Dissipative solutions to the compressible isentropic Navier-Stokes equations
aa r X i v : . [ m a t h . A P ] F e b DISSIPATIVE SOLUTIONS TO THE COMPRESSIBLE ISENTROPICNAVIER-STOKES EQUATIONS
LIANG GUO, FUCAI LI, AND CHENG YU
Abstract.
The existence of dissipative solutions to the compressible isentropic Navier-Stokes equations was established in this paper. This notion was inspired by the concept ofdissipative solutions to the incompressible Euler equations of Lions ([16], Section 4.4). Ourmethod is to recover such solutions by passing to the limits from approximated solutions,thanks to compactness argument. Introduction
This paper aims to study the existence of dissipation solutions to the compressible Navier-Stokes equations, which was inspired by the work of Lions [16]. In particular, Lions introducedthe concept of dissipation solutions to the incompressible Euler equations and proved itsexistence. This can imply the property of weak-strong uniqueness of the Euler equations.In this paper, we are particularly interested in extending these results to the compressibleNavier-Stokes equations.Thus, we consider the following compressible isentropic Navier-Stokes equations over R + × Ω(Ω ⊂ R ): ( ∂ t ρ + div ( ρ u ) = 0 ,∂ t ( ρ u ) + div ( ρ u ⊗ u ) + ∇ p( ρ ) = div S ( ∇ u ) , (1.1)where ρ denotes the density, u ∈ R the velocity and p( ρ ) = Aρ γ the pressure with theconstant A > γ >
1, respectively. The viscous stress tensor S satisfies the Newton’s rheological law: S ( ∇ u ) = µ ( ∇ u + ∇ ⊤ u ) + λ (div u ) I , where the constants µ and λ are the Lam´e viscosity coefficients of the flow satisfying µ > µ + 3 λ ≥
0, and I is the 3 × ρ, ρ u ) | t =0 = ( ρ , m ) , (1.2)and as one of the following boundary conditions:1. the periodic case Ω = T = R / ( − π, π ) ; (1.3)2. the Dirichlet boundary condition u | ∂ Ω = 0 , (1.4) Mathematics Subject Classification.
Key words and phrases.
Compressible isentropic Navier-Stokes equations, Dissipative solutions, weak-stronguniqueness. where Ω is a bounded domain in R . For the weak solutions of the compressible Navier-Stokes equations, Lions [17] introducedthe concept of renormalized solutions. This allows him to establish the global existence ofweak solutions with large data for any γ ≥ . Later, his result was improved in [12] byextending the value of the adiabatic exponent to γ > . Improving the range of γ is aninteresting and fundamental problem, which certainly is from a physical viewpoint. It is alsoa challenging problem in mathematics, since the restriction on γ > is absolutely essentialto the analysis in [17, 12]. This current paper aims to build up solutions in a weaker sensethan in the renormalized sense, which was inspired by the concept of dissipation solutions inLions [16].DiPerna and Majda [7] proposed a measure-valued solution, by a generalized Young measureand proved the global existence of such solution to the incompressible Euler equations withany initial data. However, they have not investigated the weak-strong uniqueness structure.The concept of dissipation solutions to the incompressible Euler equations was introduced andits existence was given in [16]. This solution can imply the weak-strong uniqueness property.Meanwhile, Bellout et al. [1] also proposed a very weak L solution. Later, Brenier et al.[2] established the weak-strong uniqueness of the admissible measure-valued solutions to theincompressible Euler equations, and showed that the admissible measure-valued solution isalso a dissipative solution in the sense of Lions.The admissible assumption is to say that the kinetic energy is always less than or equalto the initial energy, which plays a key role in [2]. Under the admissible assumption, DeLellis and Sz´ekelyhidi ([6], Proposition 1) proved that the weak solution of incompressibleEuler equations is a dissipative solution in the sense of Lions. Gwiazda et al. [14] extendedthe measure-valued solutions to some compressible fluid models and proved the weak-stronguniqueness of the admissible measure-valued solutions to the isentropic Euler equations in anyspace dimension. And afterwards, Feireisl et al. [9] introduced a dissipative measure-valuedsolution to the compressible barotropic Navier-Stokes system, and proved the existence forthe adiabatic exponent γ > E ( r, v ) = ∂ t r + div ( r v ) , E ( r, v ) = r∂ t v + r v · ∇ v + ∇ p( r ) − div S ( ∇ v ) , (1.5)where r (has a positive lower bound r ) and v be two smooth functions on [0 , ∞ ) × Ω. Thisallows us to derive a priori relative entropy (2.15). It is crucial to have (2.15) for giving thedefinition of dissipative solutions to the compressible Navier-Stokes equations, which impliesthe weak-strong uniqueness property.The main idea is to build up the solutions of the modified Brenner model (4.1), and to showthat this approximated solution is also a dissipative solution to (4.1). Then we can recoverour solution by passing to the limits from this dissipative solution, thanks to the compactnessargument.
ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 3
This paper is organized as follows. In Section 2, we derive some a priori estimates, introducethe definition of dissipative solutions to the compressible isentropic Navier-Stokes equationsand state our main results: the existence of dissipative solutions (Theorem 2.1) and the weaksolution is also a dissipative solution (Theorem 2.3). In Section 3, we show that the smoothfunctions r and v can be replaced by a class of functions with lower regularities. This canbe done through the regularization procedure. In Section 4, we give the proof of Theorem2.1 by using compactness analysis on the approximated solutions. Section 5 is the proof ofTheorem 2.3. Note that Corollary 2.2 and Corollary 2.4 are direct conclusions of Theorem2.1 and Theorem 2.3, respectively, thus we omit their proofs. In Appendix A, we collect someauxiliary lemmas.2. Definition of dissipative solutions and the main results
The main goal of this section is to define our dissipative solutions to the compressibleNavier-Stokes equations and address our main results. To this end, we start with derivingsome a priori estimates which are crucial to our definition. Thus, we assume the solutions aresmooth here.Multiplying the continuity equation (1.1) by | v | − P ′ ( r ) (P ′ ( r ) = Aγγ − r γ − ), integratingthe result over Ω, taking the inner product of the momentum equation (1.1) with u − v , byintegration by parts, and adding them up givesddt ˆ ρ | u − v | + P( ρ ) − P ′ ( r ) ρ d x + ˆ S ( ∇ u ) : ∇ u d x = − ˆ ρ ( u − v ) · ∇ v · ( u − v ) d x + ˆ ρ ( ∂ t v + v · ∇ v ) · ( v − u ) d x − ˆ ρ∂ t P ′ ( r ) d x − ˆ ρ u · ∇ P ′ ( r ) d x − ˆ p( ρ )div v d x + ˆ S ( ∇ u ) : ∇ v d x , (2.1)where P( s ) = Aγ − s γ . Note that P ′ ( r ) r − P( r ) = p( r ) = Ar γ and P ′′ ( r ) r = p ′ ( r ), this givesddt ˆ P ′ ( r ) r − P( r ) d x = ˆ ∂ t p( r ) + div (p( r ) v ) d x = ˆ p ′ ( r ) ∂ t r + v · ∇ p( r ) + p( r )div v d x = ˆ r∂ t P ′ ( r ) d x + ˆ r v · ∇ P ′ ( r ) + p( r )div v d x . (2.2)Using(1.5), (2.1) and (2.2), one obtains thatddt ˆ ρ | u − v | + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x + ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x ≤ − ˆ ρ ( u − v ) · D ( v ) · ( u − v ) d x − ˆ (p( ρ ) − p ′ ( r )( ρ − r ) − p( r ))div v d x + ˆ ( r − ρ )P ′′ ( r ) E ( r, v ) d x + ˆ ρr E ( r, v ) · ( v − u ) d x + ˆ ρ − rr div S ( ∇ v ) · ( v − u ) d x , (2.3) LIANG GUO, FUCAI LI, AND CHENG YU where D ( v ) = ( ∇ v + ∇ ⊤ v ), and we have used( b · ∇ ) a · b = 12 b · ( ∇ a + ∇ ⊤ a ) · b , for the vectors a and b .Next we look at the last term on the left-hand side of (2.3) for the two boundary cases(1.3) and (1.4).For the periodic case Ω = T , using Lemma A.2 in Appendix A gives k u − v k H ≤ c (cid:0) k∇ ( u − v ) k L + k√ ρ ( u − v ) k L (cid:1) , (2.4)where c > γ , and γ ≥ for n = 3 from Lemma A.2.For the bounded domain Ω with Dirichlet boundary condition, it needs the extra restriction v | ∂ Ω = 0. The Poincar´e’s inequality yields k u − v k H ≤ c k∇ ( u − v ) k L , (2.5)where c > γ > ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x = µ k∇ ( u − v ) k L + ( µ + λ ) k div ( u − v ) k L . (2.6)Using Sobolev embedding H ֒ → L with the generic constant ˆ c , and the estimates (2.4),(2.5) and (2.6), we have k v − u k L ≤ ˆ c k v − u k H ≤ cc γ k∇ ( u − v ) k L + 2ˆ cc γ k√ ρ ( u − v ) k L ≤ cc γ µ ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x + 2ˆ cc γ k√ ρ ( u − v ) k L (2.7)for the periodic case (1.3), and k v − u k L ≤ ˆ c k v − u k H ≤ cc γ µ ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x (2.8)for the Dirichlet boundary case (1.4). Here, c γ = max { c , c } .In view of the convexity of P( ρ ) with γ > r ≥ r >
0, it holds thatP( ρ ) − P ′ ( r )( ρ − r ) − P( r ) ≥ ( C (1 + | ρ − r | ) , r ≤ ρ ≤ r ,C (1 + | ρ − r | γ ) , ≤ ρ < r, ρ > r . (2.9)For the convenience of notations, letting f be a given function and the bold be charac-teristic function, we denoteΩ ess = n x ∈ Ω : | ρ − r | ≤ r o , Ω res = n x ∈ Ω : 0 ≤ ρ < r , ρ > r o ,f = [ f ] ess + [ f ] res , [ f ] ess = f Ω ess , [ f ] res = f Ω res . Utilizing the above notations, and by H¨older’s inequality, one deduces that ˆ ρ − rr div S ( ∇ v ) · ( v − u ) d x = ˆ r ( √ ρ − √ r )( √ ρ + √ r )div S ( ∇ v ) · ( v − u ) d x = ˆ Ω ess +Ω res √ ρ − √ r √ r div S ( ∇ v ) · ( v − u ) d x ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 5 + ˆ Ω ess +Ω res √ ρ − √ rr div S ( ∇ v ) · √ ρ ( v − u ) d x ≤ ˆ Ω ess +Ω res √ r | ρ − r | | div S ( ∇ v ) || v − u | d x + ˆ Ω ess +Ω res r | ρ − r | | div S ( ∇ v ) ||√ ρ ( v − u ) | d x ≤ √ k div S ( ∇ v ) k L k v − u k L + (cid:13)(cid:13)(cid:13) √ r (cid:13)(cid:13)(cid:13) L ∞ k [ ρ − r ] res k L γ k div S ( ∇ v ) k L γ γ − k v − u k L + (cid:13)(cid:13)(cid:13) √ r (cid:13)(cid:13)(cid:13) L ∞ k div S ( ∇ v ) k L k√ ρ ( v − u ) k L + (cid:13)(cid:13)(cid:13) r (cid:13)(cid:13)(cid:13) L ∞ k [ ρ − r ] res k L γ k div S ( ∇ v ) k L γγ − k√ ρ ( v − u ) k L , (2.10)where we have also employed the elementary inequality | ρ θ − r θ | ≤ | ρ − r | θ , ≤ θ ≤ , with the special one θ = . Noticing that r ≥ r >
0, with the help of (2.7), (2.8), (2.9) andYoung’s inequality, we have ˆ ρ − rr div S ( ∇ v ) · ( v − u ) d x ≤ µ cc γ k v − u k L + ˆ cc γ µ k div S ( ∇ v ) k L + µ cc γ k v − u k L + 2ˆ cc γ µr k div S ( ∇ v ) k L γ γ − k [ ρ − r ] res k L γ + 12 √ r k div S ( ∇ v ) k L (1 + k√ ρ ( v − u ) k L )+ 1 r k div S ( ∇ v ) k L γγ − ( k√ ρ ( v − u ) k L + k [ ρ − r ] res k L γ ) ≤ ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x + µ ˆ ρ | u − v | d x + C ˆ cc γ µr k div S ( ∇ v ) k L γ γ − ˆ P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x + C √ r r k div S ( ∇ v ) k L γγ − ˆ ρ | u − v | + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x (2.11)under the case (1.3), and ˆ ρ − rr div S ( ∇ v ) · ( v − u ) d x ≤ ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x + C ˆ cc γ µr k div S ( ∇ v ) k L γ γ − ˆ P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x + C √ r r k div S ( ∇ v ) k L γγ − ˆ ρ | u − v | + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x (2.12) LIANG GUO, FUCAI LI, AND CHENG YU under the case (1.4).Putting (2.11) and (2.12) into (2.3), respectively, it follows thatddt ˆ ρ | u − v | + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x + 12 ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x ≤ C Λ( v ) ˆ ρ | u − v | + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x + ˆ (cid:12)(cid:12) ( r − ρ )P ′′ ( r ) E ( r, v ) (cid:12)(cid:12) d x + ˆ (cid:12)(cid:12)(cid:12) ρr E ( r, v ) · ( v − u ) (cid:12)(cid:12)(cid:12) d x , (2.13)where C > v ) = ( (cid:0) µ + Λ ( v ) (cid:1) , for the case (1.3) , (cid:0) Λ ( v ) (cid:1) , for the case (1.4) , (2.14)with Λ ( v ) = k D ( v ) k L ∞ + ˆ cc γ µr k div S ( ∇ v ) k L γ γ − + 1 + √ r r k div S ( ∇ v ) k L γγ − . Applying Gr¨onwall’s inequality to (2.13), we obtain that, for all t ≥ ρ, u ; r, v ) := ˆ ρ | u − v | + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x + 12 ˆ t ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x d s ≤ exp (cid:16) ˆ t C Λ( v ) d s (cid:17) ˆ ρ (cid:12)(cid:12)(cid:12) m ρ − v (cid:12)(cid:12)(cid:12) + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x + ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17)(cid:12)(cid:12) ( r − ρ )P ′′ ( r ) E ( r, v ) (cid:12)(cid:12) d x d s + ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17)(cid:12)(cid:12)(cid:12) ρr E ( r, v ) · ( v − u ) (cid:12)(cid:12)(cid:12) d x d s =:RS( ρ, u ; r, v ) , (2.15)where ( r , v ) = ( r, v ) (cid:12)(cid:12) t =0 .With the above a priori estimates at hand, we are ready to define the dissipative solutionsto the compressible isentropic Navier-Stokes (1.1) in the following sense. Definition 2.1.
