A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations
aa r X i v : . [ m a t h . A P ] J a n A BEALE-KATO-MAJDA BLOW-UP CRITERION FOR THE 3-DCOMPRESSIBLE NAVIER-STOKES EQUATIONS
YONGZHONG SUN, CHAO WANG, AND ZHIFEI ZHANGA bstract . We prove a blow-up criterion in terms of the upper bound ofthe density for the strong solution to the 3-D compressible Navier-Stokesequations. The initial vacuum is allowed. The main ingredient of theproof is a priori estimate for an important quantity under the assumptionthat the density is upper bounded, whose divergence can be viewed asthe e ff ective viscous flux.
1. I ntroduction
In this paper, we consider the isentropic compressible Navier-Stokes sys-tem in three dimensional space. The system reads ( ∂ t ρ + div( ρ u ) = , in (0 , T ) × Ω ,∂ t ( ρ u ) + div( ρ u ⊗ u ) − Lu + ∇ p = , in (0 , T ) × Ω , (1.1)together with the initial-boundary conditions( ρ ( t , x ) , u ( t , x )) | t = = ( ρ ( x ) , u ( x )) , in Ω , (1.2) u ( t , x ) = , on (0 , T ) × ∂ Ω . (1.3)Here Ω is either R or a bounded domain in R , ρ and u are the density andvelocity of the fluid respectively, p = a ρ γ with γ > L is defined by Lu = µ ∆ u + ( µ + λ ) ∇ div u , with constant viscosity coe ffi cients µ and λ satisfying µ > , λ + µ ≥ . (1.4)In the absence of vacuum for the initial density, the local existence ofstrong solution as well as the global existence of strong solution and weaksolution with the initial data close to an equilibrium state were well devel-oped, see [21, 22, 24, 14, 10, 6] and references therein. The global existenceof weak solution for large initial data was first solved by P. L. Lions in [20]for γ ≥ . E. Feireisl, A. Novotn´y and H. Petzeltov´a [13] extended Lions’s Date : January 6, 2010.
Key words and phrases.
Blow-up criterion, compressible Navier-Stokes equations,Lam´e system, strong solution. result to the case of γ > . S. Jiang and P. Zhang [18, 19] proved the globalexistence of weak solution for any γ > Ω = T under the assumption that µ is a constant and λ = ρ β with β >
3. Onthe other hand, when the initial density is compactly supported, Z. Xin [28]proved that smooth solution will blow up in finite time in the whole space.To proceed we introduce some notations for the standard homogeneousand inhomogeneous Sobolev spaces. D k , r ( Ω ) def = { u ∈ L loc ( Ω ) : k∇ k u k L r ( Ω ) < ∞} , W k , r ( Ω ) def = L r ( Ω ) ∩ D k , r ( Ω ) , H k ( Ω ) = W k , ( Ω ) , D k ( Ω ) = D k , ( Ω ) , D ( Ω ) def = { u ∈ L ( Ω ) : k∇ u k L ( Ω ) < ∞ and u = on ∂ Ω } , H ( Ω ) def = L ( Ω ) ∩ D ( Ω ) , k u k D k , r ( Ω ) = k∇ k u k L r ( Ω ) . When the initial vaccuum is allowed, the local well-posedness and blow-up criterion for strong solutions to the compressible Navier-Stokes equa-tions were established in a series of papers [7, 8, 9] by Cho, Choe and Kim.Here we write down one of those results.
Theorem 1.1.
Let Ω be a bounded smooth domain or R and q ∈ (3 , ρ ≥ W , q ( Ω ) ∩ H ( Ω ) ∩ L ( Ω ), u ∈ D ( Ω ) ∩ D ( Ω ) with the following compatibility condition satisfied, − µ ∆ u − ( µ + λ ) ∇ div u + ∇ p ( ρ ) = √ ρ g , (1.5)for some vector field g ∈ L ( Ω ). Then there exist a time T ∈ (0 , ∞ ] and aunique strong solution ( ρ, u ) to (1.1) such that ρ ∈ C ([0 , T ) , H ∩ W , q ( Ω )) , u ∈ C ([0 , T ) , D ( Ω )) ∩ L (0 , T ; D , q ( Ω )) . Moreover, let T ∗ be a maximal existence time of the solution. If T ∗ < ∞ ,then there holdslim sup t ↑ T ∗ (cid:16) k ρ ( t ) k W , q ( Ω ) + k u ( t ) k D ( Ω ) (cid:17) = ∞ . (1.6)Since the initial vacuum is allowed, it is then important to investigate thepossible blow-up mechanism of the solution. In their recent works [15, 16],X. Huang and Z. Xin established a Beale-Kato-Majda blow up criterion forthe above strong solution. More precisely, OMPRESSIBLE NAVIER-STOKES EQUATIONS 3
Theorem 1.2.
