On Compressible Navier-Stokes Equations Subject to Large Potential Forces with Slip Boundary Conditions in 3D Bounded Domains
aa r X i v : . [ m a t h . A P ] F e b On Compressible Navier-Stokes Equations Subject to LargePotential Forces with Slip Boundary Conditions in 3DBounded Domains ∗ Guocai C AI a , Bin H UANG b , Xiaoding S HI b , a. School of Mathematical Sciences,Xiamen University, Xiamen 361005, P. R. China;b. College of Mathematics and Physics,Beijing University of Chemical Technology, Beijing 100029, P. R. China Abstract
We deal with the barotropic compressible Navier-Stokes equations subject tolarge external potential forces with slip boundary condition in a 3D simply con-nected bounded domain, whose smooth boundary has a finite number of 2D con-nected components. The global existence of strong or classical solutions to theinitial boundary value problem of this system is established provided the initialenergy is suitably small. Moreover, the density has large oscillations and containsvacuum states. Finally, we show that the global strong or classical solutions decayexponentially in time to the equilibrium in some Sobolev’s spaces, but the oscilla-tion of the density will grow unboundedly in the long run with an exponential ratewhen the initial density contains vacuum states.
Keywords: compressible Navier-Stokes equations; slip boundary condition; vacuum;large external forces; global existence; large-time behavior.
The motion of three-dimensional viscous compressible barotropic flows is governedby the compressible Navier-Stokes equations ( ρ t + div( ρu ) = 0 , ( ρu ) t + div( ρu ⊗ u ) − µ ∆ u − ( µ + λ ) ∇ div u + ∇ P ( ρ ) = ρf, (1.1)where ( x, t ) ∈ Ω × (0 , T ], Ω is a domain in R N , t ≥ x is the spatialcoordinate. ρ ≥ , u = ( u , · · · , u N ) and P ( ρ ) = aρ γ ( a > , γ >
1) (1.2)are the unknown fluid density, velocity and pressure, respectively. We mainly considerthe case that the external force f ( x ) is a gradient of a scalar potential, that is, f ( x ) = ∇ ψ ( x ) (1.3) ∗ Email: [email protected] (G.C.Cai); [email protected] (B. Huang); [email protected](X. D. Shi) µ and λ are the shear and bulk viscosity coefficients respectively satis-fying the following physical restrictions: µ > , µ + N λ ≥ . (1.4)In this paper, we assume that Ω is a simply connected bounded domain in R , itsboundary ∂ Ω is of class C ∞ and only has a finite number of 2-dimensional connectedcomponents. In addition, the system is studied subject to the given initial data ρ ( x,
0) = ρ ( x ) , ρu ( x,
0) = ρ u ( x ) , x ∈ Ω , (1.5)and slip boundary condition u · n = 0 and curl u × n = 0 on ∂ Ω , (1.6)where n = ( n , n , n ) is the unit outward normal vector on ∂ Ω.The first condition in (1.6) is the non-penetration boundary condition, while thesecond one is also known in the form( D ( u ) n ) τ = − κ τ u τ , (1.7)where D ( u ) = ( ∇ u + ( ∇ u ) tr ) / κ τ is the correspondingnormal curvature of ∂ Ω in the τ direction. Therefore, in the case that ∂ Ω is of constantcurvature, (1.6) is a special Navier-type slip boundary condition (see [4]), in which thereis a stagnant layer of fluid close to the wall allowing a fluid to slip, and the slip velocityis proportional to the shear stress. This type of boundary condition was originallyintroduced by Navier [27] in 1823, which was followed by many applications, numericalstudies and analysis for various fluid mechanical problems, see, for instance [8, 15, 32]and the references therein. In fact, all our results of the paper are also valid formore general boundary conditions including Navier-type slip boundary condition (seeRemark 1.4 below).The compressible Navier-Stokes system has been attracted a lot of attention andsignificant progress has been made in the analysis of the well-posedness and dynamicbehavior. we only briefly review some results related to the existence of strong orclassical solutions and large-time behavior. The local solvability in classical spacessubject to the various boundary conditions was established by Serrin [31], Nash [26],Solonnikov [34], Tani [35] early and then some further works were given by Cho-Choe-Kim (see [5] [6], [7]), Huang [12]. For the whole space R and without external forces,the global classical solutions were first obtained by Matsumura-Nishida [22] for initialdata close to a nonvacuum equilibrium in H . It is worth mentioned that their resultshave been improved by Huang-Li-Xin [14] and Li-Xin [19], in which the global existenceof classical solutions to the Cauchy problem for the barotropic compressible Navier-Stokes equations is obtained with smooth initial data that are of small energy butpossibly large oscillations with constant state as far field which could be either vacuumor nonvacuum. Very recently, for the barotropic compressible Navier-Stokes equationsin a bounded domain with slip boundary conditions, Cai-Li [4] proved that the classicalsolution of the initial-boundary-value problem in the absence of exterior forces existsglobally with vacuum and small energy but possibly large oscillations.For compressible Navier-Stokes system with external forces, some early research workfocused on small external forces (see [22–24, 33] and the references therein), mainly2ecause the large external forces have a powerful influence on the dynamic motion ofcompressible flows and bring some serious difficulties (cf. [9, 17, 25, 37]). On the onehand, there are many results regarding the large-time behavior of weak solutions tothe problem (1.1). Feireisl-Petzeltov´a [9], Novotny-Straˇskraba [30] showed that fordifferent boundary conditions, the density of any global weak solution converges tothe steady state density in L q space for some q > as time goes to infinity if thereexists a unique steady state. On the other hand, it seems that there are few resultson the global existence of classical or strong solutions to the system (1.1) in R withlarge external forces except for that of Li-Zhang-Zhao [20], where they proved thatthe Cauchy problem has a unique global strong solution with large oscillations andinterior vacuum, provided the initial data are of small energy and the unique steadystate is strictly away from vacuum. However, their method depends crucially on theCauchy problem and cannot be applied directly to the bounded domains. Therefore,our main purpose in the paper is to study the mechanism of the influence of the largepotential forces to the compressible system (1.1) with Navier-slip boundary conditionsin bounded domains.Before stating the main results, we explain the notations and conventions usedthroughout this paper. We denote Z f dx = Z Ω f dx. For integer k and 1 ≤ q < + ∞ , W k,q (Ω) is the standard Sobolev spaces and W ,q (Ω) = { u ∈ W ,q (Ω) : u is equipped with zero trace on ∂ Ω } . We also denote H k (Ω) = W k, (Ω) , H (Ω) = W , (Ω) . For simplicity, we denote L q (Ω), W k,q (Ω), H k (Ω), H (Ω) by L q , W k,q , H k , H respec-tively.For two 3 × A = { a ij } , B = { b ij } , the symbol A : B represents the traceof AB , that is, A : B , tr( AB ) = X i,j =1 a ij b ji . Finally, for v = ( v , v , v ), we denote ∇ i v , ( ∂ i v , ∂ i v , ∂ i v ) for i = 1 , , , and thematerial derivative of v by ˙ v , v t + u · ∇ v .It is natural to expect an equilibrium density ρ s = ρ s ( x ) and velocity u s = u s ( x )to the initial boundary-value problem (1.1)–(1.6), which is a solution of the rest stateequations div( ρ s u s ) = 0 in Ω , − µ ∆ u s − ( λ + µ ) ∇ div u s + ∇ P ( ρ s ) = ρ s ∇ ψ in Ω ,u s · n = 0 , curl u s × n = 0 on ∂ Ω , R ρ s dx = R ρ dx. (1.8)We have the following conclusion. 3 emma 1.1 ( [21]) Assume that Ω is a bounded domain with smooth boundary, and ψ satisfies ψ ∈ H , Z (cid:18) γ − aγ ( ψ − inf Ω ψ ) (cid:19) γ − dx < Z ρ dx, (1.9) then there exists a unique solution ( ρ s , of (1.8) such that ρ s ∈ H , < ρ ≤ inf Ω ρ s ≤ sup Ω ρ s ≤ ¯ ρ, (1.10) where ρ and ¯ ρ are positive constants depending only on a , γ , inf Ω ψ and sup Ω ψ . Inaddition, if ψ ∈ W ,q for some q ∈ (3 , , then k ρ s k W ,q ≤ C, (1.11) where C is a positive constant depending only on a , γ , inf Ω ψ and k ψ k W ,q . Remark 1.1
It should be noted that the equilibrium density ρ s is in fact a solution ofthe rest state equations ( ∇ P ( ρ s ) = ρ s ∇ ψ in Ω , R ρ s dx = R ρ dx. (1.12)We denote the initial total energy of (1.1) as C , Z Ω (cid:18) ρ | u | + G ( ρ , ρ s ) (cid:19) dx, (1.13)where G ( ρ, ρ s ) is the potential energy density given by G ( ρ, ρ s ) , ρ Z ρρ s P ( ξ ) − P ( ρ s ) ξ dξ. (1.14)Our first result is concern with the global existence of a strong solution of the problem(1.1)-(1.6) in a bounded domain. Theorem 1.1
Let Ω be a simply connected bounded domain in R and its smoothboundary ∂ Ω has a finite number of 2-dimensional connected components. For ψ ∈ H with (1.9) , ρ s is the steady state density given by (1.12) . For given positive constants M and ˆ ρ > ¯ ρ + 1 , assume that ( ρ , u ) satisfy for some q ∈ (3 , , ( ρ , P ( ρ )) ∈ W ,q , u ∈ (cid:8) f ∈ H (cid:12)(cid:12) f · n = 0 , curl f × n = 0 on ∂ Ω (cid:9) , (1.15)0 ≤ ρ ≤ ˆ ρ, k u k H ≤ M. (1.16) Then there exists a positive constant ε depending only on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω , and M such that if the initial total energy C < ε , then the initial-boundary-valueproblem (1.1) - (1.6) has a unique strong solution ( ρ, u ) in Ω × (0 , ∞ ) satisfying ≤ ρ ( x, t ) ≤ ρ, ( x, t ) ∈ Ω × (0 , ∞ ) , (1.17) and for any τ ∈ (0 , T ) , ( ρ, P ) ∈ C ([0 , T ]; W ,q ) , ρ t ∈ L ∞ (0 , T ; L ) ,ρu ∈ C ([0 , T ]; L ) , u ∈ L ∞ (0 , T ; H ) ∩ L (0 , T ; H ) ,ρ / u t ∈ L (0 , T ; L ) , ( ρ / u t , ∇ u ) ∈ L ∞ ( τ, T ; L ) u t ∈ L ( τ, T ; H ) . (1.18)4 emark 1.2 In this conclusion, we have no restrictions on the potential force ψ expectfor (1.9) . Moreover, besides the small total energy, large oscillations of the density andvacuum are also allowed. Remark 1.3
It is clear that u ∈ C ([ τ, T ]; H ) for any < τ < T . However, although u ∈ H , it seems difficult to derive that u ∈ C ([0 , T ]; H ) due to the lack of thecompatibility condition (see (1.20) ) and the presence of vacuum. The second goal of this paper is to provide the global existence of classical solutions of(1.1)-(1.6) in a bounded domain as follows:
Theorem 1.2
In addition to the conditions of Theorem 1.1, assume further that ψ ∈ H , the initial data ( ρ , u ) satisfy ( ρ , P ( ρ )) ∈ W ,q , (1.19) and the compatibility condition − µ △ u − ( µ + λ ) ∇ div u + ∇ P ( ρ ) = ρ / g, (1.20) for some g ∈ L . Then there exists a positive constant ε depending only on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω , and M such that the initial-boundary-value problem (1.1) - (1.6) has a unique classical solution ( ρ, u ) in Ω × (0 , ∞ ) satisfying (1.17) and for any < τ < T < ∞ , ( ρ, P ) ∈ C ([0 , T ]; W ,q ) , ∇ u ∈ C ([0 , T ]; H ) ∩ L ∞ ( τ, T ; W ,q ) ,u t ∈ L ∞ ( τ, T ; H ) ∩ H ( τ, T ; H ) , √ ρu t ∈ L ∞ (0 , ∞ ; L ) , (1.21) provided C ≤ ε. Moreover, there exist positive constants C and ˜ C depending only on µ, λ, γ, a , inf Ω ψ , k ψ k H , ˆ ρ , M , Ω , p , q and C such that the following large-time behavior holds for any q ∈ [1 , ∞ ) and p ∈ [1 , , (cid:0) k ρ − ρ s k L q + k u k W ,p + k√ ρ ˙ u k L (cid:1) ≤ Ce − ˜ Ct . (1.22) Remark 1.4
Similar to what have done in [4], one can get the same conclusion undermore relaxed assumption on the initial data and more wide boundary condition (see [4,Theorem 1.1] for the details).
