as in [17, Section 5.3.2], we deduce that by convergences from Section 3.1 the balanceof entropy (1.35) in the limit ε → ˆ T ˆ Ω t ̺ (cid:16) ∂ ̺ s ( ̺, ϑ ) ̺ (1) + ∂ ϑ s ( ̺, ϑ ) ϑ (1) (cid:17) ( ∂ t ϕ + U · ∇ x ϕ ) d x d t (3.32) − ˆ T ˆ Ω t κ ( ϑ ) ϑ ∇ x ϑ (1) · ∇ x ϕ d x d t = − ˆ Ω ̺ (cid:16) ∂ ̺ s ( ̺, ϑ ) ̺ (1) + ∂ ϑ s ( ̺, ϑ ) ϑ (1) (cid:17) ϕ (0 , · ) d x. for any ϕ ∈ C ( Q T ), ϕ ( T, · ) = 0.Further, we multiply the momentum equation (1.33) by ε and we let ε → ˆ T ˆ Ω t (cid:16) ∂ ̺ p ( ̺, ϑ ) ̺ (1) + ∂ ϑ p ( ̺, ϑ ) ϑ (1) (cid:17) div x ϕ d x d t = − ˆ T ˆ Ω t ̺ ∇ x F · ϕ d x d t. (3.33)for any ϕ ∈ C ( Q T ), ϕ ( T, · ) = 0, ϕ · n | Γ τ = 0.We define C ( t ) := Ω t F ( x ) d x and we can assume, without loss of generality, that C (0) = 0. The conservation of masstogether with the assumption (1.13) yields ´ Ω t ̺ (1) = 0, whereas the same property for thetemperature ´ Ω t ϑ (1) = 0 is a consequence of (3.32) and (1.13). These properties and (3.33)yield ̺ (1) = − ∂ ϑ p ( ̺, ϑ ) ∂ ̺ p ( ̺, ϑ ) ϑ (1) + ̺F∂ ̺ p ( ̺, ϑ ) − ̺C ( t ) ∂ ̺ p ( ̺, ϑ ) . (3.34)In order to simplify notation we introduce c p ( ̺, ϑ ) = ∂ ϑ e ( ̺, ϑ ) + α ( ̺, ϑ ) ϑ̺ ∂ ϑ p ( ̺, ϑ ) , c p = c p ( ̺, ϑ ) ,α ( ̺, ϑ ) = 1 ̺ ∂ ϑ p ( ̺, ϑ ) ∂ ̺ p ( ̺, ϑ ) , α = α ( ̺, ϑ ) . We plug (3.34) into (3.32), we multiply it by ϑ and we employ Maxwell and Gibbs relationsin order to get ˆ T ˆ Ω t ̺ (cid:16) c p ϑ (1) − ϑα ( F − C ) (cid:17) ( ∂ t ϕ + U · ∇ x ϕ ) d x d t (3.35) − ˆ T ˆ Ω t κ ( ϑ ) ∇ x ϑ (1) · ∇ x ϕ d x d t = − ˆ Ω ϑ̺ (cid:16) ∂ ̺ s ( ̺, ϑ ) ̺ (1)0 + ∂ ϑ s ( ̺, ϑ ) ϑ (1)0 (cid:17) ϕ (0 , · ) d x for any ϕ ∈ C ( Q T ), ϕ ( T, · ) = 0. e use the transport Theorem 3.1 and (3.31) to observe that for all ϕ ∈ C ( Q T ), ϕ ( T, · ) =0 it holds ˆ T ˆ Ω t F ( x )( ∂ t ϕ + U · ∇ x ϕ ) d x d t = ˆ T ˆ Ω t ( F ∂ t ϕ − ϕ div x ( F U )) d x d t + ˆ T ˆ Γ t F ϕ U · n d S d t = ˆ T ˆ Ω t ( ∂ t ( F ϕ ) + div x ( F ϕ V ) − ϕ U · ∇ x F ) d x d t = ˆ T (cid:18) dd t ˆ Ω t F ϕ d x (cid:19) d t − ˆ T ˆ Ω t ϕ U · ∇ x F d x d t = − ˆ Ω F ( x ) ϕ (0 , x ) d x − ˆ T ˆ Ω t ϕ U · ∇ x F d x d t, whereas for the same class of test functions ϕ we have (recalling (2.1) where we defined G ( t )) ˆ T ˆ Ω t C ( t )( ∂ t ϕ + U · ∇ x ϕ ) d x d t = ˆ T ˆ Ω t ( C∂ t ϕ − ϕ div x ( C U )) d x d t + ˆ T ˆ Γ t Cϕ U · n d S d t = ˆ T ˆ Ω t ( ∂ t ( Cϕ ) + div x ( Cϕ V ) − ∂ t Cϕ ) d x d t = ˆ T dd t ˆ Ω t C ( t ) ϕ ( t, x ) d x d t − ˆ T ˆ Ω t G ( t ) ϕ ( t, x ) d x d t = − ˆ T ˆ Ω t G ( t ) ϕ ( t, x ) d x d t. Here we have used C (0) = 0 and the observation thatdd t C ( t ) = dd t Ω t F ( x ) d x = Ω t div x ( F V ) d x = Ω t V · ∇ x F d x = G ( t ) . Taking Θ = ϑ (1) , we deduce from (3.35) the following equation ̺ c p ∂ t Θ + ̺ c p div x (Θ U ) − κ ( ϑ )∆ x Θ − α ̺ ϑ U · ∇ x F = − α ̺ ϑG, with the initial data c p Θ = ϑ (cid:16) ∂ ̺ s ( ̺, ϑ ) ̺ (1)0 + ∂ ϑ s ( ̺, ϑ ) ϑ (1)0 + αF (cid:17) . (3.36)Finally, we define r := ̺ (1) − ̺∂ ̺ p ( ̺, ϑ ) ( F − C ) . Then, (3.34) yields the Boussinesq relation r + ̺ α Θ = 0 . Since div x U = 0 we may take as a test function ϕ ∈ C c ( Q T ), ϕ ( T, · ) = 0, ϕ · n | Γ t = 0 suchthat div x ϕ = 0 in Q T when passing to the limit in the momentum equation (1.33). Relations(3.14)–(3.30) imply ˆ t ˆ Ω t ̺ U · ∂ t ϕ + ̺ U ⊗ U : ∇ x ϕ + µ ( ϑ ) ∇ x U : ∇ x ϕ − ̺ (1) ∇ x F · ϕ d x d t = ˆ Ω ̺ U · ϕ (0 , · ) d x where ̺ U ⊗ U is the weak limit of ̺ ε u ε ⊗ u ε . Note that if ˆ T ˆ Ω t ̺ U ⊗ U : ∇ x ϕ d x = ˆ T ˆ Ω t ( ̺ U ⊗ U ) : ∇ x ϕ d x (3.37) or all ϕ ∈ C c ( Q T ), ϕ ( T, · ) = 0, ϕ · n | Γ t = 0 such that div x ϕ = 0 in Q T , the Theorem 2.1 isproven. The rest of this paper is devoted to the analysis of this particular limit.We start with a simple observation, that due to the uniform bound of ̺ ε u ε ⊗ u ε in L (0 , T ; L ( R )) it suffices to show (3.37) for test functions compactly supported in Q T ∩ (cid:0) (0 , T ) × R (cid:1) . We introduce the Helmholtz decomposition v = H t [ v ] + H ⊥ t [ v ] in L (Ω t ; R ) in a standardway. Namely we define the projection H ⊥ t [ v ] = ∇ x Ψ as the unique solution to the Neumannproblem ∆Ψ = div x v in Ω t , ∇ x Ψ · n = v · n on Γ t , ˆ Ω t Ψ d x = 0 . (4.1)Hence, it is easy to observe thatdiv x H t [ v ] = 0 in Ω t and H t [ v ] · n = 0 on Γ t . Lemma 4.1.
Let z be such that ∂ t div x z ∈ W , (Ω t ) and ∂ t z ∈ W / , (Γ t ) . Then ∂ t H t [ z ] ∈ W , (Ω t ) .Proof. See [26] and [18, Section 3.1].
Although div x U = 0 a.e. in Q T , we cannot conclude that H t [ U ] = 0 due to the inhomogeneousboundary condition U · n = V · n on Γ t . Instead we have U = H t [ U ] + ∇ x W, where ∇ x W = H ⊥ t [ V ] . Note that ˆ T ˆ Ω t H t [ v ] · ϕ d x d t = ˆ T ˆ Ω t v · H t [ ϕ ] d x d t for all v , ϕ ∈ L ( Q T ), this property will be used extensively throughout the rest of this paper.Moreover, we also have k H t [ v ] k L q (Ω τ ) ≤ c ( q ) k v k L q (Ω τ ) for any 2 ≤ q < ∞ and τ ∈ [0 , T ] due to the elliptic regularity theory. Since the domains Ω τ are regular, the constant c ( q ) can be chosen independently of τ , see [9, Theorem 1.2].Convergences from Section 3.1 imply that for ε → H t [ u ε ] ⇀ H t [ U ] weakly in L ( Q T ) , ∇ x H t [ u ε ] ⇀ ∇ x H t [ U ] weakly in L ( Q T ) , H ⊥ t [ u ε ] ⇀ ∇ x W weakly in L ( Q T ) , ∇ x H ⊥ t [ u ε ] ⇀ ∇ x W weakly in L ( Q T ) . We want to prove strong convergence of H t [ u ε ] to H t [ U ] in L ( Q T ). To this end we testthe momentum equation (1.33) by H t [ ϕ ] with ϕ ∈ C c ( Q T ), ϕ ( T ) = 0, ϕ · n = 0 on Γ τ .We denote I ε ϕ ( t ) := ˆ Ω t ( ̺ ε u ε )( t, x ) · H t [ ϕ ( t, x )] d x = ˆ Ω t H t [( ̺ ε u ε )( t, x )] · ϕ ( t, x ) d x and, consequently, I ε ϕ ( t ) − I ε ϕ ( t ′ ) = ˆ t ′ t ˆ Ω τ ( ̺ ε u ε ⊗ u ε − S ( ϑ ε , ∇ x u ε )) : ∇ x H t [ ϕ ] + ̺ ε − ̺ε ∇ x F · H t [ ϕ ] d x d t + ˆ t ′ t ˆ Ω τ ̺ ε u ε · ∂ t H t [ ϕ ] d x d t. (4.2) e estimate the right hand side of (4.2) by (3.3), (3.5), (3.6), (3.8) and the regularity of thetime derivative of H t [ ϕ ]. We end up with (cid:12)(cid:12) I ε ϕ ( t ) − I ε ϕ ( t ′ ) (cid:12)(cid:12) ≤ C | t − t ′ | / . Fix a time interval [ T , T ] and an open set K ⊂ R such that [ T , T ] × K ⊂ Q T . We use theArzel´a-Ascoli theorem to conclude that I ε ϕ is precompact in C ( T , T ) and therefore H t [ ̺ ε u ε ] → H t [ ̺ U ] strongly in C w ( T , T ; L / ( K )) . This implies that H t [ ̺ ε u ε ] → H t [ ̺ U ] strongly in L p ( T , T ; W − , ( K )) , ≤ p < ∞ . We also have ( ̺ ε − ̺ ) u ε = ε ̺ ε − ̺ε u ε → L ( T , T ; L / ( K ))which yields the same property for H t [( ̺ ε − ̺ ) u ε ] and for H ⊥ t [( ̺ ε − ̺ ) u ε ]. Therefore we canwrite ̺ H t [ u ε ] u ε = ( H t [( ̺ − ̺ ε ) u ε ] + H t [ ̺ ε u ε ]) · u ε ⇀ ̺ | U | weakly in L (( T , T ) × K );in particular ˆ T T ˆ K | H t [ u ε ] | = ˆ T T ˆ K H t [ u ε ] · u ε → ˆ T T ˆ K | U | and we conclude that H t [ u ε ] → H t [ U ] strongly in L ([ T , T ] × K ) . We deduce that in order to show (3.37) it is enough to prove that ˆ T ˆ Ω t H ⊥ t [ ̺ ε ( u ε − V )] ⊗ H ⊥ t [( u ε − V )] : ∇ x ϕ d x d t → ϕ ∈ C c ( Q T ), div x ϕ = 0, ϕ (0 , · ) = ϕ ( T, · ) = 0, ϕ · n | Γ t = 0 (for detailssee [18, Section 3.3]). We rewrite the continuity equation (1.32) (with B ≡ , b ≡
0) and the momentum equation(1.33) in the form of the acoustic analogy. To this end we reformulate both in a new variables ̺ (1) ε = ̺ ε − ̺ε , z ε = ̺ ε ( u ε − V ) . (4.4)The continuity equation (1.32) reads as ˆ T ˆ Ω t ε̺ (1) ε ∂ t ϕ + z ε · ∇ x ϕ d x d t = − ˆ T ˆ Ω t ε̺ (1) ε V · ∇ x ϕ d x d t (4.5)for any ϕ ∈ C ∞ c ( Q T ), ϕ (0 , · ) = 0, ϕ ( T, · ) = 0 and the momentum equation (1.33) can bewritten in the following form ˆ T ˆ Ω t ε z ε · ∂ t ϕ + (cid:20) [ p ( ̺ ε , ϑ ε )] ess − p ( ̺, ϑ ) ε − ̺F (cid:21) div x ϕ d x d t = ε ˆ T ˆ Ω t ̺ − ̺ ε ε ∇ x F · ϕ d x d t + ε ˆ T ˆ Ω t (cid:0) H ε : ∇ x ϕ + h ε · ϕ (cid:1) d x d t (4.6)for any ϕ ∈ C ∞ c ( Q T ), ϕ (0 , · ) = 0, ϕ ( T, · ) = 0, ϕ · n | Γ t = 0 for t ∈ [0 , T ], where we set H ε = − ̺ ε u ε ⊗ u ε + S ε − [ p ( ̺ ε , ϑ ε )] res ε I + ̺ ε u ε ⊗ V , (4.7) h ε = ̺ ε ∂ t V + ̺ ε u ε · ∇ x