Compressible Navier-Stokes equations with heterogeneous pressure laws
aa r X i v : . [ m a t h . A P ] J u l Compressible Navier-Stokes equationswith heterogeneous pressure laws
Didier Bresch, Pierre–Emmanuel Jabin, and Fei Wang
Abstract.
This paper concerns the existence of global weak solutions `a la Leray forcompressible Navier-Stokes equations with a pressure law which depends on the densityand on time and space variables t and x . The assumptions on the pressure contain onlylocally Lipschitz assumption with respect to the density variable and some hypothesis withrespect to the extra time and space variables. It may be seen as a first step to considerheat-conducting Navier-Stokes equations with physical laws such as the truncated virialassumption. The paper focuses on the construction of approximate solutions througha new regularized and fixed point procedure and on the weak stability process takingadvantage of the new method introduced by the two first authors with a careful study ofan appropriate regularized quantity linked to the pressure.
1. Introduction and main result
As mentioned in [ ], the existence of global weak solutions, in the sense of J. Leray,to the non-stationary barotropic compressible Navier-Stokes system with constant shearand bulk viscosities µ and λ remained a longstanding open problem in space dimensionstrictly greater than one until the first results by P.–L. Lions (see [ ]) with P ( ρ ) = aρ γ ( γ > d/ ( d + 2)). Many important contributions followed to improve the result includingE. Feireisl–A. Novotny–H. Petzeltova ( γ > d/
2, see [ ]), P.I. Plotnikov–V.A. Weigant( γ = d/
2, see [ ]), E. Feireisl (pressure law s P ( s ) non-monotone on a compact set, see[ ]) and more recently D. Bresch-P.E. Jabin (thermodynamically unstable pressure law s P ( s ) or anisotropic viscosities, see [ ]).One of the main issue is that the weak bound of the divergence of the velocity field doesnot a priori rule out singular behaviors by the density which may oscillate, concentrate oreven vanish (vacuum state) even if this is not the case initially.Heat-conducting viscous compressible Navier-Stokes equations (Navier-Stokes-Fourier)with constant viscosities namely with a pressure law ( ρ, ϑ ) P ( ρ, ϑ ) and an extra equationon the temperature ϑ has been firstly discussed in [ ] and solved by E. Feireisl and A.Novotny for specific pressure laws, see [ ] and [ ] which in some sense are monotone withrespect to the density after a fixed value. In the present paper, we prepare the resolution ofthe heat-conducting compressible Navier-Stokes equations with a truncated virial pressurelaw P ( ρ, ϑ ) = ρ γ + ϑ [ γ/ X n =0 B n ( ϑ ) ρ n . (1.1) Such pressure law is not monotone with respect to the density after a fixed value andtherefore is not thermodynamically stable. This paper concerns the existence of globalweak solutions `a la Leray for compressible Navier-Stokes equations with a pressure lawwhich depends on the density and on time and space variables t and x . It may be seen asa first step to consider heat-conducting Navier-Stokes equations with physical laws such asthe truncated virial assumption. More precisely, we consider the compressible Navier-Stokes(CNS) equations ∂ t ρ + div( ρu ) = 0 (1.2) ∂ t ( ρu ) + div( ρu ⊗ u ) − µ ∆ u − ( µ + λ ) ∇ div u + ∇ P = 0 (1.3)with initial condition ρ | t =0 = ρ ( ρu ) | t =0 = m , (1.4)in a periodic box Ω = T d for d ≥ µ and λ two constants satisfying the physicalconstraint µ > λ + 2 µ/d >
0. The pressure P = P ( t, x, ρ ) is a given functiondepending on the time t , space x , and the density ρ . For simplicity in the redaction weconsider in the sequel that the shear viscosity µ = 1 and the bulk viscosity λ = −
1: Thisdoes not changed the mathematical proof and result.For simplicity, we consider the periodic boundary conditions in x , namely Ω = T d , evenif arguments can be adapted to the whole space case as well. As explained previously, thearticle should be seen as a first step to solve the truncated virial case where we assume thatthe temperature ϑ ( t, x ) is actually given instead of solving the temperature equation ∂ t ( ρ E ) + div x ( ρ E u ) + div x ( P ( ρ, ϑ ) u ) = div x ( ∇ x u · u ) + div x ( κ ( ϑ ) ∇ ϑ ) , (1.5)where E = | u | / e ( ρ, ϑ ) is the total energy density with e ( ρ, ϑ ) is the specific internalenergy and initial condition ρE | t =0 = ρ E . (1.6)with the virial pressure state law (1.1). The main result presented here will be used in ourupcoming article (see [ ]) to construct solutions to the full system (1.2)–(1.4) and (1.5)–(1.6)as it provides the starting point for the fixed point procedure that we adopt. If ϑ is giventhen naturally P ( t, x, ρ ) = P ( ρ, ϑ ( t, x )). But there are however several other contexts (forinstance in biology) where it is necessary to involve non spatially homogeneous pressurelaw and for this reason, it is useful to consider more general formulas for P than givenby (1.1). Note that as shown in [ ], the procedure developped here is also applicable forthe compressible Brinkman system (semi-stationary compressible Stokes system) which isstandard system that may be seen in porous media and biology.The construction of appropriate approximate solutions will be a difficulty in our paper.It is based on an original approximate system for which existence of solutions is obtainedthrough a regularization and a fixed point approach. The weak stability property on thesequence of approximate solution is obtained using the new method introduced by the twofirst authors in [ ] and taking care of the regularized term linked to the pressure state lawwhich involves serious difficulties.We assume hypothesis on the pressure law ( t, x, s ) P ( t, x, s ): Some of them are usedto ensure the propagation of energy and the others are used to garantee the propagation ofcompactness on the density. OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 3
More precisely, let us present:– Assumptions to ensure the propagation of energy .Let γ > d/ ( d + 2):( P
1) There exist q > , ≤ ¯ γ ≤ γ/ , and a smooth function P such that | P ( t, x, s ) − P ( t, x, s ) | ≤ C R ( t, x ) + C s ¯ γ for R ∈ L q ((0 , T ) × T d ) . (1.7)( P
2) There exist p < γ + 2 γd − , q > , Θ ( t, x ) ∈ L q ((0 , T ) × T d ) , such that C − s γ − Θ ( t, x ) ≤ P ( t, x, s ) ≤ Cs p + Θ ( t, x ) . (1.8)( P
3) There exist p < γ + 2 γd − , and Θ ∈ L q ([0 , T ] × T d ) with q > | ∂ t P ( t, x, s ) | ≤ Cs p + Θ ( t, x ) . (1.9)( P |∇ x P ( t, x, s ) | ≤ C s γ/ + Θ ( t, x ) , for Θ ∈ L ([0 , T ] , L d/ ( d +2) ( T d )) . (1.10)– Assumptions required for the propagation of compactness on the density.( P
5) The pressure P is locally Lipschitz in the sense of that | P ( t, x, z ) − P ( t, y, w ) | ≤ Q ( t, x, y ) + ( C ( z γ − + w γ − )+ ( e P ( t, x ) + e P ( t, y )) | z − w | , for some e P ∈ L s ([0 , T ] , T d ) and Q ∈ L s ([0 , T ] , T d ) for some s , s > . (1.11)( P
6) The functions Q, e P satisfy that for some r h → , as h → kK h k L Z T Z T d K h ( x − y ) (cid:16) | e P ( t, x ) − e P ( t, y ) | s + | Q ( t, x, y ) | s (cid:17) dx dy ds = r h . (1.12) The total energy of the CNS system.
The total energy of the system, which is the sum ofthe kinetic and the potential energies, reads E ( t, x, ρ, ρu ) = Z T d (cid:18) | ρu | ρ + ρe ( t, x, ρ ) (cid:19) dx where e ( t, x, ρ ) = Z ρρ ref P ( t, x, s ) s ds (1.13)with ρ ref a constant reference density. We also define similarly the reduced total energy E ( t, x, ρ, ρ u ) which is based on P instead of P , see assumption (1.7). Note that we assumeas usually u = m ρ when ρ = 0 and u = 0 elsewhere, (1.14) | m | ρ = 0 a.e. on { x ∈ Ω : ρ ( x ) = 0 } . (1.15)The following is our main result dealing with heterogeneous pressure laws. Theorem . Assume the initial data m and ρ ≥ with R T d ρ = M > satisfy E ( ρ , m ) = Z T d (cid:18) | m | ρ + ρ e (0 , x, ρ ) (cid:19) dx < ∞ . D. BRESCH, P. JABIN, AND F. WANG
Suppose that the pressure P satisfies (1.7) – (1.12) . Then there exists a global weak solutionto Compressible Navier–Stokes System (1.2) – (1.4) such that u ∈ L (0 , T ; H ( T d )) , | m | / ρ ∈ L ∞ (0 , T ; L ( T d )) ρ ∈ C ([0 , T ] , L γ ( T d ) weak ) ∩ L p ((0 , T ) × T d ) where < p < γ ( d + 2) / − with the heterogeneous pressure state law P satisfying the energy inequality Z T d E ( ρ, u ) dx + Z t Z T d |∇ u ( s, x ) | dx ds ≤ E ( ρ , u )+ Z t Z T d div x u ( s, x ) ( P ( s, x, ρ ( s, x )) − P ( s, x, ρ ( s, x ))) ds dx + Z t Z T d ( ρ ∂ t e ( ρ ) + ρ u · ∇ x e ( ρ )) dx ds where E ( ρ, u ) = | ρu | / ρ + ρ Z ρρ ref P ( t, x, s ) s ds. Remark . Note that u ∈ L (0 , T ; H ( T d )) comes from the control of the gradient ofthe velocity field ∇ u in L ((0 , T ) × Ω) and the control of | m | /ρ in L ((0 , T ) × T d ) usingthe fact that R Ω ρ = R Ω ρ = M > . The interested reader is referred to [ ].
2. The approximation systems with a sketch of proof and a priori estimates
We present here the approximate system upon which we rely to construct the solutionto (1.2)–(1.4) with the pressure law P given by (1.7)–(1.12). As is classical in compress-ible Fluid Mechanics, the approximation procedure is performed through several stages,involving different approximate systems. Oneof the main difficulty is to find a proper approximation of the above system so that we mayconstruct a solution of it and prove the compactness of the solutions. We propose to definethe approximating system ∂ t ρ ε,η + div( ρ ε,η u ε,η ) = 0 (2.1) ∂ t ( ρ ε,η u ε,η ) + div( ρ ε,η u ε,η ⊗ u ε ) − ∆ u ε,η + ∇ ( P art,η ( ρ ε,η ) + L ε ∗ P ) = 0 (2.2)with initial condition ρ ε,η | t =0 = ρ ,ε,η and ( ρ ε,η u ε,η ) | t =0 = m ,ε,η (2.3)where an artificial pressure term reads P art,η ( ρ ε,η ) = η ρ γ art, ε,η + . . . + η m ρ γ art,m ε,η for some fixed parameters γ art = γ art, > γ art, > · · · > γ art,m . The coefficients η , . . . , η m will later be let to converge to 0 in that order and the γ art,i will be chosen so that γ art, > γ, γ art,i +1 + 2 γ art,i +1 d − > γ art,i , γ + 2 γd − > γ art,m . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 5
In addition an appropriate regularization of the pressure state law L ε ∗ ( P ( t, · , ρ ε,η ( t, · )) hasbeen introduced. More precisely the key step is to construct a suitable mollifying operator L ε defined as follows L ε ( x ) = 1log 2 Z εε L ε ′ ( x ) dε ′ ε ′ . where L ε is a standard mollifier given by L ε ( x ) = 1 ε d L (cid:16) xε (cid:17) , with L is a non-negative smooth function such that L ∈ C ∞ ( T d ) and R T d L ( x ) dx = 1 . Then L ε → δ as ε →
0, with δ being the Dirac Delta function at 0. It is straightforward tocheck that Z T d L ε ( x ) dx = 1and L ε → δ , as ε → . We observe that we easily have the following global existence result through a fixed pointargument that will be presented in the Appendix for readers convenience
Theorem . Assume that P satisfies (1.7) with γ art > γ and that the initial data ρ ,ε,η , u ,ε,η satisfy the uniform bound sup ε,η Z T d ( η ( ρ ,ε,η ( x )) γ art, + . . . + η m ( ρ ,ε,η ( x )) γ art,m + ρ ,ε,η ( x ) | u ,εη ( x ) | ) dx < ∞ . There exist ρ ε,η ∈ L ∞ ([0 , T ] , L γ art ( T d )) ∩ L p ([0 , T ] × T d ) for any p < γ art + 2 γ art /d − , u ε,η ∈ L ([0 , T ] , H ( T d )) solution to (2.1) - (2.2) . Moreover, ρ ε,η , u ε,η satisfy the uniformin ε bounds sup ε sup t ∈ [0 , T ] Z T d ( η ρ γ art, ε,η ( t, x ) + . . . + η m ρ γ art,m ε,η ( t, x ) + ρ ε,η ( t, x ) | u ε,η ( t, x ) | ) dx < ∞ , (2.4a)sup ε Z T Z T d |∇ u ε,η | dx dt < ∞ , (2.4b)sup ε Z T Z T d η ρ pε,η ( t, x ) dx dt < ∞ for any p < γ art + 2 γ art /d − . (2.4c) Finally, we have the explicit energy inequality Z T d (cid:18) η ρ γ art, ε,η ( t, x ) γ art, − . . . + η m ρ γ art,m ε,η ( t, x ) γ art,m − ρ ε,η ( t, x ) | u ε,η ( t, x ) | (cid:19) dx + Z t Z T d |∇ u ε,η ( s, x ) | dx ds ≤ Z t Z T d div u ε,η L ε ⋆ x P dx ds + Z T d η ( ρ ε,η ) γ art, ( t, x ) γ art, − . . . + η m ( ρ ε,η ) γ art,m ( t, x ) γ art,m − ρ ε,η ( t, x ) | u ε,η ( t, x ) | ! dx. (2.5)The main difficulty and contribution of the present article is the limit passage ε → η fixed, given by the following result D. BRESCH, P. JABIN, AND F. WANG
Theorem . Assume that P satisfies (1.11) and (1.12) . Let γ art > max(2 s ∗ , s ∗ , d ) ,where s ∗ and s ∗ are the H¨older conjugate exponents of s and s respectively. Suppose thatthe initial data ρ ε , u ε of the system (2.1) – (2.2) satisfy that ρ ,ε,η → ρ ,η in L γ art ( T d ) , ρ ,ε,η u ,ε,η → ρ ,η u ,η and ρ ,ε,η | u ,ε,η | → | ρ ,η u ,η | in L ( T d ) . Let ( ρ ε,η , u ε ; η ) be the cor-responding sequence of solutions satisfying the energy estimate (2.4) . Then ρ ε,η is compactin L p ( T d ) for ≤ p < γ art as ε → . The particular form of the mollifier operator L ε is strongly used for the compactnessproperty on { ρ ε,η } ε to have enough control of terms involving the pressure terms in themethod introduced by the two first authors in [ ]. Using the previous Theorem, the limitpassage provides a sequence of global weak solutions ( ρ η , u η ) to the following system ∂ t ρ η + div( ρ η u η ) = 0 (2.6) ∂ t ( ρ η u η ) + div( ρ u η ⊗ u η ) − ∆ u η + ∇ ( P art,η ( ρ η ) + P ( t, x, ρ η )) = 0 , (2.7)for some large γ art ≥ γ with initial boundary conditions ρ η | t =0 = ρ ,η , ρ η u η | t =0 = m ,η . (2.8)Fortunately once we obtain global weak solutions to (2.6)-(2.8) then passing to the limit as η →
0, then η → η m →
0, to obtain global weak solutions to (1.2)–(1.4) is infact a straightforward consequence of [ ]. More precisely we have Theorem . Assume that P satisfies (1.7) – (1.12) . Consider any sequence ρ η ∈ L ∞ ([0 , T ] , L γ art ( T d )) with γ art,m < γ + 2 γ/d − , γ art,i < γ art,i +1 + 2 γ art,i +1 /d − and γ art > γ , any sequence u η ∈ L ([0 , T ] , H ( T d )) of solutions to (2.6) - (2.7) over [0 , T ] .Suppose moreover that ρ η → ρ in L γ ( T d ) , ρ η u η → ρ u and ρ η | u η | → ρ | u | both in L ( T d ) . Assume finally that sup η sup t ∈ [0 , T ] R T d ρ η | u η | dx < ∞ .Then ρ η is compact in L t,x , u η is compact in L t,x and converge to a global solution to (1.2) , (1.3) with Z T d E ( ρ, u ) dx + Z t Z T d |∇ u ( s, x ) | dx ds ≤ E ( ρ , u )+ Z t Z T d div x u ( s, x ) ( P ( s, x, ρ ( s, x )) − P ( s, x, ρ ( s, x ))) ds dx + Z t Z T d ( ρ ∂ t e ( ρ ) + ρ u · ∇ x e ( ρ )) dx ds. The proof of Th. 2.3 will be discussed in the appendix of the article for reader’s convenience.This will end the proof of the main theorem 1.1.
