Homogenization of the compressible Navier Stokes equations in domains with very tiny holes Yong Lu Sebastian Schwarzacher Abstract We consider the homogenization problem of the compressible Navier Stokes equations in a bounded three dimensional domain perforated with very tiny holes. As the number of holes increases to infinity, we show that, if the size of the holes is small enough, the homogenized equations are the same as the compressible Navier Stokes equations in the homogeneous domain domain without holes. This coincides with the previous studies for the Stokes equations and the stationary Navier Stokes equations. It is the first result of this kind in the instationary barotropic compressible setting. The main technical novelty is the study of the Bogovskiĭ operator in non-lipschitz domains. Keywords: Homogenization; Navier Stokes equations; perforated domains; Bogovskiĭ operator. Introduction In practice, there comes up the study of fluid flows in domains distributed with a large number of holes that represent solid obstacles. It is suitable, for instance, to model polluted underground water or oil development. The fluid flows passes between the small obstacles or in the holes in between. Such domains are usually called perforated domains and a typical example is the so-called porous media. The perforation parameters, which are mainly the size of holes and the mutual distance of the holes in the perforated domain under consideration, play a determinant role in these problems. We refer to [22] for a number of real world applications. Homogenization problems in fluid mechanics represent the study of the asymptotic behavior of fluid flows in perforated domains as the number of holes obstacles goes to infinity and the size of holes obstacles goes to zero simultaneously. The mathematical concern is the asymptotic behavior of the solutions to equations describing fluid flows with respect to perforation parameters. With an increasing number of holes within the domain of the fluid, the fluid flow approaches an effective state governed by certain homogenized equations in the full domain. With different physical backgrounds, the mathematical equations governing the fluid flows are different. There are typically Stokes equations, Navier Stokes equations or Euler equations. There also include equations describing non Newtonian fluid flows, such as p Stokes equations, Oldroyd B models, and many others. Accordingly, the mathematical study of homogenization problems in fluid mechanics includes the homogenization for various fluid models. We refer to [24, 2, 3] for Department of Mathematics, Nanjing University, 22 Hankou Road, Gulou District, Nanjing 293, China. Email: luyong@nju.edu.cn. Department of mathematical analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 86 75 Praha, Czech Republic. Email schwarz@karlin.mff.cuni.cz.
Stokes equations, [2] for incompressible Navier Stokes equations, [9] for compressible Navier Stokes equations and [3] for the complete compressible Navier Stokes Fourier equations. We also refer to the book [5] for other models, such as two phase models and non Newtonian fluid models. In this paper, we study the homogenization problem for the compressible Navier Stokes equations. We consider a bounded three dimensional domain perforated with tiny holes, where the diameters of the holes are taken to be of size Oε α with α, and the minimal mutual distances between the holes are of size Oε. In this work we will provide a size parameter α to be specified later on in Theorem.6, such that for α α a homogenization sequence of solutions converges to the solution of the compressible Navier Stokes system in the non-perforated domain. This means that if the size of the holes is small enough, then in the homogenization limit they shall not be seen anymore. This is a known phenomenon: analogous results for stationary incompressible or compressible fluids have been shown and will be discussed below in more detail. However, we would like to emphasize that up to our knowledge, this result is the first one in this direction for unsteady compressible fluids. Let us introduce the setting in more detail. Let R 3 be a C 2 bounded domain and {T ε,k } k Kε be a family of closed sets named holes or solid obstacles satisfying T ε,k = x ε,k + ε α T k Bx ε,k, δ ε α Bx ε,k, δ ε α Bx ε,k, δ 2 ε Bx ε,k, δ 3 ε,. where for each k, T k R 3 is a simply connected bounded domain of class C 2, where the C 2 constant is assumed to be uniformly bounded in k. With Bx, r we denote the open ball centered at x with radius r in R 3. Here we assume that δ, δ, δ 2, δ 3 are positive fixed constants independent of ε, such that δ < δ and δ 2 < δ 3. Moreover, we suppose that the balls control volumes {Bx ε,k, δ 3 ε} k Kε are pairwise disjoint. The corresponding ε-dependent perforated domain is then defined as ε := \ k K ε T ε,k..2 By the distribution of holes assumed above, the number of holes in ε satisfy K ε C, for some C independent of ε..3 ε3 We consider the following Navier Stokes equations in the space-time cylinder, T ε : t ϱ + div x ϱu =,.4 t ϱu + div x ϱu u + x pϱ = div x S x u + ϱf,.5 S x u = µ x u + t xu 2 3 div xui + ηdiv x ui, µ >, η..6 Here, ϱ is the fluid density, u is the velocity field, p = pϱ denotes the pressure, S = S x u stands for the Newtonian viscous stress tensor with µ, η the viscosity coefficients, f is the external force function satisfying f L,T ;R 3 <. We remark that this condition on the external force f is not optimal and can be improved. However, this improvement does not influence our argument in an essential manner, so we keep this restriction. We impose the no-slip boundary condition u = on, T ε.7 2