V . (4.8) e also need the entropy balance (1.35) in the form ˆ T ˆ Ω t ε (cid:18) ̺ ε s ( ̺ ε , ϑ ε ) − s ( ̺, ϑ ) ε (cid:19) ∂ t ϕ d x d t = ˆ T ˆ Ω t h ε · ∇ x ϕ d x d t − h σ ε ; ϕ i (4.9)for all ϕ ∈ C ∞ c ( Q T ), ϕ (0 , · ) = 0, ϕ ( T, · ) = 0, where h ε = κ ( ϑ ε ) ϑ ε ∇ x ϑ ε − ̺ ε ( s ( ̺ ε , ϑ ε ) − s ( ̺, ϑ )) u ε . (4.10)In the next step we rewrite the system (4.5), (4.6), (4.9) using new set of variables, namelywe define r ε = ̺ (1) ε + Aζ ̺ ε s ( ̺ ε , ϑ ε ) − s ( ̺, ϑ ) ε − ζ ̺F, (4.11)where we set A = ∂ ϑ p ( ̺, ϑ ) ̺∂ ϑ s ( ̺, ϑ ) , ζ = ∂ ̺ p ( ̺, ϑ ) + A ∂ ϑ p ( ̺, ϑ ) ̺ . (4.12)Theorem 3.1 yields ˆ T ˆ Ω t F ∂ t ϕ d x d t = − ˆ T ˆ Ω t div x ( F ϕ V ) d x d t = − ˆ T ˆ Ω t V · ∇ x F ϕ + F V · ∇ x ϕ d x d t. (4.13)Hence, we sum (4.5) and an appropriate multiple of (4.9) and by (4.13) we end up with ˆ T ˆ Ω t εr ε ∂ t ϕ + z ε · ∇ x ϕ d x d t = ε ˆ T ˆ Ω t (cid:0) h ε · ∇ x ϕ + h ε ϕ (cid:1) d x d t − Aζ h σ ε ; ϕ i (4.14)for all ϕ ∈ C ∞ c ( Q T ), ϕ (0 , · ) = 0, ϕ ( T, · ) = 0, where h ε = − ̺ (1) ε V + 1 ε Aζ h ε + ̺ζ F V , (4.15) h ε = ̺ζ V · ∇ x F. (4.16)The acoustic version of the momentum equation (4.6) is rewritten as ˆ T ˆ Ω t ε z ε · ∂ t ϕ + ζr ε div x ϕ d x d t = ε ˆ T ˆ Ω t ̺ − ̺ ε ε ∇ x F · ϕ d x d t + ε ˆ T ˆ Ω t (cid:0) H ε : ∇ x ϕ + h ε · ϕ + h ε div x ϕ (cid:1) d x d t (4.17)for any ϕ ∈ C ∞ c ( Q T ), ϕ (0 , · ) = 0, ϕ ( T, · ) = 0, ϕ · n | Γ t = 0 for t ∈ [0 , T ], where we use thedefinitions of A and ζ stated in (4.12) together with the notation[ p ( ̺ ε , ϑ ε )] ess − p ( ̺, ϑ ) ε = ∂ ̺ p ( ̺, ϑ ) ̺ (1) ε + ∂ ϑ p ( ̺, ϑ ) ϑ (1) ε + h ε , (cid:20) ̺ ε s ( ̺ ε , ϑ ε ) − s ( ̺, ϑ ) ε (cid:21) ess = ̺∂ ̺ s ( ̺, ϑ ) ̺ (1) ε + ̺∂ ϑ s ( ̺, ϑ ) ϑ (1) ε + h ε ,h ε = 1 ε (cid:18) A (cid:20) ̺ ε s ( ̺ ε , ϑ ε ) − s ( ̺, ϑ ) ε (cid:21) res + Ah ε − h ε (cid:19) , (4.18)and since both p and s are twice continuously differentiable we have by [17, Proposition 5.2]ess sup t ∈ (0 ,T ) ˆ Ω t (cid:12)(cid:12) h ε ( t, x ) (cid:12)(cid:12) d x ≤ Cε, (4.19)ess sup t ∈ (0 ,T ) ˆ Ω t (cid:12)(cid:12) h ε ( t, x ) (cid:12)(cid:12) d x ≤ Cε. (4.20) .4 Time lifting The right hand side of (4.14) contains a measure σ ε and thus the associated solution is notnecessarily continuous. To prevent this, we adopt the method described in [17, Section 5.4.7]- called time lifting - and we introduce a ”primitive” measure Σ ε on Q t defined as h Σ ε ; ϕ i = h σ ε ; I [ ϕ ] i , with I [ ϕ ]( t, x ) = ˆ t ϕ ( τ, e X ( τ, x ))d τ where e X ( τ, x ) is a solution to dd t e X ( τ, x ) = V (cid:16) τ, e X ( τ, x ) (cid:17) with the condition e X ( t, x ) = x . It follows that h Σ ε ; ∂ t ϕ + ∇ x ϕ · V i = h σ ε ; ϕ i and Σ ε can be also identified as a mapping [0 , T )