Important remark.
It is important to note that the requirement for having several exponents γ art,i in the artificial pressure P art,η appears from the constraints in the proofs of Theorems2.1-2.3. To recover the appropriate energy terms in Theorem 2.1, we need to treat the actualpressure P as a source term. This is only possible if div u L ε ⋆ P is integrable uniformly in ε and, as P . ρ γ , it forces that γ art > γ .On the other hand, assuming that γ art, , . . . , γ art,i − = 0, to pass to the limit in theterm η i ρ γ art,i as η i → η i +1 >
0, we again need to have ρ γ art,i integrable. From thegain of integrability detailed in the next subsection, this only appears possible if γ art,i <γ art,i +1 + 2 γ art,i +1 /d −
1. If we had only one correction in P art,η , i.e. m = 1, then we would OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 7 actually need both γ art > γ and γ art < γ + 2 γ/d −
1, which is of course not possible if d ≥
2. The introduction of several exponents γ art,i seems to be a fairly straightforwardmanner of resolving this issue. As those are used several times, we collect here thebasic energy estimates for the generic system ∂ t ρ + div( ρ u ) = 0 ,∂ t ( ρ u ) + div( ρ u ⊗ u ) − ∆ u + ∇ ( P ( t, x, ρ ) + S ( t, x )) = 0 . (2.9)There exist a well-known gain in integrability on ρ from the momentum equation. Forconvenience later, we write it in a slightly more general form. Lemma . Assume that ρ ∈ L ∞ ([0 , T ] , L γ ( T d )) solves (B.1) with a velocity field u ∈ L ([0 , T ] , H ( T d )) and source term S ∈ L ([0 , T ] , L γ ∗ ( T d ) satisfies that change L ∞ to W − , ∞ ? ∇ x P ( t, x, ρ ) ∈ L ([0 , T ] , W − ,p ( T d ))+ H − ([0 , T ] , L pd/ (2 d +2 p − pd ) ( T d )) ∩ W − , ∞ ([0 , T ] , L pd/ ( p + d ) ( T d )) . Then for any < θ < γ /p ∗ , Z T Z T d ρ θ ( s, x ) P ( s, x, ρ ( s, x )) ds dx ≤ C d k ρ k θL ∞ t L γ x (1 + k u k L t H x ) k∇ x P ( ρ ) k L t W − ,px + H − t L pd/ (2 d +2 p − pd ) x ∩ W − , ∞ L pd/ ( p + d ) + C d k ρ k θL ∞ t L γ x k S k L t L γ ∗ x . Proof.
We can rewrite the assumption simply as ∇ x ( P ( t, x, ρ ) + S ) = div x f + ∂ t g, where f ∈ L ([0 , T ] , L p ( T d )) and g ∈ L ([0 , T ] , L pd/ (2 d +2 p − pd ) ( T d )), with in addition g ∈ L ∞ ([0 , T ] , L pd/ ( p + d ) ( T d )). For a fixed exponent θ > c θ = R T d ρ θ ( t, x ) dx and B ( t, x ) = −∇ x ∆ − x ( ρ θ − c θ ). In the case of a bounded domainwith a boundary instead of the torus, one has to be more careful and use the appropriateBogovski operator (see [ ] for example).The idea is then simply for multiply by B and first notice that Z T Z T d B ( s, x ) · ∇ x ( S + P ( s, x, ρ )) dx ds = Z T Z T d ( ρ θ ( s, x ) − c θ ) ( S + P ( s, x, ρ )) dx ds ≥ − C + Z T Z T d ρ θ ( s, x ) ( S + P ( s, x, ρ ( s, x ))) dx ds. D. BRESCH, P. JABIN, AND F. WANG
The integral of ρ θ S can be bounded immediately to yield the second in the right-hand sideof the lemma. On the other hand Z T Z T d B ( s, x ) · ∇ x ( S + P ( s, x, ρ )) dx ds = − Z T Z T d ∇ x B ( s, x ) : f ( s, x ) dx ds − Z T Z T d ∂ t B ( s, x ) · g ( s, x ) dx ds + Z T d ( B (0 , x ) · g (0 , x ) − B (0 , T ) · g ( T, x )) dx. By standard Calderon-Zygmund theory, k∇ x B k L ∞ t L γ /θx ≤ C d k ρ k θL ∞ t L γ x . Hence the firstterm in the r.h.s. is directly bounded by − Z T Z T d ∇ x B ( s, x ) : f ( s, x ) dx ds ≤ k f k L t L px k∇ B k L ∞ t L p ∗ x ≤ C d k f k L t L px k ρ k θL ∞ t L γ x , since p ∗ ≤ γ /θ as θ < γ /p ∗ . By Sobolev embedding k B k L ∞ t L qx ≤ C d k ρ k θL ∞ t L γ x with1 /q = θ/γ − /d . Hence we have again that Z T d ( B (0 , x ) · g (0 , x ) − B (0 , T ) · g ( T, x )) dx ≤ C d k ρ k θL ∞ t L γ x k g k L ∞ t L pd/ ( p + d ) x , since 1 − ( p + d ) /pd = 1 − /d − /p ≥ θ/γ − /d by the same condition on θ . The secondterm in the r.h.s is handled by using the continuity equation (B.1) satisfied by ρ . Since γ ≥ ρ is a renormalized solution to (B.1) by Th. B.1 and hence we have that ∂ t ρ θ + div( ρ θ u ) = (1 − θ ) ρ θ div u. We may replace Z T Z T d ∂ t B ( s, x ) · g ( s, x ) dx ds = Z T Z T d ∇ x ∆ − x ((1 − θ ) ρ θ div u − div( ρ θ u ) − e c θ ) · g ( s, x ) dx ds, for some time dependent constant e c θ . Using that g ∈ L t L pd/ (2 d +2 p − pd ) x , we bound in asimilar manner all the terms and conclude that − Z T Z T d ∂ t B ( s, x ) · g ( s, x ) dx ds ≤ C d k ρ k θL ∞ t L γ x k u k L t H x k g k L t L p/ ( p +2) x . (cid:3)
3. Technical Preliminaries
We list here technical results and considerations, which were mostly developed in [ ]and upon which our proof relies. As is classical in compressible Fluid Mechanics,the main difficulty in obtaining existence is to prove the compactness of a sequence ofapproximations of the density ρ ε . As mentioned above, we follow here the general strategyof [ ], and we hence rely on the following criterion. Lemma . Let ρ ε be a family of functions which are bounded in some L p ([0 , T ] × T d ) with ≤ p < ∞ . Assume that K h is a family of positive bounded functions such that • sup h R | x |≥ η K h ( x ) dx < ∞ for any η > . • kK h k L → ∞ as h → . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 9
Assume that for some q ≥ ε k ∂ t ρ ε k L q ([0 ,T ] ,W − , ( T d )) < ∞ and lim h → lim sup ε Z T Z T d K h ( x − y ) kK h k L | ρ ε ( x ) − ρ ε ( y ) | p dx dy ds = 0 . Then the family of functions ρ ε is compact in L p ([0 , T ] × T d ) . Conversely if ρ ε is compactin L p ([0 , T ] × T d ) , then the above limit is . The construction of a suitable kernel function K h for the system that we are consideringagain follows [ ]. We first define a bounded, positive, and symmetric function e K h such that e K h ( x ) = 1( h + | x | ) d + a , for | x | ≤ a > e K h independent of h for | x | ≥ /
3. We will also require that e K h ∈ C ∞ ( T d \ B (0 , / e K h ⊂ B (0 , K h = e K h k e K h k L ( T d ) . we have immediately that k K h k L ( T d ) = 1and | x ||∇ K h ( x ) | . | K h ( x ) | . (3.1)For our compactness argument, we use the operator K h = Z h K h ( x ) dhh . (3.2)Note that kK h k L ( T d ) = c | log h | for some positive constant c . With the above notation, one of our main steps is to showthat lim sup ε Z T Z T d K h ( x − y ) | ρ ε ( x ) − ρ ε ( y ) | p dx dy ds → h →
0, from where the compactness of the family ρ ε follows. As our main strategy is to control differences δρ ε , whichrequires some specific lemmas. One may find proofs for these lemmas in [ ]. Our basic wayof estimating differences is through Lemma . Let u ∈ W , , we have | u ( x ) − u ( y ) | . ( D | x − y | u ( x ) + D | x − y | u ( y )) | x − y | , where D h u ( x ) = 1 h Z | z |≤ h |∇ u ( x + z ) || z | d − dz. The next lemma provides a bound for the term D h u ( x ) in term of the maximal function. Lemma . For any u ∈ W ,p with p ≥ , the following inequality D h u ( x ) . M |∇ u | ( x ) holds. Remark . By the above two lemmas we deduce immediately the classical inequality | u ( x ) − u ( y ) | . ( M ∇ u ( x ) + M ∇ u ( y )) | x − y | . (3.3)In several critical places of the proof, we need to estimate the difference D | z | u ( x ) − D | z | u ( x − z ) while relying only on the L regularity of ∇ u . Using classical harmonic analysisresults, we can get the following. Lemma . Assume that u ∈ H ( T d ) . Then for any < p < ∞ , one has Z h Z T d K h ( z ) k D | z | u ( x ) − D | z | u ( x − z ) k L px dz dhh . k u k B p, as a result of which, we further have that Z h Z T d K h ( z ) k D | z | u ( x ) − D | z | u ( x − z ) k L x dz dhh . k u k H | log h | / . Moreover, the following estimate Z h Z T d K h ( z ) K h ( ξ ) k D | z | u ( x ) − D | z | u ( x − ξ ) k L x dz dξ dhh . k u k H | log h | / holds. In most instances, the above estimate is sufficient. But in several cases, we need themore general version,
Lemma . Consider a family of kernels N r ∈ W s, ( T d ) , where s > , which satisfy • sup | ξ |≤ sup r r − s R T d | z | s | N r ( z ) − N r ( z − rξ ) | dz < ∞ , • sup r ( k N r k L + r s k N r k W s, ) < ∞ .Then the estimate Z h Z T d K h ( z ) k N h ∗ u ( x ) − N h ∗ u ( x − z ) k L px dz dhh . k u k L p | log h | / holds for any u ∈ L p with < p ≤ . We now turn to the construction of anappropriate weight function tailored for the proof of Th. 2.2. First we define the function w ε which satisfies the equations ∂ t w ε + u ε · ∇ w ε = − D ε w ε (3.4) w ε (0) = 1 (3.5)where D ε is given by D ε = λ ( M |∇ u ε | + | ρ ε | γ + K h ∗ ( | div u ε | + |L ε ∗ P | + | e P lε | )) . (3.6)Denote w ε,h = K h ∗ w ε . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 11
Then the weight function W ε,h we use is given by W x,yε,h = w ε,h ( x ) + w ε,h ( y )which could capture the feature that W ε,h is big if either one of w ε,h ( x ) and w ε,h ( y ) is big.Since the function W x,yε = w ε ( x ) + w ε ( y ) satisfies the following equation ∂ t W ε + u xε · ∇ x W ε + u yε · ∇ y W ε = − ( D xε w ε ( x ) + D yε w ε ( y )) , it follows that ∂ t W ε,h + u xε · ∇ x W ε,h + u yε · ∇ y W ε,h = − D x,yε,h + Com x,yε,h (3.7)where D x,yε,h = K h ∗ ( D ε w ε )( x ) + K h ∗ ( D ε w ε )( y ) (3.8)and Com x,yε,h = [ u ε · , K h ∗ ] ∇ w ε ( x ) + [ u ε · , K h ∗ ] ∇ w ε ( y ) . (3.9)We conclude the subsection by listing several properties of this weight function withoutgiving a proof (see again [ ] for the proof). Proposition . Assume that ( ρ ε , u ε ) solves system (2.1) – (2.2) with the bounds (2.4) satisfied. Then there exists a weight function w ε which satisfies Equation (3.4) – (3.5) with D ε given by (3.6) such that the following hold: • For any t, x , ≤ w ε ≤ . • If p ≥ γ + 1 , then we have sup t ∈ [0 ,T ] Z T d ρ ε ( t, x ) | log w ε ( t, x ) | dx ≤ C (1 + λ ) . (3.10) • For p ≥ γ , sup t ∈ [0 ,T ] Z T d ρ ε ( t, x ) K h ∗ w ε ≤ η dx ≤ C λ | log η | . (3.11) • For p > γ , we have the following commutator estimate Z h Z t k K h ∗ ( K h ∗ ( | div u ε | + |L ∗ P | + | e P lε ))) w ε − K h ∗ ( | div u ε | + |L ∗ P | + | e P lε | ) w ε,h k L q dt dhh ≤ C | log h | / (3.12) with q = min(2 , p/γ ) .