With (1.22) at hand, similar to [4, Theorem 1.2], we can obtain the following large-time behavior of the gradient of the density when vacuum states appear initially.
Theorem 1.3
In addition to the conditions of Theorem 1.2, assume further that thereexists some point x ∈ Ω such that ρ ( x ) = 0 . Then there exist positive constants ˆ C and ˆ C depending only on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω , M , r and C such that forany t > , k∇ ρ ( · , t ) k L r ≥ ˆ C e ˆ C t . (1.23)5e now comment on the analysis of this paper. Compared with [4], as indicatedby [9, 17, 25, 37], the large external forces will bring some serious difficulties due to itspowerful influence on the dynamic motion of compressible flows. To overcome thesedifficulties, we need some new ideas. More precisely, firstly, introducingcurl u , ∇ × u, F , ( λ + 2 µ ) div u − ( P − P ( ρ s )) , (1.24)we rewrite (1 . in the form ρ ˙ u − ρ ∇ ψ = ∇ F − µ ∇ × curl u, (1.25)where F is called the effective viscous flux and plays an important role in our followinganalysis. Combining (1.25) with the slip boundary condition (1.6), we obtain theestimates of ∇ F and ∇ curl u . Furthermore, together with the following inequality k∇ u k W k,q ≤ C ( k div u k W k,q + k curl u k W k,q ) for any q > , k ≥ , which is shown in [36] when Ω is simply connected, it allows us to control ∇ u bymeans of div u and curl u . Secondly, since u · n = 0 and curl u × n = 0 on ∂ Ω, denote u ⊥ , − u × n , then u = u ⊥ × n , moreover, u · ∇ u · n = − u · ∇ n · u, (1.26)and ( ˙ u + ( u · ∇ n ) × u ⊥ ) · n = 0 on ∂ Ω , which are the key to estimating the integrals on the boundary ∂ Ω. Finally, to deal withthe large external potential forces, following [11] (see also [17, 20]), we have ρ − s ( ∇ ( ρ γ − ρ γs ) − γ ( ρ − ρ s ) ρ γ − s ∇ ρ s )= ∇ ( ρ − s ( ρ γ − ρ γs )) − γ − a G ( ρ, ρ s ) ∇ ρ − s , (1.27)which indeed gives the ’small’ estimate of k ρ − ρ s k L (Ω × [0 ,T ]) (see (3.45) and (3.46)).The rest of the paper is organized as follows. In Section 2, some known facts andelementary inequalities needed in later analysis are collected. Sections 3 and 4 aredevoted to deriving the necessary a priori estimates which can guarantee the localstrong (or classical) solution to be a global strong (or classical) one. Finally, the mainresults, Theorems 1.2 and 1.3 will be proved in Section 5. In this subsection, we will recall some known theorems and facts, which are frequentlyutilized in our discussion.First, similar to the proof of [12, Theorems 1.2 and 1.4], we have the local existenceof strong and classical solutions. 6 emma 2.1
Let Ω be as in Theorem 1.2, assume that ( ρ , u ) satisfy (1.15) . Thenthere exist a small time T > and a unique strong solution ( ρ, u ) to the problem (1.1) - (1.6) on Ω × (0 , T ] satisfying for any τ ∈ (0 , T ) , ( ρ, P ) ∈ C ([0 , T ]; W ,q ) , ∇ u ∈ L ∞ (0 , T ; L ) ∩ ∈ L ∞ ( τ, T ; W ,q ) ,ρ / u t ∈ L (0 , T ; L ) ∩ L ∞ ( τ, T ; L ) ,u t ∈ L ∞ ( τ, T ; H ) . Furthermore, if the initial data ( ρ , u ) satisfy (1.19) and the compatibility conditions (1.20) , then there exist T > and a unique classical solution ( ρ, u ) to the problem (1.1) - (1.6) on Ω × (0 , T ] satisfying for any τ ∈ (0 , T ) , ( ρ, P ) ∈ C ([0 , T ]; W ,q ) , ∇ u ∈ C ([0 , T ]; H ) ∩ L ∞ ( τ, T ; W ,q ) ,u t ∈ L ∞ ( τ, T ; H ) ∩ H ( τ, T ; H ) , √ ρu t ∈ L ∞ (0 , T ; L ) . Next, the well-known Gagliardo-Nirenberg’s inequality (see [28]) will be used later.
Lemma 2.2 (Gagliardo-Nirenberg)
Assume that Ω is a bounded Lipschitz domainin R . For p ∈ [2 , , q ∈ (1 , ∞ ) , and r ∈ (3 , ∞ ) , there exist generic constants C i > i = 1 , · · · , which depend only on p , q , r , and Ω such that for any f ∈ H (Ω) and g ∈ L q (Ω) ∩ D ,r (Ω) , k f k L p (Ω) ≤ C k f k − p p L k∇ f k p − p L + C k f k L , (2.1) k g k C ( Ω ) ≤ C k g k q ( r − / (3 r + q ( r − L q k∇ g k r/ (3 r + q ( r − L r + C k g k L . (2.2) Moreover, if either f · n | ∂ Ω = 0 or ¯ f = 0 , we can choose C = 0 . Similarly, the constant C = 0 provided g · n | ∂ Ω = 0 or ¯ g = 0 . We need the following Zlotnik’s inequality, by which we can get the uniform (in time)upper bound of the density ρ. Lemma 2.3 ( [38])
Suppose the function y satisfies y ′ ( t ) = g ( y ) + b ′ ( t ) on [0 , T ] , y (0) = y , with g ∈ C ( R ) and y, b ∈ W , (0 , T ) . If g ( ∞ ) = −∞ and b ( t ) − b ( t ) ≤ N + N ( t − t ) (2.3) for all ≤ t < t ≤ T with some N ≥ and N ≥ , then y ( t ) ≤ max (cid:8) y , ζ (cid:9) + N < ∞ on [0 , T ] , where ζ is a constant such that g ( ζ ) ≤ − N for ζ ≥ ζ. (2.4)7or the Lam´e’s system ( − µ ∆ u − ( λ + µ ) ∇ div u = f, x ∈ Ω ,u · n = 0 , curl u × n = 0 , x ∈ ∂ Ω , (2.5)where u = ( u , u , u ) , f = ( f , f , f ), Ω is a bounded smooth domain in R , and µ, λ satisfy the condition (1.4), the following estimate is standard (see [1]). Lemma 2.4
Let u be a solution of the Lam´e’s equation (2.5) , there exists a positiveconstant C depending only on λ, µ, q, k and Ω such that(1) If f ∈ W k,q for some q ∈ (1 , ∞ ) , k ≥ , then u ∈ W k +2 ,q and k u k W k +2 ,q ≤ C ( k f k W k,q + k u k L q ) , (2) If f = ∇ g and g ∈ W k,q for some q ≥ , k ≥ , then u ∈ W k +1 ,q and k u k W k +1 ,q ≤ C ( k g k W k,q + k u k L q ) . Definition 2.1
Let Ω be a domain in R . If the first Betti number of Ω vanishes,namely, any simple closed curve in Ω can be contracted to a point, we say that Ω issimply connected. If the second Betti number of Ω is zero, we say that Ω has no holes. The following two lemmas can be found in [36, Theorem 3.2] and [2, Propositions2.6-2.9].
Lemma 2.5
Let k ≥ be a integer, < q < + ∞ , and assume that Ω is a simplyconnected bounded domain in R with C k +1 , boundary ∂ Ω . Then for v ∈ W k +1 ,q with v · n = 0 on ∂ Ω , there exists a constant C = C ( q, k, Ω) such that k v k W k +1 ,q ≤ C ( k div v k W k,q + k curl v k W k,q ) . (2.6) In particular, for k = 0 , we have k∇ v k L q ≤ C ( k div v k L q + k curl v k L q ) . (2.7) Lemma 2.6 ( [2])
Let k ≥ be an integer, < q < + ∞ . Suppose that Ω is a boundeddomain in R and its C k +1 , boundary ∂ Ω only has a finite number of 2-dimensionalconnected components. Then for v ∈ W k +1 ,q with v × n = 0 on ∂ Ω , there exists aconstant C = C ( q, k, Ω) such that k v k W k +1 ,q ≤ C ( k div v k W k,q + k curl v k W k,q + k v k L q ) . In particular, if Ω has no holes, then k v k W k +1 ,q ≤ C ( k div v k W k,q + k curl v k W k,q ) . The following Beale-Kato-Majda type inequality, which was first proved in [3, 16]when div u ≡ , and improved in [13], we give a similar result with respect to the slipboundary condition (1.6) to estimate k∇ u k L ∞ and k∇ ρ k L which have been provenin [4]. 8 emma 2.7 ( [4]) For < q < ∞ , assume that u · n = 0 and curl u × n = 0 on ∂ Ω , u ∈ W ,q , then there is a constant C = C ( q, Ω) such that the following estimate holds k∇ u k L ∞ ≤ C ( k div u k L ∞ + k curl u k L ∞ ) ln( e + k∇ u k L q ) + C k∇ u k L + C. Next, consider the problem ( div v = f, x ∈ Ω ,v = 0 , x ∈ ∂ Ω . (2.8)One has the following conclusion. Lemma 2.8 [10, Theorem III.3.1] There exists a linear operator B = [ B , B , B ] enjoying the properties:1) B : { f ∈ L p (Ω) | Z Ω f dx = 0 } 7→ ( W ,p (Ω)) is a bounded linear operator, that is, kB [ f ] k W ,p (Ω) ≤ C ( p ) k f k L p (Ω) , for any p ∈ (1 , ∞ ) , (2.9)
2) The function v = B [ f ] solve the problem (2.8) .3) if, moreover, f can be written in the form f = div g for a certain g ∈ L r (Ω) , g · n | ∂ Ω = 0 , then kB [ f ] k L r (Ω) ≤ C ( r ) k g k L r (Ω) , for any r ∈ (1 , ∞ ) . (2.10) F , curl u and ∇ u From now on, we always assume Ω is a simply connected bounded domain in R whosesmooth boundary ∂ Ω only has a finite number of 2-dimensional connected componentsand ψ ∈ H satisfies (1.9). For F , curl u and ∇ u , we give the following conclusion,which is often used later. Lemma 2.9