Assume that the coe ffi cients of the operator L satisfies (1.4)and moreover, λ < µ. (1.7)Let ( ρ, u ) be the strong solution constructed in Theorem 1.1 and T ∗ be amaximal existence time. If T ∗ < ∞ , thenlim T ↑ T ∗ k∇ u k L (0 , T , L ∞ ( Ω )) = ∞ . (1.8)Recently, J. Fan, S. Jiang and Y. Ou [12] also obtained a similar result forthe compressible heat-conductive flows. On the other hand, for the 2D com-pressible Navier-Stokes equations in T , B. Desjardins [11] proved moreregularity of weak solution under the assumption that the density is upperbounded; Very recently, L. Jiang and Y. Wang [17], Y. Sun and Z. Zhang[26] obtained a blow-up criterion in terms of the upper bound of the den-sity for the strong solution. In [26], the initial vacuum is allowed and thedomain includes the bounded domain. Note that the L (0 , T , L ∞ ( Ω )) boundfor ∇ u immediately implies the upper bound for the density ρ .The purpose of this paper is to obtain a Beale-Kato-Majda blow-up cri-terion in terms of the upper bound of the density for the 3-D compressibleNavier-Stokes equations. Our main result is stated as follows. Theorem 1.3.
Assume that ( ρ, u ) is the strong solution constructed in The-orem 1.1. Let µ, λ be as in Theorem 1.2 and T ∗ be a maximal existence timeof the solution. If T ∗ < ∞ , then we havelim sup T ↑ T ∗ k ρ ( t ) k L ∞ (0 , T ; L ∞ ( Ω )) = ∞ . (1.9) Remark 1.4.
This result seems surprise, if we compare with the incom-pressible Navier-Stokes equations where the density is a constant. It iswell-known that if we have some kind of control for the pressure, the Lerayweak solution is in fact smooth for the incompressible Navier-Stokes equa-tions, see [4]. For the compressible Navier-Stokes equations, the pressureis determined by the density, the bound of the density thus implies a boundfor the pressure. From this viewpoint, our result seems natural.
Remark 1.5.
In a forthcoming paper, we will extend similar result to thecompressible heat-conductive flows.Let us conclude this section by introducing the main idea of our proof.First of all, if the density is upper bounded, we can obtain a high integra-bility of the velocity, see Lemma 3.1. This bound can be used to controlthe nonlinear term. The trouble is to control the density, which satisfies atransport equation. In order to propagate the regularity of the density, it isnecessary to require that the velocity is bounded in L (0 , T ; W , ∞ ( Ω )). On YONGZHONG SUN, CHAO WANG, AND ZHIFEI ZHANG the other hand, we have to obtain a priori bound of ∇ ρ in order to prove u ∈ L (0 , T ; W , ∞ ( Ω )). To overcome this di ffi culty, we introduce an im-portant quantity w defined by w = u − v , where v is the solution of Lam´esystem ( µ ∆ v + ( λ + µ ) ∇ div v = ∇ p ( ρ ) in Ω , v ( x ) = ∂ Ω . In the case of
Ω = R , ( λ + µ )div w = ( λ + µ )div u − p def = G . It iswell known that G is called the e ff ective viscous flux, which plays an im-portant role in the existence theory of weak solution. A key point is thatwe can obtain the better regularity of w than u under the only assump-tion that the density is upper bounded. More precisely, we proved that ∇ w ∈ L (0 , T ; L ( Ω )), which combined with the bound of the density im-plies that ∇ u ∈ L (0 , T ; L ∞ ( Ω ) + L ∞ (0 , T ; BMO ( Ω )). This bound still doesnot imply that ∇ u is bounded in L (0 , T ; L ∞ ( Ω )). We need to introduce thesecond key ingredient: a logarithmic estimate for Lam´e system. Then theresult can be deduced by combining the above two estimates into the energyestimates for the density. 2. P reliminaries Consider the following boundary value problem for the Lam´e operator L ( µ ∆ U + ( µ + λ ) ∇ div U = F , in Ω , U ( x ) = , on ∂ Ω . (2.1)Here U = ( U , U , U ), F = ( F , F , F ). It is well known that under theassumption (1.4), (2.1) is a strongly elliptic system. If F ∈ W − , ( Ω ), thenthere exists an unique weak solution U ∈ D ( Ω ). We begin with recallingvarious estimates for this system in L q ( Ω ) spaces. Proposition 2.1.