7→ M + (Ω t ), where Σ ε ( t ) is defined by theduality with a function e ϕ ∈ C (Ω t ) as follows h Σ ε ( t ); e ϕ i = lim δ → + h σ ε ; ψ δ e ϕ ext i , for almost all t ∈ [0 , T )with ψ δ ( τ ) = τ ≤ t δ ( τ − t ) for τ ∈ ( t, t + δ )1 for τ ≥ t + δ and e ϕ ext ( τ, x ) is the extension of the function e ϕ ( x ) given as e ϕ ext ( τ, e X ( τ, x )) = e ϕ ( x ) . It holds that ess sup t ∈ (0 ,T ) k Σ ε ( t ) k M + (Ω τ ) ≤ k σ ε k M + ( Q T ) ≤ ε c. (4.21)We use a notation h Σ ε ( t ); ϕ i := ˆ Ω t Σ ε ( t ) ϕ d x. We define a new variable Z ε = r ε + Aεζ Σ ε . (4.22)and we rewrite (4.14) and (4.17) as ˆ T ˆ Ω t εZ ε ∂ t ϕ + z ε ·∇ x ϕ d x d t = ε ˆ T ˆ Ω t (cid:18) h ε · ∇ x ϕ + h ε ϕ − Aεζ Σ ε V · ∇ x ϕ (cid:19) d x d t (4.23)and ˆ T ˆ Ω t ε z ε · ∂ t ϕ + ζZ ε div x ϕ d x d t = ε ˆ T ˆ Ω t ̺ − ̺ ε ε ∇ x F · ϕ d x d t + ε ˆ T ˆ Ω t (cid:18) H ε : ∇ x ϕ + h ε · ϕ + (cid:18) h ε + Aε Σ ε (cid:19) div x ϕ (cid:19) d x d t. (4.24) We want to reduce our problem and study it only after projecting it into finite number ofmodes. To this end we follow the strategy developed in [18, Section 4] and we introduce theeigenvalue problem ∇ x ω = − λ ( t ) a , div x a = − λ ( t ) ω in Ω t , a · n = 0 on Γ t , hich admits solutions in the form a j ( t, x ) = i p Λ j ( t ) ∇ x ω j ( t, x ) , λ j ( t ) = i p Λ j ( t ) , j = 1 , , . . . . Here Λ j ( t ), ω j ( t, · ) are eigenvalues and eigenfunctions of the Neumann Laplace problem − ∆ x ω = Λ( t ) ω in Ω t , ∇ x ω · n = 0 on Γ t , which admits real eigenvalues 0 = Λ ( t ) < Λ ( t ) ≤ Λ ( t ) ≤ ... We have that { a j ( t, · ) } ∞ j =1 forms an orthonormal basis in H ⊥ t ( L (Ω t )) = n span { i a j ( t, · ) } ∞ j =1 o L (Ω t ) and the eigenspace of λ ( t ) = 0 is H t ( L (Ω t )).Since V is smooth enough we use [3, Theorem 4.3] to conclude (cid:12)(cid:12)(cid:12)(cid:12) j ( t ) − j ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | t − t | , for t , t ∈ [0 , T ], however no such property holds for the eigenfunctions ω j and a j . Thereforewe work with the projections on the eigenspaces spanned by finite number of eigenfunctions.More precisely, fixing an integer M > P M [ ϕ ]( t, · ) := M X j =1 ω j ( t, · ) ˆ Ω t ϕ ( t, y ) ω j ( t, y ) d y, ϕ ∈ L ( Q T ) , Q M [ ϕ ]( t, · ) := M X j =1 a j ( t, · ) ˆ Ω t ϕ ( t, y ) · a j ( t, y ) d y, ϕ ∈ L ( Q T ) . As explained in [18, Section 4] the Lipschitz continuity of projections P M , Q M cannotbe expected in general on the whole time interval [0 , T ] even if V is smooth, because suchproperty holds only under the assumptionΛ M +1 = Λ M . (4.25)It may happen that there is no M > t ∈ [0 , T ]. This is