and impose the standard technical hypothesis imposed on the pressure in order to ensure the existence of global-in-time weak solutions to the primitive Navier Stokes system: p C[, C,, p =, p ϱ > for ϱ >, p ϱ lim ϱ ϱ γ = p >.8 with γ for which the value range that we can handle is given precisely later in Theorem.6. In the sequel, for a function f defined in ε, the notation f or Ef stands for the zero-extension of f in R 3 : f = f in ε, f = in R 3 \ ε.. Known results We introduce some known results concerning the homogenization problems in the framework of fluid mechanics. For the case α =, meaning that the size of holes is proportional to their mutual distance, Tartar [24] recovered the Darcy law from the homogenization of the Stokes equations. In [2, 3], Allaire gave a systematic study for the homogenization problems of the Stokes equations and stationary incompressible Navier Stokes equations. He showed that the homogenization process crucially depends on the perforation parameters of the domain. Specifically, in three dimensions, Allaire showed that when α < 3 corresponding to large holes, the behavior of the limit fluid is governed by the classical Darcy s law, as shown by Tartar in [24] for the case α = ; when α > 3 corresponding to small holes, the equations do not change in the homogenization process and the limit homogenized system is the same system as the orginal Stokes or Navier Stokes equations in homogeneous domain domain without holes; when α = 3 corresponding to the critical size of holes, in the limit it yields Brinkman s law which is a combination of Darcy s law and the original equations. Later on, in the case α =, the results have been extended to the incompressible instationary Navier Stokes equations by Mikelić [2], to the compressible Navier Stokes system by Masmoudi [9], and to the complete Navier Stokes Fourier system in [3]. In all the aforementioned cases, the homogenization limit gives rise to Darcy s law. Recently, the case α > 3 was considered in [, 7] for the stationary compressible Navier Stokes equation. If in addition γ > 2 satisfies αγ 2 2γ 3 >, it was shown that the limit homogenized equations remain unchanged. This coincides with the results obtained by Allaire [2, 3] for Stokes equations. Up to our knowledge, the result in this paper is the first analytical result in the study of homogenization for the instationary compressible Navier Stokes equations when the perforation parameter α..2 Difficulties In this section, we illustrate some difficulties in the study of homogenization problems for compressible Navier Stokes equations. In particular, we point out the main new difficulties compared to the previous study [, 7] concerning stationary compressible Navier Stokes equations..2. A review of difficulties in the compressible fluid setting In our setting with very small holes, the result is formally similar as for incompressible equations obtained by Allaire [2, 3], where the limit equations remain unchanged. However, the techniques for 3
compressible system are rather different. The main difference lies within the pressure term, where in compressible setting, the pressure is generally a nonlinear function of the density. To obtain the same pressure form in the limit, one needs to show strong convergence of the density. While in the incompressible case, the equations are usually treated in divergence free spaces testing by functions that are divergence free or applying the Helmholtz decomposition operator, and the pressure is reconstructed through the equation by using Ne cas theorem on negative Sobolev norms. For compressible Navier Stokes equations, a priori we only have L, T ; L ε uniform estimate for the pressure pϱ, that is only L bound with respect to the spatial variables. However, a uniform bounded family in L is not weakly pre-compact in L. It was observed in the study of compressible Navier Stokes equations in [6, 9, 2] that one can improve the integrability by employing some kind of divergence inverse operator. For equations in bounded domains, the socalled Bogovskiĭ operator, which is the famous inverse operator of div x to trace free functions [4, 5], was applied in [2] to obtain the higher integrability of pϱ. However, the operator norm of the classical Bogovskiĭ operator depends on the Lipschitz character of the domain which is perforated in our case. We refer to [4, Chapter 3] for a construction, as well as a norm estimate, of the Bogovskiĭ operator in Lipschitz domains and star-shaped domains. It can be shown that the Lipschitz norm of the perforated domain ε is at least of order /ε which is unbounded as ε. Thus by employing the classical Bogovskiĭ operator one cannot get uniform higher integrability of pϱ. However, for any fixed ε >, the existence theory of finite energy weak solutions to the compressible Navier Stokes equations is well developed for the case γ > 3/2. To be able to pass to the limit ε, we have to establish uniform pressure estimates despite the fact that the domains we consider are not uniform Lipschitz. Since the higher integrability of the pressure is a key step to prove the convergence with ε, we introduce the respective stationary technique of [] in more detail as an introduction to the problems that we have to handle in the instationary case. The method is based on finding a convenient inverse of the div x operator for non Lipschitz domains. In [] it was considered the case γ 3 which guarantees the L 2 integrability of the pressure. Indeed, there holds ϱ L 3γ ε, if 3/2 < γ 3; ϱ L 2γ ε, if γ 3..9 The construction of an inverse of div x is done by employing an Restriction operator R ε : W,2 ; R 3 W,2 ε ; R 3, that was introduced by Allaire [2]. For any f L 2 ε which is the set of L 2 ε functions with zero mean value, the authors first considered its zero-extension Ef := f in and then employ the classical Bogovskiĭ operator B : L 2,2 W on the unperforated domain and find that the operator defined as satisfies B ε := R ε B E : L 2 ε W,2 ε ; R 3. B ε f W,2 ε, div x B ε f = f in ε.. Observe that the operator norm of B ε relies on ε only through the operator norm of R ε from W,2 ; R 3 to W,2 ε ; R 3. By Allaire s construction, the operator norm of R ε partially depends on the uniform estimates of a Stokes system in a ball with a shrinking hole. Let B = B, be the unit ball and T B be a model obstacle in the fluid which is assumed to be a simply connected C 2 domain. In accordance to the construction in [2], the operator norm of R ε depends on ε partially through the W,2 estimate v L 2 B \ε α T + q L 2 B \ε α T C ε g L 2 B \ε α T.2 4