Let ( ρ, u ) be a smooth solution of (1.1) with slip boundary condition (1.6) .Then for any p ∈ [2 , , there exists a positive constant C depending only on p , q , µ , λ , Ω and k ψ k H such that k∇ u k L p ≤ C ( k div u k L p + k curl u k L p ) , (2.11) k∇ F k L p ≤ C ( k ρ ˙ u k L p + k ρ − ρ s k L (6 p ) / (6 − p ) ) , (2.12) k∇ curl u k L p ≤ C ( k ρ ˙ u k L p + k∇ u k L + k ρ − ρ s k L (6 p ) / (6 − p ) ) , (2.13) k F k L p ≤ C k ρ ˙ u k (3 p − / (2 p ) L ( k∇ u k L + k ρ − ρ s k L ) (6 − p ) / (2 p ) + C ( k∇ u k L + k ρ − ρ s k L ) , (2.14) k curl u k L p ≤ C k ρ ˙ u k (3 p − / (2 p ) L k∇ u k (6 − p ) / (2 p ) L + C ( k∇ u k L + k ρ − ρ s k L ) . (2.15) Moreover, k F k L p + k curl u k L p ≤ C ( k ρ ˙ u k L + k∇ u k L + k ρ − ρ s k L ) , (2.16) k∇ u k L p ≤ C k ρ ˙ u k (3 p − / (2 p ) L k∇ u k (6 − p ) / (2 p ) L + C ( k∇ u k L + k ρ − ρ s k L + k ρ − ρ s k L p ) . (2.17)9 emark 2.1 If p = 6 , then k ρ − ρ s k L (6 p ) / (6 − p ) , k ρ − ρ s k L ∞ . Proof . The inequality (2.11) is a direct result of Lemma 2.5, since u · n = 0 on ∂ Ω.By (1 . , one can find that the viscous flux F satisfies Z ∇ F · ∇ ηdx = Z ( ρ ˙ u − ( ρ − ρ s ) ∇ ψ ) · ∇ ηdx, ∀ η ∈ C ∞ ( R ) , i.e., ( ∆ F = div( ρ ˙ u − ( ρ − ρ s ) ∇ ψ ) , x ∈ Ω , ∂F∂n = ( ρ ˙ u − ( ρ − ρ s ) ∇ ψ ) · n, x ∈ ∂ Ω . It follows from Lemma 4.27 in [29] that for any p ∈ [2 , k∇ F k L p ≤ C k ( ρ ˙ u − ( ρ − ρ s ) ∇ ψ ) k L p ≤ C ( k ρ ˙ u k L p + k ρ − ρ s k L (6 p ) / (6 − p ) | ∇ ψ k L ) ≤ C ( k ρ ˙ u k L p + k ρ − ρ s k L (6 p ) / (6 − p ) ) (2.18)On the other hand, one can rewrite (1 . as µ ∇ × curl u = ∇ F − ρ ˙ u. Notice thatcurl u × n = 0 on ∂ Ω and div( ∇ × curl u ) = 0, by Lemma 2.6, we get k∇ curl u k L p ≤ C ( k∇ × curl u k L p + k curl u k L p ) ≤ C ( k ρ ˙ u k L p + k ρ − ρ s k L (6 p ) / (6 − p ) + k curl u k L p ) . (2.19)By Sobolev’s inequality and (2.19), k∇ curl u k L p ≤ C ( k ρ ˙ u k L p + k curl u k L p + k ρ − ρ s k L (6 p ) / (6 − p ) ) ≤ C ( k ρ ˙ u k L p + k∇ curl u k L + k curl u k L + k ρ − ρ s k L (6 p ) / (6 − p ) ) ≤ C ( k ρ ˙ u k L p + k ρ ˙ u k L + k curl u k L + k ρ − ρ s k L (6 p ) / (6 − p ) ) ≤ C ( k ρ ˙ u k L p + k∇ u k L + k ρ − ρ s k L (6 p ) / (6 − p ) ) , so that (2.13) holds.Furthermore, one can deduce from (2.1) and (2.12) that for p ∈ [2 , k F k L p ≤ C k F k (6 − p ) / (2 p ) L k∇ F k (3 p − / (2 p ) L + C k F k L ≤ C ( k ρ ˙ u k L + k ρ − ρ s k L ) (3 p − / (2 p ) ( k∇ u k L + k ρ − ρ s k L ) (6 − p ) / (2 p ) + C ( k∇ u k L + k ρ − ρ s k L ) ≤ C k ρ ˙ u k (3 p − / (2 p ) L ( k∇ u k L + k ρ − ρ s k L ) (6 − p ) / (2 p ) + C ( k∇ u k L + k ρ − ρ s k L ) , which also implies that k F k L p ≤ C ( k ρ ˙ u k L + k∇ u k L + k ρ − ρ s k L ) . Similarly, k curl u k L p ≤ C k curl u k (6 − p ) / (2 p ) L k∇ curl u k (3 p − / (2 p ) L + C k curl u k L ≤ C ( k ρ ˙ u k L + k ρ − ρ s k L + k∇ u k L ) (3 p − / (2 p ) k∇ u k (6 − p ) / (2 p ) L + C k∇ u k L ≤ C k ρ ˙ u k (3 p − / (2 p ) L k∇ u k (6 − p ) / (2 p ) L + C ( k∇ u k L + k ρ − ρ s k L ) , k curl u k L p ≤ C ( k ρ ˙ u k L + k∇ u k L + k ρ − ρ s k L ) . Hence, (2.14) and (2.16) are established.By virtue of (2.11) and (2.14), it indicates that k∇ u k L p ≤ C ( k div u k L p + k curl u k L p ) ≤ C ( k F k L p + k curl u k L p + k P − P ( ρ s ) k L p ) ≤ C ( k ρ ˙ u k (3 p − / (2 p ) L k∇ u k (6 − p ) / (2 p ) L + k∇ u k L + k ρ − ρ s k L + k ρ − ρ s k L p ) . This completes the proof.
Remark 2.2
Consider the following Lam´e’s system ( − µ ∆ u − ( λ + µ ) ∇ div u = − ρ ˙ u − ∇ ( P − P ( ρ s )) + ( ρ − ρ s ) ∇ ψ, x ∈ Ω ,u · n = 0 and curl u × n = 0 , x ∈ ∂ Ω , (2.20) by Lemma 2.4 and Gagliardo-Nirenberg’s inequality, k∇ u k L p ≤ C ( k ρ ˙ u k L p + k P − P ( ρ s ) k W ,p + k ( ρ − ρ s ) ∇ ψ k L p + k u k L p ) ≤ C ( k ρ ˙ u k L p + k∇ u k L + k∇ P k L p + k P − P ( ρ s ) k L + k P − P ( ρ s ) k L p )+ C k ( ρ − ρ s ) ∇ ψ k L p , (2.21) and k∇ u k L p ≤ C ( k ρ ˙ u k W ,p + k P − P ( ρ s ) k W ,p + C k ( ρ − ρ s ) ∇ ψ k W ,p + k u k L p ) ≤ C ( k ρ ˙ u k L p + k∇ u k L + k∇ ( ρ ˙ u ) k L p + k∇ P k L p ) + C k ( ρ − ρ s ) ∇ ψ k W ,p + C ( k∇ P k L p + k P − P ( ρ s ) k L + k P − P ( ρ s ) k L p ) . (2.22) Let
T > ρ, u ) be a smooth solution to (1.1)-(1.6) on Ω × (0 , T ]with smooth initial data ( ρ , u ) satisfying (1.15) and (1.16). We will derive somenecessary a priori bounds for smooth solutions to the problem (1.1)-(1.6) which canextend the local strong or classical solution guaranteed by Lemma 2.1 to be a globalone.Setting σ = σ ( t ) , min { , t } , we define A ( T ) , sup ≤ t ≤ T (cid:0) σ k∇ u k L (cid:1) + Z T σ k ρ / ˙ u k L dt, (3.1) A ( T ) , sup ≤ t ≤ T σ k ρ / ˙ u k L dx + Z T σ k∇ ˙ u k L dt, (3.2)and A ( T ) , sup ≤ t ≤ T k∇ u k L . (3.3)This section is entirely devoted to prove the following conclusion.11 roposition 3.1 Under the conditions of Theorem 1.2, for δ , s − s ∈ (0 , ] , thereexist positive constant ε and K depending on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω , and M such that if ( ρ, u ) is a smooth solution of (1.1) – (1.6) on Ω × (0 , T ] satisfying sup Ω × [0 ,T ] ρ ≤ ρ, A ( T ) + A ( T ) ≤ C / , A ( σ ( T )) ≤ K, (3.4) then the following estimates hold sup Ω × [0 ,T ] ρ ≤ ρ/ , A ( T ) + A ( T ) ≤ C / , A ( σ ( T )) ≤ K, (3.5) provided C ≤ ε. Proof . Proposition 3.1 is a consequence of the following Lemmas 3.4–3.8.One can extend the function n to Ω such that n ∈ C ( ¯Ω), and in the followingdiscussion we still denote the extended function by n .The first lemma in this section, which depends on u · n | ∂ Ω = 0, is proven in [4, Lemma3.2]. Lemma 3.1 ( [4]) If ( ρ, u ) is a smooth solution of (1.1) with slip boundary condition (1.6) , then there exists a positive constant C depending only on Ω such that k ˙ u k L ≤ C ( k∇ ˙ u k L + k∇ u k L ) , (3.6) k∇ ˙ u k L ≤ C ( k div ˙ u k L + k curl ˙ u k L + k∇ u k L ) . (3.7)In the following, we will use the convention that C denotes a generic positive constantdepending on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω, M , and use C ( α ) to emphasize that C depends on α .We begin with the following standard energy estimate for ( ρ, u ). Lemma 3.2
Let ( ρ, u ) be a smooth solution of (1.1) – (1.6) on Ω × (0 , T ] satisfying ρ ≤ ρ. Then there is a positive constant C depending only on µ , λ , γ , ˆ ρ , and Ω suchthat sup ≤ t ≤ T Z (cid:0) ρ | u | + ( ρ − ρ s ) (cid:1) dx + Z T k∇ u k L dt ≤ C (ˆ ρ ) C . (3.8) Proof . First, since − ∆ u = −∇ div u + ∇ × curl u, using (1.12), we rewrite (1 . as ρ ˙ u − ( λ + 2 µ ) ∇ div u + µ ∇ × curl u + ∇ P = aγγ − ρ ∇ ρ γ − s . (3.9)Multiplying (3.9) by u and integrating the resulting equality over Ω lead to12 (cid:18)Z ρ | u | dx (cid:19) t + ( λ + 2 µ ) Z (div u ) dx + µ Z | curl u | dx = Z P div udx − aγγ − Z ρ γ − s div( ρu ) dx. (3.10)12ultiplying (1 . by G ′ ( ρ ) with G ( ρ ) as in (1.14), integrating over Ω and applying slipboundary condition (1.6), we have (cid:18)Z G ( ρ ) dx (cid:19) t + Z P div udx − aγγ − Z ρ γ − s div( ρu ) dx = 0 . (3.11)Finally, it is easy to check that there exists a positive constant C = C ( ρ, ¯ ρ ) such that C − ( ρ − ρ s ) ≤ G ( ρ ) ≤ C ( ρ − ρ s ) , which together with (3.10), (3.11), and (2.11) gives (3.8).The following conclusion shows preliminary L bounds for ∇ u and ρ / ˙ u . Lemma 3.3
Let ( ρ, u ) be a smooth solution of (1.1) - (1.6) on Ω × (0 , T ] satisfying ρ ≤ ρ . Then there is a positive constant C depending only on µ , λ , a , γ , inf Ω ψ , k ψ k H , ˆ ρ and Ω such that A ( T ) ≤ CC + C Z T Z σ |∇ u | dxdt, (3.12) and A ( T ) ≤ CC + CA ( T ) + C Z T Z σ |∇ u | dxdt. (3.13) Proof . The proof is similar to that of [4, Lemma 3.4] expect for some modificationswhich are caused by the external force term. For convenience, we still write down theproof completely.Let m ≥ . can be rewritten as ρ ˙ u − ( λ + 2 µ ) ∇ div u + µ ∇ × curl u + ∇ ( P − P ( ρ s )) = ( ρ − ρ s ) ∇ ψ. (3.14)Multiplying it by σ m ˙ u and then integrating the resulting equality over Ω lead to Z σ m ρ | ˙ u | dx = − Z σ m ˙ u · ∇ ( P − P ( ρ s )) dx + ( λ + 2 µ ) Z σ m ∇ div u · ˙ udx − µ Z σ m ∇ × curl u · ˙ udx + Z σ m ( ρ − ρ s ) ∇ ψ · ˙ udx , I + I + I + I . (3.15)We will estimate I , I , I and I one by one. Firstly, by (1 . , one can check that P t + div( P u ) + ( γ − P div u = 0 , (3.16)or P t + ∇ P · u + γP div u = 0 . (3.17)13 direct calculation together with (3 .