Let ρ ∈ L ∞ (0 , T ; L γ ) ∩ C w ([0 , T ]; L γ ) , √ ρ u ∈ L ∞ (0 , T ; L ) and u ∈ L (0 , T ; H ) for any fixed T > , ( ρ, u ) is a dissipative solution of the problem (1.1) - (1.3) or (1.1) , (1.2) and (1.4) , if (2.15) holds for ( r, v ) satisfying r ∈ C ([0 , T ]; L γ ) , r ≥ r > , v ∈ C ([0 , T ]; L γγ − ) , D ( v ) ∈ L (0 , T ; L ∞ ) , ∇ v ∈ L (0 , T ; L ) , div S ( ∇ v ) ∈ L (0 , T ; L γ γ − ) ∩ L (0 , T ; L γγ − ) , P ′′ ( r ) E ( r, v ) ∈ L (0 , T ; L γγ − ) , r E ( r, v ) ∈ L (0 , T ; L γγ − ) , (2.16) where r is a positive constant. Note that it needs the extra condition v | ∂ Ω = 0 for the boundeddomain case (1.4) . ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 7
Remark 2.1. If ( r, v ) satisfies (2.16) , then (2.15) is well-defined. We observe that v ∈ L (0 , T ; W ,p ) for any < p ≤ ∞ from (2.16) . Indeed, in view of the Korn’s inequality ([11] ,Theorem 11.21 ) k v k W ,p ≤ C (cid:16) k D ( v ) k L p + ˆ | v | d x (cid:17) , < p < ∞ , it follows that k v k L (0 ,T ; W ,p ) ≤ C (cid:0) k D ( v ) k L (0 ,T ; L ∞ ) + k v k C ([0 ,T ]; L γγ − ) (cid:1) , < p < ∞ . For the case p = ∞ , notice that on the right-hand side above is independent of p , by LemmaA.3, then k v k L (0 ,T ; W ,p ) → k v k L (0 ,T ; W , ∞ ) as p → ∞ . Furthermore, it infers that v · ∇ v ∈ L (0 , T ; L γγ − ) .We also find that ∇ v ∈ L (0 , T ; L ∪ L γγ − ) from (2.16) . Indeed, by the elliptic regularityresult ( see (50) in page of [5]) : k D v k L p ≤ C (cid:0) kL v k L p + k∇ v k L p (cid:1) , < p < ∞ , where L v = − µ ∆ v − ( µ + λ ) ∇ div v = − div S ( ∇ v ) , together with div S ( ∇ v ) ∈ L (0 , T ; L γ γ − ) and ∇ v ∈ L (0 , T ; L ) in (2.16) , then, D v ∈ L (0 , T ; L ) for < γ < and D v ∈ L (0 , T ; L γ γ − ) for γ ≥ . By the Sobolev embedding H ֒ → L for the case < γ < and W , γ γ − ֒ → L q (1 ≤ q ≤ γγ − ) for the case γ ≥ , it follows that ∇ v ∈ L (0 , T ; L ∪ L γγ − ) . Remark 2.2.
We observe that the conditions P ′′ ( r ) E ( r, v ) ∈ L (0 , T ; L γγ − ) and r E ( r, v ) ∈ L (0 , T ; L γγ − ) are equivalent to ∂ t P ′ ( r ) ∈ L (0 , T ; L γγ − ) and ∇ P ′ ( r ) ∈ L (0 , T ; L γγ − ) if ∂ t v ∈ L (0 , T ; L γγ − ) . Indeed, D ( v ) ∈ L (0 , T ; L ∞ ) implies div v ∈ L (0 , T ; L ∞ ) . Notice that P ′′ ( r ) E ( r, v ) = ∂ t P ′ ( r ) + p ′ ( r )div v + v · ∇ P ′ ( r ) , r E ( r, v ) = ∂ t v + v · ∇ v + ∇ P ′ ( r ) − r div S ( ∇ v ) , along with r ∈ C (0 , T ; L γ ) , r ≥ r > , v ∈ C ([0 , T ]; L γγ − ) , v · ∇ v ∈ L (0 , T ; L γγ − ) and div S ( ∇ v ) ∈ L (0 , T ; L γγ − ) , it infers the observation. Here we address our first main result on the existence of dissipative solution in the senseof Definition 2.1.
Theorem 2.1.
Suppose that the assumptions of Definition hold. In addition, we assumethat r ≤ r if γ > , ∂ t r ∈ L (0 , T ; L γγ − ) ∩ L (0 , T ; L γγ +1 ∩ L ) , ∇ r ∈ L ∞ (0 , T ; L γγ − ) , ∂ t v ∈ L (0 , T ; L γγ − ) ∩ L (0 , T ; L γγ +1 ∩ L ) , ∇ v ∈ L (0 , T ; L ∞ ) , (2.17) where r is a positive constant.1. For the periodic case (1.3) : Let γ ≥ . Assume that the initial data (1.2) satisfy ρ ∈ L γ , ρ ≥ , ˆ ρ d x ≥ C ρ , m = 0 a.e. in { x ∈ Ω : ρ = 0 } , m ρ ∈ L , (2.18) LIANG GUO, FUCAI LI, AND CHENG YU for some positive constant C ρ . Then, there exists a dissipative solution of the compressibleisentropic Navier-Stokes system (1.1) with (1.2) and (1.3) .2. For the Dirichlet boundary case (1.4) : Let γ > . Assume the initial data (1.2) satisfy ρ ∈ L γ , ρ ≥ , m = 0 a.e. in { x ∈ Ω : ρ = 0 } , m ρ ∈ L . (2.19) Then, there exists a dissipative solution of the compressible isentropic Navier-Stokes system (1.1) with (1.2) and (1.4) . Remark 2.3.