4. Proof of Theorem 2.2
In this section, we give a proof of the Theorem 2.2 using the compactness argumentprovided in Lemma 3.1. Because all coefficients η i are fixed for this section, we drop theindex η in our notations to keep them simple.In order to carry out our approach, we introduce a smooth function χ ( ξ ) ∈ C ( R ) givenby χ ( ξ ) = | ξ | l (4.1)where 0 < l < / ε → Z T d K h ( x − y ) χ ( δρ ε ) dxdy → h → . (4.2) To close the estimate, it is convenient to consider the the following quantity instead: T h ,ε ( t ) = Z h Z T d K h ( x − y ) W x,yε,h χ ( δρ ε ) dxdy dhh where W x,yε,h = w xε,h + w yε,h . The proof of statement (4.2) is divided into the following several lemmas. Before statingthe lemma, we recall some notation used in subsection 3.3. The penalization term is definedas D x,yε,h = K h ∗ ( D ε w ε )( x ) + K h ∗ ( D ε w ε )( y )and commutator term is given asCom x,yε,h = [ u ε · , K h ∗ ] ∇ w ε ( x ) + [ u ε · , K h ∗ ] ∇ w ε ( y ) . Compared with [ ], we have a different approximation system (2.1) and (2.2). The maininnovation in this paper is the treatment of the pressure term, which is in subsection 4.5.For the estimate of the terms I , I , and I defined below in Lemma 4.1, we use similarideas as in [ ]. T h ,ε ( t ) . Lemma . Let ρ ε and u ε be a sequence of solutions to the system (2.1) - (2.2) satisfyingthe bound (2.4) with γ art ≥ d/ ( d + 2) . Assume that the pressure P satisfies (1.7) , (1.8) ,and (1.11) . Then we have the estimate T h ,ε ( t ) . T h ,ε (0) + I + I + I + I + I , (4.3) where the terms I – I are given by I = Z t Z h Z T d δu ε ∇ x K h ( x − y ) W x,yε,h χ ( δρ ) dxdy dhh ds (4.4) I = − Z t Z h Z T d K h ( x − y ) D x,yε,h χ ( δρ ) dxdy dhh ds (4.5) I = Z t Z h Z T d K h ( x − y ) Com x,yε,h χ ( δρ ) dxdy dhh ds (4.6) I = − Z t Z h Z T d K h ( x − y ) W x,yε,h χ ′ ( δρ ) ρδ (div u ε )( x ) dxdy dhh ds (4.7) I = Z t Z h Z T d K h ( x − y ) W x,yε,h (cid:18) χ ( δρ ) − χ ′ ( δρ ) δρ (cid:19) div x u ε ( x ) dxdy dhh ds. (4.8) Proof.
From (2.1), one gets an equation for δρ ε ∂ t δρ ε + div x ( ρ ε u ε )( x ) − div y ( ρ ε u ε )( y ) = 0 , which may be rewritten as ∂ t δρ ε + div x ( δρ ε u ε )( x ) + div y ( δρ ε u ε )( y ) + ρ ε ( y ) div x u ε ( x ) − ρ ε ( x ) div y u ε ( y ) = 0 . (4.9) OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 13
Note that the terms ρ ε ( y ) div x u ε ( x ) and ρ ε ( x ) div y u ε ( y ) are well-defined since ρ ε ∈ L anddiv x u ε ∈ L . By (2.4), we have ρ lε ∈ L for γ art > l ) and ∇ x u ε ∈ L . Hence, byTheorem B.1, δρ ε is a renormalized solution for the system(4.9). Noticing that − ρ ε ( y ) div x u ε ( x ) + ρ ε ( x ) div y u ε ( y )= 12 ( δρ ε (div x u ε ( x ) + div y u ε ( y ) − ¯ ρ ε (div x u ε ( x ) − div y u ε ( y )))we arrive at ∂ t χ ( δρ ε ) + div x ( χ ( δρ ε ) u ε )( x ) + div y ( χ ( δρ ε ) u ε )( y ) (4.10)= (cid:18) χ ( δρ ε ) − χ ′ ( δρ ε ) δρ ε (cid:19) (div x u ε ( x ) + div y u ε ( y ) − χ ′ ( δρ ε )¯ ρ (div x u ε ( x ) − div y u ε ( y )) . From the definition of χ in (4.1), it follows easily χ ( δρ ε ) and χ ′ ( δρ ε ) δρ ε ≤ Cρ lε , which implies that χ ( δρ ε ) , χ ′ ( δρ ε )¯ ρ ∈ L . Since ∇ x u ε ∈ L , all the terms on the right sideof (4.10) make sense. By (3.7), we obtain ∂ t ( K h ( x − y ) W x,yε,h χ ( δρ ε )) = K h ( x − y ) ∂ t W x,yε,h χ + K h ( x − y ) W x,yε,h ∂ t χ = − K h ( x − y ) u ε ( x ) ∇ x W x,yε,h χ − K h ( x − y ) u ε ( y ) ∇ y W x,yε,h χ − K h ( x − y ) D ε,h χ + K h ( x − y )Com ε,h χ + K h ( x − y ) W x,yε,h ( χ − χ ′ δρ ε )(div x u ε ( x ) + div y u ε ( y ))+ 12 K h ( x − y ) W x,yε,h χ ′ δρ ε div u ε ( x ) − K h ( x − y ) W x,yε,h χ ′ ρ ε δ div u ε ( x ) − K h ( x − y ) W x,yε,h div x ( χu ε ( x )) − K h ( x − y ) W x,yε,h div y ( χu ε ( y )) . (4.11)The above equation may be justified as the following. First, in order to show K h ( x − y ) u ε ( x ) ∇ x W x,yε,h χ ∈ L x,y , we just need to prove K h ( x − y ) u ε ( x ) χ ∈ L x,y since ∇ x W x,yε,h ∈ L ∞ .In fact we note Z T d K h ( x − y ) | u ε ( x ) | χ dxdy = Z T d K h ( y ) Z T d | u ε ( x ) | χ ( ρ ε ( x ) − ρ ε ( x − y )) dxdy . Z T d K h ( y ) dy . . Therefore, the term K h ( x − y ) u ε ( x ) ∇ x W x,yε,h χ is well-defined. Similar arguments could showthat K h ( x − y ) u ε ( y ) ∇ y W x,yε,h χ ∈ L x,y . Second, noting that K h ( x − y ) W x,yε,h ≤ K h ( x − y ) , the terms K h ( x − y ) W x,yε,h ( χ − χ ′ δρ ε )(div x u ε ( x ) + div y u ε ( y )), K h ( x − y ) W x,yε,h χ ′ δρ ε div u ε ( x ),and K h ( x − y ) W x,yε,h χ ′ ρ ε δ div u ε ( x ) belong to L x,y by similar arguments as for the first term.Third, we note that D ε,h is smooth and belongs to L ∞ . Hence, K h ( x − y ) D ε,h χ makessense since χ ( δρ ) ∈ L x . One may check easily that ρ l u ε ∈ L for γ art ≥ d/ ( d + 2)and thus K h ( x − y )Com ε,h χ ∈ L x,y . Lastly, div x ( χu ε ( x )) ∈ W − ,r for some r > K h ( x − y ) W x,yε,h ∈ W ,r ′ where r ′ is the H¨older conjugate exponent of r . Therefore, theterms K h ( x − y ) W x,yε,h div x ( χu ε ( x )) and K h ( x − y ) W x,yε,h div y ( χu ε ( y )) make sense. Using the product rule, we further rewrite (4.11) as ∂ t ( K h ( x − y ) W x,yε,h χ ( δρ ε )) = − div x (cid:16) u ( x ) K h ( x − y ) W x,yε,h χ (cid:17) − div y (cid:16) u ( y ) K h ( x − y ) W x,yε,h χ (cid:17) + δu ε ( x ) ∇ x K h ( x − y ) W x,yε,h χ − K h ( x − y ) D ε,h χ + K h ( x − y )Com ε,h χ + K h ( x − y ) W x,yε,h ( χ − χ ′ δρ ε )div x u ε ( x ) + 12 K h ( x − y ) W x,yε,h χ ′ δρ ε div u ε ( x ) − K h ( x − y ) W x,yε,h χ ′ ρ ε δ div u ε ( x ) , which could be justified similarly as the equation (4.11). Integrating the time derivative of T h ,ε ( t ) from 0 to t gives (4.3), concluding the proof. (cid:3) I . In this subsection, we estimate the terms I in the followinglemma. Lemma . Let I be given by (4.4) . Under the assumptions in Lemma 4.1, the estimate I ≤ C | log h | / + Cλ − D holds with the penalization D defined by D = λ Z t Z h Z T d K h ( x − y )( K h ∗ (( M |∇ u ε | + | ρ ε | γ ) w ε )( x ) χ ( δρ ε ) dxdy dhh ds (4.12) for t ≤ T , where T can be any positive number and the constant C may depend on time T . Proof.
We first recall I = Z t Z h Z T d δu ε ∇ x K h ( x − y ) W x,yε,h χ ( δρ ) dxdy dhh ds. By Lemma 3.2, it follows | δu ε ( x ) | = | u ε ( x ) − u ε ( y ) | . | x − y | ( D | x − y | u ε ( x ) + D | x − y | u ε ( y )) , with D h u ε ( x ) given by D h u ε ( x ) = 1 h Z | z |≤ h |∇ u ε ( x + z ) || z | d − dz. Hence, in view of (3.1), we obtain I . Z t Z h Z T d K h ( x − y )( D | x − y | u ε ( x ) + D | x − y | u ε ( y )) W x,yε,h χ ( δρ ε ) dxdy dhh ds = 2 Z t Z h Z T d K h ( x − y )( D | x − y | u ε ( x ) + D | x − y | u ε ( y )) w xε,h χ ( δρ ε ) dxdy dhh ds where we used symmetry in x and y of the integral bound in the last step. Since we onlyhave k u ε k L H . k ρ k L γart . , we can not expect the last integral to be much smaller than (cid:13)(cid:13)(cid:13)(cid:13)Z h K h dhh (cid:13)(cid:13)(cid:13)(cid:13) L = | log h | . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 15
Instead, we use the penalty defined in (3.6) to absorb the main contribution of I and provethe remainder is of the size of | log h | / . In order to proceed, we rewrite Z t Z h Z T d K h ( x − y )( D | x − y | u ε ( x ) + D | x − y | u ε ( y )) w xε,h χ ( δρ ε ) dxdy dhh ds = Z t Z h Z T d K h ( x − y )( D | x − y | u ε ( y ) − D | x − y | u ε ( x )) w xε,h χ ( δρ ε ) dxdy dhh ds + 2 Z t Z h Z T d K h ( x − y ) D | x − y | u ε ( x ) w xε,h χ ( δρ ε ) dxdy dhh ds = I , + I , . (4.13)To estimate the term I , , we change the variable to arrive at I , = Z t Z h Z T d K h ( x − y )( D | x − y | u ε ( y ) − D | x − y | u ε ( x )) w xε,h ρ lε ( x ) dxdy dhh ds = Z t Z h Z T d K h ( z )( D | z | u ε ( x − z ) − D | z | u ε ( x )) w xε,h ρ lε ( x ) dx dz dhh ds. From Proposition 3.7, we know 0 ≤ w ε ≤
1, which implies0 ≤ w ε,h ≤ h > k K h k L = 1 . By H¨older’s inequality, Lemma 3.5, we obtain Z t Z h Z T d K h ( z )( D | z | u ε ( x − z ) − D | z | u ε ( x )) w xε,h ρ lε ( x ) dx dz dhh ds . Z t Z h Z T d K h ( z ) k| D | z | u ε ( x − z ) − D | z | u ε ( x ) |k L x dz dhh ds . | log h | / k u ε k L H . While for the second integral I , , it is not in a form to which we could directly applyLemma 3.5. Instead, we rewrite it as I , = 2 Z t Z h Z T d K h ( x − y ) K h ( x − z )( D | x − y | u ε ( x ) − D | x − y | u ε ( z )) w zε χ ( δρ ε ) dxdy dhh ds + Z t Z h Z T d K h ( x − y ) K h ( x − z ) D | x − y | u ε ( z ) w zε χ ( δρ ε ) dx dy dz dhh ds ≤ Z t Z h Z T d K h ( x − y ) K h ( x − z )( D | x − y | u ε ( x ) − D | x − y | u ε ( z )) w zε χ ( δρ ε ) dx dy dz dhh ds + C Z t Z h Z T d K h ( x − y ) K h ( x − z ) M ( ∇ u ε )( z ) w zε χ ( δρ ε ) dx dy dz dhh ds where we used Lemma 3.3 in the last step. By Lemma 3.5 and the uniform boundedness of ρ ε in L γ art , we further get Z t Z h Z T d K h ( x − y ) K h ( x − z )( D | x − y | u ε ( x ) − D | x − y | u ε ( z )) w zε χ ( δρ ε ) dx dy dz dhh ds . Z t Z h Z T d K h ( y ) K h ( z ) k| D | y | u ε ( x − z ) − D | y | u ε ( x ) |k L x dy dz dhh ds . | log h | / k u ε k L H . (4.14)Collecting the estimates of I , with I , and applying them to (4.13) gives I . | log h | / + Z t Z h Z T d K h ( x − y ) K h ( x − z ) M ( ∇ u ε )( z ) w zε χ ( δρ ε ) dx dy dz dhh ds (4.15)where the last integral could be bounded by Cλ − D and the proof is completed. (cid:3) I . We denote D ( x ) = | div u ε | ( x ) + |L ε ∗ P | ( x ) + | e P ε | l ( x )and the estimate for I is provided in the lemma below. Lemma . Let I be as in (4.5) . Under the assumptions in Lemma 4.1, then we havethat I ≤ C | log h | θ − D − D holds for some > θ > with the penalization D defined in (4.12) and D given by D = λ Z t Z h Z T d K h ( x − y ) K h ∗ D ( x ) w ε,h ( x ) χ ( δρ ε ) dxdy dhh ds (4.16) for t ≤ T , where T can be any positive number and the constant C may depend on time T . Proof.
The term I is negative and helps us in controlling other terms. We pull outthe penalization terms D with D and the error is bounded by C | log h | / . To be morespecific, we have I = − Z t Z h Z T d K h ( x − y ) D x,yε,h χ ( δρ ε ) dxdy dhh ds = − λ Z t Z h Z T d K h ( x − y )( K h ∗ (( M |∇ u ε | + | ρ ε | γ ) w ε )( x ) χ ( δρ ε ) dxdy dhh ds − λ Z t Z h Z T d K h ( x − y )( K h ∗ ( K h ∗ D w ε )( x ) χ ( δρ ε ) dxdy dhh ds By the symmetry in x and y of the above expression, we further get I = − λ Z t Z h Z T d K h ( x − y )( K h ∗ (( M |∇ u ε | + | ρ ε | γ ) w ε )( x ) χ ( δρ ε ) dxdy dhh ds − λ Z t Z h Z T d K h ( x − y )( K h ∗ ( K h ∗ D w ε )( x ) χ ( δρ ε ) dxdy dhh ds = − D + I , . (4.17) OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 17
We extract the second penalization D from I , as I , = − λ Z t Z h Z T d K h ( x − y ) K h ∗ D ( x ) w ε,h ( x ) χ ( δρ ε ) dxdy dhh ds + 2 λ Z t Z h Z T d K h ( x − y ) (cid:16) K h ∗ D ( x ) w ε,h ( x ) − K h ∗ ( K h ∗ D w ε )( x ) (cid:17) χ ( δρ ε ) dxdy dhh ds. Noting w ε,h ( x ) = K h ∗ w ε ( x ), in view of (3.12), we may bound the last commutator integralin the above equality by C | log h | θ for some 1 > θ >
0. Therefore, we arrive at I , ≤ − D + C | log h | θ . Hence, from (4.17) we get I ≤ − D − D + C | log h | θ (4.18)concluding the proof. (cid:3) I . We bound the term I in this subsection. Lemma . Let I be given by (4.6) . Under the assumptions in Lemma 4.1, the estimate I ≤ C | log h | / − Cλ − D holds with the penalization D defined by (4.12) for t ≤ T , where T can be any positivenumber and the implicit constant may depend on time T . Proof.
In view of (3.9), we may write I as I = Z t Z h Z T d K h ( x − y )Com x,yε,h χ ( δρ ε ) dxdy dhh ds = Z t Z h Z T d K h ( x − y ) ([ u ε · , K h ∗ ] ∇ w ε ( x ) + [ u ε · , K h ∗ ] ∇ w ε ( y )) χ ( δρ ε ) dxdy dhh ds = 2 Z t Z h Z T d K h ( x − y )[ u ε · , K h ∗ ] ∇ w ε ( x ) χ ( δρ ε ) dxdy dhh ds where we used the symmetry in x and y in the last step. Expanding the commutator andusing the identity u ε · ∇ w ε ( x ) = div( u ε w ε ( x )) − div( u ε ) w ε ( x ) , we arrive at I =2 Z t Z h Z T d K h ( x − y )( u xε · ∇ K h ( x − z ) w zε − u zε · ∇ K h ( x − z ) w zε ) χ ( δρ ε ) dx dy dz dhh ds + 2 Z t Z h Z T d K h ( x − y ) K h ∗ (div u ε w ε )( x ) χ ( δρ ε ) dxdy dhh ds =2 Z t Z h Z T d K h ( x − y )( u xε − u zε ) · ∇ K h ( x − z ) w zε χ ( δρ ε ) dx dy dz dhh ds + 2 Z t Z h Z T d K h ( x − y ) K h ∗ (div u ε w ε )( x ) χ ( δρ ε ) dxdy dhh ds where the second integral in the last equality of the above expression is bounded by Cλ − D since | div u ε | ≤ |∇ u ε | ≤ M |∇ u ε | . By Lemma 3.2 and the inequality (3.1), the first integral is estimated as Z t Z h Z T d K h ( x − y )( u xε − u zε ) · ∇ K h ( x − z ) w zε χ ( δρ ε ) dx dy dz dhh ds . Z t Z h Z T d K h ( x − y )( D | x − z | u ε ( x ) + D | x − z | u ε ( z )) × | ( x − z ) · ∇ K h ( x − z ) | w zε χ ( δρ ε ) dx dy dz dhh ds . Z t Z h Z T d K h ( x − y ) K h ( x − z ) | D | x − z | u ε ( x ) − D | x − z | u ε ( z ) | w zε χ ( δρ ε ) dx dy dz dhh ds + 2 Z t Z h Z T d K h ( x − y ) K h ( x − z ) D | x − z | u ε ( z ) w zε χ ( δρ ε ) dx dy dz dhh ds (4.19)where the second integral in the last inequality is bounded by Cλ − D by Lemma 3.3. Bythe definition of χ in (4.1), we change the variable to get Z t Z h Z T d K h ( x − y ) K h ( x − z ) | D | x − z | u ε ( x ) − D | x − z | u ε ( z ) | w zε χ ( δρ ε ) dx dy dz dhh ds = Z t Z h Z T d K h ( y ) K h ( z ) | D | z | u ε ( x ) − D | z | u ε ( x − z ) | w x − zε χ ( ρ xε − ρ x − zε ) dx dy dz dhh ds . Z t Z h Z T d K h ( y ) K h ( z ) | D | z | u ε ( x ) − D | z | u ε ( x − z ) | w x − zε × ( ρ lε ( x ) + ρ lε ( x − z )) dx dy dz dhh ds, from where by H¨older’s inequality and Lemma 3.5 we obtain a further bound of the aboveintegral Z t Z h Z T d K h ( z ) K h ( y ) k D | z | u ε ( x ) − D | z | u ε ( x − z ) k L x dz dy dhh ds . | log h | / k u ε k L H . | log h | / . Collecting the estimates for the two terms in (4.19), we arrive at I ≤ C | log h | / + Cλ − D (4.20)proving the lemma. (cid:3) In this section, we treat the terms involving the pressure. Actu-ally the pressure term appears in both I and I in slightly different forms. We introducean abstract function to give the estimate in a more general form and the corresponding OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 19 bounds in terms I and I follow easily. We define the following integral I P = − Z t Z εε Z h Z T d K h ( x − y ) K h ( x − e y ) f ( x, y, e y ) w ε,h ( x ) × ( L ε ′ ∗ P ( x ) − L ε ′ ∗ P ( y )) dx dy d e y dhh dε ′ ε ′ ds (4.21)and establish the estimate of I P in the Lemma 4.5 below.In the estimate of the first three terms I , I , and I , the argument is still true even ifwe replace the mollifying kernel L ε by L ε , i.e., we may have an upper bound point-wise in ε . The kernel L ε is only necessary in the treatment of the pressure term. In fact for thepressure term, it is very difficult to obtain an estimate uniform in ε (using the mollifier L ε )since when ε is relatively big compared to h , the error term Diff defined by (4.26) is outof control because L ε ∗ P can not approximate P precisely enough. Therefore, instead ofconsider a L ∞ ε topology, we consider L ε ( dε/ε ). In order to treat the term I P , we need tostudy two cases separately, i.e., h ≤ ε ′ and ε ′ ≤ h . The case h ≤ ε ′ is easy. We bound theterm δ ( L ε ′ ∗ P ) by the H¨older norm of L ε ′ , which is under our control since ε ′ is relativelybig. For the case ε ′ ≤ h , it is much more difficult. Roughly speaking, we use the fact thatthe smoothing effect of K h is dominant since the scaling of L ε ′ is smaller. Therefore, wetreat L ε ′ ∗ P as an approximation of P which is bounded by P in any L p for p ∈ [1 , ∞ ] suchthat P ∈ L p . The main difficulty of executing this idea is that we can not control L ε ′ ∗ P directly with our penalization. Instead, we need to consider the quantity L ε ′ ∗ ( w θ P ) forsome θ > L ε to close the estimate. Lemma . Let I P be defined by (4.21) and ( ρ ε , u ε ) be a sequence of solutions to thesystem (2.1) - (2.2) satisfying the bound (2.4) with γ art ≥ max(2 s ∗ , s ∗ , d/ ( d + 2)) where s ∗ and s ∗ are the H¨older conjugate exponent of s and s respectively. Assume the pressure P satisfies (1.7) , (1.8) , (1.11) , and (1.12) . Let f ( x, y, e y ) be such that | f ( x, x − y, x − e y ) | ≤ C ( χ ′ ( δρ ε ( x, y )) ρ ε ( x, y ) + χ ′ ( δρ ε ( x, e y )) ρ ε ( x, e y )) . (4.22) Let r h be defined as in (1.12) . We have | I P | ≤ C + C (cid:18)Z εε r max( h ,ε ′ ) dε ′ ε ′ (cid:19) ¯ θ | log( h ) | θ + C Z t T h ,ε ( s ) ds + Cλ − D + 3 D with D given by (4.16) and D by D = η (1 + l ) Z t Z h Z T d K h ( x − y ) W x,yε,h χ ( δρ ε ) ρ γ art ε ( x ) dxdy dhh ds, (4.23) for some < ¯ θ , < θ < , and t ≤ T , where T can be any positive number and the implicitconstant may depend on time T . Proof.
Here we give a uniform estimate in ε of this term, which may be divided intotwo cases: ε ′ < h and ε ′ ≥ h : I P = − Z t Z εε Z h Z T d K h ( x − y ) K h ( x − e y ) f w ε,h ( x ) δ ( L ε ′ ∗ P ) dx dy d e y dhh dε ′ ε ′ ds = − Z t Z εε Z h Z T d ( ε ′ ≥ h + ε ′
0. By (1.7) and (1.8), we get Z T d P ( t, z, ρ ε ( z )) dz . Z T d R ( t, z ) + Θ ( z ) + ρ p ( z ) dz . γ art ≥ p . Therefore, by (4.22), using the uniform integrability of ρ ε and the fact that k K h k L = 1 , we arrive at | I b | . Z εε Z max( h ,ε ′ ) h Z T d K h ( y ) | y | θ ε ′ θ dy dhh dε ′ ε ′ . Z εε Z max( h ,ε ′ ) h h θ ε ′ θ dhh dε ′ ε ′ . . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 21
Next we treat the difficult term I s . Denoting e ε = max( h , ε ′ ), by assumptions (1.11), weobtain | I s | ≤ C Z t Z εε Z e ε Z T d K h ( x − y ) K h ( x − e y ) f ( x, y, e y ) w ε,h ( x ) L ε ′ ( z ) | ρ ε ( x − z ) − ρ ε ( y − z ) | ( ρ γ − ε ( x − z ) + ρ γ − ε ( y − z )) dx dy d e ydz dhh dε ′ ε ′ ds + C Z t Z εε Z e ε Z T d K h ( x − y ) K h ( x − e y ) f ( x, y, e y ) w ε,h ( x ) L ε ′ ( z )( Q x − z,y − zε + ( e P x − zε + e P y − zε ) | ρ ε ( t, x − z ) − ρ ε ( t, y − z ) | ) dx dy d e ydz dhh dε ′ ε ′ ds = I s, + I s, + I s, (4.24)where I s, is the first integral with I s, and I s, corresponding to the integrals containing Q x − z,y − zε and ( e P x − zε + e P y − zε ) respectively. For the sake of simplicity, we suppress theconstant C in I s, , I s, , and I s, . By making constants in the following estimates bigger ifnecessary, we may recover the bound for I s . The first integral I s, is the most difficult oneamong the three. In order to estimate this term, we need to use the penalization term D as well as the regularity of the weight function w ε,h . To be more specific, we have I s, = Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) f ( x, x − y, x − e y ) w ε,h ( x ) L ε ′ ( z ) | ρ ε ( x − z ) − ρ ε ( x − y − z ) | ( ρ γ − ε ( x − z ) + ρ γ − ε ( x − y − z )) dx dy d e ydz dhh dε ′ ε ′ ds = ¯ I s, + Diffwhere we denoted using the notation in Subsection B.1¯ I s, = Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) f ( x, x − y, x − e y ) w /γ art ε,h ( x ) w − /γ art ε,h ( x − z ) L ε ′ ( z ) × | δρ ε ( x − z, y ) | ρ γ − ε ( x − z, y ) dx dy dz dhh dε ′ ε ′ ds (4.25)and Diff = Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) f ( x, x − y, x − e y ) w /γ art ε,h ( x ) L ε ′ ( z ) | δρ ε ( x − z, y ) |× ( w − /γ art ε,h ( x ) − w − /γ art ε,h ( x − z )) ρ γ − ε ( x − z, y ) dx dy dz dhh dε ′ ε ′ ds. (4.26) As we see below, the term ¯ I s, is the leading order term and Diff is a perturbation of constantsize. Using H¨older’s inequality, the term ¯ I s, is bounded by Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) k L ε ′ ∗ ( | δρ ε ( x, y ) | ρ γ − ε ( x, y ) w − /γ art ε,h ) k L γ ′ artx × k f ( x, x − y, x − e y ) w /γ art ε,h k L γartx dy d e y dhh dε ′ ε ′ ds ≤ C Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) k δρ ε ( x, y ) ρ γ − ε ( x, y ) w − /γ art ε,h k L γ ′ artx × k ( | χ ′ | ρ ε ( x, y ) + | χ ′ | ρ ε ( x, e y )) w /γ art ε,h k L γartx dy d e y dhh dε ′ ε ′ ds ≤ C Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) k ( δρ ε ) σ ( x, y ) w − γ/γ art ε,h k L α x k ( δρ ε ) − σ ( x, y ) ρ γ − ε ( x, y ) w ( γ − /γ art ε,h k L α x k ( | χ ′ | ρ ε ( x, y ) + | χ ′ | ρ ε ( x, e y )) w /γ art ε,h k L γartx dy d e y dhh dε ′ ε ′ ds (4.27)where α , α , and σ are given by α = γ art γ art − γ α = γ art γ − , σ = 1 − ( γ − l ) γ art . We also require lγ art = 1 + l. Using Young’s inequality, one further gets¯ I s, ≤ Z t Z h Z T d K h ( y ) K h ( e y ) (cid:18) Cη Z | δρ ε | l ( x, y ) w ε,h dx + η Z | δρ ε | l ( x, y ) ρ γ art ε ( x, z ) w ε,h dx + η Z | δρ ε | l ( x, e y ) ρ γ art ε ( x, y ) w ε,h dx (cid:19) dy d e y dhh ds = Cη Z t Z h Z T d K h ( x − y ) | δρ ε | l w ε,h dxdy dhh ds + η Z t Z h Z T d K h ( x − y ) | δρ ε | l ρ γ art ε w ε,h dxdy dhh ds where we used k K h k L = 1 and the last integral may be bounded by D /
8. Next we turnto the term Diff. Noting w − /γ art ε,h ( x ) − w − /γ art ε,h ( x − z ) ≤ C | z | − /γ art h − /γ art , OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 23 we obtainDiff ≤ Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) f ( x, x − y, x − e y ) w /γ art ε,h ( x ) | z | − /γ art h − /γ art L ε ′ ( z ) × | δρ ε ( x − z, y ) | ρ γ − ε ( x − z, y ) dx dy dz d e y dhh dε ′ ε ′ ds ≤ C Z εε Z e ε Z T d K h ( y ) K h ( e y ) | z | − /γ art h − /γ art L ε ′ ( z ) k f ( x, x − y, x − e y ) w /γ art ε,h k L γartx × k| δρ ε ( x, y ) | ρ γ − ε ( x, y ) k L γ ′ artx , dy dz d e y dhh dε ′ ε ′ (4.28)from where using (4.22) and Young’s inequality, by the uniform integrability of ρ ε and k K h k L = 1, we further getDiff ≤ Cνη Z t Z εε Z e ε Z T d | z | − /γ art h − /γ art L ε ′ ( z ) dz Z T d K h ( x − y ) | δρ ε | l ρ γ art ε w ε,h dxdy dhh dε ′ ε ′ ds + Cν Z εε Z e ε Z T d | z | − /γ art h − /γ art L ε ′ ( z ) dz dhh dε ′ ε ′ for a small parameter ν >
0. For the second integral in the right side of the above inequality,we have Cν Z εε Z e ε Z T d | z | − /γ art h − /γ art L ε ′ ( z ) dz dhh dε ′ ε ′ ≤ Cν Z εε Z e ε ( ε ′ ) − /γ art h − /γ art dhh dε ′ ε ′ ≤ Cν .