16) gives I = − Z σ m ˙ u · ∇ ( P − P ( ρ s )) dx = Z σ m ( P − P ( ρ s )) div u t dx − Z σ m u · ∇ u · ∇ ( P − P ( ρ s )) dx = (cid:18)Z σ m ( P − P ( ρ s )) div u dx (cid:19) t − mσ m − σ ′ Z ( P − P ( ρ s )) div u dx + Z σ m P ∇ u : ∇ udx + ( γ − Z σ m P (div u ) dx + Z σ m u · ∇ u · ∇ P ( ρ s ) dx − Z ∂ Ω σ m P u · ∇ u · nds ≤ (cid:18)Z σ m ( P − P ( ρ s )) div u dx (cid:19) t + C k∇ u k L + Cmσ m − σ ′ C − Z ∂ Ω σ m P u · ∇ u · nds ≤ (cid:18)Z σ m ( P − P ( ρ s )) div u dx (cid:19) t + C k∇ u k L + Cmσ m − σ ′ C . (3.18)where in the last inequality, we have utilized the fact that − Z ∂ Ω σ m P u · ∇ u · nds = Z ∂ Ω σ m P u · ∇ n · uds ≤ C Z ∂ Ω σ m | u | ds ≤ Cσ m k∇ u k L , (3.19)due to (1.26). Similarly, it indicates that I = ( λ + 2 µ ) Z σ m ∇ div u · ˙ udx = ( λ + 2 µ ) Z ∂ Ω σ m div u ( ˙ u · n ) ds − ( λ + 2 µ ) Z σ m div u div ˙ udx = − ( λ + 2 µ ) Z ∂ Ω σ m div u ( u · ∇ n · u ) ds − λ + 2 µ (cid:18)Z σ m (div u ) dx (cid:19) t + λ + 2 µ Z σ m (div u ) dx − ( λ + 2 µ ) Z σ m div u ∇ u : ∇ udx + m ( λ + 2 µ )2 σ m − σ ′ Z (div u ) dx. (3.20)Notice that (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω div u ( u · ∇ n · u ) ds (cid:12)(cid:12)(cid:12)(cid:12) = 1 λ + 2 µ (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω ( F + P − P ( ρ s ))( u · ∇ n · u ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18)Z ∂ Ω | F || u | ds + Z ∂ Ω | u | ds (cid:19) ≤ C ( k F k H k u k H + k∇ u k L ) ≤ C ( k∇ F k L + k∇ u k L + 1) k∇ u k L ≤ k ρ ˙ u k L + C ( k∇ u k L + k∇ u k L ) . I ≤ − λ + 2 µ (cid:18)Z σ m (div u ) dx (cid:19) t + Cσ m k∇ u k L + 12 σ m k ρ ˙ u k L + Cσ m k∇ u k L + C k∇ u k L . (3.21)By (1.6), we have I = − µ Z σ m ∇ × curl u · ˙ udx = − µ Z σ m curl u · curl ˙ udx = − µ (cid:18)Z σ m | curl u | dx (cid:19) t + µm σ m − σ ′ Z | curl u | dx − µ Z σ m curl u · curl( u · ∇ u ) dx = − µ (cid:18)Z σ m | curl u | dx (cid:19) t + µm σ m − σ ′ Z | curl u | dx − µ Z σ m ( ∇ u i × ∇ i u ) · curl udx + µ Z σ m | curl u | div udx ≤ − µ (cid:18)Z σ m | curl u | dx (cid:19) t + C k∇ u k L + Cσ m k∇ u k L . (3.22)Finally, I = Z σ m ( ρ − ρ s ) ∇ ψ · ˙ udx = (cid:18)Z σ m ( ρ − ρ s ) ∇ ψ · u dx (cid:19) t − mσ m − σ ′ Z ( ρ − ρ s ) ∇ ψ · u dx Z σ m ρu · ∇ ( ∇ ψ · u ) dx + Z σ m ( ρ − ρ s ) ∇ ψ · ( u · ∇ u ) dx ≤ (cid:18)Z σ m ( ρ − ρ s ) ∇ ψ · u dx (cid:19) t + Cmσ m − σ ′ C + C k u k L k∇ ψ k L + C k u k L k∇ u k L k∇ ψ k L ≤ (cid:18)Z σ m ( ρ − ρ s ) ∇ ψ · u dx (cid:19) t + C k∇ u k L + Cmσ m − σ ′ C . (3.23)It follows from (3.15) and (3.18)-(3.23) that (cid:18) ( λ + 2 µ ) Z σ m (div u ) dx + µ Z σ m | curl u | dx (cid:19) t + Z σ m ρ | ˙ u | dx ≤ (cid:18) Z σ m ( P − P ( ρ s )) div udx + 2 Z σ m ( ρ − ρ s ) ∇ ψ · u dx (cid:19) t + Cmσ m − σ ′ C + Cσ m k∇ u k L + C k∇ u k L + Cσ m k∇ u k L , (3.24)which together with (2.11), Lemma 3.2 and Young’s inequality, yields that for any m ≥ σ m k∇ u k L + Z T Z σ m ρ | ˙ u | dxdt ≤ CC + C Z T σ m k∇ u k L dt + C Z T σ m k∇ u k L dt. (3.25)15hoosing m = 1 , and by virtue of the assumption (3.4) and (3.8), we obtain (3.12).It remains to prove (3.13). In the discussion, we will utilize the following facts morethan once, which are given (3.6) and (3.7), that is, k F k H + k curl u k H ≤ C ( k ρ ˙ u k L + k∇ u k L + k ρ − ρ s k L ) (3.26)and k∇ F k L + k∇ curl u k L ≤ C ( k ˙ u k H + k∇ u k L + k ρ − ρ s k L ∞ ) ≤ C ( k∇ ˙ u k L + k∇ u k L + k∇ u k L + k∇ u k L + 1) . (3.27)Rewrite (3.14) as ρ ˙ u = ∇ F − µ ∇ × curl u + ( ρ − ρ s ) ∇ ψ. (3.28)Operating σ m ˙ u j [ ∂/∂t + div( u · )] to (3 . j , summing with respect to j , and integratingover Ω , together with (1 . , we get (cid:18) σ m Z ρ | ˙ u | dx (cid:19) t − m σ m − σ ′ Z ρ | ˙ u | dx = Z σ m ( ˙ u · ∇ F t + ˙ u j div( u∂ j F )) dx + µ Z σ m ( − ˙ u · ∇ × curl u t − ˙ u j div(( ∇ × curl u ) j u )) dx + Z σ m ( ρ t ˙ u · ∇ ψ + ˙ u j div(( ρ − ρ s ) ∂ j ψu )) dx , J + µJ + J . (3.29)For J , by (1.6) and (3.17), a direct computation yields J = Z σ m ˙ u · ∇ F t dx + Z σ m ˙ u j div( u∂ j F ) dx = Z ∂ Ω σ m F t ˙ u · nds − Z σ m F t div ˙ udx − Z σ m u · ∇ ˙ u j ∂ j F dx = Z ∂ Ω σ m F t ˙ u · nds − (2 µ + λ ) Z σ m (div ˙ u ) dx + (2 µ + λ ) Z σ m div ˙ u ∇ u : ∇ udx + Z σ m div ˙ uu · ∇ F dx − γ Z σ m P div u div ˙ udx − Z σ m u · ∇ ˙ u j ∂ j F dx ≤ Z ∂ Ω σ m F t ˙ u · nds − (2 µ + λ ) Z σ m (div ˙ u ) dx + δ σ m k∇ ˙ u k L + σ m k∇ F k L k∇ F k L k∇ ˙ u k L k∇ u k L + C ( δ ) σ m ( k∇ u k L + k∇ u k L ) ≤ Z ∂ Ω σ m F t ˙ u · nds − (2 µ + λ ) Z σ m (div ˙ u ) dx + δσ m k∇ ˙ u k L + C ( δ ) σ m (cid:0) k∇ u k L k ρ ˙ u k L + k∇ u k L + k∇ u k L (cid:1) (3.30)where in the third equality we have used F t = (2 µ + λ )div u t − P t = (2 µ + λ )div ˙ u − (2 µ + λ )div( u · ∇ u ) + u · ∇ P + γP div u = (2 µ + λ )div ˙ u − (2 µ + λ ) ∇ u : ∇ u − u · ∇ F + γP div u. Z ∂ Ω σ m F t ˙ u · nds = − Z ∂ Ω σ m F t ( u · ∇ n · u ) ds = − (cid:18)Z ∂ Ω σ m ( u · ∇ n · u ) F ds (cid:19) t + mσ m − σ ′ Z ∂ Ω ( u · ∇ n · u ) F ds + σ m Z ∂ Ω F ˙ u · ∇ n · uds + σ m Z ∂ Ω F u · ∇ n · ˙ uds − σ m Z ∂ Ω F ( u · ∇ ) u · ∇ n · uds − σ m Z ∂ Ω F u · ∇ n · ( u · ∇ ) uds ≤ − (cid:18)Z ∂ Ω σ m ( u · ∇ n · u ) F ds (cid:19) t + Cmσ ′ σ m − k∇ u k L k F k H + δσ m k ˙ u k H + C ( δ ) σ m k∇ u k L k F k H − σ m Z ∂ Ω F ( u · ∇ ) u · ∇ n · uds − σ m Z ∂ Ω F u · ∇ n · ( u · ∇ ) uds, (3.31)where in the last inequality we have used (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω ( u · ∇ n · u ) F ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ u k L k F k H . (3.32)Since u · n | ∂ Ω = 0 , we have u = − ( u × n ) × n = u ⊥ × n on ∂ Ω , (3.33)with u ⊥ , − u × n. And then − Z ∂ Ω F ( u · ∇ ) u · ∇ n · uds = − Z ∂ Ω u ⊥ × n · ∇ u i ∇ i n · uF ds = − Z ∂ Ω n · ( ∇ u i × u ⊥ ) ∇ i n · uF ds = − Z Ω div(( ∇ u i × u ⊥ ) ∇ i n · uF ) dx = − Z Ω ∇ ( ∇ i n · uF ) · ( ∇ u i × u ⊥ ) dx + Z Ω ∇ u i · ∇ × u ⊥ ∇ i n · uF dx ≤ C Z Ω |∇ F ||∇ u || u | dx + C Z Ω | F | ( |∇ u | | u | + |∇ u || u | ) dx ≤ C k∇ F k L k∇ u k L + C k F k H k∇ u k L (cid:0) k∇ u k L + k∇ u k L (cid:1) ≤ δ k∇ ˙ u k L + C ( δ ) k∇ u k L + C k∇ u k L + C k∇ u k L + C k ρ ˙ u k L (cid:0) k∇ u k L + 1 (cid:1) , (3.34)where in the fourth equality we have useddiv( ∇ u i × u ⊥ ) = −∇ u i · ∇ × u ⊥ . − Z ∂ Ω F u · ∇ n · ( u · ∇ ) uds ≤ C k∇ F k L k∇ u k L + C k F k H k∇ u k L (cid:0) k∇ u k L + k∇ u k L (cid:1) ≤ δ k∇ ˙ u k L + C ( δ ) k∇ u k L + C k∇ u k L + C k∇ u k L + C k ρ ˙ u k L (cid:0) k∇ u k L + 1 (cid:1) . (3.35)It follows from (3.30), (3.6), (3.8), (3.31), (3.34), and (3.35) that J ≤ Cmσ m − σ ′ ( k ρ ˙ u k L + k∇ u k L + k∇ u k L ) − (cid:18)Z ∂ Ω σ m ( u · ∇ n · u ) F ds (cid:19) t + Cδσ m k∇ ˙ u k L + C ( δ ) σ m k ρ ˙ u k L ( k∇ u k L + 1) − ( λ + 2 µ ) Z σ m (div ˙ u ) dx + C ( δ ) σ m ( k∇ u k L + k∇ u k L + k∇ u k L ) . (3.36)Observing that curl u t = curl ˙ u − u · ∇ curl u − ∇ u i × ∇ i u, we get J = − Z σ m | curl ˙ u | dx + Z σ m curl ˙ u · ( ∇ u i × ∇ i u ) dx + Z σ m u · ∇ curl u · curl ˙ udx + Z σ m u · ∇ ˙ u · ( ∇ × curl u ) dx ≤ − Z σ m | curl ˙ u | dx + δσ m k∇ ˙ u k L + δσ m k∇ curl u k L + C ( δ ) σ m k∇ u k L + C ( δ ) σ m k∇ u k L k∇ curl u k L . (3.37)Finally, J = − Z σ m ( ρu · ∇ ( ˙ u · ∇ ψ ) + ( ρ − ρ s )( u · ∇ ˙ u ) · ∇ ψ ) dx ≤ σ m k u k L k∇ ˙ u k L k∇ ψ k L + σ m k u k L k ˙ u k L k∇ ψ k L ≤ δσ m k∇ ˙ u k L + C ( δ ) k∇ u k L . (3.38)Putting (3.36)-(3.38) into (3.29) gives (cid:18) σ m k ρ ˙ u k L (cid:19) t + ( λ + 2 µ ) σ m k div ˙ u k L + µσ m k curl ˙ u k L ≤ Cmσ m − σ ′ ( k ρ ˙ u k L + k∇ u k L + k∇ u k L ) + 2 δσ m k∇ ˙ u k L − (cid:18)Z ∂ Ω σ m ( u · ∇ n · u ) F ds (cid:19) t + Cσ m k ρ ˙ u k L ( k∇ u k L + 1)+ C ( δ ) σ m ( k∇ u k L + k∇ u k L + k∇ u k L ) , (3.39)which together with (3.7) leads to (cid:16) σ m k ρ ˙ u k L (cid:17) t + ( λ + 2 µ ) σ m k div ˙ u k L + µσ m k curl ˙ u k L ≤ Cmσ m − σ ′ ( k ρ ˙ u k L + k∇ u k L + k∇ u k L ) − (cid:18)Z ∂ Ω σ m ( u · ∇ n · u ) F ds (cid:19) t + Cσ m k ρ ˙ u k L ( k∇ u k L + 1)+ Cσ m ( k∇ u k L + k∇ u k L + k∇ u k L ) . (3.40)18ombining this, (3.32), and (3.4) gives (3.13) by taking m = 3 in (3.40). We finish theproof of Lemma 3.3. Lemma 3.4
Assume that ( ρ, u ) is a smooth solution of (1.1) - (1.6) satisfying ρ ≤ ρ and the initial data condition k u k H ≤ M in (1.16) , then there exist positive constants K and ε depending only on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω and M such that A ( σ ( T )) + Z σ ( T )0 k ρ / ˙ u k L dt ≤ K, (3.41) provide that C < ε . Proof . Choosing m = 0 in (3.24) and integrating over (0 , σ ( T )), we deduce from(2.11), (2.17) and (3.8) that A ( σ ( T )) + Z σ ( T )0 k ρ / ˙ u k L dxdt ≤ C ( C + M ) + 12 Z σ ( T )0 k ρ / ˙ u k L dxdt + 12 C C A ( σ ( T ))( A ( σ ( T )) + 1) , where we have used the fact that Z σ ( T )0 k∇ u k L dt ≤ C Z σ ( T )0 (cid:18) k∇ u k L k ρ / ˙ u k L + k∇ u k L + k∇ u k L + k ρ − ρ s k L (cid:19) dt ≤ CC + CC A ( σ ( T ))( A ( σ ( T )) + 1) + 12 Z σ ( T )0 k ρ / ˙ u k L dxdt. Hence, A ( σ ( T )) + Z σ ( T )0 k ρ / ˙ u k L dxdt ≤ C ( C + M ) + C C A ( σ ( T ))( A ( σ ( T )) + 1) . Now we can choose a positive constant K such that K ≥ C ( M + 1), as a result, if A ( σ ( T )) < K and C < ε = min { , / (8( K + 1) C ) } , then we establish (3.41). Lemma 3.5
Assume that ( ρ, u ) is a smooth solution of (1.1) - (1.6) satisfying (3.4) with K given by Lemma 3.4 and the initial data condition k u k H ≤ M in (1.16) . Thenthere exists a positive constant C depending only on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω and M such that sup ≤ t ≤ T k∇ u k L + Z T k ρ / ˙ u k L dt ≤ C, (3.42)sup ≤ t ≤ T σ k ρ / ˙ u k L + Z T σ k∇ ˙ u k L dt ≤ C, (3.43) provide that C < ε . roof . (3.42) is an immediate result of (3.41) and (3.4). It remains to prove (3.43).Choosing m = 1 in (3.40), by (3.4), (3.7), (3.8), (3.26), (3.32) and (3.42), σ k ρ / ˙ u k L + Z T σ k∇ ˙ u k L dtx ≤ C + 12 σ k ρ / ˙ u k L + Z T σ k∇ u k L dt ≤ C + C Z T σ k ρ / ˙ u k L k∇ u k L dt + 12 σ k ρ / ˙ u k L ≤ C + C sup ≤ t ≤ T ( σ k ρ / ˙ u k L ) + 12 σ k ρ / ˙ u k L , which gives (3.43). Lemma 3.6
Let ( ρ, u ) be a smooth solution of (1.1) - (1.6) on Ω × (0 , T ] satisfying (3.4) .Then there exists a positive constant ε depending only on µ , λ , γ , a , ρ , ρ and ˆ ρ and Ω such that Z T k ρ − ρ s k L dt ≤ CC . (3.44) Proof . Using (1.27), we can rewrite (1.1) as − ∇ (cid:0) ρ − s ( P − P ( ρ s )) (cid:1) = ρ − s ( ρ ˙ u − ( λ + µ ) ∇ div u − µ ∆ u ) − γ − a ∇ ρ − s G ( ρ, ρ s ) . (3.45)Multiplying (3.45) by B [ ρ − ρ s ] and integrating over Ω , by (2.8), one has Z ρ − s ( P − P ( ρ s ))( ρ − ρ s ) dx = (cid:18)Z ρ − s ρu · B [ ρ − ρ s ] dx (cid:19) t − Z ρu · B [ ρ t ] dx − Z ρ − s ρu i u j ∂ j B i ( ρ − ρ s ) dx − Z ρu i u j ( ∂ i ρ − s B j ( ρ − ρ s )) dx + µ Z ρ − s ∇ u : ∇B ( ρ − ρ s ) dx + µ Z ∂ i u j ∂ i ρ − s B j ( ρ − ρ s ) dx + ( λ + µ ) Z (cid:0) ρ − s ( ρ − ρ s ) + ∇ ρ − s · B ( ρ − ρ s ) (cid:1) div u dx − γ − a Z G ( ρ, ρ s ) ∇ ρ − s · B ( ρ − ρ s ) dx ≤ (cid:18)Z ρ − s ρu · B [ ρ − ρ s ] dx (cid:19) t + C k∇ u k L + C k u k L k∇ ρ − s k L kB [ ρ − ρ s ] k L + C k∇ u k L kB [ ρ − ρ s ] k L + C k∇ u k L k∇ ρ − s k L kB [ ρ − ρ s ] k L + C k∇ u k L k ρ − ρ s k L + C k G ( ρ, ρ s ) k L k G ( ρ, ρ s ) k L k∇ ρ − s k L kB [ ρ − ρ s ] k L ≤ (cid:18)Z ρu · B [ ρ − ρ s ] dx (cid:19) t + δ k ρ − ρ s k L + C ( δ ) k∇ u k L + C C k ρ − ρ s k L . Now choosing δ = and using (3.8), we obtain (3.44) provided C < ε = ( C ) .20 emma 3.7 Let ( ρ, u ) be a smooth solution of (1.1) - (1.6) on Ω × (0 , T ] satisfying (3.4) and the initial data condition k u k H ≤ M in (1.16) . Then there exists a positiveconstant ε depending only on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω and M such that A ( T ) + A ( T ) ≤ C , (3.46) provided C ≤ ε . Proof . By (2.17), (3.4), (3.44), and Lemmas 3.2 and 3.5, we get Z T σ k∇ u k L dt ≤ C Z T σ k ρ ˙ u k L k∇ u k L dt + C Z T σ ( k∇ u k L + k ρ − ρ s k L ) dt ≤ C (cid:18)Z T ( σ k ρ ˙ u k L )( σ k ρ ˙ u k L )( σ k∇ u k L ) dt (cid:19) + C (cid:18)Z T ( σ k∇ u k L ) k∇ u k L dt + Z T σ k ρ − ρ s k L dt (cid:19) ≤ C (cid:20) ( A ( T ) + C ) A ( T ) A ( T ) + C (cid:21) ≤ CC , (3.47)which along with (3.12) and (3.13) gives A ( T ) + A ( T ) ≤ CC + C Z T σ k∇ u k L dt. (3.48)For the last term on the righthand side of (3.48), on the one hand, we deduce from(2.17), (3.4), (3.44) and Lemmas 3.2 that Z σ ( T )0 σ k∇ u k L dt ≤ C Z σ ( T )0 σ k ρ ˙ u k L k∇ u k L dt + C Z σ ( T )0 σ ( k∇ u k L + k ρ − ρ s k L ) dt ≤ C Z σ ( T )0 k∇ u k L k∇ u k L ( σ k ρ ˙ u k L ) dt + CC ≤ C ( sup ≤ t ≤ σ ( T ) k∇ u k L ) Z σ ( T )0 k∇ u k L dt ! Z σ ( T )0 σ k ρ ˙ u k L dt ! + CC ≤ C ( A ( T )) C + CC ≤ CC , (3.49)provided C < b ε = min { ε , ε } .On the other hand, by (3.47) and (3.8), Z Tσ ( T ) σ k∇ u k L dt ≤ Z Tσ ( T ) σ k∇ u k L dt + Z Tσ ( T ) σ k∇ u k L dt ≤ CC . (3.50)21ence, by (3.48)-(3.50), A ( T ) + A ( T ) ≤ C C , which yields that (3.46) holds, provided that C < ε , min { b ε , ( C ) } .To give a uniform (in time) upper bound for the density, which is crucial to get allthe higher order estimates and thus to extend the classical solution globally. We willadopt an approach motivated by the work of [18], see also [14]. Lemma 3.8