Feireisl, Novotn´y and Sun [13] introduced a class of suitable weak solutions tothe compressible barotropic Navier-Stokes equations and proved that the solution satisfies therelative entropy inequality for γ > . Compared with [13] , we define the dissipative solutionsatisfying the inequality (2.15) . We give the process of the density arguments at Section 3. Inaddition, the adiabatic exponent can reduce to γ ≥ for the periodic case and γ > for theDirichlet boundary case. Applying Theorem 2.1 directly, we have the following result on the relationship of the strongsolution and dissipative solution.
Corollary 2.2.
Assume that ( r, v ) is a strong solution of (1.1) and the initial data ( r, v ) | t =0 =( r , v ) together with (1.3) or (1.4) , satisfying the regularities r ∈ C ([0 , T ]; L γ ) , r ≤ r ≤ r , ∂ t r ∈ L (0 , T ; L γγ − ) ∩ L (0 , T ; L γγ +1 ∩ L ) , v ∈ C ([0 , T ]; L γγ − ) , ∂ t v ∈ L (0 , T ; L γγ − ) ∩ L (0 , T ; L γγ +1 ∩ L ) , ∇ r ∈ L ∞ (0 , T ; L γγ − ) , div S ( ∇ v ) ∈ L (0 , T ; L γ γ − ) ∩ L (0 , T ; L γγ − ) , ∇ v ∈ L (0 , T ; L ∞ ) , (2.20) for some positive constants r and r , and any T ∈ (0 , T max ) , where T max is the maximalexistence time. Let ( ρ, u ) be a dissipative solution of the system (1.1) - (1.2) together with (1.3) or (1.4) , and the initial data satisfy ˆ ρ (cid:12)(cid:12)(cid:12) m ρ − v (cid:12)(cid:12)(cid:12) + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x = 0 . Then, the dissipative solution ( ρ, u ) is equal to ( r, v ) on a.e. ( t, x ) ∈ [0 , T ] × Ω . Remark 2.4.
The condition ∇ v ∈ L (0 , T ; L ∞ ) implies that D ( v ) = ( ∇ v + ∇ ⊤ v ) ∈ L (0 , T ; L ∞ ) , which is corresponding to the blow up criteria of the Navier-Stokes equationsgiven by Huang et al. [15] . Under the condition D ( v ) ∈ L (0 , T ; L ∞ ) , the regularities (2.20) can be replaced by making restriction on the initial data and γ in some situations. For exam-ple, endowing the initial data < r ≤ r ≤ r , r ∈ W ,p for p > , v ∈ H , (2.21) for some positive constants r and r , and making use of the condition D ( v ) ∈ L (0 , T ; L ∞ ) , adirect conclusion from [15] shows that there exists a global strong solution ( r, v ) with ( r ≤ r ≤ r , r ∈ C ([0 , T ); W , ) , ∂ t r ∈ C ([0 , T ); L ) , v ∈ C ([0 , T ); H ) ∩ L (0 , T ; W , ) , ∂ t v ∈ L ∞ (0 , T ; L ) ∩ L (0 , T ; H ) , (2.22) for any T ∈ (0 , ∞ ) . We see that ( r, v ) with (2.22) meets the requirement of (2.20) for γ ≥ .It says that (2.20) can be substituted by (2.21) for γ ≥ . ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 9
Our next goal is to show the weak solution of the compressible isentropic Navier-Stokesequations is also a dissipative solution. We first recall the definition of weak solution asfollows.
Definition 2.2.
A pair ( ρ, u ) is a weak solution of the problem (1.1) - (1.3) or (1.1) , (1.2) and (1.4) provided that, for any fixed T > , ρ ∈ L ∞ (0 , T ; L γ ) , √ ρ u ∈ L ∞ (0 , T ; L ) , u ∈ L (0 , T ; H ) , (2.23) and the continuity equation (1.1) and the momentum equation (1.1) are satisfied in D ′ ([0 , T ] × Ω) , that is, for Ψ ∈ C ∞ c ([0 , T ] × Ω) , Φ ∈ C ∞ c ([0 , T ] × Ω) , ˆ ρ ( T, x )Ψ(
T, x ) d x − ˆ ρ Ψ(0 , x ) d x = ˆ T ˆ ρ∂ t Ψ + ρ u · ∇ Ψ d x d t , (2.24) ˆ ρ ( T, x ) u ( T, x ) · Φ( T, x ) d x − ˆ m · Φ(0 , x ) d x = ˆ T ˆ ρ u · ∂ t Φ + ρ u ⊗ u : ∇ Φ+ Aρ γ div Φ − S ( ∇ u ) : ∇ Φ d x d t , (2.25) and the energy inequality holds ˆ ρ | u | + Aγ − ρ γ d x + ˆ t ˆ S ( ∇ u ) : ∇ u d x d t ≤ ˆ | m | ρ + Aγ − ρ γ d x , (2.26) for almost every t ∈ [0 , T ] . Then, we write our next theorem in the following.
Theorem 2.3.
Suppose that the assumptions of Theorem 2.1 hold. If ( ρ, u ) is a weak solutionof the problem (1.1) - (1.3) or (1.1) , (1.2) and (1.4) , then, the weak solution is a dissipativesolution in the sense of Definition 2.1. The weak solution of the compressible isentropic Navier-Stokes equations also has the weak-strong uniqueness property, see [10]. By means of Theorem 2.3, it can directly give anotherversion proof of the weak-strong uniqueness property for the weak solution of the compressibleisentropic Navier-Stokes equations. We state the conclusion by a corollary as follows.
Corollary 2.4.