Let q ∈ (1 , ∞ ) and U be a solution of (2.1). There exists aconstant C depending only on λ, µ, q and Ω such that the following estimateshold.(1) If F ∈ L q ( Ω ), then k D U k L q ( R ) ≤ C k F k L q ( R ) , k U k W , q ( Ω ) ≤ C k F k L q ( Ω ) ; if Ω is a bounded domain . (2.2)(2) If F ∈ W − , q ( Ω )(i.e., F = div f with f = ( f i j ) × , f i j ∈ L q ( Ω )), then ( k DU k L q ( R ) ≤ C k f k L q ( R ) , k U k W , q ( Ω ) ≤ C k f k L q ( Ω ) ; if Ω is a bounded domain . (2.3) OMPRESSIBLE NAVIER-STOKES EQUATIONS 5 (3) If F = div f with f i j = ∂ k h ki j and h ki j ∈ W , q ( Ω ) for i , j , k = , ,
3, then k U k L q ( Ω ) ≤ C k h k L q ( Ω ) . (2.4) Proof.
In the case when Ω is a bounded domain, the estimates (2.2) and(2.3) are classical for strongly elliptic systems, see for example [3]. Theestimate (2.4) can be proved by a duality argument with the help of (2.2).In the case of Ω = R , one can give an explicit representation formula forthe solution as follows. Taking divergence on both sides of (2.1), one findsdiv U = λ + µ ∆ − div F . Substituting this into (2.1) gives us ∆ U = µ F − λ + µµ ( λ + µ ) ∇ ∆ − div F . Denote the Riesz transform R = ( R , R , R ) = ∇ ∆ − / . Then ∆ U = µ F − λ + µµ ( λ + µ ) R ( R · F ) . Hence for i , j , k = , , ∂ i j U k = µ R i R j F k − λ + µµ ( λ + µ ) R i R j R k ( R · F ) . The classical L q ( R )-boundedness for Riesz transform gives k D U k L q ( R ) ≤ C ( q ) 2 λ + µµ ( λ + µ ) k F k L q ( R ) . Similar argument gives the estimates (2.3) and (2.4). (cid:3)
We need an endpoint estimate for L in the case q = ∞ . Let BMO ( Ω )stand for the John-Nirenberg’s space of bounded mean oscillation whosenorm is defined by k f k BMO ( Ω ) def = k f k L ( Ω ) + [ f ] BMO , with [ f ] BMO ( Ω ) def = sup x ∈ Ω , r ∈ (0 , d ) ? Ω r ( x ) | f ( y ) − f Ω r ( x ) | dy , f Ω r ( x ) = ? Ω r ( x ) f ( y ) dy = | Ω r ( x ) | Z Ω r ( x ) f ( y ) dy . Here Ω r ( x ) = B r ( x ) ∩ Ω , B r ( x ) is the ball with center x and radius r and d is the diameter of Ω . | Ω r ( x ) | denotes the Lebesque measure of Ω r ( x ). Notethat [ f ] BMO ( Ω ) ≤ k f k L ∞ ( Ω ) . YONGZHONG SUN, CHAO WANG, AND ZHIFEI ZHANG
Proposition 2.2. If F = div f with f = ( f i j ) × , f i j ∈ L ∞ ( Ω ) ∩ L ( Ω ), then ∇ U ∈ BMO ( Ω ) and there exists a constant C depending only on λ, µ and Ω such that k∇ U k BMO ( Ω ) ≤ C (cid:16) k f k L ∞ ( Ω ) + k f k L ( Ω ) (cid:17) . (2.5) Proof.
When Ω is a bounded domain,the estimate (2.5) can be found in[1] for a more general setting. Now if Ω = R we use the representationformula for ∇ U . Since ∆ U = µ div f − λ + µµ ( λ + µ ) ∇ ∆ − div div f = µ div f − λ + µµ ( λ + µ ) ∇ G , with G = P i , j = R i R j f i j . For k , l = , , ∂ k U l = µ R k X j = R j f l j − λ + µµ ( λ + µ ) R k R l G . By the Fe ff erman-Stein’s classical result on BMO -boundedness of singularintegral operators [25], there exists an absolute constant C > ∇ U ] BMO ( R ) ≤ C λ + µµ ( λ + µ ) k f k L ∞ ( Ω ) . This inequality combined with (2.3) with q = (cid:3) In the next lemma, we will give a variant of the Brezis-Waigner’s inequal-ity [5]. To our knowledge, such a kind of inequality was first establishedin [23] in the case of
Ω = R . For the reader’s convenience, we will give aproof in the case when Ω is a bounded Lipschitz domain, see also [26]. Lemma 2.3.