There exists a positive constant ε depending on µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω and M such that, if ( ρ, u ) is a smooth solution of (1.1) - (1.6) on Ω × (0 , T ] satisfying (3.4) and the initial data condition k u k H s ≤ M in (1.16) , then sup ≤ t ≤ T k ρ ( t ) k L ∞ ≤ ρ , (3.51) provided C ≤ ε. Proof . Denote D t ρ , ρ t + u · ∇ ρ, g ( ρ ) , − ρ ( P − P ( ρ s ))2 µ + λ , b ( t ) , − µ + λ Z t ρF dt, then (1 . can be rewritten as D t ρ = g ( ρ ) + b ′ ( t ) . (3.52)By Lemma 2.3, it is sufficient to check that the function b ( t ) must verify (2.3) withsome suitable constants N , N .First, it follows from (2.2), (2.12), (2.16), (3.6), (3.4), (3.8) and Lemma 3.5 that for22 ≤ t ≤ t ≤ σ ( T ), | b ( t ) − b ( t ) |≤ C Z σ ( T )0 k ( ρF )( · , t ) k L ∞ dt ≤ C Z σ ( T )0 k F k L k∇ F k L dt + C Z σ ( T )0 k F k L dt ≤ C Z σ ( T )0 ( k ρ ˙ u k L + k∇ u k L + k ρ − ρ s k L )( k ρ ˙ u k L + k ρ − ρ s k L ) dt + C Z σ ( T )0 ( k∇ u k L + k ρ − ρ s k L ) dt ≤ C Z σ ( T )0 ( k ρ ˙ u k L + k∇ u k L + C )( k∇ ˙ u k L + k∇ u k L + C ) dt + CC ≤ C Z σ ( T )0 σ − ( σ k ρ ˙ u k L ) ( σ k ρ ˙ u k L ) ( σ k∇ ˙ u k L ) dt + C Z σ ( T )0 σ − ( k∇ u k L + C )( σ k∇ ˙ u k L ) dt + CC + C Z σ ( T )0 ( k ρ ˙ u k L ) ( k∇ u k L ) dt + C Z σ ( T )0 ( σ k ρ ˙ u k L ) C σ − dt ≤ C Z σ ( T )0 σ − ( σ k ρ ˙ u k L ) dt ! + C Z σ ( T )0 t − ( k∇ u k L ) dt ! + CC ≤ C Z σ ( T )0 t − dt ! Z σ ( T )0 t k ρ ˙ u k L dt ! + C Z σ ( T )0 t − dt ! Z σ ( T )0 k∇ u k L dt ! + CC ≤ CA ( σ ( T )) + CC ≤ C C , (3.53)provided C < ε .By (3.53) and (3.52), choosing N = 0, N = C C , ¯ ζ = ˆ ρ in Lemma 2.3 givessup t ∈ [0 ,σ ( T )] k ρ k L ∞ ≤ ˆ ρ + C C ≤ ρ , (3.54)provided C ≤ ˆ ε , min { ε , ( ˆ ρ C ) } . On the other hand, for σ ( T ) ≤ t ≤ t ≤ T, we deduce from (2.12), (2.14), (3.6),233.4), (3.8) and (3.44) that | b ( t ) − b ( t ) | ≤ C Z t t k F k L ∞ dt ≤ aλ + 2 µ ( t − t ) + C Z t t k F k L ∞ dt ≤ aλ + 2 µ ( t − t ) + C Z t t ( k F k L k∇ F k L + k F k L ) dt ≤ aλ + 2 µ ( t − t ) + CC Z Tσ ( T ) k∇ ˙ u k L dt + CC ≤ aλ + 2 µ ( t − t ) + C C . (3.55)Now we choose N = C C , N = aλ +2 µ in (2.3) and set ¯ ζ = ρ in (2.4). Notice thatfor all ζ ≥ ¯ ζ = ρ > ¯ ρ + 1, g ( ζ ) = − aζ µ + λ ( ζ γ − ρ γs ) ≤ − aλ + 2 µ = − N . Consequently, by (3.52), (3.55) and Lemma 2.3, we havesup t ∈ [ σ ( T ) ,T ] k ρ k L ∞ ≤ ρ C C ≤ ρ , (3.56)provided C ≤ ε , min { ˆ ε , ( ˆ ρ C ) } . (3.57)The combination of (3.54) with (3.56) completes the proof of Lemma 3.8. Let ( ρ, u ) be a smooth solution of (1.1)-(1.6). This section is devoted to derivingsome necessary higher order estimates, which play an important role in proving thatthe classical or strong solution exists globally in time. We will adopt some ideas of thearticle [14, 19] with slight modifications. In this section, we always assume that theinitial energy C satisfies (3.57). Lemma 4.1
For q ∈ (3 , as in Theorem 1.1, it holds that for r = (9 q − / (10 q − ≤ t ≤ T ( k∇ ρ k L q + σ k u k H ) + Z T ( k∇ u k rL ∞ + k∇ u k rL q + k∇ u k L ) dt ≤ C, (4.1) where and in what follows, C is a positive constant depending on T, k ρ − ρ s k W ,q , µ , λ , γ , a , inf Ω ψ , k ψ k H , ˆ ρ , Ω and M . roof . By (1.1) , it is clear that |∇ ρ | p , p ∈ [2 ,
6] satisfies( |∇ ρ | p ) t + div( |∇ ρ | p u ) + ( p − |∇ ρ | p div u + p |∇ ρ | p − ( ∇ ρ ) tr ∇ u ( ∇ ρ ) + pρ |∇ ρ | p − ∇ ρ · ∇ div u = 0 , where ( ∇ ρ ) tr is the transpose of ∇ ρ .Therefore, by (2.12), (3.6) and (3.42),( k∇ ρ k L p ) t ≤ C (1 + k∇ u k L ∞ ) k∇ ρ k L p + C k∇ F k L p ≤ C (1 + k∇ u k L ∞ ) k∇ ρ k L p + C ( k ρ ˙ u k L p + 1) . (4.2)By Lemma 2.7, (2.21) and (3.6), it indicates that k∇ u k L ∞ ≤ C ( k div u k L ∞ + k curl u k L ∞ ) ln( e + k∇ u k L p ) + C k∇ u k L + C ≤ C (1 + k ρ ˙ u k L p ) ln( e + k ρ ˙ u k L p + k∇ ρ k L p ) . (4.3)where in the second inequality, we have taken advantage of the fact k div u k L ∞ + k curl u k L ∞ ≤ C ( k F k L ∞ + k P − P ( ρ s ) k L ∞ ) + k curl u k L ∞ ≤ C ( k F k L + k∇ F k L p + k curl u k L + k∇ curl u k L p + k P − P ( ρ s ) k L ∞ ) ≤ C ( k ρ ˙ u k L p + 1) , (4.4)which is due to Gagliardo-Nirenberg’s inequality, (3.6), (2.12), (2.13) and (3.42).Consequently,( e + k∇ ρ k L p ) t ≤ C (1 + k ρ ˙ u k L p ) ln( e + k ρ ˙ u k L p ) ln( e + k∇ ρ k L p )) ( e + k∇ ρ k L p ) , (4.5)which implies that(ln( e + k∇ ρ k L p )) t ≤ C (1 + k ρ ˙ u k L p ) ln( e + k ρ ˙ u k L p ) ln( e + k∇ ρ k L p ) . (4.6)Notice that, by Lemma 3.5, Z T k ρ ˙ u k rL q dt ≤ Z T k ρ / ˙ u k (6 − q ) r/ qL ( k∇ ˙ u k L + k∇ u k L ) (3 q − r/ q dt ≤ Z T σ − / ( σ k ρ / ˙ u k L ) (6 − q ) r q ( σ k∇ ˙ u k L + k∇ u k L ) (3 q − r q dt ≤ ( Z T σ − qr q − qr +6 r dt ) q − qr +6 r q ( Z T ( σ k∇ ˙ u k L + k∇ u k L ) dt ) q qr − r ≤ C, (4.7)which together with (4.7) and Gronwall’s inequality showssup ≤ t ≤ T k∇ ρ k L q ≤ C. Combining this with (4.3), (4.7), (4.13), (2.21) and Lemma 3.5 proves (4.1).25 emma 4.2
There exists a positive constant C such that sup ≤ t ≤ T ( k ρ t k L + σ k ρ u t k L ) + Z T ( k ρ u t k L + σ k∇ u t k L ) dt ≤ C. (4.8) Proof . First, by (1.1) and (4.1), k ρ t k L ≤ k u k L k∇ ρ k L + k∇ u k L ≤ C. Next, a direct calculation gives k ρ u t k L ≤ C ( k ρ ˙ u k L + k∇ u k H )which together with (4.1) leads tosup ≤ t ≤ T σ k ρ u t k L + Z T k ρ u t k L dt ≤ C. Similarly, Z T σ k∇ u t k L dt ≤ C Z T σ ( k∇ ˙ u k L + k∇ ( u · ∇ u ) k L ) dt ≤ C Z T σ ( k∇ ˙ u k L + k∇ u k H ) dt. In this section, we always assume that the initial energy C satisfies (3.57), ψ ∈ H ,and that the positive constant C may depend on T, k g k L , k∇ u k H , k ρ − ρ s k W ,q , k P ( ρ ) − P ( ρ s ) k W ,q , k ψ k W ,q , besides µ , λ , γ , a , inf Ω ψ , ˆ ρ , Ω and M , where q ∈ (3 ,
6) and g ∈ L is given as in (1.20). Lemma 4.3
There exists a positive constant C, such that sup ≤ t ≤ T k ρ ˙ u k L + Z T k∇ ˙ u k L dt ≤ C, (4.9)sup ≤ t ≤ T ( k∇ ρ k L + k u k H ) + Z T ( k∇ u k L ∞ + k∇ u k L ) dt ≤ C. (4.10) Proof . First, choosing m = 0 in (3.40), by (2.17), we have (cid:16) k ρ ˙ u k L (cid:17) t + k∇ ˙ u k L ≤ − (cid:18)Z ∂ Ω ( u · ∇ n · u ) F ds (cid:19) t + C k ρ ˙ u k L ( k∇ u k L + 1)+ C ( k∇ u k L + k∇ u k L + k∇ u k L ) ≤ − (cid:18)Z ∂ Ω ( u · ∇ n · u ) F ds (cid:19) t + C k ρ ˙ u k L ( k ρ ˙ u k L + k∇ u k L + 1)+ C ( k∇ u k L + k∇ u k L + k P − P ( ρ s ) k L + k P − P ( ρ s ) k L + k ρ − ρ s k L ) . (4.11)26y Gronwall’s inequality and the compatibility condition (1.20), we deduce (4.9) from(4.11), (3.42) and (3.32). Furthermore, by (3.6), Z T k ρ ˙ u k L dt ≤ Z T k∇ ˙ u k L + k∇ u k L dt ≤ C. (4.12)As a result, setting p = 6 in (4.6), and by Gronwall’s inequality again, we havesup ≤ t ≤ T k∇ ρ k L ≤ C. (4.13)Finally, by (4.3) and (2.21), together with (3.6), (3.42) and (4.9), we have Z T k∇ u k L ∞ dt ≤ C, Z T k∇ u k L dt ≤ C and sup ≤ t ≤ T k u k H ≤ C. This completes the proof of Lemma 4.3.