Under the same assumptions of Corollary 2.2, if we assume that ( ρ, u ) is aweak solution of the problem (1.1) - (1.3) or (1.1) , (1.2) and (1.4) , then ( ρ, u ) = ( r, v ) on a.e. ( t, x ) ∈ [0 , T ] × Ω . regularization of r and v This section devotes to show that the smooth function r and v can be replaced by thefunctions with regularities given in (2.16) and (2.17). Thus, the following is main result inthis section. Proposition 3.1. If (2.15) holds for smooth functions r and v , then it also holds for thefunctions satisfying (2.16) and (2.17) .Proof. We first extend r and v on [ − δ , T + δ ] × e Ω with small δ > ⊂⊂ e Ω, whichstill satisfy the regularities (2.16) and (2.17). Let η ( x ) ∈ C ∞ c ( R ) , supp η = { x ∈ R : | x | ≤ } , ˆ R η ( x ) d x = 1 , η δ ( x ) = 1 δ η ( xδ ) , (3.1) ˜ η ( t ) ∈ C ∞ c ( R ) , supp ˜ η = { t ∈ R : | t | ≤ } , ˆ R ˜ η ( t ) d t = 1 , ˜ η δ ( t ) = 1 δ ˜ η ( tδ ) , (3.2)where 0 < δ ≤
1. We mollify r and v with respect to space and time in the following way: r δt,x ( t, x ) = ( r ∗ η δ ) ∗ ˜ η δ ( t, x ) = ˆ R ˆ R n r ( t − s, x − y ) η δ ( y ) d y ˜ η δ ( s ) d s , v δt,x ( t, x ) = ( v ∗ η δ ) ∗ ˜ η δ ( t, x ) = ˆ R ˆ R n v ( t − s, x − y ) η δ ( y ) d y ˜ η δ ( s ) d s . Since ∇ r ∈ L ( − δ , T + δ ; L γγ − ( e Ω)) , ∂ t r ∈ L ( − δ , T + δ ; L γγ − ( e Ω)) , ∇ v ∈ L ( − δ , T + δ ; L ( e Ω)) , div v ∈ L ( − δ , T + δ ; L ( e Ω)) , by Lemmas A.6 and A.7, then, as δ → k∇ r δt,x − ∇ r k L (0 ,T ; L γγ − ) = k ( ∇ r ) δt,x − ∇ r k L (0 ,T ; L γγ − ) → , (3.3) k ∂ t r δt,x − ∂ t r k L (0 ,T ; L γγ − ) = k ( ∂ t r ) δt,x − ∂ t r k L (0 ,T ; L γγ − ) → , (3.4) k∇ v δt,x − ∇ v k L (0 ,T ; L ) = k ( ∇ v ) δt,x − ∇ v k L (0 ,T ; L ) → , (3.5) k div v δt,x − div v k L (0 ,T ; L ) = k (div v ) δt,x − div v k L (0 ,T ; L ) → . (3.6)For v ∈ C ([ − δ , T + δ ]; L γγ − ( e Ω)), ∂ t v ∈ L ( − δ , T + δ ; L γγ − ( e Ω)), by Lemma A.8, then,as δ → k v δt,x − v k L ∞ (0 ,T ; L γγ − ) → . (3.7)For ∂ t r ∈ L ( − δ , T + δ ; L γγ − ( e Ω)) and ∇ r ∈ L ∞ ( − δ , T + δ ; L γγ − ( e Ω)), by Lemma A.8,we know that, as δ → k r δt,x − r k L ∞ (0 ,T ; L ∞ ) → . (3.8)In addition, since r ≥ r if 1 < γ ≤ r ≤ r ≤ r if γ >
2, and noticing that ´ R η δ ( y ) d y = 1 and ´ R ˜ η δ ( s ) d s = 1, then we have r δt,x ≥ r if 1 < γ ≤ , r ≤ r δt,x ( t, x ) ≤ r if γ > . (3.9)By Lagrange mean value theorem, there exists a θ ∈ (0 ,
1) such thatP ′′ ( r δt,x ) − P ′′ ( r ) =P ′′′ ( θ r δt,x + (1 − θ ) r )( r δt,x − r )= Aγ ( γ − γ − θ r δt,x + (1 − θ ) r ] γ − ( r δt,x − r ) . Combining (3.8) and (3.9), it follows from the above equality that, as δ → k P ′′ ( r δt,x ) − P ′′ ( r ) k L ∞ (0 ,T ; L ∞ ) → . (3.10)By the same argument, we have k p ′ ( r δt,x ) − p ′ ( r ) k L ∞ (0 ,T ; L γγ − ) → . (3.11)Since ∇ P ′ ( r δt,x ) − ∇ P ′ ( r ) = P ′′ ( r δt,x )( ∇ r δt,x − ∇ r ) + [P ′′ ( r δt,x ) − P ′′ ( r )] ∇ r , along with (3.3), (3.9) and (3.10), then it holds, as δ → k∇ P ′ ( r δt,x ) − ∇ P ′ ( r ) k L (0 ,T ; L γγ − ) → . (3.12) ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 11
Next, we will show that P ′′ ( r ) E ( r, v ) can be approximated by P ′′ ( r δt,x ) E ( r δt,x , v δt,x ). Recallthe definition E ( r, v ) in (1.5) and the relation r P ′′ ( r ) = p ′ ( r ), then,P ′′ ( r δt,x ) E ( r δt,x , v δt,x ) = ∂ t P ′ ( r δt,x ) + v δt,x · ∇ P ′′ ( r δt,x ) + p ′ ( r δt,x )div v δt,x =(P ′′ ( r ) E ( r, v )) δt,x + ∂ t P ′ ( r δt,x ) − ( ∂ t P ′ ( r )) δt,x + v δt,x · ∇ P ′ ( r δt,x ) − ( v · ∇ P ′ ( r )) δt,x + p ′ ( r δt,x )div v δt,x − (p ′ ( r )div v ) δt,x . (3.13)By Lemma A.7, we have, as δ → ′′ ( r ) E ( r, v )) δt,x → P ′′ ( r ) E ( r, v ) strongly in L (0 , T ; L γγ − ) . (3.14)For the term ∂ t P ′ ( r δt,x ) − ( ∂ t P ′ ( r )) δt,x , it can be rewritten as ∂ t P ′ ( r δt,x ) − ( ∂ t P ′ ( r )) δt,x =P ′′ ( r )( ∂ t r δt,x − ∂ t r ) + [P ′′ ( r δt,x ) − P ′′ ( r )] ∂ t r δt,x + ∂ t P ′ ( r ) − ( ∂ t P ′ ( r )) δt,x . (3.15)By (3.4), (3.10) and Lemma A.7,we have, as δ → ∂ t P ′ ( r δt,x ) − ( ∂ t P ′ ( r )) δt,x → L (0 , T ; L γγ − ) . (3.16)Similarly, we have v δt,x · ∇ P ′ ( r