Let
Ω = R or be a bounded Lipschitz domain and f ∈ W , q ( Ω ) with q ∈ (3 , ∞ ). There exists a constant C depending on q andthe Lipshitz property of Ω such that k f k L ∞ ( Ω ) ≤ C (cid:0) + k f k BMO ( Ω ) ln (cid:0) e + k∇ f k L q ( Ω ) (cid:1)(cid:1) . (2.6) Proof.
First note that for a Lipschitz domain, the following so-called A -property holds:There exist two constants A ≥ r ∈ (0 , d ) such that for any r ∈ (0 , r ) and x ∈ Ω , | Ω r ( x ) | ≤ | B r ( x ) | ≤ A | Ω r ( x ) | . Without loss of generality we assume r ≤ | f Ω r ( x ) | with 0 < r < r and x ∈ Ω . If r ≥ r ,then (cid:12)(cid:12)(cid:12) f Ω r ( x ) (cid:12)(cid:12)(cid:12) ≤ | Ω r ( x ) | Z Ω r ( x ) | f ( y ) | dy ≤ C k f k L ( Ω ) . OMPRESSIBLE NAVIER-STOKES EQUATIONS 7 If r < r , then there exists some integer k ≥ r k + ≤ r < r k , k ≤ C (1 + | ln r | ) . Denoting Ω j = Ω j r ( x ) for j = , , · · · , k , we have (cid:12)(cid:12)(cid:12) f Ω r ( x ) (cid:12)(cid:12)(cid:12) ≤ k X j = (cid:12)(cid:12)(cid:12) f Ω j − − f Ω j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f Ω k (cid:12)(cid:12)(cid:12) ≤ k X j = ? Ω j − (cid:12)(cid:12)(cid:12) f ( y ) − f Ω j (cid:12)(cid:12)(cid:12) dy + C k f k L ( Ω ) ≤ N A k X j = ? Ω j (cid:12)(cid:12)(cid:12) f ( y ) − f Ω j (cid:12)(cid:12)(cid:12) dy + C k f k L ( Ω ) ≤ Ck [ f ] BMO ( Ω ) + C k f k L ( Ω ) ≤ C (1 + | ln r | ) k f k BMO ( Ω ) . We conclude that there exists a constant C = C ( A , r , N ) such that (cid:12)(cid:12)(cid:12) f Ω r ( x ) (cid:12)(cid:12)(cid:12) ≤ C (1 + | ln r | ) k f k BMO ( Ω ) , which together with Sobolev embedding theorem in a Lipschitz domain [2]ensures that for any fixed x ∈ Ω and small enough ε > | f ( x ) | ≤ | f ( x ) − f Ω ε ( x ) | + | f Ω ε ( x ) | ≤ C (cid:16) ε − Nq k f k W , q ( Ω ) + (1 + | ln ε | ) k f k BMO ( Ω ) (cid:17) . A suitable choice of ε yields the inequality (2.6). (cid:3) In the subsequent context we will use L − F to denote the unique solution U of the Lam´e system (2.1).3. A priori estimates for the effective viscous flux In what follows, we assume that ( ρ, u ) is a strong solution of (1.1) in[0 , T ) with the regularity stated in Theorem 1.1.Standard energy estimates yields that for any t ∈ [0 , T ), k ρ ( t ) k L ( Ω ) ≤ k ρ k L ( Ω ) , k ρ ( t ) k γ L γ ( Ω ) + k ρ | u | ( t ) k L ( Ω ) + k∇ u k L ((0 , t ) × Ω ) ≤ C (cid:0) k ρ k γ L γ ( Ω ) + k ρ | u | k L ( Ω ) (cid:1) . Note that by the assumption on ρ , u , k ρ k γ L γ ( Ω ) ≤ k ρ k γ − L ∞ ( Ω ) k ρ k L ( Ω ) , k ρ | u | k L ( Ω ) ≤ k ρ k L / ( Ω ) k u k L ( Ω ) . We thus have the following bounds k ρ k L ∞ (0 , T ; L ( Ω )) , k √ ρ u k L ∞ (0 , T ; L ( Ω )) , k∇ u k L (0 , T ; L ( Ω )) ≤ C . (3.1)Here C depends only on µ, λ, γ, a and ρ , u . YONGZHONG SUN, CHAO WANG, AND ZHIFEI ZHANG
In what follows the dependence of the constant C on µ, λ, γ, a and Ω willnot be mentioned.The following lemma is the first key step, whose argument comes from[14] and [16]. Lemma 3.1.
Assume that µ < λ and the density ρ satisfies k ρ k L ∞ (0 , T ; L ∞ ( Ω )) ≤ M . (3.2)There exists r ∈ (3 ,
6) such that ρ | u | r ∈ L ∞ (0 , T ; L ( Ω )) with k ρ | u | r k L ∞ (0 , T ; L ( Ω )) ≤ C . Here C depends on T , k ρ k L ∞ ( Ω ) , k∇ u k L ( Ω ) , M . Proof.