Using ε ′ ≤ h and choosing ν sufficiently small, we arrive at Cνη Z t Z εε Z e ε Z T d | z | − /γ art h − /γ art L ε ′ ( z ) dz Z T d K h ( x − y ) | δρ ε | l ρ γ art ε w ε,h dxdy dhh dε ′ ε ′ ds ≤ Cνη Z t Z εε Z e ε Z T d ( ε ′ ) − /γ art h − /γ art K h ( x − y ) | δρ ε | l ρ γ art ε w ε,h dxdy dhh dε ′ ε ′ ds ≤ η Z t Z e ε Z T d K h ( x − y ) | δρ ε | l ρ γ art ε w ε,h dxdy dhh ds which may be bounded by D /
16. Therefore, we obtainDiff ≤ C + D . Next we turn to the treatment of the term I s, . By changing the variables, we rewriteit as I s, = Z t Z εε Z e ε Z T d K h ( x − y ) K h ( x − e y ) f ( x, y, e y ) w ε,h ( x ) L ε ′ ( z ) Q x − z,y − zε (4.29) dx dy dz d e y dhh dε ′ ε ′ ds = Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) f ( x, x − y, x − e y ) w ε,h ( x ) L ε ′ ( z ) Q x − z,x − y − zε (4.30) dx dy dz d e y dhh dε ′ ε ′ ds. (4.31) In view of w ε,h ( x ) ≤
1, we get I s, ≤ Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) | f ( x, x − y, x − e y ) | w /γ art ε,h ( x ) L ε ′ ( z ) Q ε dx dy dz dhh dε ′ ε ′ ds where Q ε = Q x − z,x − y − zε . Using (4.22), H¨older’s inequality, and that k L ε ′ k L = 1, we arriveat I s, ≤ Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) k f ( x, x − y, x − e y ) w /γ art ε,h k L γartx × (cid:13)(cid:13)(cid:13)(cid:13)Z T d L ε ′ ( z ) Q ε dz (cid:13)(cid:13)(cid:13)(cid:13) L γ ′ artx dy d e y dhh dε ′ ε ′ ds ≤ C Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) k ( | χ ′ | ρ ε ( x, y ) + | χ ′ | ρ ε ( x, e y )) w /γ art ε,h k L γartx × (cid:13)(cid:13) Q x,x − yε (cid:13)(cid:13) L γ ′ artx dy d e y dhh dε ′ ε ′ ds where γ ′ art is the H¨older conjugate exponent of γ art . By Young’s inequality, we further get I s, ≤ η Z t Z h Z T d K h ( x − y ) | δρ ε | l ρ γ art ε w ε,h dxdy dhh ds + Cη Z t Z h Z T d K h ( x − y ) | Q x,x − yε | γ ′ art dxdy dhh ds where the first integral on the right side is bounded by D /
8. Using H¨older’s inequalities,the second integral may be estimated as Cη Z t Z h Z T d K h ( x − y ) | Q x,x − yε | γ ′ art dxdy dhh ds ≤ Cη (cid:18)Z t Z h Z T d K h ( x − y ) dxdy dhh ds (cid:19) ( s − γ ′ art ) /s × (cid:18)Z t Z h Z T d K h ( x − y ) | Q x,x − yε | s dxdy dhh ds (cid:19) γ ′ art /s with s − γ ′ art ≥ γ art ≥ s ′ . From (1.12), the above expression may be furtherbounded by C (cid:18)Z εε r e ε dε ′ ε ′ (cid:19) γ ′ art /s | log h | ( s − γ ′ art ) /s . Therefore, we obtain I s, ≤ D C (cid:18)Z εε r e ε dε ′ ε ′ (cid:19) γ ′ art /s | log h | ( s − γ ′ art ) /s . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 25
We estimate the term I s, next and rewrite it as I s, = Z t Z εε Z e ε Z T d K h ( x − y ) K h ( x − e y ) f ( x, y, e y ) w ε,h ( x ) L ε ′ ( z )( e P y − zε − e P x − zε ) | ρ ε ( t, x − z ) − ρ ε ( t, y − z ) | dx dy dz d e y dhh dε ′ ε ′ ds + 2 Z t Z εε Z e ε Z T d K h ( x − y ) K h ( x − e y ) f ( x, y, e y ) w ε,h ( x ) L ε ′ ( z ) e P x − zε | ρ ε ( t, x − z ) − ρ ε ( t, y − z ) | dx dy dz d e y dhh dε ′ ε ′ ds. (4.32)For the first term, we perform the change of variables and use H¨older’s inequality to arriveat Z t Z εε Z e ε Z T d K h ( x − y ) K h ( x − e y ) f ( x, y, e y ) w ε,h ( x ) L ε ′ ( z )( e P y − zε − e P x − zε ) × | ρ ε ( t, x − z ) − ρ ε ( t, y − z ) | dx dy dz d e y dhh dε ′ ε ′ ds ≤ C Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) (cid:13)(cid:13)(cid:13) | χ ′ | ρ ε w /γ art ε,h ( x, y ) + | χ ′ | ρ ε w /γ art ε,h ( x, e y ) (cid:13)(cid:13)(cid:13) L γartx × (cid:13)(cid:13)(cid:13) ( δ e P x,yε ) | δρ ε ( x, y ) | (cid:13)(cid:13)(cid:13) L γ ′ artx dy d e y dhh dε ′ ε ′ ds where we also used the bound w ε,h ( x ) ≤ k L ε ′ k L = 1 for any ε ′ >
0. Using Young’sinequality and Minkowsky’s inequality, we get a further bound for the above term η Z t Z εε Z e ε Z T d K h ( y ) (cid:13)(cid:13) | χ ′ | ρ ε w ε,h ( x ) (cid:13)(cid:13) γ art L γartx dy dhh dε ′ ε ′ ds + Cη Z t Z εε Z e ε Z T d K h ( y ) (cid:13)(cid:13)(cid:13) ( δ e P x,yε ) | δρ ε ( x, y ) | (cid:13)(cid:13)(cid:13) γ ′ art L γ ′ artx dy dhh dε ′ ε ′ ds. The first integral in the above bound is bounded by D /
16. In order to estimate the secondintegral, we introduce the truncation function e φ Mε ( x, y ) = ¯ φ ( ρ xε /M ) ¯ φ ( ρ yε /M )where ¯ φ is a smooth function such that¯ φ ( s ) = , ≤ s ≤ , , s ≥ ∈ [0 , , otherwise . (4.33)Then we have Cη Z t Z εε Z e ε Z T d K h ( y ) (cid:13)(cid:13)(cid:13) ( δ e P x,yε ) | δρ ε ( x, y ) | (cid:13)(cid:13)(cid:13) γ ′ art L γ ′ artx dy dhh dε ′ ε ′ ds ≤ C Z t Z εε Z e ε Z T d K h ( y ) | δ e P x,yε | γ ′ art e φ Mε ( x, x − y ) | δρ ε ( x, y ) | γ ′ art dxdy dhh dε ′ ε ′ ds + C Z t Z εε Z e ε Z T d K h ( y ) | δ e P x,yε | γ ′ art (1 − e φ Mε ( x, x − y )) | δρ ε ( x, y ) | γ ′ art dxdy dhh dε ′ ε ′ ds Applying H¨older’s inequality and using (1.12), we bound the truncated term as Cη Z t Z εε Z e ε Z T d K h ( y ) | δ e P x,yε | γ ′ art φ Mε ( x, x − y ) | δρ ε ( x, y ) | γ ′ art dxdy dhh dε ′ ε ′ ds ≤ CM γ ′ art Z t Z εε Z e ε Z T d K h ( y ) dxdy dhh dε ′ ε ′ ds ! − γ ′ art /s × Z t Z εε Z e ε Z T d K h ( y ) | e P x − yε − e P xε | s dxdy dhh dε ′ ε ′ ds ! γ ′ art /s ≤ CM γ ′ art | log h | − γ ′ art /s Z εε r e ε dε ′ ε ′ ! γ ′ art /s . For the remainder term, (i.e., the term involving 1 − φ Mε ), we use the simple relation( { ρ ( x ) ≥ M } ∩ { ρ ( z ) ≥ M } ) c = { ρ ( x ) ≥ M } c ∪ { ρ ( z ) ≥ M } c to obtain Z t Z εε Z e ε Z T d K h ( y ) | δ e P x,yε | γ ′ art (1 − e φ Mε ( x, x − y )) | δρ ε ( x, y ) | γ ′ art dxdy dhh dε ′ ε ′ ds ≤ Z t Z εε Z e ε Z T d K h ( y ) | δ e P x,yε | γ ′ art ( { ρ x ≥ M } + { ρ x − y ≥ M } ) | δρ ε ( x, y ) | γ ′ art dxdy dhh dε ′ ε ′ ds By H¨older’s and Young’s inequalities, we get Cη Z t Z εε Z e ε Z T d K h ( y ) | δ e P x,yε | γ ′ art (1 − φ Mε ( x, x − y )) | δρ ε ( x, y ) | γ ′ art dxdy dhh dε ′ ε ′ ds . Z t Z εε Z e ε Z T d K h ( y ) | e P x − yε − e P xε | s dxdy dhh dε ′ ε ′ ds + Z t Z εε Z e ε Z T d K h ( y ) { ρ x ≥ M } ρ ε ( t, x ) s γ ′ art / ( s − γ ′ art ) dxdy dhh dε ′ ε ′ ds . r h + M − ( γ art − s γ ′ art / ( s − γ ′ art )) | log h | . Note for γ art ≥ s ′ , one can easily check that γ art − s γ ′ art / ( s − γ ′ art ) >
0. For the secondterm in (4.32), we need to use the penalty function defined in (3.6). More specifically,we need to extract an integral involving K h ∗ e P and estimate the remainder term with aquantity converging to 0. To proceed, we rewrite this integral as2 Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) f ( x, x − y, x − e y ) w /γ art ε,h ( x ) L ε ′ ( z ) e P x − zε w / (1+ l ) ε,h ( x − z ) × | ρ ε ( t, x − z ) − ρ ε ( t, x − y − z ) | dx dy dz dhh dε ′ ε ′ ds + 2 Z t Z εε Z e ε Z T d K h ( y ) K h ( e y ) f ( x, x − y, x − e y ) w /γ art ε,h ( x ) L ε ′ ( z )( w / (1+ l ) ε,h ( x ) − w / (1+ l ) ε,h ( x − z )) e P x − zε | ρ ε ( t, x − z ) − ρ ε ( t, x − y − z ) | dx dy dz dhh dε ′ ε ′ ds = I G + Diff . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 27
The treatment of I G is slightly difficult. Similar to previous calculations in (4.27), we changevariable and use H¨older’s inequality to obtain | I G | ≤ η Z t Z εε Z e ε Z T d K h ( y ) χρ γ art ε w ε,h ( x ) dxdy dhh dε ′ ε ′ ds + Cη Z t Z εε Z e ε Z T d K h ( y ) e P lε ( x ) w ε,h ( x ) | δρ ε ( x, y ) | l dxdy dhh dε ′ ε ′ ds. The first term in the above inequality is bounded by D /
16. To estimate the second term,we need to introduce K h ∗ G to use the penalty function: Cη Z t Z εε Z e ε Z T d K h ( y ) e P lε ( x ) w ε,h ( x ) χ dxdy dhh dε ′ ε ′ ds ≤ Cη Z t Z εε Z e ε Z T d K h ( y ) K h ( z ) (cid:12)(cid:12)(cid:12) e P ε ( x ) − e P ε ( x − z ) (cid:12)(cid:12)(cid:12) l w ε,h ( x ) χ dx dy dz dhh dε ′ ε ′ ds + Cη Z t Z εε Z e ε Z T d K h ( y ) K h ( z ) | e P ε | l ( x − z ) w ε,h ( x ) χ dx dy dz dhh dε ′ ε ′ ds where the last term may be bounded by Cλ − D with Cλ − being arbitrarily small provided λ is sufficiently large. By H¨older we bound the first term as Cη Z t Z εε Z e ε Z T d K h ( y ) K h ( z ) (cid:12)(cid:12)(cid:12) e P ε ( x ) − e P ε ( x − z ) (cid:12)(cid:12)(cid:12) l w ε,h ( x ) χ dx dy dz dhh dε ′ ε ′ ds ≤ C Z t Z εε Z e ε Z T d K h ( y ) K h ( z ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) e P ε ( x ) − e P ε ( x − z ) (cid:12)(cid:12)(cid:12) l (cid:13)(cid:13)(cid:13)(cid:13) L s / (1+ l ) x × k χ k L s / ( s − (1+ l )) x dz dy dhh dε ′ ε ′ ds Note that for γ art ≥ s ′ we always have s (1 + l ) / ( s − (1 + l )) ≤ γ art . Hence, we get k χ k L s / ( s − (1+ l )) x ≤ C. Therefore, we have a further bound Cη Z t Z εε Z e ε Z T d K h ( y ) K h ( z ) (cid:12)(cid:12)(cid:12) e P ε ( x ) − e P ε ( x − z ) (cid:12)(cid:12)(cid:12) l w ε,h ( x ) χ dx dy dz dhh dε ′ ε ′ ds ≤ C Z t Z εε Z e ε Z T d K h ( z ) (cid:13)(cid:13)(cid:13) e P ε ( x ) − e P ε ( x − z ) (cid:13)(cid:13)(cid:13) lL s x dz dhh dε ′ ε ′ ds ≤ C (cid:18)Z εε r e ε dε ′ ε ′ (cid:19) (1+ l ) /s | log h | ( s − − l ) /s . By H¨older’s inequality, the Diff term is estimated similarly to (4.28) asDiff ≤ η Z t Z εε Z e ε Z T d K h ( y ) χρ γ art ε w ε,h ( x ) dxdy dhh dε ′ ε ′ ds + C Z t Z εε Z e ε Z T d K h ( y ) (cid:13)(cid:13)(cid:13) ρ l (cid:13)(cid:13)(cid:13) L s / ( s − (1+ l )) x L ε ′ ( z ) (cid:16) zh (cid:17) / (1+ l ) × (cid:13)(cid:13)(cid:13) | e P x − zε | l (cid:13)(cid:13)(cid:13) L s / (1+ l ) x dy dz dhh dε ′ ε ′ ds ≤ D + C Z t Z εε Z e ε Z T d K h ( y ) L ε ′ ( z ) (cid:16) zh (cid:17) / (1+ l ) dy dz dhh dε ′ ε ′ ds ≤ D + C provided γ art > s ′ . Collecting all the estimates of I s, , I s, , with I s, and optimizing in M concludes the proof. (cid:3) I . Before giving the bound for the integral terms I and I , we intro-duce the following lemma needed for the treatment of the effective viscous flux F =∆ − div( ∂ t ( ρ ε u ε ) + div( ρ ε u ε ⊗ u ε )). We refer the readers to [ ] for a proof of this result. Lemma . Let F be the effective viscous flux introduced above. Assume that ( ρ ε , u ε ) is a solution of the system (2.6) – (2.7) satisfying the bound (2.4) with γ art > d/ . Supposethat Φ ∈ L ∞ ([0 , T ] × T d ) and that C Φ : = (cid:13)(cid:13)(cid:13)(cid:13)Z T d K h ( x − y )Φ( t, x, y ) dy (cid:13)(cid:13)(cid:13)(cid:13) W , (0 ,T ; W − , x ( T d )) + (cid:13)(cid:13)(cid:13)(cid:13)Z T d K h ( x − y )Φ( t, x, y ) dx (cid:13)(cid:13)(cid:13)(cid:13) W , (0 ,T ; W − , y ( T d )) < ∞ , then there exists θ > such that Z t Z T d K h ( x − y )Φ( t, x, y )( F ( t, x ) − F ( t, y )) dx dy dt . h θ ( C Φ + k Φ k L ∞ ((0 ,T ) × T d ) ) holds, where the implicit constant in . is independent of ε . Next we estimate I in the lemma below. We use θ to denote a parameter between 0and 1 which may be different from line to line. Lemma . Let I be defined by (4.7) . Under the assumptions of Lemma 4.5, it follows I ≤ C + C (cid:18)Z εε r max( h ,ε ′ ) dε ′ ε ′ (cid:19) ¯ θ | log( h ) | θ + C Z t T h ,ε ( s ) ds − D − D − D . with D , D , and D given by (4.12) , (4.16) , and (4.23) respectively. Here < ¯ θ , < θ < ,and t ≤ T , where T can be any positive number and the implicit constant may depend ontime T . Proof.
We first recall I = − Z t Z h Z T d K h ( x − y ) W x,yε,h χ ′ ρδ (div u ε )( x ) dxdy dhh ds. OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 29
We proceed by getting a representation formula for div u ε from (2.7)div u ε = ηρ γ art ε + L ε ∗ P + F (4.34)where F is the effective viscous flux: F = ∆ − div F ( ∂ t ( ρ ε u ε ) + div( ρ ε u ε ⊗ u ε )) . Then the term I may be rewritten as I = − Z t Z h Z T d K h ( x − y ) W x,yε,h χ ′ ρ ε δ ( ηρ γ art ε + L ε ∗ P + F )( x ) dxdy dhh ds = I , + I , + I , with I , , I , , and I , being the integrals corresponding to the three terms in the paren-theses of the above formula. Noting that ηχ ′ ρ ε δ ( ρ γ art ε ) ≥ ηχ ′ ρ ε ( ρ ε ( x ) − ρ ε ( y ))( ρ γ art − ε ( x ) + ρ γ art − ε ( y ))= η (1 + l ) χ ( δρ ε ) ρ ε ( ρ γ art − ε ( x ) + ρ γ art − ε ( y )) ≥ η (1 + l ) χ ( δρ ε ) ρ γ art ε we arrive at I , ≤ − η (1 + l ) Z t Z h Z T d K h ( x − y ) W x,yε,h χ ( δρ ε ) ρ γ art ε ( x ) dxdy dhh ds (4.35)which serves as a penalization. To bound the term I , , we rewrite it as I , = − Z t Z h Z T d K h ( x − y ) W x,yε,h χ ′ ρ ε δ ( L ε ∗ P )( x ) dxdy dhh ds = − Z t Z h Z T d K h ( x − y ) K h ( x − e y ) w xε,h χ ′ ρ ε δ ( L ε ∗ P )( x ) dx dy d e y dhh ds. Let f ( x, y, e y ) = χ ′ ( δρ ( x, x − y )) ρ ε ( x, x − y ), then it is straightforward to check that f satisfiesthe condition (4.22). Appealing to the Lemma 4.5, we arrive at | I , | ≤ C + C (cid:18)Z εε r max( h ,ε ′ ) dε ′ ε ′ (cid:19) ¯ θ | log( h ) | θ + C Z t T h ,ε ( s ) ds + Cλ − D + 3 D . Finally, we deal with the effective viscous flux term I , , which is rewritten as I , = − Z t Z h Z T d φ Mε K h ( x − y ) W x,yε,h χ ′ ρδF ( x ) dxdy dhh ds − Z t Z h Z T d (1 − φ Mε ) K h ( x − y ) W x,yε,h χ ′ ρδF ( x ) dxdy dhh ds. (4.36) For the second integral, we use the uniform integrability of ρ ε and div u ε to obtain (cid:12)(cid:12)(cid:12)(cid:12)Z t Z h Z T d (1 − φ Mε ) K h ( x − y ) W x,yε,h χ ′ ρδF ( x ) dxdy dhh ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z t Z h Z T d (1 − φ Mε ) K h ( y ) W x,x − yε,h χ ′ ρδF ( x ) dxdy dhh ds (cid:12)(cid:12)(cid:12)(cid:12) . Z t Z h Z T d K h ( y ) k (1 − φ Mε ) χ ′ ρ k L p/ ( p − γart ) x k δF ( x ) k L p/γartx dy dhh ds . | log h | M − θ with some 1 > θ > p = γ art + 2 γ art /d − − /λ for a sufficiently large constant λ .Note here (1 + l ) p/ ( p − γ art ) < γ art since we require γ art > d. While for the first integralin (4.36), we need to use Lemma 4.6 withΦ = W x,yε,h χ ′ ρφ Mε . Obviously we have that k Φ k L ∞ . M l . In view of the system (2.6)–(2.7), we get anequation for Φ as ∂ t Φ + div x (Φ u xε ) + div y (Φ u yε ) = f x,yε, div x u xε + f x,yε, div y u yε + f x,yε, λ D xε + f x,yε, λ D yε where D ε is the penalization introduced in (3.6) and f x,yε,i are polynomials of ρ ε , w ε , φ Mε ,and derivatives of φ Mε for i = 1 , , ,
4. Noting that k f x,yε,i k L ∞ . M l for i = 1 , , , , it is not difficult to get that C Φ . M l where C Φ is defined in Lemma 4.6. Hence Lemma 4.6 implies (cid:12)(cid:12)(cid:12)(cid:12)Z t Z h Z T d φ Mε K h ( x − y ) W x,yε,h χ ′ ρδF ( x ) dxdy dhh ds (cid:12)(cid:12)(cid:12)(cid:12) . M l . Optimizing the bound in M gives I , . | log h | θ for some 0 < θ <
1. The proof is concluded by collecting the estimates for I , , I , , and I , . (cid:3) I . We give the estimate for I in this subsection. Lemma . Let I be defined by (4.8) . Under the assumptions in Lemma 4.5, we have I ≤ C + C (cid:18)Z εε r max( h ,ε ′ ) dε ′ ε ′ (cid:19) ¯ θ | log( h ) | θ + C Z t T h ,ε ( s ) ds + Cλ − D + D with D and D given by (4.16) and (4.23) respectively, for some < ¯ θ , < θ < , and t ≤ T , where T can be any positive number and the implicit constant may depend on time T . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 31
Proof.
We recall I = Z t Z h Z T d K h ( x − y ) W x,yε,h (cid:18) χ − χ ′ δρ (cid:19) div x u ε ( x ) dxdy dhh ds. By the definition of χ in (4.1), the term I may be rewritten as I = 1 − l Z t Z h Z T d K h ( x − y ) W x,yε,h χ div x u ε ( x ) dxdy dhh ds = 1 − l Z t Z h Z T d K h ( x − y ) W x,yε,h χ ( P art,η ( ρ ε ( x )) + L ε ∗ P ( x ) + F ( x )) dxdy dhh ds = I , + I , + I , . Note that since P art,η ( ρ ) ≤ C ρ γ art , the term I , may be absorbed by the term D / I , as I , = 1 − l Z t Z h Z T d K h ( x − y ) w ε,h ( x ) χ ( δρ )( L ε ∗ P ( x ) + L ε ∗ P ( y )) dxdy dhh ds = (1 − l ) Z t Z h Z T d K h ( x − y ) w ε,h ( x ) χ ( δρ ) L ε ∗ P ( x ) dxdy dhh ds − − l Z t Z h Z T d K h ( x − y ) w ε,h ( x ) χ ( δρ ) δ ( L ε ∗ P )( x ) dxdy dhh ds. Since the second integral in the right side of the last equality is already estimated in I , weonly need to consider the first integral. We need to use the penalization D defined in (4.16)to control the main contribution of this term. Note Z t Z h Z T d K h ( x − y ) w ε,h ( x ) χ ( δρ ) L ε ∗ P ( x ) dxdy dhh ds = Z t Z h Z T d K h ( x − y ) K h ( x − z ) w ε,h ( x ) χ ( δρ ) L ε ∗ P ( x ) dx dy dz dhh ds = Z t Z h Z T d K h ( x − y ) K h ( x − z ) w ε,h ( x ) χ ( δρ )( L ε ∗ P ( x ) − L ε ∗ P ( z )) dx dy dz dhh ds + Z t Z h Z T d K h ( x − y ) K h ( x − z ) w ε,h ( x ) χ ( δρ ) L ε ∗ P ( z ) dx dy dz dhh ds where the last integral is bounded by Cλ − D . We switch variables to rewrite the firstintegral as Z t Z h Z T d K h ( x − y ) K h ( x − z ) w ε,h ( x ) χ ( δρ )( L ε ∗ P ( x ) − L ε ∗ P ( z )) dx dy dz dhh ds = Z t Z h Z T d K h ( x − y ) K h ( x − e y ) w ε,h ( x ) χ ( δρ ( x, x − e y ))( L ε ∗ P ( x ) − L ε ∗ P ( y )) dx dy d e y dhh ds. Let f ( x, y, e y ) = χ ( δρ ( x, x − e y )), then it is easy to check that (4.22) holds. Using Lemma 4.5,we arrive at Z t Z h Z T d K h ( x − y ) K h ( x − e y ) w ε,h ( x ) χ ( δρ ( x, x − e y ))( L ε ∗ P ( x ) − L ε ∗ P ( y )) dx dy d e y dhh ds ≤ C + C (cid:18)Z εε r max( h ,ε ′ ) dε ′ ε ′ (cid:19) ¯ θ | log( h ) | θ + C Z t T h ,ε ( s ) ds + Cλ − D + 3 D . At last, we treat the effective viscous flux term as I , = (1 − l ) Z t Z h Z T d K h ( x − y ) w xε,h χF ( x ) dxdy dhh ds = (1 − l ) Z t Z h Z T d K h ( x − y ) w xε,h χ ( F ( y ) − F ( x )) dxdy dhh ds + 2(1 − l ) Z t Z h Z T d K h ( x − y ) w xε,h χF ( x ) dxdy dhh ds. Note that the first integral is already treated in I , , and we now deal with the secondintegral as Z t Z h Z T d K h ( x − y ) w xε,h χF ( x ) dxdy dhh ds = Z t Z h Z T d K h ( x − y ) K h ( x − z ) w xε,h χ ( F ( x ) − F ( z )) dx dy dz dhh ds + Z t Z h Z T d K h ( x − y ) K h ( x − z ) w xε,h χF ( z ) dx dy dz dhh ds. For the first integral, by similar argument as in the treatment of I , , we arrive at Z t Z h Z T d K h ( x − y ) K h ( x − z ) w xε,h χ ( F ( x ) − ∆ − div F ( z )) dx dy dz dhh ds . | log h | θ for some 0 < θ <
1. While for the second integral, we use the formula (4.34) to obtain Z t Z h Z T d K h ( x − y ) K h ( x − z ) w xε,h χF ( z ) dx dy dz dhh ds ≤ Z t Z h Z T d K h ( x − y ) K h ( x − z ) w xε,h χ ( δρ ε ) | div u ε | ( z ) dx dy dz dhh ds which is bounded by Cλ − D . Collecting all the estimate and optimizing in M concludesthe proof. (cid:3) Proof of Theorem 2.2.