solvedby introducing a finite cover of [0 , T ] formed by intervals { I l } nl =1 , where for every l ∈ { , ..., n } there exists M l > M for some fixed M such thatΛ M l +1 = Λ M l for all t ∈ I l . (4.26)We take ψ ∈ C ∞ c ( I l ) and ϕ ∈ C ∞ c ((0 , T ) × R ) and use ψ ( t ) P M l [ ϕ ]( t, x ) as a test functionin (4.23) to obtain ˆ I l ψ ˆ Ω t ( ε∂ t P M l [ Z ε ] + div x Q M l [ z ε ]) ϕ d x d t (4.27)= − ε ˆ I l ψ ˆ Ω t (cid:18) h ε · ∇ x P M l [ ϕ ] + h ε P M l [ ϕ ] − Aεζ Σ ε V · ∇ x P M l [ ϕ ] (cid:19) d x d t + ε ˆ I l ψ ˆ Ω t ( Z ε ∂ t P M l [ ϕ ] − V · ∇ x ( P M l [ Z ε ] ϕ ) − P M l [ Z ε ] ∂ t ϕ ) d x d t. We take ϕ ∈ C ∞ c ((0 , T ) × R ), ϕ · n = 0 on Γ t , and use ψ ( t ) Q M l [ ϕ ]( t, x ) as a test functionin (4.24) to obtain ˆ I l ψ ˆ Ω t ( ε∂ t Q M l [ z ε ] + ζ ∇ x P M l [ Z ε ]) · ϕ d x d t (4.28)= − ε ˆ I l ψ ˆ Ω t (cid:18) ̺ − ̺ ε ε ∇ x F · Q M l [ ϕ ]+ H ε : ∇ x Q M l [ ϕ ] + h ε · Q M l [ ϕ ] + (cid:18) h ε + Aε Σ ε (cid:19) div x Q M l [ ϕ ] (cid:19) d x d t + ε ˆ I l ψ ˆ Ω t ( z ε · ∂ t Q M l [ ϕ ] − V · ∇ x ( Q M l [ z ε ] · ϕ ) − Q M l [ z ε ] · ∂ t ϕ ) d x d t. e observe that introducing d ε,l := P M l [ Z ε ] and ∇ x Ψ ε,l := Q M l [ H ⊥ [ z ε ]], the system ofequations (4.27)-(4.28) can be formally written as ε∂ t d ε,l + ∆ x Ψ ε,l = εf ε,l (4.29) ε∂ t ∇ x Ψ ε,l + ζ ∇ x d ε,l = ε f ε,l (4.30)for some f ε,l , f ε,l , which is to be satisfied in { ( t, x ) : t ∈ I l , x ∈ Ω t } . However, we will ratherwork with the weak formulation which reads as ˆ I l ψ ˆ Ω t ( ε∂ t d ε,l + ∆ x Ψ ε,l ) ϕ d x d t (4.31)= − ε ˆ I l ψ ˆ Ω t (cid:18) h ε · ∇ x P M l [ ϕ ] + h ε P M l [ ϕ ] − Aεζ Σ ε V · ∇ x P M l [ ϕ ] (cid:19) d x d t + ε ˆ I l ψ ˆ Ω t ( Z ε ∂ t P M l [ ϕ ] − V · ∇ x ( d ε,l ϕ ) − d ε,l ∂ t ϕ ) d x d t =: εf ε,l [ ψ, ϕ ]for all ψ ∈ C ∞ c ( I l ) and ϕ ∈ C ∞ c ((0 , T ) × R ) and ˆ I l ψ ˆ Ω t ( ε∂ t ∇ x Ψ ε,l + ζ ∇ x d ε,l ) · ϕ d x d t (4.32)= − ε ˆ I l ψ ˆ Ω t (cid:18) ̺ − ̺ ε ε ∇ x F · Q M l [ ϕ ] + H ε : ∇ x Q M l [ ϕ ]+ h ε · Q M l [ ϕ ] + (cid:18) h ε + Aε Σ ε (cid:19) div x Q M l [ ϕ ] (cid:19) d x d t + ε ˆ I l ψ ˆ Ω t ( z ε · ∂ t Q M l [ ϕ ] − V · ∇ x ( ∇ x Ψ ε,l · ϕ ) − ∇ x Ψ ε,l · ∂ t ϕ ) d x d t =: ε f ε,l [ ψ, ϕ ] . for all ψ ∈ C ∞ c ( I l ) and ϕ ∈ C ∞ c ((0 , T ) × R ), ϕ · n = 0. Let us remind that we want to prove (4.3). For this order we introduce the following partitionof unity on a time interval [0 , T ] n X l =1 ψ l ( t ) = 1 for all t ∈ [0 , T ] , where ψ l ∈ C ∞ c ( I l ) l = 1 , ..., n, where I l are intervals introduced in the previous section such that (4.26) holds and we write ˆ T ˆ Ω t H ⊥ t [ ̺ ε ( u ε − V )] ⊗ H ⊥ t [( u ε − V )] : ∇ x ϕ d x d t = n X l =1 ˆ I l