δt,x ) − ( v · ∇ P ′ ( r )) δt,x → L (0 , T ; L γγ − ) . (3.17)Now, we turn to deal with the term p ′ ( r δt,x )div v δt,x − (p ′ ( r )div v ) δt,x , which can be rewrittenas p ′ ( r δt,x )div v δt,x − (p ′ ( r )div v ) δt,x = (cid:16) p ′ ( r δt,x ) − [p ′ ( r )] δt,x (cid:17) div v δt,x + [p ′ ( r )] δt,x div v δt,x − (p ′ ( r )div v ) δt,x = (cid:16) p ′ ( r δt,x ) − [p ′ ( r )] δt,x (cid:17) div v δt,x + (cid:16) [p ′ ( r )] δt,x − p ′ ( r ) (cid:17) (div v δt,x − div v ) − ˆ R ˆ R n (cid:16) [p ′ ( r )]( t, x ) − [p ′ ( r )]( t − s, x − y ) (cid:17) div ( v ( t, x ) − v ( t − s, x − y )) η δ ( y ) d y ˜ η δ ( s ) d s = (cid:16) p ′ ( r δt,x ) − p ′ ( r ) (cid:17) div v δt,x − (cid:16) [p ′ ( r )] δt,x − p ′ ( r ) (cid:17) div v − ˆ R ˆ R n (cid:16) [p ′ ( r )]( t, x ) − [p ′ ( r )]( t − δs, x − δy ) (cid:17) div ( v ( t, x ) − v ( t − δs, x − δy )) η ( y ) d y ˜ η ( s ) d s =: I + I + I . For the term I , by (3.11), we get, as δ → k I k L (0 ,T ; L γγ − ) ≤ C k div v k L (0 ,T ; L ∞ ) k p ′ ( r δt,x ) − p ′ ( r ) k L ∞ (0 ,T ; L γγ − ) → . For the term I , thanks to p ′ ( r ) = Aγr γ − ∈ L ∞ ( − δ , T + δ ; L γγ − ( e Ω)) and ∂ t p ′ ( r ) = Aγ ( γ − r γ − ∂ t r ∈ L ( − δ , T + δ ; L γγ − ( e Ω)), by Lemma A.8, we get, as δ → k I k L (0 ,T ; L γγ − ) ≤ C k div v k L (0 ,T ; L ∞ ) k [p ′ ( r t,x )] δ − p ′ ( r ) k L ∞ (0 ,T ; L γγ − ) → . For the term I , taking a similar way to (A.1) and (A.2) at Lemma A.8 in Appendix A, wehave I = − ˆ R ˆ R n (cid:16) [p ′ ( r )]( t, x ) − [p ′ ( r )]( t − δs, x ) (cid:17) div ( v ( t, x ) − v ( t − δs, x − δy )) η ( y ) d y ˜ η ( s ) d s − ˆ R ˆ R n (cid:16) [p ′ ( r )]( t − δs, x ) − [p ′ ( r )]( t − δs, x − δy ) (cid:17) div ( v ( t, x ) − v ( t − δs, x − δy )) η ( y ) d y ˜ η ( s ) d s ≤ Cδ ˆ T ˆ | ∂ t p ′ ( r ) || div v | d x d t + Cδ ˆ T ˆ |∇ p ′ ( r ) || div v | d x d t . Thanks to r ≥ r if 1 < γ ≤ r ≤ r ≤ r if γ > ∂ t r ∈ L ( − δ , T + δ ; L γγ +1 ( e Ω) ∩ L ( e Ω)), ∇ r ∈ L ( − δ , T + δ ; L γγ +1 ( e Ω) ∩ L ( e Ω)) and ∇ v ∈ L ( − δ , T + δ ; L γγ − ( e Ω) ∪ L ( e Ω)), and byH¨older’s inequality, it follows that, as δ → k I k L (0 ,T ; L γγ − ) → . Therefore, we have, as δ → ′ ( r δt,x )div v δt,x − (p ′ ( r )div v ) δt,x → L (0 , T ; L γγ − ) . (3.18)Putting (3.13), (3.14), (3.16), (3.17) and (3.18) together, it confirms thatP ′′ ( r δt,x ) E ( r δt,x , v δt,x ) → P ′′ ( r ) E ( r, v ) strongly in L (0 , T ; L γγ − ) , as δ → . (3.19)Using the definition of E ( r, v ) in (1.5), by Lemma A.6, we express r δt,x E ( r δt,x , v δt,x ) as1 r δt,x E ( r δt,x , v δt,x ) = ∂ t v δt,x + v δt,x · ∇ v δt,x + 1 r δt,x ∇ p ( r δt,x ) − r δt,x div S ( ∇ v δt,x )= (cid:16) r E ( r, v ) (cid:17) δt,x + P ′′ ( r δt,x ) ∇ r δt,x − ( P ′′ ( r ) ∇ r ) δt,x + v δt,x · ∇ v δt,x − ( v · ∇ v ) δt,x − r δt,x div S ( ∇ v δt,x ) + 1 r div S ( ∇ v ) . Taking a similar argument to (3.19), we have1 r δt,x E ( r δt,x , v δt,x ) → r E ( r, v ) strongly in L (0 , T ; L γγ − ) . (3.20)Recalling the expression Λ( · ) in (2.14), noticing that k η δ k L ( R ) = 1 and k ˜ η δ k L ( R ) = 1, andby Lemmas A.5 and A.6, it yields Λ( v δt,x ) ≤ Λ( v ) . (3.21)By (3.5), (3.6), (3.7) and (3.8), and note the definition of LS( ρ, u ; · , · ) in (2.15), we haveLS( ρ, u ; r δt,x , v δt,x ) → LS( ρ, u ; r, v ) as δ → . In view of (3.7), (3.8), (3.19), (3.20) and (3.21), and note that RS( ρ, u ; · , · ) in (2.15), we haveRS( ρ, u ; r δt,x , v δt,x ) ≤ RS( ρ, u ; r, v ) as δ → . So we proved it. (cid:3)
ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 13 Proof of Theorem 2.1
This section aims to present the proof of Theorem 2.1 with focusing on the periodic case.We will also point out the difference between the Dirichlet boundary case and the periodiccase.Our proof is starting with the following modified Brenner model: ( ∂ t ρ ǫ + div ( ρ ǫ u ǫ ) = ǫ ∆ ρ ǫ ,∂ t ( ρ ǫ u ǫ ) + div ( ρ ǫ u ǫ ⊗ u ǫ ) + ∇ p( ρ ǫ ) + ǫ a ∇ ( ρ ǫ ) β = div S ( ∇ u ǫ ) + ǫ div ( u ǫ ⊗ ∇ ρ ǫ ) , (4.1)where ǫ ∈ (0 ,