Collecting the estimates from Lemmas 4.1, 4.2–4.4, and 4.7–4.8, choosing λ sufficiently large, and dropping the extra penalization D , D , and D , we OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 33 have T h ,ε ( t ) . T h ,ε (0) + C Z t T h ,ε ( s ) ds + | log h | θ for some 0 < θ <
1. A Gronwall inequality implies T h ,ε ( t ) . e CT | log h | θ for t ≤ T . Recalling the definition of T h ,ε , in order to get the compactness of the solution ρ ε , we need to get rid of the weight function. Note that Z T d K h ( x − y ) χ ( δρ ε ) dxdy = Z T d K h ( x − y ) χ ( δρ ε ) w xε,h ≤ η w yε,h ≤ η dxdy + Z T d K h ( x − y ) χ ( δρ ε )(1 − w xε,h ≤ η w yε,h ≤ η ) dxdy where η > h to be chosen later. For the first integral,in view of (3.11), we have Z T d K h ( x − y ) χ ( δρ ε ) w xε,h ≤ η w yε,h ≤ η dxdy . Z T d K h ( x − y ) ρ lε ( x ) w xε,h ≤ η dxdy + Z T d K h ( x − y ) ρ lε ( y ) w yε,h ≤ η dxdy . Z T d ρ lε ( x ) w xε,h ≤ η dx | log h | . | log h || log η | α for some 0 < α <
1. For the second integral, we use T h ,ε to get Z T d K h ( x − y ) χ ( δρ ε )(1 − w xε,h ≤ η w yε,h ≤ η ) dxdy ≤ η T h ,ε ( t ) . η | log h | θ . By choosing η = | log h | , we arrive at Z T d K h ( x − y ) χ ( δρ ε ) dxdy . | log h | log | log h | , which implies the compactness of the solution ρ ε by Lemma 3.1. (cid:3) Appendix A. Proof of Theorems 2.3 and 2.1A.1. Proof of Th. 2.3.
The proof is performed by taking several consecutive limits,first η →
0, then η → η m →
0. The generic step is hence, once wealready have η = · · · = η i = 0, to pass to the limit η i +1 →
0. For this reason, we introducethe notation ρ η,i , u η,i which is obtained by taking the first i − η → η i − →
0. More precisely, after extracting subsequences, we have that ρ η, = ρ η , u η, = u η and ρ η,i +1 = w − lim η i → ρ η,i , u η,i +1 = w − lim η i → u η,i . The final solution that we will obtain is simply ρ = ρ η,m +1 , u = u η,m +1 which is independentof all η i . Assuming that ρ η,i is a weak solution to the system ∂ t ρ η i + div( ρ η i u η i ) = 0 ,∂ t ( ρ η i u η i ) + div( ρ η i u η i ⊗ u η i ) − ∆ u η i + ∇ ( η i ρ γ art,i η i + . . . + η m ρ γ art,m η i + P ( t, x, ρ η i )) = 0 , (A.1)then we have to show that ρ η,i +1 solves the same system with η i = 0. Step 1: Basic energy inequality for ρ η , u η . We observe that ρ η , u η solves (A.1) directlyfrom Theorem 2.1. However the a priori estimates provided by Theorem 2.1 are not uni-form in η so that our first step consists in deriving such estimate starting from the energyinequality (2.5).The first point is to pass to the limit as ε → ρ ε,η , u ε,η so it handled in the usual manner. We have that div u ε,η is uniformlybounded in L t,x so div u ε,η → div u η in w − L t,x .On the other hand by (1.7)-(1.8), we have that | P ( t, x, ρ ε,η ) | ≤ R + Θ + C ρ pε,η with p ≤ γ + γd − R + Θ ∈ L qt,x with q >
2. By Theorem 2.1, we have that ρ ε,η ∈ L p art t,x uniformly in ε for any p art ≤ γ art + 2 γ art /d −
1. Observe that 2 ( γ + γd − < γ + γd − ≤ γ art + 2 γ art /d − γ ≤ γ art . This is the first place where the assumption 2 γ ≤ γ art is critical.Hence P ( t, x, ρ ε,η ) is uniformly bounded in ε in L qt,x for some q >
2. By the compactnessof ρ ε,η provided by Theorem 2.2, we obtain that P ( t, x, ρ ε,η ) → P ( t, x, ρ η ) strongly in L t,x .Therefore this provides a solution ρ η , u η to the system (2.6)-(2.7) with, for a fixed η ,the bounds ρ η ∈ L ∞ t L γ art x , ρ η ∈ L pt,x for any p ≤ γ art + 2 γ art /d − u η ∈ L t H x , and thebasic energy inequality Z T d ( η ρ γ art, η ( t, x ) γ art, − . . . + η m ρ γ art,m η ( t, x ) γ art,m − ρ η ( t, x ) | u η ( t, x ) | ) dx + Z t Z T d |∇ u η ( s, x ) | dx ds ≤ Z t Z T d div u η P dx ds + Z T d η ( ρ ε,η ) γ art, ( t, x ) γ art, − . . . + η m ( ρ ε,η ) γ art,m ( t, x ) γ art,m − ρ ε,η ( t, x ) | u ε,η ( t, x ) | ! dx. (A.2) Step 2: Modified energy inequality.
Our next step is to work with (A.2) to obtain a formthat is more suitable to the derivation of a priori estimates.We recall that E = ρ η ( | u η | / e ( ρ η )) with e ( t, x, ρ ) = R ρρ ref P ( t, x, s ) /s ds .We have that ddt Z T d ρ η e ( t, x, ρ η ) dx = Z T d ( ρ η ∂ t e ( ρ η ) + ρ η u η · ∇ x e ( ρ η )) dx + Z T d div u η ( ρ η e ( ρ η ) − ρ η ∂ ρ ( ρ η e ( ρ η ))) dx. From the definition of e , we get that ddt Z T d ρ η e ( t, x, ρ η ) dx = Z T d ( ρ η ∂ t e ( ρ η ) + ρ η u η · ∇ x e ( ρ η )) dx − Z T d div u η P ( ρ η ) dx. Note that from (1.8), (1.9), (1.10), we have that for a fixed η , P ∈ L while ρ η ∂ t e ( ρ η ) ∈ L t,x and ρ η ∇ x e ( ρ η ) ∈ L t L d/ ( d +2) x so that ρ η u η · ∇ x e ( ρ η ) ∈ L t,x as well. Therefore all termsmake sense and this is again due to the assumption γ art ≥ γ . OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 35
Adding this to (A.2) yields the more precise energy inequality Z T d ( E ( ρ η , u η ) + η ρ γ art, η γ art, − . . . + η m ρ γ art,m η γ art,m − dx + Z t Z T d |∇ u η ( s, x ) | dx ds ≤ Z t Z T d div x u η ( s, x ) ( P ( s, x, ρ η ( s, x )) − P ( s, x, ρ η ( s, x ))) ds dx + Z t Z T d ( ρ η ∂ t e ( ρ η ) + ρ η u η · ∇ x e ( ρ η )) dx ds + Z T d η ( ρ ε,η ) γ art, ( t, x ) γ art, − . . . + η m ( ρ ε,η ) γ art,m ( t, x ) γ art,m − E ( ρ η , u η ) ! dx, (A.3)which we will use to obtain our a priori estimates. Step 3: A priori estimates on ρ η , u η . From (A.3), we first observe that from (1.7) since¯ γ ≤ γ/ Z t Z T d div x u η ( s, x ) ( P ( s, x, ρ η ( s, x )) − P ( s, x, ρ η ( s, x ))) ds dx ≤ C + 14 Z t Z T d |∇ u η ( s, x ) | dx ds + C Z t Z T d | ρ η ( s, x ) | γ dx ds. Similarly by (1.9)-(1.10), we can bound Z t Z T d ( ∂ t e ( ρ η ) + ρ η u η · ∇ x e ( ρ η )) dx ds ≤ C + 14 Z t Z T d |∇ u η ( s, x ) | dx ds + C Z t Z T d | ρ η ( s, x ) | γ dx ds. By (1.8), we hence obtain that Z T d ( ρ γη C + η ρ γ art, η γ art, − . . . + η m ρ γ art,m η γ art,m − ρ η | u η | dx + 12 Z t Z T d |∇ u η ( s, x ) | dx ds ≤ C + C Z t Z T d | ρ η ( s, x ) | γ dx ds. By Gronwall’s lemma, we deduce the first main estimate on ρ η and u η , for some constant C independent of η Z T d ( ρ γη C + η ρ γ art, η γ art, − . . . + η m ρ γ art,m η γ art,m − ρ η | u η | dx ≤ C e
C t , Z t Z T d |∇ u η ( s, x ) | dx ds ≤ C e
C t . (A.4)Those estimates are convex in ρ η and u η . Hence by the definition of the ρ η,i , u η,i , wetrivially have as well that Z T d ( ρ γη,i C + η i ρ γ art,i η,i γ art,i − . . . + η m ρ γ art,m η,i γ art,m − ρ η,i | u η,i | dx ≤ C e
C t , Z t Z T d |∇ u η,i ( s, x ) | dx ds ≤ C e
C t . (A.5) When considering the limit η i → ρ η,i , u η,i , we have that η i +1 , . . . , η m >
0. We hencehave all the bounds needed to apply Lemma 2.4 with S = P − P , γ = γ art,i +1 and1 /p = 1 + 1 /γ art,i +1 − /d or γ art,i +1 /p ∗ = 2 γ art,i +1 /d −
1. This lets us obtain our last apriori estimatesup η i Z T Z T d ρ qη,i ( t, x ) dx dt < ∞ , ∀ q < γ art,i +1 + 2 γ art,i +1 /d − . (A.6) Step 4: Passing to the limit.
Equipped with those bounds, we have the weakly convergingsubsequences as η i → ρ η,i → ρ η,i +1 in w − L ∞ t L γ art,i +1 x and w − L qt,x for any q < γ art,i +1 +2 γ art,i +1 /d −
1, and u η,i → u η,i +1 in w − L t H x .As usual, this is also enough to show the weak limits ρ η,i u η,i → ρ η,i +1 u η,i +1 and ρ η,i u η,i ⊗ u η,i → ρ η,i +1 u η,i +1 ⊗ u η,i +1 . Those bounds also provides equi-integrability on P ( t, x, ρ η,i ) by the upper bounds following from (1.7)-(1.8). Equi-integrability also holds on η i ρ γ art,i η,i γ art,i − . . . + η m ρ γ art,m η,i γ art,m − , since γ art,i < γ art,i +1 +2 γ art,i +1 /d − γ art,i .The main remaining question is to prove the compactness of ρ η,i in L t,x . This is ingeneral the difficult question for compressible Navier-Stokes but, fortunately in this case,we may directly apply the result of [ ].Specifically we invoke Th. 5.1, case (ii) in that article (page 613). Our sequence ρ η,i , u η,i solves the continuity equation (denoted (5.1) in the article). The momentum equationimplies that u η,i solves equation (5.2) in the article with constant viscosity and R k = 0.Our a priori estimates directly ensures the bounds (5.3)-(5.7) that are required by Th.5.1 in [ ]. Finally the assumption on the pressure law for this theorem is identical to ourassumptions (1.11)-(1.12).We hence deduce the compactness of ρ η,i and hence the convergence of P ( t, x, ρ η,i ) + η i ρ γ art,i η,i + . . . + η m ρ γ art,m η,i to P ( t, x, ρ η,i +1 ) + η i +1 ρ γ art,i +1 η,i +1 + . . . + η m ρ γ art,m η,i +1 . This implies that ρ η,i +1 , u η,i +1 solves (A.1) with η i = 0 and finally that ρ, u is indeed a global solution to thesystem (1.2)-(1.3) as claimed with the corresponding estimates for i = m + 1 following from(A.5) and (A.6). Finally the energy inequality is directly obtained from (A.3) by taking thesuccessive limits. A.2. Proof of Th. 2.1.