Multiplying the second equation of (1.1) by r | u | r − u , and integratingthe resulting equation on Ω to obtain ddt Z Ω ρ | u | r dx + Z Ω r | u | r − (cid:16) µ |∇ u | + ( λ + µ )(div u ) (cid:17) + r ( r − (cid:16) µ | u | r − |∇| u || + ( λ + µ )(div u ) | u | r − u · ∇| u | (cid:17) dx = Z Ω r p ( ρ ) div( | u | r − u ) dx (3.3)By using the fact |∇ u | ≥ |∇| u || , the term in the second integrand can beestimated from below by r | u | r − h µ |∇ u | + ( λ + µ )(div u ) + ( r − µ |∇| u || − ( λ + µ )( r − |∇| u ||| div u | i ≥ r | u | r − (cid:2) µ |∇ u | + ( λ + µ ) (cid:0) div u − r − |∇| u || (cid:1) − ( λ + µ ) ( r − |∇| u || + ( r − µ |∇| u || (cid:3) ≥ r | u | r − (cid:2) µ |∇ u | + ( r − (cid:0) µ − ( λ + µ ) r − (cid:1) |∇| u || (cid:3) Recalling that λ < µ , there exists r ∈ (3 ,
6) such that the last term is greaterthan c | u | r − |∇ u | . On the other hand, because of k ρ k L ∞ ≤ M , we find that the right-hand sideof (3.3) is controlled by C Z Ω ρ r − r | u | r − |∇ u | dx ≤ ǫ Z Ω | u | r − |∇ u | dx + C ǫ Z Ω ρ | u | r dx r − r . OMPRESSIBLE NAVIER-STOKES EQUATIONS 9
Taking ǫ = c to yield that ddt Z Ω ρ | u | r dx ≤ C Z Ω ρ | u | r dx r − r , which together with the following bound k ρ | u | r k L ( Ω ) ≤ k ρ k L − r ( Ω ) k u k rL ( Ω ) ≤ C k ρ k L − r ( Ω ) k∇ u k rL ( Ω ) , implies the desired estimate. (cid:3) Now for each t ∈ [0 , T ), we denote v ( t , x ) def = L − ∇ p ( ρ ). That is, v ( t ) isthe solution of ( µ ∆ v + ( λ + µ ) ∇ div v = ∇ p ( ρ ) in Ω , v ( t , x ) = ∂ Ω . (3.4)Thanks to Proposition 2.1, for any q ∈ (1 , ∞ ), there exists a constant C independent of t such that k∇ v ( t ) k L q ( Ω ) ≤ C k p ( ρ ( t )) k L q ( Ω ) , k∇ v ( t ) k L q ( Ω ) ≤ C k∇ p ( ρ ( t )) k L q ( Ω ) . (3.5)Now let us introduce an important quantity w = u − v , whose divergence can be viewed as the e ff ective viscous flux.An important observation is that this quantity possesses more regular-ity information than u does under the assumption that the density is upperbounded. More precisely, Proposition 3.2.
Under the assumption (3.2), we have k∇ w k L ∞ (0 , T ; L ( Ω )) , k ρ ∂ t w k L ((0 , T ) × Ω ) , k∇ w k L ((0 , T ) × Ω ) ≤ C . (3.6)Here the constant C depends on k ρ k L ∞ ( Ω ) , k∇ u k L ( Ω ) , M , T . Proof.