Lemma 4.4
There exists a positive constant C such that sup ≤ t ≤ T k ρ u t k L + Z T Z |∇ u t | dxdt ≤ C, (4.14)sup ≤ t ≤ T ( k ρ − ρ s k H + k P − P ( ρ s ) k H ) ≤ C. (4.15) Proof . By Lemma 4.3, a straightforward calculation shows that k ρ u t k L ≤ k ρ ˙ u k L + k ρ u · ∇ u k L ≤ C + C k u k L k∇ u k L ≤ C + C k∇ u k L k u k H ≤ C, (4.16)and that Z T k∇ u t k L dt ≤ Z T k∇ ˙ u k L dt + Z T k∇ ( u · ∇ u ) k L dt ≤ C + Z T k∇ u k L + k u k L ∞ k∇ u k L dt ≤ C + C Z T ( k∇ u k L + k∇ u k H k∇ u k L ) dt ≤ C, (4.17)and then (4.14) holds.Using (3.17), (1 . , (2.21), (2.22) and Lemma 4.3, we have ddt (cid:0) k∇ P k L + k∇ ρ k L (cid:1) ≤ C (1 + k∇ u k L ∞ )( k∇ P k L + k∇ ρ k L ) + C k∇ ˙ u k L + C. (4.18)Combining this with Gronwall’s inequality and Lemma 4.3 implies thatsup ≤ t ≤ T (cid:0) k∇ P k L + k∇ ρ k L (cid:1) ≤ C. Thus we finish the proof of Lemma 4.4. 27 emma 4.5
There exists a positive constant C, such that sup ≤ t ≤ T ( k ρ t k H + k P t k H ) + Z T (cid:0) k ρ tt k L + k P tt k L (cid:1) dt ≤ C, (4.19)sup ≤ t ≤ T σ k∇ u t k L + Z T σ k ρ u tt k L dt ≤ C. (4.20) Proof . By (3.17) and Lemma 4.3, k P t k L ≤ C k u k L ∞ k∇ P k L + C k∇ u k L ≤ C. (4.21)Differentiating (3.17) yields ∇ P t + u · ∇∇ P + ∇ u · ∇ P + γ ∇ P div u + γP ∇ div u = 0 . Hence, by Lemmas 4.3 and 4.4, k∇ P t k L ≤ C k u k L ∞ k∇ P k L + C k∇ u k L k∇ P k L + C k∇ u k L ≤ C, (4.22)which together with (4.21) yieldssup ≤ t ≤ T k P t k H ≤ C. (4.23)By (3.17) again, we find that P tt satisfies P tt + γP t div u + γP div u t + u t · ∇ P + u · ∇ P t = 0 . (4.24)Multiplying (4.24) by P tt and integrating over Ω × [0 , T ] , by (4.23), Lemmas 4.3 and4.4, we obtain that Z T k P tt k L dt = − Z T Z γP tt P t div udxdt − Z T Z γP tt P div u t dxdt − Z T Z P tt u t · ∇ P dxdt − Z T Z P tt u · ∇ P t dxdt ≤ C Z T k P tt k L ( k P t k L k∇ u k L + k∇ u t k L + k u t k L k∇ P k L + k u k L ∞ k∇ P t k L ) dt ≤ C Z T k P tt k L (1 + k∇ u t k L ) dt ≤ Z T k P tt k L dt + C, which gives Z T k P tt k L dt ≤ C. We can deal with ρ t and ρ tt similarly and get (4.19).Finally, we will prove (4.20). Introducing the function H ( t ) = ( λ + 2 µ ) Z (div u t ) dx + µ Z | curl u t | dx, u t · n = 0 on ∂ Ω and Lemma 2.5, one has k∇ u t k L ≤ CH ( t ) . (4.25)Differentiating (1 . with respect to t and multiplying by u tt , we have ddt H ( t ) + 2 Z ρ | u tt | dx = ddt (cid:18) − Z ρ t | u t | dx − Z ρ t u · ∇ u · u t dx + 2 Z P t div u t dx + Z ρ t ∇ ψ · u t dx (cid:19) + Z ρ tt | u t | dx + 2 Z ( ρ t u · ∇ u ) t · u t dx − Z ρu t · ∇ u · u tt dx − Z ρu · ∇ u t · u tt dx − Z P tt div u t dx − Z ρ tt ∇ ψ · u t dx , ddt I + X i =1 I i . (4.26)We have to estimate I i ( i = 0 , , · · · ,
6) one by one. It follows from (1 . , (3.6), (4.9),(4.10), (3.42), (4.14), (4.19), (4.25) and Sobolev’s and Poincar´e’s inequalities that | I | = (cid:12)(cid:12)(cid:12)(cid:12) − Z ρ t | u t | dx − Z ρ t u · ∇ u · u t dx + 2 Z P t div u t + Z ρ t ∇ ψ · u t dxdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z div( ρu ) | u t | dx (cid:12)(cid:12)(cid:12)(cid:12) + C k ρ t k L k u k L ∞ k∇ u k L k u t k L + C k P t k L k∇ u t k L + C k ρ t k L |∇ ψ k L k u t k L ≤ C Z | u || ρu t ||∇ u t | dx + C k∇ u t k L ≤ C k u k L k ρ / u t k / L k u t k / L k∇ u t k L + C k∇ u t k L ≤ C k∇ u k L k ρ / u t k / L k∇ u t k / L + C k∇ u t k L ≤ H ( t ) + C, (4.27) | I | = (cid:12)(cid:12)(cid:12)(cid:12)Z ρ tt | u t | dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z div( ρu ) t | u t | dx (cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12)Z ( ρ t u + ρu t ) · ∇ u t · u t dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) k ρ t k H k u k H + k ρ / u t k L k∇ u t k L (cid:19) k∇ u t k L ≤ C k∇ u t k L + C k∇ u t k L + C ≤ C k∇ u t k L H ( t ) + C k∇ u t k L + C, (4.28) | I | = 2 (cid:12)(cid:12)(cid:12)(cid:12)Z ( ρ t u · ∇ u ) t · u t dx (cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12)Z ( ρ tt u · ∇ u · u t + ρ t u t · ∇ u · u t + ρ t u · ∇ u t · u t ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ρ tt k L k u · ∇ u k L k u t k L + k ρ t k L k u t k L k∇ u k L + k ρ t k L k u k L ∞ k∇ u t k L k u t k L ≤ C k ρ tt k L + C k∇ u t k L , (4.29)29 I | + | I | = 2 (cid:12)(cid:12)(cid:12)(cid:12)Z ρu t · ∇ u · u tt dx (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z ρu · ∇ u t · u tt dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ρ / u tt k L ( k u t k L k∇ u k L + k u k L ∞ k∇ u t k L ) ≤ k ρ / u tt k L + C k∇ u t k L , (4.30)and | I | + | I | = 2 (cid:12)(cid:12)(cid:12)(cid:12)Z P tt div u t dx (cid:12)(cid:12)(cid:12)(cid:12) + | Z ρ tt ∇ ψ · u t dx |≤ C k P tt k L k div u t k L + + k ρ t k L k∇ u t k L k∇ ψ k L ≤ C k P tt k L + C k ρ tt k L + C k∇ u t k L . (4.31)Consequently, along with (4.28)-(4.31), and by (4.26), we have ddt ( σH ( t ) − σI ) + σ Z ρ | u tt | dx ≤ C (1 + k∇ u t k L ) σH ( t ) + C (1 + k∇ u t k L + k ρ tt k L + k P tt k L ) . By Gronwall’s inequality, (4.14), (4.19) and (4.27),sup ≤ t ≤ T ( σH ( t )) + Z T σ k ρ u tt k L dt ≤ C. which together with (4.25), gives (4.20). Lemma 4.6
There exists a positive constant C so that for any q ∈ (3 , , sup t ∈ [0 ,T ] σ k∇ u k H + Z T (cid:0) k∇ u k H + k∇ u k p W ,q + σ k∇ u t k H (cid:1) dt ≤ C, (4.32)sup t ∈ [0 ,T ] ( k ρ − ρ s k W ,q + k P − P ( ρ s ) k W ,q ) ≤ C, (4.33) where p = q − q − ∈ (1 , ) . Proof . It follows from Lemma 4.3, Poincar´e’s and Sobolev’s inequalities that k∇ ( ρ ˙ u ) k L ≤ k|∇ ρ | | u t |k L + k ρ ∇ u t k L + k|∇ ρ | | u | |∇ u |k L + k ρ |∇ u | k L + k ρ | u | |∇ u |k L ≤ C + C k∇ u t k L , (4.34)which together with (4.15) and Lemma 4.3 yields k∇ u k H ≤ C ( k ρ ˙ u k H + k P − P ( ρ s ) k H + k u k L ) ≤ C + C k∇ u t k L . (4.35)And by (4.35), (4.10), (4.14) and (4.20),sup ≤ t ≤ T σ k∇ u k H + Z T k∇ u k H dt ≤ C. (4.36)30e deduce from Lemma 4.3, (4.15) and (4.19) that k∇ u t k L ≤ C ( k ( ρ ˙ u ) t k L + k P t k H + k u t k L + k ρ t ∇ ψ k L ) ≤ C ( k ρu tt + ρ t u t + ρ t u · ∇ u + ρu t · ∇ u + ρu · ∇ u t k L )+ C ( k∇ P t k L + k u t k L ) + C ≤ C ( k ρu tt k L + k ρ t k L k u t k L + k ρ t k L k u k L ∞ k∇ u k L ) + C + C ( k u t k L k∇ u k L + k u k L ∞ k∇ u t k L + k∇ P t k L + k u t k L ) ≤ C k ρ u tt k L + C k∇ u t k L + C, (4.37)where in the first inequality, we have applied Lemma 2.4 to the system ( µ ∆ u t + ( λ + µ ) ∇ div u t = ( ρ ˙ u ) t + ∇ P t − ρ t ∇ ψ in Ω ,u t · n = 0 and curl u t × n = 0 on ∂ Ω . (4.38)By (4.37) and (4.20), we get Z T σ k∇ u t k H dt ≤ C. (4.39)By Sobolev’s inequality, (3.6), (4.10), (4.15) and (4.20), we check that for any q ∈ (3 , k∇ ( ρ ˙ u ) k L q ≤ C k∇ ρ k L q ( k∇ ˙ u k L q + k∇ ˙ u k L + k∇ u k L ) + C k∇ ˙ u k L q ≤ C ( k∇ ˙ u k L + k∇ u k L ) + C ( k∇ u t k L q + k∇ ( u · ∇ u ) k L q ) ≤ C ( k∇ u t k L + 1) + C k∇ u t k − q q L k∇ u t k q − q L + C ( k u k L ∞ k∇ u k L q + k∇ u k L ∞ k∇ u k L q ) ≤ Cσ − + C k∇ u k H + Cσ − ( σ k∇ u t k H ) q − q + C, (4.40)which along with (4.9) and (4.39), leads to Z T k∇ ( ρ ˙ u ) k p L q dt ≤ C. (4.41)On the other hand, notice that, by (2.21), (2.22), (4.9) and (4.15), k∇ u k W ,q ≤ C ( k ρ ˙ u k L q + k∇ ( ρ ˙ u ) k L q + k∇ P k L q + k∇ P k L q + k∇ u k L + k P − P ( ρ s ) k L + k P − P ( ρ s ) k L q + C k ( ρ − ρ s ) ∇ ψ k W ,q ) ≤ C (1 + k∇ u t k L + k∇ ( ρ ˙ u ) k L q + k∇ P k L q ) , (4.42)which together with (3.17) and (4.15) yields( k∇ P k L q ) t ≤ C k∇ u k L ∞ k∇ P k L q + C k∇ u k W ,q ≤ C (1 + k∇ u k L ∞ ) k∇ P k L q + C (1 + k∇ u t k L )+ C k∇ ( ρ ˙ u ) k L q . (4.43)Now by Gronwall’s inequality, (4.10), (4.14) and (4.41), we derive thatsup t ∈ [0 ,T ] k∇ P k L q ≤ C, (4.44)31hich along with (4.14), (4.15), (4.42) and (4.41) also givessup t ∈ [0 ,T ] k P − P ( ρ s ) k W ,q + Z T k∇ u k p W ,q dt ≤ C. (4.45)Similarly, sup ≤ t ≤ T k ρ − ρ s k W ,q ≤ C, As a result, we obtain (4.33) and finish the proof of Lemma 4.6.