We can obtain solutions to (2.1)-(2.2) through a fixed pointtheorem. Given any S ∈ L ([0 , T ] × T d )), we define N S , U S as a global weak solution to ∂ t N S + div( N S U S ) = 0 , N S ( t = 0) = ρ ε ,∂ t ( N S U S ) + div( N S U S ⊗ U S ) − ∆ U S + ∇ ( P η ( N S ) + S ) = 0 , U ρ ( t = 0) = u ε . (A.7)System (A.7) is in fact the classical compressible Navier-Stokes system with barotropicpressure law P η ( ρ ) = η ρ γ art, + . . . + η m ρ γ art,m and a source term. Provided that γ art +2 γ art /d − > γ art = γ art, = max i γ art,i , which we assumed, existence of global solu-tion to this system is guaranteed by [ ] and moreover such solutions satisfy the following OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 37 energy estimate for some constant C sup t ∈ [0 , T ] Z T d (( N S ( t, x )) γ art + N S | U S | ) dx + Z T Z T d |∇ U S | dx ≤ C Z T d (( ρ ε ( t, x )) γ art + ρ ε | u ε | ) dx + C k S k L t,x . (A.8)and the following energy inequality Z T d (cid:18) η N γ art, S ( t, x ) γ art, − . . . + η m N γ art,m S ( t, x ) γ art,m − N S ( t, x ) | U S ( t, x ) | (cid:19) dx + Z t Z T d |∇ U S ( s, x ) | dx ds ≤ Z t Z T d div U S · S dx ds + Z T d η ( ρ ε,η ) γ art, ( t, x ) γ art, − . . . + η m ( ρ ε,η ) γ art,m ( t, x ) γ art,m − ρ ε,η ( t, x ) | u ε,η ( t, x ) | ! dx. (A.9)We are now using Lemma 2.4 with P = P η ( N S ) = η N γ art, S + . . . + η m N γ art,m s .One has that N S ∈ L ∞ t L γ art x solves (B.1) with u ε ∈ L t H x . Since γ art > S ∈ L t,x ⊂ L t L γ ∗ art x trivially. On the other handsup ε k ∆ U S k L t H − < ∞ , and using Sobolev embeddings U S ∈ L t L qx with 1 /q = 1 / − /d so thatsup ε k N S U S ⊗ U S k L t L px < ∞ , p = 1 γ art + 2 q = 1 γ art + 1 − d . Similarly sup ε k N S U S k L t L rx < ∞ , r = 1 γ art + 1 q = 1 γ art + 12 − d , and one notes that 2 pd/ (2 d + 2 p − pd ) = r or 1 /r = 1 /p + 1 /d − / R N S | U S | dx , we also have that Z T d N sS | U S | s dx ≤ (cid:18)Z T d N S | U S | (cid:19) s/ (cid:18)Z T d N s/ (2 − s ) S (cid:19) − s/ . Note that s/ (2 − s ) = γ art iff s = 2 γ art / (1 + γ art ), implying thatsup ε k N S U S k L ∞ t L sx < ∞ , s = 2 γ art / (1 + γ art ) , with in particular s = 2 pd/ ( d + 2 p ) ≥ pd/ ( p + d ).We hence deduce that for θ < γ art /p ∗ or θ < γ art /d − ε Z T Z T d N θS P ( N S ) dt dx < ∞ , or, in other words, Lemma 2.4 implies that Z T Z T d N qS ( t, x ) dx dt ≤ C Z T d (( ρ ε ( t, x )) γ art + ρ ε | u ε | ) dx + C k S k L t,x . (A.10)This leads to defining the following operator F : S −→ F ( S )( t, x ) = L ε ⋆ P ( N S ) . From the definition, we have that k F ( S ) k L t,x ≤ C ε − d k P ( N S ) k L t L x ≤ C ε − d k R k L + C ε − d k N pS k L t L x , for some p < γ + 2 γ/d −
1, by using assumptions (1.7)-(1.8) on P . Since R ∈ L t,x ,, wededuce that k F ( S ) k L t,x ≤ C ε − d + C ε − d k N S k pL pt,x . Finally, γ art + 2 γ art /d − ≥ γ + 4 γ/d − > p since γ art ≥ γ , we have by (A.10) k F ( S ) k L t,x ≤ C ε − d + C ε − d T k S k θL t,x , for some exponent θ < ε bound onsup ε Z T d (( ρ ε ( t, x )) γ art + ρ ε | u ε | ) < ∞ . As θ <
1, there exists a ball B ⊂ L t,x with large enough radius such that F ( B ) ⊂ B .Moreover F ( S ) ∈ L t H x for any S ∈ B thanks to the convolution in x giving com-pactness in the space variable. To prove the time compactness, one could observe that theargument in [ ] or the quantitative estimates from [ ] provide full compactness on thedensity provided that the source term is compact in space ( i.e. without time compactnessbeing required).However, since it is possible to obtain the time compactness in a straightforward mannerand for the sake of completeness, we present the argument here. We need to introducevarious regularization and truncations. First of all (1.7) implies that P/ (1 + s p ) is in L t,x uniformly in s . Hence we can choose P η ( t, x, s ) a regularization of P in t and x with forexample | ∂ t P η ( t, x, s ) | + |∇ x P η ( t, x, s ) | + | ∂ s P η ( t, x, s ) | ≤ Cη (1 + s p ) , k P ( ., ., s ) − P η ( ., ., s ) k L t,x ≤ f ( η ) (1 + s p ) , for some continuous function f with f (0) = 0.By (1.7) again and since (A.8) shows that N S ∈ L ∞ t L γ art x with γ art > γ , we mayimmediately deduce from the last point that there exists e f continuous with e f (0) = 0 suchthat for any S ∈ B k P ( ., ., N S ) − P η ( ., ., N S ) k L t,x ≤ e f ( η ) . (A.11)Now choosing any standard convolution kernel L , we may write L ε ⋆ P η ( N S )( t, x ) = Z T d L ε ( x − y ) P η ( t, y, N S ( t, y )) dy = Z T d L ε ( x − y ) L √ η ( y − z ) P η ( t, z, N S ( t, y )) dy dz + Z T d L ε ( x − y ) L √ η ( y − z ) ( P η ( t, y, N S ( t, y )) − P η ( t, z, N S ( t, y ))) dy dz. Therefore (cid:13)(cid:13)(cid:13)(cid:13) L ε ⋆ P η ( N S ) − Z T d L ε ( x − y ) L √ η ( y − z ) P η ( t, z, N S ( t, y )) dy dz (cid:13)(cid:13)(cid:13)(cid:13) L t,x ≤ C √ η. (A.12) OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 39
Since N S solves the continuity equation (B.1) and U S ∈ L t H x , we have by Th. B.1 that forany fixed z∂ t ( P η ( t, z, N S ( t, x ))) = ∂ t P η ( t, z, N S ( t, x )) + div x ( P η ( t, z, N S ( t, x )) U S ( t, x ))= ( P η ( t, z, N S ( t, x )) − N S ∂ s P η ( t, z, N S ( t, x ))) div U S . From this, we obtain that ddt Z T d L ε ( x − y ) L √ η ( y − z ) P η ( t, z, N S ( t, y )) dy dz = Z T d L ε ( x − y ) L √ η ( y − z ) ∂ t P η ( t, z, N S ( t, y )) dy dz + Z T d L ε ( x − y ) ∇ y L √ η ( y − z ) P η ( t, z, N S ( t, y )) U S ( t, y ) dy dz + Z T d L ε ( x − y ) L √ η ( y − z ) ( P η ( t, z, N S ( t, x )) − N S ∂ s P η ( t, z, N S ( t, x ))) div U S dy dz. Bounding directly each term, this implies that (cid:12)(cid:12)(cid:12)(cid:12) ddt Z T d L ε ( x − y ) L √ η ( y − z ) P η ( t, z, N S ( t, y )) dy dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε η − k , (A.13)for some exponent k > L ε ⋆ P ( N S ) and hence the compactness in L t,x of F ( B ). By the Schauder fixed point, F has a fixed point S in B ⊂ L t,x . We now simply choose ρ ε = N S and u ε = U S and since N S , U S solve (A.7) with S = F ( S ) = L ε ⋆ P ( N S ) = L ε ⋆ P ( ρ ε ), we obtain a solution to(2.1)-(2.2). The energy bound (A.8) provides all uniform in ε bounds on ρ ε while the energyinequality (A.9) of course leads to the corresponding inequality in the theorem. Estimate(A.10) provides the extra-integrability on ρ ε,η . Appendix B.B.1. Notations.
Because we use functions at various points and differences of func-tions, we introduce specific notations. First, the symbol f x stands for a function of x , i.e., f x = f ( x ). Next, we also denote δf ( x, ξ ) = f ( x ) − f ( x − ξ )and ¯ f ( x, ξ ) = f ( x ) + f ( x − ξ ) . If the argument is not mentioned explicitly or only the x variable is mentioned in the abovenotation, then we set ξ = x − y , i.e., δf = δf ( x ) = δf ( x, x − y ) = f ( x ) − f ( y )and f = f ( x ) = f ( x, x − y ) = f ( x ) + f ( y ) . We denote the maximum operator by
M f ( x ) = sup r> | B r | Z B r | f ( x ) | dx. Recall that k M f k L p . k f k L p for p > f . g stands for that f ≤ Cg for some constant C > f, T ] g = f T g − T ( f g )where f and g are smooth functions and T is an operator. B.2. Renormalized solutions.
We rely on the concept of renormalized solution tojustify several a priori formal calculations in the article. For this reason, we recall herethe main definitions. Given our system, we naturally focus on the conservative transportequation ∂ t ρ + div( ρu ) = 0 . (B.1)Given a weak solution ρ to the above, it is not a priori possible to calculate non-linearfunctions of ρ which is precisely what we need here. Hence one introduces the notion ofrenormalized solutions Definition
B.1 . A weak solution ρ ∈ L pt,x to (B.1) with u ∈ L qt,x for 1 /p + 1 /q = 1 isa renormalized solution iff for any χ ∈ C ( R ) with | χ ′ ( s ) | ≤ C (1 + | s | p − ), one has that ∂ t χ ( ρ ) + div( χ ( ρ ) u ) = ( χ ( ρ ) − ρχ ′ ( ρ )) div u (B.2)in the sense of distributions.Renormalized solutions were first introduced in the famous [ ] which in particularproved that if u belongs to the right Sobolev space then all weak solutions are renormalized. Theorem
B.1 . Assume that ρ ∈ L pt,x is a solution to (B.1) in the sense of distributions.Suppose that u ∈ L qt W ,qx with /p + 1 /q = 1 , then ρ is a renormalized solution to (B.1) . For linear equations, i.e. when u is given in (B.1), then the theory of renormalizedsolutions immediately provides many key properties such as the compactness for a sequenceor the uniqueness of a solution. For example, assume there are two solutions ρ and ρ to(B.1) for the same u . Applying Theorem B.1 to the function ρ = ρ − ρ with χ ( x ) = | x | and integrating in time and space gives ddt Z T d χ ( ρ ) dx = 0which immediately implies that ρ = ρ .Observe however that in general and unless div u ∈ L ∞ , it is not possible to have ageneral existence result for (B.1) for a given u ∈ L qt W ,qx . A solution with only div u ∈ L may for example concentrate, by forming Dirac masses.Following [ ] and the BV extensions in [ ] for the kinetic case and the seminal [ ] inthe general case, the theory of renormalized solutions is now an extensive field for which werefer for example to the reviews [
2, 9 ].In the context of compressible Fluid Mechanics, renormalized solutions have been criticalto obtaining the compactness of the density since the first breakthrough in [ ] and theyalso form the basis of the extension introduced in [
11, 15 ]. We in particular cite thestraightforward compactness result from [ ] Theorem
B.2 . Consider a sequence u n converging strongly to u in L ([0 , T ] , L q ( T d )) s.t div u n converges to div u in L ([0 , T ] , L q ( T d )) as well. Consider any sequence ρ n suchthat ρ n , u n satisfies Eq. (B.1) and ρ n uniformly bounded in L ∞ ([0 , T ] , L p ( T d )) with OMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NON-HOMEGENEOUS PRESSURE 41 /p + 1 /q < . Assume finally that u ∈ L ([0 , T ] , W ,p ∗ ( T d ) with /p + 1 /p ∗ = 1 . Thenthe sequence ρ n is compact in L ([0 , T ] × T d ) . Th. B.2 can be deduced from Th. B.1. The proof of Th. B.1 itself relies on a so-called qualitative commutator estimate and in several respects, the method introduced in[ ] consists in quantifying this commutator estimate. Acknowledgments.
The first author is partially supported by the SingFlows project,grant ANR-18-CE40-0027. The second author is partially supported by NSF DMS Grant161453, 1908739, and NSF Grant RNMS (Ki-Net) 1107444.
References [1] L. Ambrosio. Transport equation and Cauchy problem for BV vector fields. Invent. Math. , 227–260(2004).[2] L. Ambrosio, G. Crippa, Continuity equations and ODE flows with non-smooth velocity.
Proc. Roy.Soc. Edinburgh Sect. A (2014), no. 6, 1191–1244.[3] F. Bouchut. Renormalized solutions to the Vlasov equation with coefficients of bounded variation.
Arch. Ration. Mech. Anal. (2001), 75–90.[4] D. Bresch, P.–E. Jabin. Global existence of weak solutions for compressible Navier-Stokes equations:thermodynamically unstable pressure and anisotropic viscous stress tensor.
Ann. of Math. (2) 188, no.2, 577–684 (2018).[5] D. Bresch, P.–E. Jabin. Quantitative regularity estimates for compressible transport equations. Newtrends and results in mathematical description of fluid flows.
Birkhauser
In-vent. Math. (1989), 511–547.[11] E. Feireisl. Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and itsApplications,
26. Oxford University Press, Oxford, 2004.[12] E. Feireisl, A. Novotny. Singular limits in thermodynamics of viscous fluids. Advanced in Math FluidMech, Birkhauser, 2017.[13] E. Feireisl, A. Novotn´y, H. Petzeltov´a. On the existence of globally defined weak solutions to theNavier-Stokes equations.
J. Math. Fluid Mech. (2001), 358–392.[14] E. Feireisl, T. Karper, M. Pokorny. Mathematical Theory of Compressible Viscous Fluids: Analysisand Numerics. Birkhauser-Verlag, Basel, 2016.[15] E. Feireisl. Compressible Navier–Stokes Equations with a Non-Monotone Pressure Law. J. Diff. Eqs W , velocities and applications. Ann. Mat. Pura Appl. (2004), 97–130.[17] J. Leray. Sur le mouvement d’un fluide visqueux remplissant l’espace,
Acta Math.
63, 193–248, (1934).[18] P.-L. Lions.
Mathematical topics in fluid mechanics. Vol. 2. Compressible models.
Oxford LectureSeries in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press,Oxford University Press, New York, 1998.[19] A. Novotny, I. Straskraba.
Introduction to the mathematical theory of compressible flow.
Oxford LectureSeries in Mathematics and its Applications. Oxford Science publications. The Clarendon press, OxfordUniversity press, New York, 2004. [20] P.I. Plotnikov, W. Weigant. Isothermal Navier-Stokes equations and Radon transform.
SIAM J. Math.Anal.
47 (2015), no. 1, 626–653.[21] E.M. Stein
Harmonic Analysis.
Princeton Univ. Press 1995 (second edition).
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chamb´ery, France
E-mail address : [email protected] Department of Mathematics, University of Maryland, College Park, MD 20740, USA
E-mail address : [email protected] Department of Mathematics, University of Maryland, College Park, MD 20740, USA
E-mail address ::