By using the continuity equation, we find that w satisfies ( ρ∂ t w − µ ∆ w − ( λ + µ ) ∇ div w = ρ F , in (0 , T ) × Ω , w ( t , x ) = , T ) × ∂ Ω , w (0 , x ) = w ( x ) , in Ω , (3.7)with w ( x ) = u ( x ) + v ( x ) and F = − u · ∇ u − L − ∇ ( ∂ t p ( ρ )) = − u · ∇ u + L − ∇ div[ p ( ρ ) u ] + L − ∇ [( ρ p ′ ( ρ ) − p ( ρ )) div u ] . Multiplying the first equation of (3.7) by ∂ t w and integrating the resultingequation over Ω to obtain , ddt Z Ω µ |∇ w | + ( λ + µ ) | div w | dx + Z Ω ρ | ∂ t w | dx = Z Ω ρ F · ∂ t wdx , which together with H¨older inequality and Young’s inequality gives ddt Z Ω µ |∇ w | + ( λ + µ ) | div w | dx + Z Ω ρ | ∂ t w | dx ≤ k √ ρ F k L ( Ω ) . (3.8)Now let us estimate k √ ρ F k L ( Ω ) . We get by Lemma 3.1 and (3.5) that k √ ρ u · ∇ u k L ( Ω ) ≤ C k ρ r u k L r ( Ω ) k∇ u k L rr − ( Ω ) ≤ C k ρ r u k L r ( Ω ) (cid:0) k∇ w k L rr − ( Ω ) + k∇ v k L rr − ( Ω ) (cid:1) ≤ C ǫ k∇ w k L ( Ω ) + ǫ k∇ w k L ( Ω ) + C . Here we use the interpolation inequality k f k L q ( Ω ) ≤ C ǫ k f k L ( Ω ) + ǫ k∇ f k L ( Ω ) , ≤ q < . We infer from Proposition 2.1 that k √ ρ L − ∇ div[ p ( ρ ) u ] k L ( Ω ) ≤ C k p ( ρ ) u k L ( Ω ) ≤ C k √ ρ u k L ( Ω ) ≤ C , k √ ρ L − ∇ ( ρ p ′ − p ) div u k L ( Ω ) ≤ k √ ρ k L ( Ω ) k L − ∇ ( ρ p ′ − p ) div u k L ( Ω ) ≤ C k∇ L − ∇ ( ρ p ′ − p ) div u k L ( Ω ) ≤ C k∇ u k L ( Ω ) . Consequently, for ǫ > k √ ρ F k L ( Ω ) ≤ ǫ k∇ w k L ( Ω ) + C ǫ (cid:0) + k∇ w k L ( Ω ) + k∇ u k L ( Ω ) (cid:1) . (3.9)Noting that Lw = ρ∂ t w − ρ F , we get by using Proposition 2.1 again that k∇ w k L ( Ω ) ≤ C (cid:0) k ρ∂ t w k L ( Ω ) + k ρ F k L ( Ω ) (cid:1) ≤ C (cid:0) k √ ρ∂ t w k L ( Ω ) + k √ ρ F k L ( Ω ) (cid:1) , which implies by taking ǫ = C in (3.9) that k √ ρ F k L ( Ω ) ≤ k √ ρ∂ t w k L ( Ω ) + C (cid:0) + k∇ w k L ( Ω ) + k∇ u k L ( Ω ) (cid:1) . Substituting this estimate into (3.8) and noting that k∇ u ( t ) k L ( Ω ) ∈ L (0 , T ),the estimate (3.6) follows from Gronwall’s inequality. (cid:3) OMPRESSIBLE NAVIER-STOKES EQUATIONS 11
Corollary 3.3.
Under the assumption (3.2), we have k∇ u k L ∞ (0 , T ; L ( Ω )) , k u k L ∞ (0 , T ; L ( Ω )) , k∇ u k L (0 , T ; L q ( Ω )) ≤ C , for any q ∈ [2 , Proof.
This can be deduced from Proposition 3.2, (3.5) and Sobolev em-bedding theorem. (cid:3)
4. H igh order a priori estimates for the effective viscous flux
In this section, we will give high order regularity estimates for w . This ispossible if the initial data ( ρ , u ) satisfies the compatibility condition (1.5).We still assume that ( ρ, u ) is a strong solution of (1.1) in [0 , T ) and satisfies(3.2).The energy estimates in this section are motivated by the calculationsof D. Ho ff [14].We begin by introducing some notations. For a function or vector field(oreven a 3 × f ( t , x ), the material derivative ˙ f is defined by˙ f def = f t + u · ∇ f , and div( f ⊗ u ) def = P j = ∂ j ( f u j ). For two matrices A = ( a i j ) × and B = ( b i j ) × , we use the notation A : B = P i , j = a i j b i j and AB is as usual themultiplication of matrix.We rewrite the second equation of (1.1) as ρ ˙ u + ∇ p ( ρ ) − Lu = . By taking the material derivative to the above equation and using the fact˙ f = f t + div( f u ) − f div u , we obtain ρ ˙ u t + ρ u · ∇ ˙ u + ∇ p t + div( ∇ p ⊗ u ) = µ (cid:2) ∆ u t + div( ∆ u ⊗ u ) (cid:3) + ( λ + µ ) (cid:2) ∇ div u t + div(( ∇ div u ) ⊗ u ) (cid:3) . (4.1)Multiplying (4.1) by ˙ u and integrating on Ω to obtain ddt Z Ω ρ | ˙ u | dx − µ Z Ω ˙ u · (cid:0) ∆ u t + div( ∆ u ⊗ u ) (cid:1) dx − ( λ + µ ) Z Ω ˙ u · (cid:0) ( ∇ div u t ) + div(( ∇ div u ) ⊗ u )) (cid:1) dx = Z Ω p t div ˙ u + ( ˙ u · ∇ u ) · ∇ pdx . (4.2) The µ -term can be calculated as follows. − Z Ω ˙ u · (cid:0) ∆ u t + div( ∆ u ⊗ u ) (cid:1) dx = Z Ω [ ∇ ˙ u : ∇ u t + u ⊗ ∆ u : ∇ ˙ u ] dx = Z Ω h |∇ ˙ u | − ∇ ( u · ∇ u ) : ∇ ˙ u + u ⊗ ∆ u : ∇ ˙ u i dx = Z Ω h |∇ ˙ u | − (cid:0) ( ∇ u ∇ u ) + ( u · ∇ ) ∇ u (cid:1) : ∇ ˙ u − ∇ ( u · ∇ ˙ u ) : ∇ u i dx = Z Ω h |∇ ˙ u | − ( ∇ u ∇ u ) : ∇ ˙ u − div( ∇ u ⊗ u ) : ∇ ˙ u − ( ∇ u ∇ ˙ u ) : ∇ u − (( u · ∇ ) ∇ ˙ u ) : ∇ u i dx = Z Ω h |∇ ˙ u | − ( ∇ u ∇ u ) : ∇ ˙ u + (( u · ∇ ) ∇ ˙ u ) : ∇ u − ( ∇ u ∇ ˙ u ) : ∇ u − (( u · ∇ ) ∇ ˙ u ) : ∇ u i dx ≥ Z Ω " |∇ ˙ u | − C |∇ u | dx . To estimate the ( λ + µ )-term of (4.2), note thatdiv(( ∇ div u ) ⊗ u ) = ∇ ( u · ∇ div u ) − div(div u ∇ ⊗ u ) + ∇ (div u ) , div ˙ u = div u t + div( u · ∇ u ) = div u t + u · ∇ div u + ∇ u : ( ∇ u ) ′ . Here A ′ means the transpose of matrix A . We have − Z Ω ˙ u · h ∇ div u t + div(( ∇ div u ) ⊗ u ) i dx = Z Ω h div ˙ u div u t + div ˙ u ( u · ∇ div u ) − div u ( ∇ ˙ u ) ′ : ∇ u + div ˙ u (div u ) i dx = Z Ω h | div ˙ u | − div ˙ u ∇ u : ( ∇ u ) ′ − div u ( ∇ ˙ u ) ′ : ∇ u + div ˙ u (div u ) i dx ≥ Z Ω h | div ˙ u | − |∇ ˙ u | − C |∇ u | i dx . OMPRESSIBLE NAVIER-STOKES EQUATIONS 13
We continue to estimate the pressure term. Z Ω p t div ˙ u + ( u · ∇ ˙ u ) · ∇ pdx = Z Ω p ′ ( ρ ) ρ t div ˙ u + ( u · ∇ ˙ u ) · ∇ pdx = Z Ω − ρ p ′ ( ρ ) div u div ˙ u − ( u · ∇ p ( ρ )) div ˙ u + ( u · ∇ ˙ u ) · ∇ pdx = Z Ω − ρ p ′ ( ρ ) div u div ˙ u + p h div((div ˙ u ) u ) − div(( u · ∇ ˙ u )) i dx = Z Ω − ρ p ′ ( ρ ) div u div ˙ u + p h div u div ˙ u − ( ∇ u ) ′ : ∇ ˙ u i dx ≤ C k∇ u k L ( Ω ) k∇ ˙ u k L ( Ω ) ≤ C k∇ ˙ u k L ( Ω ) , where we used the assumption (3.2) and Corollary 3.3 in the last two in-equalities.Substituting those estimates into (4.2) yields ddt Z Ω ρ | ˙ u | dx + µ Z Ω |∇ ˙ u | dx + ( λ + µ ) Z Ω | div ˙ u | dx ≤ C Z Ω |∇ u | dx + C k∇ ˙ u k L ( Ω ) . (4.3)To conclude the estimate by Gronwall’s inequality, we will use the term k √ ρ ˙ u k L ( Ω ) to control k∇ u k L ( Ω ) . Thanks to the definition of w , we know that w satisfies µ ∆ w + ( λ + µ ) ∇ div w = ρ ˙ u in Ω , (4.4)with the zero boundary condition. We get by Proposition 2.1 that k∇ w k L ( Ω ) ≤ C k ρ ˙ u k L ( Ω ) ≤ C k √ ρ ˙ u k L ( Ω ) , which together with the interpolation inequality, Corollary 3.3, and Propo-sition 2.1 leads to k∇ u k L ( Ω ) ≤ k∇ u k L ( Ω ) k∇ u k L ( Ω ) ≤ C k∇ u k L ( Ω ) k∇ u k L ( Ω ) ≤ C k∇ u || L ( Ω ) (cid:0) k∇ w k L ( Ω ) + k∇ v k L ( Ω ) (cid:1) ≤ C k∇ u || L ( Ω ) (cid:0) + k∇ w k L ( Ω ) (cid:1) ≤ C k∇ u k L ( Ω ) (cid:16) + k √ ρ ˙ u k L ( Ω ) (cid:17) . Substituting this estimate into (4.3)and noting that ||∇ u ( t ) || L ( Ω ) ∈ L (0 , T )by Corollary 3.3, we get by Gronwall’s inequality that Z Ω ρ | ˙ u | dx + Z T Z Ω |∇ ˙ u | dxdt ≤ C , (4.5)with C depending only on T , M and ρ , u , g . Here we used the compatibil-ity condition (1.5).With the help of Sobolev embedding theorem and using the equation (4.4)again, we deduce from (4.5) that Proposition 4.1.