of the following Dirichlet problem of the Stokes equations v + q = div x g, div x v = in = B \ ε α T ; v = on B ε α T..3 Due to the L 2 framework, which represents the natural existence framework for the Stokes operator, the constant C ε in.2 is and hence independent of ε. Furthermore, under the condition α 3, it can be shown that the operator norm of B ε defined in. is independent of ε. This implies the uniform L 2 estimate of the pressure. However, the condition γ 3 is crucially needed in [] in order to obtain L 2 integrability of the pressure. If 3/2 < γ < 3, the so far known results give only L 3γ integrability for the density ϱ which means L 3γ γ integrability for the pressure pϱ, even in the standard case of unperforated domains. Clearly 3γ γ = 3 3 γ, 2 if γ 3/2, 3. Hence, to consider value range γ < 3, inspired by the construction. in [], a natural thinking is to employ this construction of Bogovskiĭ operator and generalize it to the L r ε and W,r ε; R 3 framework for general r. It means a generalization of the restriction operator R ε to an operator from W,r ; R3 W,r ε; R 3. It was shown in [8] that this generalization can be done in a rather direct way by observing the well-posedness in W,r ε; R 3, < r < of.3. The issue is that the operator norm of R ε : W,r ; R3 W,r ε; R 3 depends on the W,r estimate: v L r B \ε α T + q L r B \ε α T C ε g L r B \ε α T,.4 for v a solution to.3. However, for r 2 the constant C ε depends on the Lipschitz character of the domain B \ ε α T s and is not uniform in ε when α >. For the well-posedness result and estimates, we refer to [6] for details and the proof. It was the main motivation of the study in [8] see also [7] for the respective result for the Laplace operator, where it was shown that the estimate constant C ε in.4 is uniformly bounded in ε for the range 3/2 < r < 3. This allows us to construct a Bogovskiĭ operator that is uniformly bounded from L r ε W,r ε; R 3 for any r 3/2, 3 by using the same construction as. in []. Observe that the restriction r 3/2, 3 corresponds to 3 3 γ 3/2, 3. This indicates that the existence frame for the stationary compressible setting can be extended to γ > 2 such that 3 3 γ > 3/2. A new line of approach was developed to study the case γ < 3. Indeed, exactly for γ > 2 it was shown in [7] that an explicit construction of Bogovskiĭ type operator can be achieved. Compared to [8], the construction in [7] is more straightforward and works for a larger range of r. This builds the fundament of the Bogovskiĭ operator constructed in this article. For that reason the growth restriction on the pressure in this work is analogous to the restriction γ > 2 in the stationary case. See Remark.7 for more details..2.2 New difficulties and a technical theorem In this paper, we turn to consider instationary compressible Navier Stokes equations. We will see that our main result obtained in this paper is formally the same as for the stationary case; the techniques are rather different and there arise new difficulties. The first difficulty comes from the even lower integrability of the pressure. When γ > 3/2, the so far best integrability of the pressure one can obtain is pϱ L 5 3 γ see [2, 9], where the range of the integrability component is 5 3 γ, 5 3. This is much worse than in the stationary setting, and this is the deep reason that we have to impose the severe restriction γ > 6 in Theorem.6: to 5
make sure that 5/3 /γ > 3/2. More explanations on the restriction on γ related to the hole-size α are given in Remark.7. Technically, the main new difficulty is the absence of uniform estimates for the restricted Bogovskiĭ operator B ε in negative Sobolev spaces. Indeed, due to the instationary setting, to obtain higher integrability of the pressure by using the Bogovskiĭ operator, one needs to handle terms of the form B ε div x ϱ θ u which comes from the term B ε t ϱ θ via the renormalized continuity equation. Here θ is an exponent in the range, 2γ 3 ]. This enforces us to find uniform estimates of the following type B ε div x g L r ε;r 3 C g L r ε;r 3, for any g L r ε ; R 3, g n = on ε,.5 with a constant C independent of ε. If we would like to employ the construction of B ε in [8] and to obtain such an estimate.5, one would need to obtain uniform estimates for very weak solutions to.3: v L r B \ε α T + q W,r B \ε α T C g W,r B \ε α T,.6 with constant independent of ε. This is unknown according to the authors knowledge and we think that the estimate is not valid for any r > 3/2. Our approach is different to the approach in [] and more general, following the idea of [7]. It is possible due to a further and new refinement to the plethora of results related to the Bogovkiĭ operator. Namely uniform estimates in negative spaces for so called John domains. John domain is a typical known generalization of Lipschitz domain on which the Bogovskiĭ type operator can also be constructed. We refer to [, 8] for the definition of John domains and the construction of such Bogovskiĭ operators. As pointed out in [8], a bounded domain is a John domain if and only if it satisfies the emanating chain condition see [8, Definition 3.5, Remark 3.7]. Since this emanating chain condition will be involved in the proof of Theorem., for the sake of readers, we represent it below: Definition emanating chain condition Let R d be a bounded domain and let σ, σ 2. Then we say that satisfies the emanating chain condition with constants σ, σ 2 if there exists a covering W = {W i } i= of consisting of open cubes or balls such that: We have σ W i for all i and i= χ σ W i σ 2 χ on R d. For every W i W, there exists a chain of pairwise different W i,, W i,,, W i,mi from W such that W i, = W i,, W i,mi = W, and W i,k σ 2 W i,k2 for every k k 2 m i. Moreover, W i,k W i,k+, k < m i, contains a ball B i,k such that W i,k W i,k+ σ 2 B i,k. The chain W i,, W i,,, W i,mi is called chain emanating from W i. The number m i is called the length of this chain. The set {i : W i K } is finite for every compact subset K. The family W is called the chain-covering of. The cube W is called the central cube or ball, since every chain ends in W. We are able to prove that the very same Bogovskiĭ operator constructed in [8] actually is bounded in negative Sobolev spaces as well. Since we believe this result is of independent interest we state the theorem here as a technical tool for further use. 6