1] is a small parameter, β > max { , γ } , and a is any positive constant. Here,we put the artificial pressure term ǫ a ∇ ( ρ ǫ ) β and the artificial diffusion term ǫ ∆ ρ ǫ at the samelevel, which differs from the approximation model in [12]. We consider the approximate initialdata ( ρ ǫ , u ǫ ) | t =0 = ( ρ ǫ , u ǫ ) , (4.2)satisfying ρ ǫ ∈ C (Ω) , ˆ ρ ǫ d x ≥ C > , < ǫ ≤ ρ ǫ ≤ ǫ − a β , u ǫ ∈ C (Ω) ,ρ ǫ → ρ strongly in L γ , p ρ ǫ u ǫ → m √ ρ strongly in L , as ǫ → , (4.3)where the positive constant C ≤ C ρ is independent of ǫ , and ( ρ , m ) satisfies (2.18). Whenwe consider the Dirichlet boundary case, the condition ´ ρ ǫ d x ≥ C > ρ , m ) satisfies (2.19). It needs to add the boundary conditions ∇ ρ ǫ · n | ∂ Ω = 0and u ǫ | ∂ Ω = 0.For any fixed ǫ > T ∈ (0 , ∞ ), by the Faedo-Galerkin approximation adopted byFeireisl et al. ([12], Proposition 2.1), the system (4.1)-(4.2) has a global weak solution ( ρ ǫ , u ǫ ),which satisfies the energy differential inequalityddt ˆ ρ ǫ | u ǫ | + P( ρ ǫ ) + ǫ a Q( ρ ǫ ) d x + ˆ S ( ∇ u ǫ ) : ∇ u ǫ d x + ˆ ǫ P ′′ ( ρ ǫ ) |∇ ρ ǫ | + ǫ a Q ′′ ( ρ ǫ ) |∇ ρ ǫ | d x ≤ , (4.4)for any t ∈ [0 , T ], where P( ρ ǫ ) = Aγ − ( ρ ǫ ) γ and Q( ρ ǫ ) = β − ( ρ ǫ ) β .By (4.3) and Lemma A.2, it follows from (4.4) thatsup t ∈ [0 ,T ] k√ ρ ǫ u ǫ k L ≤ C , (4.5)sup t ∈ [0 ,T ] k ρ ǫ k L γ ≤ C , (4.6) ˆ T k u ǫ k H d t ≤ C , (4.7) ǫ ˆ T ˆ | ρ ǫ | γ − |∇ ρ ǫ | d x d t ≤ C . (4.8)Here, the estimate (4.7) requires γ ≥ from Lemma A.2. (For the Dirichlet boundary case,the estimate (4.7) is hold for γ > ∂ t ρ ǫ + div ( ρ ǫ u ǫ ) = ǫ ∆ ρ ǫ a.e. x ∈ (0 , T ) × Ω . Multiplying B ′ ( ρ ǫ ) on the both sides of above equation, we arrive at ∂ t B ( ρ ǫ ) + div ( B ( ρ ǫ ) u ǫ ) + ( B ′ ( ρ ǫ ) ρ ǫ − B ( ρ ǫ ))div u ǫ = ǫ div ( B ′ ( ρ ǫ ) ∇ ρ ǫ ) − ǫB ′′ ( ρ ǫ ) |∇ ρ ǫ | , (4.9)where B ∈ C ([0 , ∞ )) ∩ C ((0 , ∞ )) with B ′ ( z ) = 0 for large z ∈ R + .On the one hand, taking B ( z ) = z ln z for z ∈ [0 ,
1] in (4.9), and integrating the result over(0 , T ) × { x ∈ Ω : 0 < ρ ǫ ≤ } , it follows from (4.6) and (4.7) that ǫ ˆ T ˆ { x :0 <ρ ǫ ≤ } ( ρ ǫ ) − |∇ ρ ǫ | d x d t = − ˆ { x : ρ ǫ ≤ } ρ ǫ ln ρ ǫ d x (cid:12)(cid:12) T − ˆ T ˆ { x : ρ ǫ ≤ } ρ ǫ div u ǫ d x d t ≤ C .
On the other hand, by (4.8), it infers that ǫ ˆ T ˆ { x : ρ ǫ ≥ } ( ρ ǫ ) − |∇ ρ ǫ | d x d t ≤ ǫ ˆ T ˆ ( ρ ǫ ) γ − (cid:12)(cid:12)(cid:12) ∇ ρ ǫ ( ρ ǫ ) (cid:12)(cid:12)(cid:12) d x d t ≤ C .
Then, it implies that ǫ ( ρ ǫ ) − ∇ ρ ǫ ∈ L (0 , T ; L ) . (4.10)Similar to (4.10), if we take B ( z ) = − √ z for z ∈ [0 , ǫ ( ρ ǫ ) − ∇ ρ ǫ ∈ L (0 , T ; L ) . (4.11)By the estimates (4.6) and (4.10), and H¨older’s inequality, one has ǫ ∇ ρ ǫ ∈ L (0 , T ; L γγ +1 ) . (4.12)Using the energy estimates (4.6) and (4.7), up to a subsequence ( ρ ǫ , u ǫ ) without relabeled,there exists a weak limit ( ρ, u ) such that ρ ǫ ⇀ ρ weakly- ⋆ in L ∞ (0 , T ; L γ ) , (4.13) u ǫ ⇀ u weakly in L (0 , T ; H ) . (4.14)Notice that √ ρ ǫ ∈ L ∞ (0 , T ; L γ ) from (4.6), there exists a function ˜ ρ such that √ ρ ǫ ⇀ p ˜ ρ weakly- ⋆ in L ∞ (0 , T ; L γ ) . (4.15)Taking B ( ρ ǫ ) = √ ρ ǫ in (4.9), and by the estimates (4.5), (4.6), (4.7), (4.10) and (4.11), thenwe have ∂ t ( √ ρ ǫ ) = − div ( √ ρ ǫ u ǫ ) + 12 √ ρ ǫ div u ǫ + ǫ div (cid:16) √ ρ ǫ ∇ ρ ǫ (cid:17) + 14 ǫ ( ρ ǫ ) − |∇ ρ ǫ | ∈ L ∞ (0 , T ; W − , ) + L (0 , T ; L γγ +1 ) + L (0 , T ; W − , ) + L (0 , T ; L ) ⊂ L (0 , T ; W − , γγ +1 ) . (4.16)With the help of Lemma A.10, we have ρ ǫ = √ ρ ǫ √ ρ ǫ → ˜ ρ in D ′ ((0 , T ) × Ω) . (4.17)Combining (4.13) and (4.17), the uniqueness of limit implies, for a.e. ( t, x ) ∈ (0 , T ) × Ω, ρ = ˜ ρ . (4.18) ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 15