Lemma 4.7
There exists a positive constant C such that sup ≤ t ≤ T σ ( k∇ u t k H + k∇ u k W ,q ) + Z T σ k∇ u tt k dt ≤ C, (4.46) for q ∈ (3 , . Proof . Differentiating (1 . with respect to t twice leads to ρu ttt + ρu · ∇ u tt − ( λ + 2 µ ) ∇ div u tt + µ ∇ × curl u tt = 2div( ρu ) u tt + div( ρu ) t u t − ρu ) t · ∇ u t − ( ρ tt u + 2 ρ t u t ) · ∇ u − ρu tt · ∇ u − ∇ P tt + ρ tt ∇ ψ. (4.47)Then, multiplying by 2 u tt and integrating over Ω, we have ddt Z ρ | u tt | dx + 2( λ + 2 µ ) Z (div u tt ) dx + 2 µ Z | curl u tt | dx = − Z ρu itt u · ∇ u itt dx − Z ( ρu ) t · [ ∇ ( u t · u tt ) + 2 ∇ u t · u tt ] dx − Z ( ρ tt u + 2 ρ t u t ) · ∇ u · u tt dx − Z ρu tt · ∇ u · u tt dx + 2 Z P tt div u tt dx + 2 Z ρ tt ∇ ψ · u tt dx , X i =1 J i . (4.48)Due to (4.10), (4.9), (4.14), (4.19) and (4.20), we have | J | ≤ C k ρ / u tt k L k∇ u tt k L k u k L ∞ ≤ δ k∇ u tt k L + C ( δ ) k ρ / u tt k L . (4.49) | J | ≤ C ( k ρu t k L + k ρ t u k L ) ( k u tt k L k∇ u t k L + k∇ u tt k L k u t k L ) ≤ C (cid:16) k ρ / u t k / L k u t k / L + k ρ t k L k u k L (cid:17) k∇ u tt k L k∇ u t k L ≤ δ k∇ u tt k L + C ( δ ) σ − / , (4.50) | J | + | J | ≤ C ( k ρ tt k L k u k L ∞ k∇ u k L + k ρ t k L k u t k L k∇ u k L + k ρ tt k L ) k u tt k L ≤ δ k∇ u tt k L + C ( δ ) k ρ tt k L + C ( δ ) σ − , (4.51)32nd | J | + | J | ≤ C k ρu tt k L k∇ u k L k u tt k L + C k P tt k L k∇ u tt k L ≤ δ k∇ u tt k L + C ( δ ) k ρ / u tt k L + C ( δ ) k P tt k L . (4.52)Now choosing δ small enough, it follows from (4.48) that ddt k ρ / u tt k L + k∇ u tt k L ≤ C ( k ρ / u tt k L + k ρ tt k L + k P tt k L ) + Cσ − / , (4.53)where we have utlized the fact that k∇ u tt k L ≤ C ( k div u tt k L + k curl u tt k L ) , (4.54)due to u tt · n = 0 on ∂ Ω . Together with (4.19), (4.20), and by Gronwall’s inequality, we getsup ≤ t ≤ T σ k ρ / u tt k L + Z T σ k∇ u tt k L dt ≤ C. (4.55)Furthermore, by (4.37) and (4.20),sup ≤ t ≤ T σ k∇ u t k H ≤ C. (4.56)Finally, by(4.42), (4.40), (4.20), (4.33), (4.32), (4.55) and (4.56), we have σ k∇ u k W ,q ≤ C (cid:0) σ + σ k∇ u t k L + σ k∇ ( ρ ˙ u ) k L q + σ k∇ P k L q (cid:1) ≤ C (cid:18) σ + σ + σ k∇ u k H + σ ( σ k∇ u t k H ) q − q (cid:19) ≤ Cσ + Cσ ( σ − ) q − q ≤ C, which, together with (4.55) and (4.56) yields (4.46). With all the a priori estimates in Section 3 and Section 4 at hand, we are going toprove Theorems 1.1-1.2 in this section.
Proof of Theorem 1.1.
By Lemma 2.1, there exists a T ∗ > ρ, u ) on Ω × (0 , T ∗ ]. In order to extend the localstrong solution globally in time, first, by the definition of A ( T ), A ( T ) and A ( T ) (see(3.1), (3.2), (3.3)), the assumption of the initial data (1.16) , one immediately checksthat A (0) + A (0) = 0 , ≤ ρ ≤ ¯ ρ, A (0) ≤ M. Therefore, there exists a T ∈ (0 , T ∗ ] such that0 ≤ ρ ≤ ρ, A ( T ) + A ( T ) ≤ C , A ( σ ( T )) ≤ K (5.1)hold for T = T . T ∗ = sup { T | (5 .
1) holds } . (5.2)Then T ∗ ≥ T >
0. Therefore, for any 0 < τ < T ≤ T ∗ with T finite, by Lemmas4.1-4.2, we have ( ρ, P ) ∈ L ∞ ([0 , T ]; W ,q ) , ρ t ∈ L ∞ (0 , T ; L ) ,u ∈ L ∞ (0 , T ; H ) ∩ L (0 , T ; H ) ,ρ / u t ∈ L (0 , T ; L ) , ( ρ / u t , ∇ u ) ∈ L ∞ ( τ, T ; L ) u t ∈ L ( τ, T ; H ) , which immediately implies for any r ≥ ρ, P ) ∈ C ([0 , T ]; L r ) , ρu ∈ C ([0 , T ]; L ) , where we have used the standard embedding L ∞ (0 , T ; H ) ∩ H (0 , T ; H − ) ֒ → C ([0 , T ]; L q ) , for any q ∈ [2 , . By (1.1) , for any T > t > s ≥ k∇ ρ ( t ) k L q ≤ (cid:18) k∇ ρ ( s ) k L q + C Z ts k∇ u ( ς ) k L q dς (cid:19) exp( C Z ts k∇ u ( ς ) k L ∞ dς ) , which leads to lim sup t → s + k∇ ρ ( t ) k L q ≤ k∇ ρ ( s ) k L q . (5.3)On the other hand, by (4.1), ∇ ρ ∈ C ([0 , T ]; L q − weak), together with (5.3), we concludethat for any q ∈ [2 , ∇ ρ ∈ C ([0 , T ]; L q ) . Finally, we claim that T ∗ = ∞ . Otherwise, T ∗ < ∞ . Then by Proposition 3.1, it holds that0 ≤ ρ ≤
74 ˆ ρ, A ( T ∗ ) + A ( T ∗ ) ≤ C , A ( σ ( T ∗ )) ≤ K. (5.4)and ( ρ ( x, T ∗ ) , u ( x, T ∗ )) satisfy the initial data condition (1.15)-(1.17). Thus, Lemma2.1 implies that there exists some T ∗∗ > T ∗ such that (5.1) holds for T = T ∗∗ , whichcontradicts the definition of T ∗ . As a result, 0 < T < T ∗ = ∞ .By (2.1) and (1.18), it indicates that ( ρ, u ) is really the unique strong solution definedon Ω × (0 , T ] for any 0 < T < T ∗ = ∞ . Proof of Theorem 1.2.
By Theorem 1.1, we only need to prove that the uniquestrong solution is a classical one under the assumption of Theorem 1.2. for any 0 <τ < T ≤ T ∗ with T finite, it follows from Lemmas 4.5-4.7 that ( ρ − ρ s ∈ C ([0 , T ]; W ,q ) , ∇ u t ∈ C ([ τ, T ]; L q ) , ∇ u, ∇ u ∈ C (cid:0) [ τ, T ]; C ( ¯Ω) (cid:1) , (5.5)34here one has taken advantage of the standard embedding L ∞ ( τ, T ; H ) ∩ H ( τ, T ; H − ) ֒ → C ([ τ, T ]; L q ) , for any q ∈ [2 , . By Lemmas 2.1 and 4.5-4.7, ( ρ, u ) is in fact the unique classical solution defined onΩ × (0 , T ] for any 0 < T < ∞ . It remains to prove (1.22). By (3.10) and (3.11), we get (cid:18)Z ρ | u | + G ( ρ, ρ s ) dx (cid:19) t + φ ( t ) = 0 , (5.6)where φ ( t ) , ( λ + 2 µ ) k div u k L + µ k curl u k L . Notice that there exists a positive constant ˜ C < γ , ρ and ¯ ρ suchthat for any ρ ≥ C ( ρ − ρ s ) ≤ ˜ C G ( ρ, ρ s ) ≤ ( ρ γ − ρ γs )( ρ − ρ s ) , and by (3.46) and (2.11), a ˜ C Z G ( ρ, ρ s ) dx ≤ a Z ( ρ γ − ρ γs )( ρ − ρ s ) dx ≤ (cid:18)Z ρu · B [ ρ − ρ s ] dx (cid:19) t + C φ ( t ) . (5.7)Introducing the function W ( t ) = Z (cid:18) ρ | u | + G ( ρ, ρ s ) (cid:19) dx − δ Z ρu · B [ ρ − ρ s ] dx, where δ = min { C , C } , and noticing that (cid:12)(cid:12)(cid:12)(cid:12)Z ρu · B [ ρ − ρ s ] dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) k√ ρu k L + Z G ( ρ, ρ s ) dx (cid:19) , we have12 k√ ρu k L + Z G ( ρ, ρ s ) dx ≤ W ( t ) ≤ (cid:18) k√ ρu k L + Z G ( ρ, ρ s ) dx (cid:19) . (5.8)Noticing that Z ρ | u | dx ≤ C k∇ u k L ≤ C φ ( t ) , setting δ = min { aδ ˜ C , C } and adding (5.7) multiplied by δ to (5.6) yields W ′ ( t ) + δ W ( t ) ≤ , which together with (5.8) leads to Z (cid:18) ρ | u | + G ( ρ, ρ s ) (cid:19) dx ≤ C e − δ t (5.9)35or any t >
0. Moveover, by (5.6), for any 0 < δ < δ , Z ∞ φ ( t ) e δ t dt ≤ C. (5.10)Choose m = 0 in (3.24), along with (2.11), (2.17) and (3.42), a direct calculationshows that (cid:18) φ ( t ) − Z ( P − P ( ρ s )) div udx (cid:19) t + 12 k√ ρ ˙ u k L ≤ C ( k ρ − ρ s k L + φ ( t ) , (5.11)Multiplying (5.11) by e δ t , and using the fact (cid:12)(cid:12)(cid:12)(cid:12)Z ( P − P ( ρ s )) div udx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ρ − ρ s k L + 12 φ ( t ) , we get (cid:18) e δ t φ ( t ) − e δ t Z ( P − P ( ρ s )) div udx (cid:19) t + 12 e δ t k√ ρ ˙ u k L ≤ Ce δ t ( k ρ − ρ s k L + φ ( t )) , which, together with (5.9) and (5.10), yields that for any t > k∇ u k L ≤ Ce − δ t , and Z ∞ e δ t k√ ρ ˙ u k L dt ≤ C. (5.12)A similar analysis based on (3.40) and (5.12) shows k√ ρ ˙ u k L ≤ Ce − δ t . As a result, (1.22) is established with some ˜ C depending only on on µ, λ, γ, a , inf Ω ψ , k ψ k H , ˆ ρ , M , Ω, p , q and C and we complete the proof. References [1] Agmon, S., Douglis, A., Nirenberg, L. Estimates near the boundary for solutionsof elliptic partial differential equations satisfying general boundary conditions II.Commun. Pure Appl. Math. (1964), 35-92.[2] Aramaki, J. L p theory for the div-curl system. Int. J. Math. Anal. (2014), 259-271.[3] Beale, J. T., Kato, T., Majda. A. J. Remarks on the breakdown of smooth solutionsfor the 3-D Euler equations. Comm. Math. Phys. (1), 61-66 (1984)[4] Cai,G. C., Li,J. Existence and exponential growth of global classical solutionsto the compressible Navier-Stokes equations with slip boundary conditions in 3Dbounded domains. arXiv:2102.06348. 365] Cho, Y., Choe, H. J., Kim, H. Unique solvability of the initial boundary valueproblems for compressible viscous fluids. J. Math. Pures Appl. (2), 243-275(2004)[6] Cho, Y., Kim, H. On classical solutions of the compressible Navier-Stokes equationswith nonnegative initial densities. Manuscripta Mathematica. (2006), 91–129.[7] Choe, H. J., Kim, H. Strong solutions of the Navier-Stokes equations for isentropiccompressible fluids. J. Differ. Eqs. (2003), 504–523.[8] Constantin, P., Foias, C. Navier-Stokes Equations (Chicago Lectures in Mathemat-ics. University of Chicago Press, Chicago, 1988)[9] Feireisl, E., Petzeltov´a, H. Large-time behaviour of solutions to the Navier-Stokesequations of compressible flow, Arch. Ration. Mech. Anal., (1999), 77–96.[10] Galdi, G. P.
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