Theorem.. Let be a bounded John-domain. In particular let it satisfy the emanating chain condition. Let Cc, be the space of smooth and compactly supported functions with zero mean values in. Then for any < q < there exists a linear operator B : Cc, C c ; R 3 such that: If f L q with f dx =, there holds div x Bf = f in, Bf W,q ;R 3 C f L q.7 for some constant C dependent only on the emanating chain constants. If f W,r L r = {g W,r : g, = }, then Bf, φ = f, φ for φ W,r, Bf L r ;R 3 C f W,r..8 Observe, that.7 was shown in [8]. The second bound.7 is shown in section 2 below. Remark.2. Let be a John-domain, then we find for f L q ; R 3 with f n = on in the weak sense, that div x Bdiv x f = div x f. More explicit Bdiv x f, φ = f, φ for φ W,q, Bdiv x f L q ;R 3 C f L q ;R 3..9 This is due to the fact that f L q ; R 3 and f n = in the weak sense implies div x f {g W,r : g, = }. Below we will use the theorem to construct a restriction operator which is able to control the weak time derivative of the density. As a consequence, a higher integrability of the density can be deduced..3 Weak solutions Definition.3 Finite energy weak solution. We say that ϱ, u is a finite energy weak solution of the Navier Stokes equations.4-.6 supplemented with the boundary condition.7, the assumption on pressure.8, and the following initial conditions in the space-time cylinder, T ε if: There holds: ϱ, = ϱ L γ ε, u, = u L 2γ γ ε ; R 3,.2 ϱ a.e. in, T ε, ϱ C weak, T ; L γ ε, ϱu C weak, T ; L 2γ γ+ ε ; R 3, u L 2, T ; W,2 ε ; R 3. For any τ T and any test function ϕ C c [, T ] ε : τ [ϱ t ϕ + ϱu x ϕ] dx dt = ε ϱτ, ϕτ, dx ε ϱ ϕ, dx. ε.2 For any τ T and any test function ϕ Cc [, T ] ε ; R 3 : = τ τ [ϱu t ϕ + ϱu u : x ϕ + pϱdiv x ϕ] dx dt.22 ε S x u : x ϕ ϱfϕ dx dt + ϱuτ, ϕτ, dx ϱ u ϕ, dx. ε ε ε 7
The energy inequality [ ε 2 ϱ u 2 +P ϱ ] τ [ τ, dx+ S x u : x u dx dt ε ε 2 ϱ u 2 +P ϱ ] dx.23 holds for a.a. τ, T, where we have set P ϱ := ϱ ϱ Moreover, a finite energy weak solution ϱ, u is said to be a renormalized weak solution if t bϱ + div x bϱu + b ϱϱ bϱ div x u = in D, T R 3,.24 for any b C [,. In.24 ϱ, u are extended to be zero outside ε. We give some remark concerning the definition and the existence of finite energy weak solutions. Remark.4. The integrability for the initial data in.2 is imposed such that the initial energy on the right-hand side of.23 is bounded. Remark.5. For any fiexed ε >, it can be shown by the theory developed by Lions [6] and Feireisl-Novontný-Petzeltová [9] that the Navier Stokes equations.4-.6 admit a global-in-time finite energy weak solution ϱ, u for any finite energy initial data and pressure satisfying.8 with γ > 3/2. Moreover, such solutions are also renormalized in the sense of.24..4 Main results Our main result is the following: Theorem.6. Let ϱ ε, u ε <ε< be a family of finite energy weak solutions for the no-slip compressible Navier Stokes equations.4-.7 in, T ε under the pressure condition.8 with γ > 6 in the sense of Definition.3 with initial data satisfying ϱ ε, = ϱ,ε, u, = u,ε, ϱ,ε L γ ε + u,ε L γ 2γ = D <,.25 ε;r 3 sup <ε< and the strong convergence for the zero extensions as ε : pz z 2 dz. ϱ,ε ϱ strongly in L γ, ũ,ε u strongly in L 2γ γ ; R 3. Then there holds the uniform estimates for the solution family: ϱε L,T ;L γ ε + ϱ ε 5γ + u L ε 3,T ε L 2,T ;W,2 ε;r 3 sup <ε< CD <,.26 where CD depends only on D. This implies, up to a substraction of a subsequence, that ϱ ε ϱ weakly-* in L, T ; L γ, ũ ε u weakly in L 2, T ; W,2 ; R 3..27 If the growth parameter γ in.8 and size parameter α of the holes in. satisfy γ 6 α > 3,.28 2γ 3 then the couple ϱ, u is the finite energy weak solution to.4-.7 in with initial data ϱ, = ϱ, u, = u. 8
We give some remarks on the technical condition.28. Remark.7. This condition is needed due to the capacity estimate of the holes and the low integrability of the pressure pϱ ε. By.8 and.26, we have the uniform bound for the pressure: pϱ ε 5 L 3 γ C..29,T ε To extend the Navier Stokes equations in particular the momentum equation from spatial domain ε to, an idea is to find a family of functions {g ε } ε> vanishing on the holes and converges to in some Sobolev space W,q with q determined by the integrability of the pressure, and then to decompose any test function ϕ Cc, T ; R 3 as ϕ = g ε ϕ + g ε ϕ..3 Then g ε ϕ can be treated as a test function for the momentum equation.5 in ε provided all the terms in.22 make sense, where some condition on the largeness of q shall be needed. For the terms related to the part g ε ϕ, we show that they are small and converge to zero. Firstly, such a function family {g ε } ε> exists provided the following condition satisfied see Lemma 2.: 3 qα 3 >..3 The quantity on the left-hand side of.3 is determined by the uniform q-capacity assumptions of the union of all the holes: Cap q T ε,k Cap q T ε,k Cε 3 ε α3 q..32 k K ε k K ε We recall that the q-capacity of a set D R d is defined as { } Cap q D := inf f q dx : f W,q R d, D {f }. R d In.32, the following property is used: let B, r R d be a ball with radius r,, then Cap q B, r Cr d q, for some C indepedent of r. In the weak formulation of the momentum equations, by testing g ε ϕ, there arises the term τ p ϱ ε ϕ g ε dx dt. To make sure the above term makes sense, due to.29, it is necessary to impose the condition 5 3 +..33 γ q The condition.28 is sufficient and also necessary to make sure that there exists a positive q such that conditions.3 and.33 are satisfied. We also remark that condition.33 implies γ + 3 + q.34 for any γ > 3/2 because 5 3 γ γ + 3 2γ 32 which is always true. This is needed later in the proof of Proposition 3.4. 9