Under the assumption (3.2), we have for all ≤ q ≤ , (4.6) k∇ w k L (0 , T ; L ∞ ( Ω )) , k∇ w k L (0 , T ; L q ( Ω )) ≤ C , with the constant C depending on q , M , T and ρ , u , g.
5. P roof of T heorem T ∗ < ∞ andsup s ∈ [0 , T ∗ ) k ρ ( s ) k L ∞ ( Ω ) < ∞ . By Theorem 1.1, it su ffi ces to show thatsup s ∈ [0 , T ∗ ) k∇ ρ ( s ) k L q ( Ω ) < ∞ . (5.1)Taking the derivative with respect to x for the first equation of (1.1) toobtain ∂ t ∇ ρ + ( u · ∇ ) ∇ ρ + ∇ u ∇ ρ + div u ∇ ρ + ρ ∇ div u = . (5.2)In the following estimates we will use k∇ v k L q ( Ω ) ≤ C k∇ ρ k L q ( Ω ) , (5.3) k∇ v k L ∞ ( Ω ) ≤ C (cid:16) + k∇ v k BMO ( Ω ) ln( e + k∇ v k L q ( Ω ) ) (cid:17) ≤ C (cid:16) + k ρ k L ∞ ∩ L ( Ω ) ln( e + k∇ ρ k L q ( Ω ) ) (cid:17) ≤ C (cid:16) + ln( e + k∇ ρ k L q ( Ω ) ) (cid:17) (5.4)with the second estimate followed from Proposition 2.1, 2.2 and Lemma2.3. OMPRESSIBLE NAVIER-STOKES EQUATIONS 15
Multiplying (5.2) by q |∇ ρ | q − ∇ ρ and integrating the resulting equation on Ω , we obtain ddt Z Ω |∇ ρ | q dx ≤ C Z Ω |∇ u ||∇ ρ | q dx + q Z Ω ρ |∇ div u ||∇ ρ | q − dx ≤ C k∇ u k L ∞ ( Ω ) k∇ ρ k qL q ( Ω ) + C k∇ u k L q ( Ω ) k∇ ρ k q − L q ( Ω ) ≤ C (cid:0) k∇ w k L ∞ ( Ω ) + k∇ v k L ∞ ( Ω ) (cid:1) k∇ ρ k qL q ( Ω ) + C (cid:0) k∇ w k L q ( Ω ) + k∇ v k L q ( Ω ) (cid:1) k∇ ρ k q − L q ( Ω ) , from which and (5.3)-(5.4), we infer that ddt Z Ω |∇ ρ | q dx ≤ C (cid:0) + k∇ v k L ∞ ( Ω ) + k∇ w k L ∞ ( Ω ) (cid:1) k∇ ρ k qL q ( Ω ) + C k∇ w k L q ( Ω ) k∇ ρ k q − L q ( Ω ) ≤ C (cid:0) + k∇ w k W , q ( Ω ) + ln( e + k∇ ρ k L q ( Ω ) ) (cid:1) k∇ ρ k qL q ( Ω ) + k∇ w k L q ( Ω ) k∇ ρ k q − L q ( Ω ) . Note that k∇ w k W , q ( Ω ) ∈ L (0 , T ∗ ) by Proposition 4.1. Then by Gronwall’sinequality, we conclude the proof of (5.1) and hence Theorem 1.3. 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Math., 51(1998), 229-240.D epartment of M athematics , N anjing U niversity , 210093, P. R. C hina E-mail address : [email protected] A cademy of M athematics & S ystems S cience , CAS, B eijing hina E-mail address : [email protected] S chool of M athematical S ciences , P eking U niversity , 100871, P. R. C hina E-mail address ::