ψ l ˆ Ω t H ⊥ t [ ̺ ε ( u ε − V )] ⊗ H ⊥ t [( u ε − V )] : ∇ x ϕ d x d t for test functions ϕ ∈ C c ( Q T ), div x ϕ = 0, ϕ (0 , · ) = ϕ ( T, · ) = 0, ϕ · n | Γ t = 0.We split both terms of the product into the finite mode part and the remainder part H ⊥ t [ z ε ] ⊗ H ⊥ t [( u ε − V )] = ( Q M l [ H ⊥ t [ z ε ]] + ( H ⊥ t [ z ε ] − Q M l [ H ⊥ t [ z ε ]])) ⊗ ( Q M l [ H ⊥ t [ u ε − V ]] + ( H ⊥ t [ u ε − V ] − Q M l [ H ⊥ t [ u ε − V ]])) . Moreover, we also have H ⊥ t [ z ε ] − Q M l [ H ⊥ t [ z ε ]] = H ⊥ t [( ̺ ε − ̺ )( u ε − V )] − Q M l [ H ⊥ t [( ̺ ε − ̺ )( u ε − V )]]+ ̺ ( H ⊥ t [ u ε − V ] − Q M l [ H ⊥ t [ u ε − V ]])and we recall that for ε → ̺ ε − ̺ ) u ε → L (0 , T, L / (Ω t )) , o the same holds also for H ⊥ t [( ̺ ε − ̺ )( u ε − V )] − Q M l [ H ⊥ t [( ̺ ε − ̺ )( u ε − V )]].We want to show that the remainder terms are small if we choose M l large enough. Westart with a useful expression for the L -norm of div x u ε k div x u ε k L (Ω t ) = k div x ( u ε − V ) k L (Ω t ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =1 div x a j ˆ Ω t ( u ε − V ) · a j d x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω t ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j =1 p Λ j ω j ˆ Ω t ( u ε − V ) · a j d x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (Ω t ) = ∞ X j =1 Λ j (cid:18) ˆ Ω t ( u ε − V ) · a j d x (cid:19) and we use it as follows (cid:13)(cid:13)(cid:13) H ⊥ t [ u ε − V ] − Q M l [ H ⊥ t [ u ε − V ]] (cid:13)(cid:13)(cid:13) L (Ω t ) = X j>M l (cid:18) ˆ Ω t ( u ε − V ) · a j d x (cid:19) ≤ j>M l Λ j ( t ) k div x u ε k L (Ω t ) ≤ j>M Λ j ( t ) k div x u ε k L (Ω t ) . We observe that the quantity 1inf t ∈ [0 ,T ] ,j>M Λ j ( t )can be made as small as we want by the choice of M .We are left with the goal of estimating the product of finite modes terms, namely ˆ I l ψ l ˆ Ω t Q M l [ H ⊥ t [ z ε ]] ⊗ Q M l [ H ⊥ t [( u ε − V )]] : ∇ x ϕ d x d t → ε → ϕ ∈ C c ( Q T ), div x ϕ = 0, ϕ (0 , · ) = ϕ ( T, · ) = 0, ϕ · n | Γ t = 0 and for l = 1 , ..., n , which can be rewritten equivalently to ˆ I l ψ l ˆ Ω t Q M l [ H ⊥ t [ z ε ]] ⊗ Q M l [ H ⊥ t [ z ε ]] : ∇ x ϕ d x d t → ε → . We recall that we denoted ∇ x Ψ ε,l = Q M l [ H ⊥ t [ z ε ]] and we have equations (4.29)-(4.30), ormore precisely their weak formulations (4.31)-(4.32) at our disposal. Integrating by parts wehave ˆ I l ψ l ˆ Ω t ( ∇ x Ψ ε,l ⊗ ∇ x Ψ ε,l ) : ∇ x ϕ d x d t = − ˆ I l ψ l ˆ Ω t ∆ x Ψ ε,l ∇ x Ψ ε,l · ϕ d x d t − ˆ I l ψ l ˆ Ω t ∇ x |∇ x Ψ ε,l | · ϕ d x d t = − ˆ I l ψ l ˆ Ω t ∆ x Ψ ε,l ∇ x Ψ ε,l · ϕ d x d t where we used that the second term on the middle line is zero due to the fact that div x ϕ = 0.We use the equation (4.31) with ϕ = ∇ x Ψ ε,l · ϕ and (4.32) with d ε,l ϕ as a test functiontogether with the transport theorem to obtain − ˆ I l ψ l ˆ Ω t ∆ x Ψ ε,l ∇ x Ψ ε,l · ϕ d x d t = ε ˆ I l ψ l ˆ Ω t ∂ t d ε,l ∇ x Ψ ε,l · ϕ d x d t − εf ε,l [ ψ l , ∇ x Ψ ε,l · ϕ ]= ε ˆ I l ψ l ˆ Ω t ( ∂ t ( d ε,l ∇ x Ψ ε,l ) − d ε,l ∂ t ∇ x Ψ ε,l ) · ϕ d x d t − εf ε,l [ ψ l , ∇ x Ψ ε,l · ϕ ]= ε ˆ I l ˆ Ω t ( ∂ t ( ψ l d ε,l ∇ x Ψ ε,l · ϕ ) − d ε,l ∇ x Ψ ε,l · ∂ t ( ψ l ϕ )) d x d t + ˆ I l ψ l ˆ Ω t ζ ∇ x d ε,l · ϕ d x d t − ε f ε,l [ ψ l , d ε,l ϕ ] − εf ε,l [ ψ l , ∇ x Ψ ε,l · ϕ ]= − ε ˆ I l ψ l ˆ Ω t V · ∇ x ( d ε,l ∇ x Ψ ε,l · ϕ ) d x d t − ε ˆ I l ˆ Ω t d ε,l ∇ x Ψ ε,l · ∂ t ( ψ l ϕ ) d x d t − ε f ε,l [ ψ l , d ε,l ϕ ] − εf ε,l [ ψ l , ∇ x Ψ ε,l · ϕ ] , (4.33) here the last equality is true because div x ϕ = 0. All the terms on the right hand side aremultiplied by ε , so to conclude our proof it is enough to show that the integrals contained inthe right hand side are bounded independently of ε and l . The first two integrals on the righthand side of (4.33) contain only smooth functions and therefore are obviously bounded, sowe focus only on the terms f ε,l and f ε,l . By (4.31) we have f ε,l [ ψ, ϕ ] = − ˆ I l ψ ˆ Ω t (cid:18) h ε · ∇ x P M l [ ϕ ] + h ε P M l [ ϕ ] − Aεζ Σ ε V · ∇ x P M l [ ϕ ] (cid:19) d x d t (4.34)+ ˆ I l ψ ˆ Ω t Z ε ∂ t P M l [ ϕ ] d x d t + I [ ψ, ϕ ] , and one checks that I [ ψ l , ∇ x Ψ ε,l · ϕ ] contains again only smooth functions and therefore isbounded. Similarly f ε,l [ ψ, ϕ ] = − ˆ I l ψ ˆ Ω t (cid:18) ̺ − ̺ ε ε ∇ x F · Q M l [ ϕ ] + H ε : ∇ x Q M l [ ϕ ] + h ε · Q M l [ ϕ ] (cid:19) d x d t (4.35) − ˆ I l ψ ˆ Ω t (cid:18)(cid:18) h ε + Aε Σ ε (cid:19) div x Q M l [ ϕ ] − z ε · ∂ t Q M l [ ϕ ] (cid:19) d x d t + I [ ψ, ϕ ] , with I [ ψ l , d ε,l ϕ ] containing only smooth functions and therefore bounded. Recalling (4.4),(4.7), (4.8), (4.10), (4.11), (4.15), (4.16), (4.18) and (4.22) we have z ε = ̺ ε ( u ε − V ) Z ε = ̺ (1) ε + Aζ ̺ ε s ( ̺ ε , ϑ ε ) − s ( ̺, ϑ ) ε − ζ ̺F + Aεζ Σ ε H ε = − ̺ ε u ε ⊗ u ε + S ε − [ p ( ̺ ε , ϑ ε )] res ε I + ̺ ε u ε ⊗ Vh ε = ̺ ε ∂ t V + ̺ ε u ε · ∇ x Vh ε = − ̺ (1) ε V + 1 ε Aζ (cid:18) κ ( ϑ ε ) ϑ ε ∇ x ϑ ε − ̺ ε ( s ( ̺ ε , ϑ ε ) − s ( ̺, ϑ )) u ε (cid:19) + ̺ζ F V h ε = ̺ζ V · ∇ x Fh ε = 1 ε (cid:18) A (cid:20) ̺ ε s ( ̺ ε , ϑ ε ) − s ( ̺, ϑ ) ε (cid:21) res + Ah ε − h ε (cid:19) with h ε , h ε satisfying (4.19)-(4.20). By Lemma 3.1 and (4.21) we conclude that (cid:12)(cid:12) f ε,l [ ψ l , ∇ x Ψ ε,l · ϕ ] (cid:12)(cid:12) + (cid:12)(cid:12) f ε,l [ ψ l , d ε,l ϕ ] (cid:12)(cid:12) ≤ c. The proof of Theorem 2.1 is finished.
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