2 Model test functions and the Bogovskiĭ operator In this section, we introduce two basic tools that will be needed in our proof of the main theorem. 2. Test functions vanishing on the holes We introduce the following Lemma: Lemma 2.. For any < r < 3 such that 3 rα 3 >, there exists a family of functions {g ε } ε> W,r such that g ε = on k K ε T ε,k, g ε in W,r. 2. Moreover, there holds the estimate for some C independent of ε: g ε W,r Cε σ, σ := 3 rα 3/r. 2.2 Proof of Lemma 2.. By., there exists g ε C R 3 such that g ε = on T ε,k, g ε = on Bx ε,k, δ ε α c, gε L R 3 Cε α. k K ε k K ε Recall that the counting measure of K ε satisfies.3. Then direct calculation gives g ε L r Cε 3α 3/r, g ε L r Cε 3α 3/r α = Cε σ. Such g ε fulfill the request in Lemma 2.. 2.2 The Bogovskiĭ operator The aim of this section is to prove the following proposition: Proposition 2.2. Let ε defined as in.-.2 with α, then for any < q <, there exists a linear operator B ε : L q ε W,q ε; R 3 such that for any f L q ε with ε f dx =, there holds div x B ε f = f in ε, B ε f W,q ε;r 3 C + ε3 qα 3/q f L q ε 2.3 for some constant C independent of ε. For any r > 3/2, the linear operator B ε can be extended as a linear operator from {div x g : g L r ε ; R 3, g n = on ε } to L r ε ; R 3 satisfying for all some constant C independent of ε. B ε div x g L r ε;r 3 C g L r ε;r 3, 2.4 The existence of an operator such that the first equation in 2.3 is satisfied is classical, known as Bogovskiĭ operator. The key point of this lemma is to give an estimate for the operator norm, in particular, to show the dependency on ε. The analytical tool to be able to get estimates uniformly in ε is Theorem.. So we prove Theorem. first and then prove Proposition 2.2. Actually, Theorem. is an extension of [8, Theorem 5.2]. The strategy is biased on the following technical theorem [8, Theorem 4.2]. For the sake of clarity, we summarize the parts of the theorem that we will use in our argument in the following lemma
Lemma 2.3 Decomposition Theorem. Let R n be a bounded domain satisfying the emanating chain condition, i.e. Definition.2.2. Then there exists an at most countable family of linear operators T i : C c, C c, W i such that for all q, the following properties are satisfied: i The operators have a bounded extension T i : L q Lq W i. ii For all f L q we have f = i N T if. iii For all f L q we have f L q c i T if L q W i c 2 f L q, with constant c, c 2 just depending on the constants of Definition.2.2. iv For f C c, the set {i N : T if } is finite. Proof of Theorem.. The first statement.7 has been shown in [8, Theorem 5.2]. For us it is left to show that the same operator also satisfies.8. We start by using the fact that compactly supported functions are dense in {f W,r : f, = }. Indeed, as can be seen in [, Lemma.4] there exists w L r, such that f, φ := w φ dx, and w L r = f W,r. We approximate using the density of test functions in Lebesgue spaces with functions w δ Cc ; R 3, such that w δ w in L r and w δ L r 2 f W,r. By partial integration we find div x w δ Cc,. This allows us to apply the decomposition theorem [8, Theorem 4.2]. That is for the chain covering W introduced in Definition.2.2, we have the family of linear operators T i : Cc, C c, W i, for all W i W. By the definition of the reference paper we find that Bdiv x w δ := W i W B it i div x w δ, which is well defined due to the calculations above. Observe, that B i is the standard Bogovskij operator on the cube W i. Since, T i div x w δ Cc, W i we find by [, Theorem.], Lemma 2.3 and by the finite intersection property of the W i, that Bdiv x w δ r L r C W i W B i div x T i w δ r L r C C w δ r L r ;R 3 C f r W,r. W i W T i w δ r L r ;R 3 Moreover, again by Lemma 2.3 as well as the properties of B i, we find div x Bw δ = div x B i T i w δ = T i w δ = w δ. 2.5 W i W W i W This implies the desired properties for smooth functions. Since the related bounds are uniform in δ, the operator B can be extended accordingly to {f W,r :, f, = } by letting δ. Proof of Proposition 2.2. Let f L q ε with ε f dx =, we consider the extension function f = Ef defined as f = f in ε, f = on \ ε = T ε,k. 2.6 k K ε
Then by employing the classical Bogovsii s operator, there exists u = B f W,q ; R3 such that div x u = f in and u W,q ;R 3 C f L q = C f L q ε 2.7 for some constant c depends only on and q. In each ball Bx ε,k, δ 3 ε, we introduce two cut-off functions χ k and φ k such that: χ k C c Bx ε,k, δ 3 ε, χ k = on Bx ε,k, δ 2 ε, x χ k Cε ; φ k C c Bx ε,k, δ ε α, φ k = on Bx ε,k, δ ε α, x φ k Cε α. 2.8 We denote D ε,k := Bx ε,k, δ 3 ε \ Bx ε,k, δ 2 ε, E ε,k := Bx ε,k, δ 3 ε \ T ε,k. 2.9 We then define two localizations of u near the holes: b k u := χ k u u Dε,k W,q Bx ε,k, δ 3 ε, β k u := φ k u Dε,k W,q Bx ε,k, δ ε α, 2. where u Dε,k := u dx. D ε,k D ε,k Since the E ε,k are uniform John domains, the following corollary follows immediately by Theorem. and.9. Corollary 2.4. For any < q <, there exists a linear operator B Eε,k : L q E ε,k W,q E ε,k; R 3 such that for any f L q E ε,k div x B Eε,k f = f, B Eε,k f W,q E ε,k ;R 3 C f L q E ε,k, 2. for some constant C independent of ε. For any g L q E ε,k ; R 3 with g n = on E ε,k, we find Moreover, for some constant C independent of ε. B Eε,k div x g, φ = g, φ for all φ C E ε,k. B Eε,k div x g L q E ε,k ;R 3 C g L q E ε,k ;R 3, 2.2 We now define the restriction operator in the following way: R ε u := u bk u + β k u B Eε,k div x b k u + β k u. 2.3 k K ε We first check that R ε u is well defined. Since b k, β k W,q E ε,k to this end, it is sufficient to show that E ε,k div x b k + β k dx =. 2.4 Indeed, on one hand, by the zero trace 2., we have div x b k + β k dx =. 2.5 Bx ε,k,δ 3 ε 2
On the other hand, by 2.6-2., we have div x b k + β k = χ k div x u + x χ k u u Dε,k + x φ k u Dε,k =, on T ε,k. 2.6 Equations 2.5 and 2.6 imply 2.4. Hence, we can apply Corollary 2.4 on each member and 2.3 is well defined. Moreover, applying Poincaré inequality gives R ε u q dx C u q dx + C χ k u q + u u Dε,k q χ q dx + C C C C k K ε Bx ε,k,δ ε α k K ε Bx ε,k,δ 3 ε x φ k q u Dε,k q dx u q dx + C u u Dε,k q dx + C u Dε,k q Bx ε,k, δ ε α k K D ε,k ε ε k K ε u q dx + C u q dx + C u Dε,k q ε 3 qα dx k K D ε,k ε k K ε u q dx + C u q dx ε 3 qα 3 D ε,k k K ε C + ε 3 qα 3 u q W,q. We claim that the operator B ε defined in the following way fulfills the desired properties in Proposition 2.2: B ε f := R ε u = R ε B f = R ε B Ef. By the definition of R ε u in 2.3 and the property of u in 2.7, we have R ε u W,q ; R3, div x R ε u = div x u = f in. Moreover, for any x T ε,k for some k K ε, we have by using 2.8 and 2.9 that R ε ux = ux χ k x ux u Dε,k φk x u Dε,k =. Thus, R ε u W,q ε; R 3, div x R ε u = f in ε. Now we prove the second part of Proposition 2.2. Let r > 3/2 and g L r ε ; R 3 with g n = on ε. Let u := Bdiv x g L r ; R 3 with B be the standard Bogovskiĭ operator in. Moreover, there holds u L r ;R 3 C g L r ε;r 3, 2.7 where the constant C only depends on r and the Lipschitz character of the domain. We assume div x g L q ε for some q >. This ensures, together with the assumption g n = on, that u W,q ; R3, then R ε u W,q ε; R 3 is well defined where R ε is the restriction operator constructed by 2.3. Our goal is to show the following uniform estimates for some constant C independent of ε and div x g L q ε. R ε u L r ε;r 3 C g L r ε;r 3, 2.8 3
By 2. and 2.8, we have b k u r dx β k u r dx Bx ε,k,δ 3 ε Bx ε,k,δ ε α u u Dε,k r dx C u Dε,k r dx Cε 3α Bx ε,k,δ 3 ε u r dx, Bx ε,k,δ 3 ε u r dx, 2.9 where we used the fact u Dε,k r u r Dε,k. Direct calculation gives, using the fact that div x g = div x u, that div x b k u + β k u = χ k div x u + x χ k u u Dε,k + x φ k u Dε,k = χ k div x g + x χ k u u Dε,k + x φ k u Dε,k = div x χ k g x χ k g + x χ k u u Dε,k + x φ k u Dε,k = div x χ k g + div x χ k u g + div x φ k χ k u Dε,k. 2.2 We will estimate the operator B Eε,k on each three divergences separately. Observing and applying Corollary 2.4 implies χ k g = on Bx ε,k, δ 3 ε, χ k g n = on T ε,k, B Eε,k div x χ k g L r E ε,k ;R 3 C χ k g L r E ε,k ;R 3 C g L r E ε,k ;R 3. 2.2 For the other two terms we will use the W,q -bounds of the operator B Eε,k. We find by the support properties of χ, φ and the fact div x u = div x g that div x χ k u g dx = div x χ k u g dx =, E ε,k Bx ε,k,δ 3 ε div x φ k χ k u Dε,k dx = div x φ k χ k u Dε,k dx =. E ε,k Bx ε,k,δ 3 ε Since r > 3/2, we let r > such that / r = /r + /3. Then applying Corollary 2.4, together with the Sobolev embedding inequality, implies B Eε,k x χ k g u L r E ε,k ;R 3 C B Eε,k x χ k g u W, r E ε,k ;R 3 C x χ k g u L r E ε,k C x χ k L 3 E ε,k ;R 3 g L r E ε,k ;R 3 + u L r E ε,k ;R 3, B Eε,k x φ k χ k u Dε,k L r E ε,k ;R 3 C x χ k L 3 E ε,k ;R 3 + x φ k L 3 E ε,k ;R 3 u L r E ε,k ;R 3 C u L r D ε,k ;R 3. 2.22 Summarizing 2.9 2.22, using 2.7 and the fact x χ k L 3 E ε,k ;R 3 + x φ k L 3 E ε,k ;R 3 C, we obtain R ε u r L r ε;r 3 C g r L r ε;r 3 + u r L r ;R 3 C g r L r ε;r 3. for some constant C independent of ε and div x g L q ε. This implies our desired estimate 2.8. Since this estimate constant C is independent of div x g L q, by a density argument we can eliminate the assumption div x g L q. The proof is completed. 4
3 Proof of Theorem.6 3. Uniform bounds By the uniform bounds of the initial data in.25 and the energy inequality.23, together with the pressure property.8, Korn s inequality and Poincaré inequality, we have { ϱ ε } ε> uniformly bounded in L, T ; L γ, 3. { ϱ ε ũ ε 2 } ε> uniformly bounded in L, T ; L, 3.2 {ũ ε } ε> uniformly bounded in L 2, T ; W,2 ; R 3. 3.3 Then up to a substraction of subsequence, we derive the weak limit ϱ ε ϱ weakly-* in L, T ; L γ, ũ ε u weakly in L 2, T ; W,2 ; R 3. 3.2 Improved integrability on the density In this subsection, we will show the following improved uniform integrability of the density: { ϱ ε } ε> uniformly bounded in L 5γ 3, T. 3.4 This is equivalent to following integrability on the pressure: {p ϱ ε } ε> uniformly bounded in L 5 3 γ, T. 3.5 We recall that, for any fixed ε >, the result ϱ ε L 5γ 3, T is known, see for instance Theorem 7.7 in [2]. However, the estimate for ϱ ε 5γ shown in [2] is not uniform in L 3,T ε. Indeed, it depends on the Lipschitz norm of the spatial domain, which is the perforated domain ε in our setting. Our task in this section is to show such an estimate is uniform in ε. The idea as in [2, 2] is to test the the momentum equation by functions of the form ϕt, x := ψtb ε ϱ θ ε ϱ θ ε, ψ Cc, T, ϱ θ ε := ε ε ϱ θ ε dx, θ >, 3.6 where B ε is the Bogovskiĭ operator given in Proposition 2.2. To prove 3.4, a nature choice is θ = 2γ/3 in 3.6, but due to the restriction 3/2 < q 2 on the uniform bound of B ε, there arise terms that cannot be controlled by the known uniform estimates in 3. 3.3. We will improve the integrability of ϱ ε step by step by taking θ from smaller number γ/2 to our desired number 2γ/3. The following two lemmas are needed: Lemma 3.. Under the assumption in Theorem., the extension functions ϱ ε, ũ ε satisfy t ϱ ε + div x ϱ ε ũ ε =, in D, T R 3. 3.7 Lemma 3.2. Let 2 β < and ϱ L β, T ; L β loc R3, ϱ a.e. L 2, T ; W,2 loc R3 ; R 3. Suppose that in, T R 3, u t ϱ + div x ϱu =, in D, T R 3. 5
Then for any b C [, C, such that b s Cs λ for s, ], b s Cs λ for s [,, with c >, λ <, λ β 2, there holds the following renormalized equation t bϱ + div x bϱu + b ϱϱ bϱ div x u = in D, T R 3. The proof of Lemma 3. is the same as Proposition 3.3 and the proof is postponed. For the proof of Lemma 3.2, we refer to Lemma 6.9 in [2] By the estimate 3. with γ > 6 and the estimate 3.3, we apply the above two lemmas to obtain t b ϱ ε + div x b ϱε ũ ε + b ϱ ε ϱ ε b ϱ ε div x ũ ε = in D, T R 3, 3.8 for any b as in Lemma 3.2; in particular, we can take bs = s θ with < θ γ 2. We first use the function 3.6 with θ := γ/2 as a test function in.22. Thus, by uniform estimate 3. and Proposition 2.2, we have for any r, 2] that B ε ϱ θ ϱ θ L,T ;W,r ε;r 3 C ϱθ ϱ θ L,T ;L r ε C ϱ γ 2 L,T ;L γ ε C. 3.9 where I := Then direct calculation gives T I 3 := I 5 := T T T ε ψtpϱ ε ϱ θ ε dx dt = T 6 I j, ψpϱ ε ϱ θ ε dx dt, I 2 := t ψtϱ ε u ε B ε ϱ θ ε ϱ θ ε dx dt, ε ε ψϱ ε u ε u ε : x B ε ϱ θ ε ϱ θ ε T dx dt, I 4 := ψs x u ε : x B ε ϱ θ ε ϱ θ ε dx dt, ε ε ψϱ ε u ε B ε t ϱ θ ε t ϱ θ ε T dx dt, I 6 := ψϱ ε f B ε ϱ θ ε ϱ θ ε dx dt. ε ε We estimate using the choice θ = γ/2 and Hölder s inequality I C sup ϱ θ εt pϱ ε t dx C ϱ ε 3γ/2 L,T ;L γ C. ε t [,T ] ε By the fact γ + 6 + 6, by using estimate 3.9 and Hölder s inequality, we can obtain I 2 C ϱ ε L,T ;L γ ε u ε L 2,T ;L 6 ε B ε ϱ θ ε ϱε θ L C.,T ;L 6 ε Again by the fact γ > 6 and γ + 3 + 2, together with estimate 3.9 and Hölder s inequality, we obtain I 3 C ϱ ε L,T ;L γ ε u ε 2 L 2,T ;L 6 ε x B ε ϱ θ ε ϱε θ L C.,T ;L 2 ε j= 6
Similarly, for I 4, we have I 4 C x u ε L 2,T ;L 2 ε x B ε ϱ θ ε ϱ θ ε C ϱ θ ε L,T ;L 2 ε C ϱ ε θ L,T ;L γ ε C. L,T ;L2ε The most challenging term is the term which involves the weak time derivative. To estimate this term we had to introduce the theory for Bogovskiĭ type operator on negative spaces for perforated domains, namely the second part of Proposition 2.2, for which the proof needs the theory for Bogovskiĭ type operator on negative spaces for John domains, namely Theorem.. First we may use Lemma 3.2 with β = γ to obtain t ϱ θ + div x ϱ θ u + θ ϱ θ div x u = in D, T R 3, θ := γ 2. Then, direct calculation gives T I 5 = ψϱ ε u ε B ε ε T + θ ψϱ ε u ε B ε ε =: I 7 + I 8. div x ϱ θ ε u ε dx dt ϱ θ εdiv x u ε ϱ θ εdiv x u ε dx dt 3. By estimates 3. 3.3 and the fact γ > 6, we have ϱ ε u ε L 2,T ;L 3 ε ϱ ε L,T ;L 6 ε u ε L 2,T ;L 6 ε C, ϱ ε u ε L,T ;L 2 7 ε ϱ ε L,T ;L 2 ε ϱ ε u ε L,T ;L 2 ε C. 3. This implies, by interpolation, that ϱ ε u ε L 6,T ;L 2 ε ϱ ε u ε 3 L 2,T ;L 3 ϱ ε εu ε 2 3 C. 3.2 L,T ;L 2 7 ε We apply Proposition 2.2 to get I 7 C ϱ ε u ε L 6,T ;L 2 ε B ε div x ϱ θ ε u ε L 6 5,T ;L 2 C ϱθ εu ε 6 ε L 5,T ;L 2 ε C u ε L 2,T ;L 6 ε ϱ θ ε L 3,T ;L 3 ε C x u ε L 2,T ;L 2 ε ϱ θ ε L 3,T ;L 3 ε. 3.3 For I 8, by Proposition 2.2 and Sobolev embedding, together with 3. and 3.2, we have I 8 C ϱ ε u ε L 6,T ;L 2 ε B ε ϱ θ εdiv x u ε ϱ θ L εdiv x u ε 6 5,T ;L 2 ε C B ε ϱ θ εdiv x u ε ϱ θ L εdiv x u ε 6 5,T ;W, 6 3.4 5 ε C ϱ θ εdiv x u ε 6 6 C div L 5,T ;L 5 ε xu ε L 2,T ;L 2 ε ϱ θ ε L 3,T ;L 3 ε. Thus, by 3.3 and 3.4, we obtain I 5 I 7 + I 8 C ϱ θ ε L 3,T ;L 3 ε = C ϱ ε γ 2 7 L 3γ 2,T ;L 3γ 2 ε.
Finally, for I 6, it is direct to obtain I 6 ϱ ε L,T ;L 2 ε ϱ θ ε L,T ;L 2 ε C. Hence, summing up above estimates for I j, j 8, by passing ψ in L, T, we have T ε pϱ ε ϱ γ 2 ε dx dt C + C ϱ ε γ 2 L 3γ 2,T ;L 3γ 2 ε Thus, by assumption on pressure in.8, we obtain ϱ ε 3γ T 2 C + C pϱ L 3γ 2,T ;L 3γ ε ϱ 2 ε ε γ 2 ε dx dt C + C ϱ ε γ 2. L 3γ 2,T ;L 3γ 2 ε This implies { ϱ ε } ε> uniformly bounded in L 3γ 2, T. 3.5 Next, based on estimates obtained above in 3.5, we choose a bigger value θ = 2γ 3 in 3.6 to obtained our desired estimates 3.4 3.5. To this end, we estimate all I j with θ := 2γ 3. For I, estimate 3. and Hölder s inequality implies C I C sup ϱ θ εt pϱ ε t dx C ϱ ε t [,T ] ε 5γ 3 L,T ;L γ C. ε For I 2, by Proposition 2.2, estimates 3., 3.3, 3.5 and Sobolev embedding, we have I 2 C ϱ ε L,T ;L γ ε u ε L 2,T ;L 6 ε B ε ϱ θ ε ϱ θ ε L,T ;L3ε B ε ϱ θ ε ϱ θ ε L,T ;W, 3 2 ε C ϱ θ ε L,T ;L 3 2 ε C ϱ ε 2γ 3 L,T ;L γ ε. For I 3, by the fact γ > 6, θ = 2γ 3 and γ + 3 + θ γ =, we obtain by using Proposition 2.2 and Hölder s inequality I 3 C ϱ ε L,T ;L γ ε u ε 2 L 2,T ;L 6 ε x B ε ϱ θ ε ϱε θ L,T ;L γ θ ε C ϱ θ ε L,T ;L γ θ ε C ϱ ε θ L,T ;L γ ε. For I 4, by 3.3 and 3.5, and the fact 2θ < 3γ 2 for θ = 2γ 3, we obtain I 4 C x u ε L 2,T ;L 2 ε x B ε ϱ θ ε ϱ θ ε For I 6, it is direct to obtain L2,T ;L2ε C ϱ θ ε L 2,T ;L 2 ε C ϱ ε θ L 2θ,T ;L 2θ ε C. I 6 ϱ ε L,T ;L 2 ε ϱ θ ε L 2,T ;L 2 ε C. Now we estimate I 5 which is the most difficult one. By estimates in 3.5, we use Lemma 3.2 with β = 3γ/2 to obtain t ϱ θ + div x ϱ θ u + θ ϱ θ div x u = in D, T R 3, θ := 2γ 3.. 8
We then split I 5 = I 7 + I 8 the same way as 3.: T I 7 := ψϱ ε u ε B ε ε T I 8 := θ ψϱ ε u ε B ε ε div x ϱ θ ε u ε dx dt, ϱ θ εdiv x u ε ϱ θ εdiv x u ε We first estimate ϱ ε u ε. By the fact 5γ 2 3 + γ 6 2 = + 3 2 = 5γ γ 6, and Hölder s inequality, we have dx dt. ϱ ε u ε L γ 6 3,T ;L γ 6 5γ ε = ϱ ε ϱε u ε L γ 6 3,T ;L γ 6 5γ ε ϱ ε ϱ L γ 6 3,T ;L γ 6 3 ε u L ε ε,t ;L 2 C ϱ ε ε 2. L 5γ 3 3,T ;L 5γ 3 3 ε 3.6 Similarly, by the fact 5γ 3 + 2 = 5γ + 3 γ 6, and Hölder s inequality, we have 5γ 3 + 5γ + 5 = 6 65γ 3, ϱ ε u ε L γ 6 65γ 3 5γ+3,T ;L 5γ+5 ε ϱ ε L 5γ 3 C ϱ ε 5γ 3. L 3,T ;L 5γ 3 3 ε 3 u,t ;L 5γ 3 ε 3 L 2,T ;L 6 ε ε 3.7 By 3.6, 3.7, and interpolations between Lebesgue spaces, we have, for any α [, ] and r, r 2 such that there holds 3 = α r γ 6 + α 5γ + 3 γ 6, 5γ = α r 2 γ 6 ϱ ε u ε L r,t ;L r C ϱ 2 ε ε +α 2. L 5γ 3 3,T ;L 5γ 3 3 ε + α 5γ + 5 65γ 3, 3.8 By choosing α = 5 This implies in 3.8, we have r = γ + 3 γ 6, r 2 = 3γ + 3 65γ 3. ϱ ε u ε L γ 6 65γ 3 C ϱ ε 3 5. γ+3,t ;L 3γ+3 ε L 5γ 3 3,T ;L 5γ 3 3 ε 3.9 For I 7, by 3.9 and the fact 2 + 2γ 3 5γ 3 = 9γ 9 γ 6, 6 + 2γ 3 7γ 2 = 5γ 3 65γ 3 < 2 3 9
and we can apply Proposition 2.2 to obtain γ + 3 γ 6 + 9γ 9 γ 6 =, 3γ + 3 7γ 2 + 65γ 3 65γ 3 =, I 7 C ϱ ε u ε L γ 6 65γ 3 γ+3,t ;L 3γ+3 ε C ϱ ε u ε L γ 6 65γ 3 γ+3,t ;L 3γ+3 ε C ϱ ε 3 5 L 5γ 3 3 C ϱ ε 3 5 +θ L 5γ 3 3,T ;L 5γ 3 3 ε,t ;L 5γ 3 3 ε It is left to estimate I 8. Since θ = 2γ 3 ϱ θ ε B ε div x ϱ θ ε u ε L γ 6 9γ 9 ϱ θ εu ε L γ 6 9γ 9 L 5γ 3 2γ 3 C ϱ ε 2γ 3 2 5 and 65γ 3,T ;L 7γ 2 ε 65γ 3,T ;L 7γ 2 ε,t ;L 5γ 3 2γ 3 ε u ε L 2,T ;L 6 ε L 5γ 3 3,T ε 2 + 2γ 3 5γ 3 = 9γ 9 γ 6, Hölder s inequality implies ϱ θ L εdiv x u ε γ 6 div x u ε L 9γ 9,T 2,T ε ϱθ ε ε L 5γ 3 C ϱ ε θ. 2γ 3,T ε L 5γ 3 3,T ε Thus, by Proposition 2.2 and Sobolev embedding, we have B ε ϱ θ εdiv x u ε ϱ θ εdiv x u ε L γ 6 9γ 9 C C 65γ 3,T ;L 7γ 2 ε B ε ϱ θ εdiv x u ε ϱ θ εdiv x u ε L γ 6 9γ 9 By estimates 3.2 and 3.9, we obtain I 8 C B ε ϱ θ εdiv x u ε ϱ θ εdiv x u ε L γ 6 9γ 9. γ 6, 9γ 9,T ;W ε ϱ θ L εdiv x u ε γ 6 C ϱ ε θ. 9γ 9,T ε L 5γ 3 3,T ε C ϱ ε θ L 5γ 3 3,T ε ϱ ε 3 5 L 5γ 3 3,T ε 65γ 3,T ;L 7γ 2 ε ϱ ε u ε C ϱ ε 2γ 3 2 5. L 5γ 3 3,T ε L γ 6 γ+3,t ;L 3.2 65γ 3 3γ+3 ε Hence, summing up above estimates for I j, j 8, by passing ψ in L, T, we have T ε pϱ ε ϱ 2γ 3 ε Thus, by assumption on pressure in.8, we have ϱ ε 5γ 3 3 C + C L 5γ 3 3,T ε T dx dt C + C ϱ ε 2γ 3 2 5. L 5γ 3 3,T ε ε pϱ ε ϱ 2γ 3 ε This implies our desired uniform estimates in 3.4 and 3.5. dx dt C + C ϱ ε 2γ 3 2 5. L 5γ 3 3,T ε 2
3.3 Equations in homogeneous domains In this section, we derive the equations in ϱ ε, ũ ε in, T. 3.3. Continuity equation For the continuity equation, we have the following Proposition: Proposition 3.3. Under the assumption in Theorem., the extension functions ϱ ε, ũ ε satisfy t ϱ ε + div x ϱ ε ũ ε =, in D, T R 3. 3.2 Proof of Proposition 3.3. It is sufficient to prove T ϱ ε t ψ + ϱ ε ũ ε x ψ dxdt =, for any ψ C c, T R 3. Let {φ n } n C c ε such that φ n, x φ n 4n and Then T φ n = on {x distx, ε n }, φ n = on { x distx, ε 2n }. T ϱ ε t ψ + ϱ ε ũ ε x ψ dxdt = ϱ ε t ψφ n + ϱ ε u ε x ψφ n dxdt R 3 ε T + ϱ ε t ψ φ n + ϱ ε u ε x ψ φ n ϱ ε u ε ψ x φ n dxdt. ε 3.22 By the estimates obtained in 3.-3.3 and Sobolev embedding, we have where we used the fact T lim ϱ ε t ψ φ n + ϱ ε u ε x ψ φ n dxdt =, n ε φ n in L q ε, for any < q <, as n. By virtue of 3.3, we have Then by 3. and the fact distx, ε u ε L 2, T ε. we have distx, ε x φ n in L q ε, for any < q <, as n T lim ϱ ε u ε ψ x φ n dxdt =. n ε We complete the proof by passing n in 3.22 2
3.3.2 Momentum equation By Remark.7, the condition.28 implies that there exists 5/2 < q < 3 such that.3,.33 and.34 are all satisfied. We now prove the following proposition Proposition 3.4. Under the assumption in Theorem., there holds the equation t ϱ ε ũ ε + div x ϱ ε ũ ε ũ ε + x p ϱ ε = div x S x ũ ε + ϱ ε f + F ε, in D, T ; R 3, 3.23 where F ε D, T ; R 3 satisfies F ε, ϕ Cε σ t ϕ L 2,T ;L 2 ;R 3 + x ϕ L r,t ;L 3 ;R 9 + ϕ L r,t ;L r ;R 3, 3.24 for any ϕ C c, T ; R 3 and some constant C, some < r <, with σ := 3 qα 3 >. Here q 5/2, 3 satisfying.3,.33 and.34, C and r are independent of ε. Proof of Proposition 3.4. Let ϕ Cc, T ; R 3 be any test function. It is sufficient to show: T I ε : = ϱ ε ũ ε t ϕ + ϱ ε ũ ε ũ ε : x ϕ + p ϱ ε div x ϕ S x ũ ε : x ϕ + ϱf ϕ dxdt C ε σ t ϕ L 2,T ;L 2 ;R 3 + x ϕ L r,t ;L 3 ;R 9 + ϕ L r,t ;L r ;R 3, for some r <. The condition.3 makes sure that we can apply Lemma 2. to find {g ε } ε> W,q such that 2. and 2.2 are satisfied; the conditions.33 as well as.34 make sure all the terms appeared in the following equation make sense: T I ε = ϱ ε u ε t g ε ϕ + ϱ ε u ε u ε : x g ε ϕ + pϱ ε div x g ε ϕ ε 5 5 S x u ε : x g ε ϕ + ϱ ε f g ε ϕ dxdt + I j = I j, where I := I 2 := I 3 := I 4 := I 5 := T T T T T ϱ ε ũ ε g ε t ϕ dxdt, j= j= ϱ ε ũ ε ũ ε : g ε x ϕ ϱ ε ũ ε ũ ε : x g ε ϕ dxdt p ϱ ε g ε div x ϕ p ϱ ε x g ε ϕ dxdt, S x ũ ε : g ε x ϕ + S x ũ ε : x g ε ϕ dxdt, ϱ ε f gε ϕ dxdt. We now estimate I j one by one. By 2.2 in Lemma 2. and Sobolev embedding, we have g ε L q C εσ, q = q 3. 3.25 We further observe that the condition.28 is sufficient to make the inequality.33 and.34 be strict. Precisely, under condition.28, there exists 5/2 < q < 3 such that.3 and the following two inequalities are satisfied: 5 3 γ + q <, γ + 3 + q 22 <. 3.26
In the rest of this proof, we use a simpler notation L q L r to denote spaces L q, T ; L r or L q, T ; L r ; R 3 or L q, T ; L r ; R 9. We now consider I. By 3.25 and 3.26, there holds γ + 6 + q < 2. 3.27 By the uniform estimates in 3.-3.4 and Hölder s inequality, we obtain I ϱ ε L L γ ũ ε L 2 L 6 g ε L q tϕ L 2 L 2 C εσ t ϕ L 2 L 2. For I 2, by 3.25 and 3.26, using Sobolev embedding and Hölder inequality, we have { ϱ ε ũ ε 2 } ε> uniformly bounded in L L L L 3γ 3+γ. By interpolation and the inequalities in 3.26 and 3.27, we have { ϱ ε ũ ε 2 } ε> uniformly bounded in L q L q 2 for some < q < and < q 2 < 3γ/3 + γ satisfying Then Hölder s inequality implies where q 2 + q <, q 2 + q < 2 3. I 2 C ϱ ε ũ ε 2 L q L q 2 g ε L q xϕ L r L 3 + C ϱ ε ũ ε 2 L q L q 2 g ε W,q ϕ L r L r 2 Cε σ x ϕ L r L 3 + ϕ L r L r 2, r := q >, r 2 := q 2 q >. For I 3, by 3.4, similar argument as the estimate for I 2 gives I 3 C p ϱ ε L 5 3 γ L 5 3 γ g ε L q xϕ L r 3 L r 4 + C p ϱ ε L 5 3 γ L 5 3 γ g ε W,q ϕ L r 5 L r 6 where Cε σ x ϕ L r 3 L r 4 + ϕ L r 5 L r 6, 5 := r 3 r 5 := 3 γ 5 3 γ Similarly, we have for I 4, we have I 4 Cε σ x ϕ L 2 L r 7 + ϕ L 2 L r 8, Finally for I 5 : I 5 Cε σ ϕ L 2 L r 9, > We thus complete the proof by taking 3, r 4 := > 3, r 6 := 5 3 γ q > 3, 5 3 γ q >. r 7 := 2 q > 3 +, r 8 := 2 q >. r 9 := q γ > 2 5 6 > 2 5. r := max{r j j 9},. 23
3.4 Passing to the limit We need to show the weak convergence of the nonlinear terms in the sense of contribution: ϱ ε ũ ε ϱu, ϱ ε ũ ε ũ ε ϱu u, p ϱ ε pϱ. 3.4. Time derivative estimates and weak limits We would like to prove ϱ ε ũ ε ϱu, ϱ ε ũ ε ũ ε ϱu u in the sense of distribution. A key point is that we can obtain some uniform estimate for t ϱ and t ϱũ by Proposition 3.3 and Proposition 3.4 we just proved. This allows us to use the Aubin-Lions type argument to prove the weak limit of the product is the product of the weak limits. By Proposition 3.3 and the uniform estimates 3.-3.3, we have { t ϱ ε } ε> uniformly bounded in L 2 6γ,, T ; W 6+γ. 3.28 By Aubin-Lions type argument see for example [23], or directly using Lemma 5. in [6], we have By 3. and 3.2, there holds together with 3.28, we have ϱ ε ũ ε ϱu in D, T. 3.29 { ϱ ε ũ ε } ε> uniformly bounded in L, T ; L 2γ +γ ; R 3, ϱ ε ϱ in C w [, T ]; L γ ; R 3, ϱ ε ũ ε ϱu in C w [, T ]; L 2γ +γ ; R 3. By Proposition 3.4 and the uniform estimates 3.-3.3, have ϱ ε ũ ε = ϱ ε ũ ε + ε σ ϱ ε ũ ε 2, 3.3 where By observing { t ϱ ε ũ ε } ε> uniformly bounded in L, T ; W, ; R 3, 3.3 { ϱ ε ũ ε 2 } ε> uniformly bounded in L 2, T ; R 3. 3.32 we have { ϱ ε ũ ε } ε> uniformly bounded in L 2, T ; L 6γ 6+γ ; R 3 L 2, T ; R 3, { ϱ ε ũ ε } ε> uniformly bounded in L 2, T ; R 3. 3.33 By 3.3 and 3.33, together with Aubin Lions type argument or using directly Lemma 5. in [6], we have ϱ ε ũ ε ũ ε ϱ ε ũ ε u in D, T. By 3.29, 3.3, 3.32, the weak limit satisfies ϱ ε ũ ε = ϱ ε ũ ε = ϱu. 24
Finally, by observing we obtain ε σ ϱ ε ũ ε 2 u ε L,T ;R 3 R 3 Cε σ, as ε, ϱ ε ũ ε ũ ε ϱu u in D, T. 3.34 Thus, by passing ε in 3.2 and 3.23, we obtain the equations in ϱ, u: t ϱ + div x ϱu =, in D, T R 3, 3.35 t ϱu + div x ϱu u + x pϱ = div x S x u + ϱf, in D, T ; R 3. 3.36 Here in 3.36, pϱ denotes the weak convergence of p ϱ ε. Moreover, by Lemma 3.2, there holds t bϱ + div x bϱu + b ϱϱ bϱ div x u = in D, T R 3, 3.37 for any b fulfilling properties stated in Lemma 3.2. 3.4.2 Strong convergence of the density The next step is to show pϱ = pϱ, 3.38 which is a consequence of the strong convergence ϱ ε ϱ a.e. in, T. As in the existence theory of weak solutions for compressible Navier Stokes equations, the strong convergence for the density approximate solution family is a tricky part. P.-L. Lions [6] introduced the so-called effective viscous flux which enjoys an additional compactness, and this property plays a crucial role in the existence theory of weak solutions for the compressible Navier Stokes equations. This is the following lemma: Lemma 3.5. Up to a substraction of subsequence, there holds for any ψ Cc and any φ Cc, T : T lim φtψx p ϱ ε 4µ3 + ηdiv xũ ε ϱ ε dx dt ε T = φtψx pϱ 4µ 3 + ηdiv xu ϱ dx dt. Proof of Lemma 3.5. The proof of Lemma 3.5 is quite tedious but nowadays well understood. In this proof, the notations and are all associated with spatial variable x. The main idea is to employ the following test functions in the weak formulations of momentum equations: φψ ϱ ε, φψ ϱ, 3.39 where is the Fourier multiplier on R 3 with symbol ξ 2. In 3.39, ϱ ε and ϱ are treated as functions in R 3 with respect to spatial variable, where there holds ϱ ε = ϱ = on c. Notice that = R i,j i,j 3 are the classical Riesz operators. We refer to Section.3.7.2 in [2] for the concepts of Fourier multiplier and Riesz operators. We recall the classical property for Riesz operators: for any f L q R 3, < r < : f L r R 3 ;R 9 C f L r R 3. 3.4 25