By (4.15), (4.16) and (4.18), and using Lemma A.9 gives √ ρ ǫ → √ ρ in C w ([0 , T ]; L γ ) . Since 2 γ > > , by the interpolation relation L γ ֒ → ֒ → H − , we know √ ρ ǫ → √ ρ in C ([0 , T ]; H − ) . (4.19)It follows from (4.14) and (4.19) that √ ρ ǫ u ǫ → √ ρ u in D ′ ((0 , T ) × Ω) . Since √ ρ ǫ u ǫ ∈ L ∞ (0 , T ; L ) from (4.5), one has √ ρ ǫ u ǫ ⇀ √ ρ u weakly- ⋆ in L ∞ (0 , T ; L ) . (4.20)By (4.13), (4.16) and (4.20), in view of Lemma A.10, we have ρ ǫ u ǫ = √ ρ ǫ √ ρ ǫ u ǫ → ρ u in D ′ ((0 , T ) × Ω) . Since ρ ǫ u ǫ ∈ L ∞ (0 , T ; L γγ +1 ) from (4.5) and (4.6), it implies ρ ǫ u ǫ ⇀ ρ u weakly- ⋆ in L ∞ (0 , T ; L γγ +1 ) . (4.21)From the discussion in Section 3, we can choose smooth functions r > v satisfyingthe regularities (2.16) and (2.17). Taking the inner production with (4.1) by − v , multiplying(4.1) by | v | − P ′ ( r ), integrating the result over Ω with respect to x , and by integration byparts, it deduces thatddt ˆ − ρ ǫ u ǫ · v + 12 ρ ǫ | v | − P ′ ( r ) ρ ǫ d x = ˆ ρ ǫ ∂ t v · ( v − u ǫ ) d x + ˆ ρ ǫ ( u ǫ · ∇ ) v · ( v − u ǫ ) d x − ˆ p( ρ ǫ )div v d x − ǫ a ˆ ( ρ ǫ ) β div v d x + ˆ S ( ∇ u ǫ ) : ∇ v d x + ǫ ˆ ( ∇ ρ ǫ · ∇ ) v · ( u ǫ − v ) d x − ˆ ρ ǫ ∂ t P ′ ( r ) d x − ˆ ρ ǫ u ǫ · ∇ P ′ ( r ) d x + ǫ ˆ ∇ ρ ǫ · ∇ P ′ ( r ) d x . (4.22)Note that (2.2), we rewrite it asddt ˆ P ′ ( r ) r − P( r ) d x = ˆ r∂ t P ′ ( r ) d x + ˆ r v · ∇ P ′ ( r ) + p( r )div v d x . (4.23)Adding up (4.4), (4.22) and (4.23), and by means of the definitions of E ( r, v ) and E ( r, v )in (1.5), we arrive atddt ˆ ρ ǫ | u ǫ − v | + P( ρ ǫ ) − P ′ ( r )( ρ ǫ − r ) − P( r ) + ǫ a Q( ρ ǫ ) d x + ˆ S ( ∇ u ǫ − ∇ v ) : ∇ ( u ǫ − v ) d x + ˆ ǫ P ′′ ( ρ ǫ ) |∇ ρ ǫ | + ǫ a Q ′′ ( ρ ǫ ) |∇ ρ ǫ | d x ≤ − ˆ ρ ( u ǫ − v ) · D ( v ) · ( u ǫ − v ) d x − ˆ (p( ρ ǫ ) − p ′ ( r )( ρ ǫ − r ) − p( r ))div v d x − ǫ a ˆ ( ρ ǫ ) β div v d x + ˆ ( r − ρ ǫ )P ′′ ( r ) E ( r, v ) d x + ˆ ρ ǫ r E ( r, v ) · ( v − u ǫ ) d x + ˆ ρ ǫ − rr div S ( ∇ v ) · ( v − u ǫ ) d x + ǫ ˆ ( ∇ ρ ǫ · ∇ ) v · ( u ǫ − v ) d x + ǫ ˆ ∇ ρ ǫ · ∇ P ′ ( r ) d x . Using the same way as (2.11) to deal with the third term from bottom on the right-hand sideof the above equality, then applying Gr¨onwall’s inequality to the result gives ˆ ρ ǫ | u ǫ − v | + P( ρ ǫ ) − P ′ ( r )( ρ ǫ − r ) − P( r ) + ǫ a Q( ρ ǫ ) d x + 12 ˆ t ˆ S ( ∇ u ǫ − ∇ v ) : ∇ ( u ǫ − v ) d x d s ≤ R + X i =1 R i , (4.24)where R = exp( ˆ t C Λ( v ) d s ) ˆ (cid:0) ρ ǫ | u ǫ − v | + P( ρ ǫ ) − P ′ ( r )( ρ ǫ − r ) − P( r ) + ǫ a Q( ρ ǫ ) (cid:1) d x , R = ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ( r − ρ ǫ )P ′′ ( r ) E ( r, v ) d x d s , R = ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ρ ǫ r E ( r, v ) · ( v − u ǫ ) d x d s , R = ǫ ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ( ∇ ρ ǫ · ∇ ) v · ( u ǫ − v ) d x d s , R = ǫ ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ∇ ρ ǫ · ∇ P ′ ( r ) d x d s . The next step is to recover a dissipative solution by passing to the limits in (4.24) as ε tends to zero. We first deal with the left-hand side of (4.24). By the weak convergences (4.20)and (4.14), and in view of the low semi-continuous of L -norm, we get, as ǫ → ˆ ρ ǫ | u ǫ | d x = ˆ |√ ρ ǫ u ǫ | d x ≥ ˆ ρ | u | d x , ˆ t ˆ S ( ∇ u ǫ ) : ∇ ( u ǫ ) d x d s ≥ ˆ t ˆ S ( ∇ u ) : ∇ ( u ) d x d s . It follows from (4.21), (4.13) and (4.14) that, as ǫ → − ˆ ρ ǫ u ǫ · v d x → − ˆ ρ u · v d x , ˆ ρ ǫ | v | d x → ˆ ρ | v | d x , − ˆ t ˆ S ( ∇ u ǫ ) : ∇ v + S ( ∇ v ) : ∇ u ǫ d x d s → − ˆ t ˆ S ( ∇ u ) : ∇ v + S ( ∇ v ) : ∇ u d x d s . By the convexity of P( · ) and (4.13), one has, as ǫ → ˆ P( ρ ǫ ) − P ′ ( r )( ρ ǫ − r ) − P( r ) d x ≥ ˆ P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x . Then, as ǫ → ˆ ρ ǫ | u ǫ − v | + P( ρ ǫ ) − P ′ ( r )( ρ ǫ − r ) − P( r ) + ǫ a Q( ρ ǫ ) d x + 12 ˆ t ˆ S ( ∇ u ǫ − ∇ v ) : ∇ ( u ǫ − v ) d x d s ≥ ˆ ρ | u − v | + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x + 12 ˆ t ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x d s . (4.25) ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 17
Next, we tackle with the reminder terms R i ( i = 0 , , , ,
4) on the right-hand side of (4.24).Before this, recalling the definition Λ( v ) in (2.14) and the regularities D ( v ) ∈ L (0 , T ; L ∞ )and div S ( ∇ v ) ∈ L (0 , T ; L γ γ − ) ∩ L (0 , T ; L γγ − ) in (2.16), we know thatexp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ≤ exp (cid:16) ˆ t C Λ( v ) d τ (cid:17) ≤ C .
Now, we begin to deal with the terms R i . By the initial conditions (4.3), we have, as ǫ → R → exp (cid:16) ˆ t C Λ( v ) d s (cid:17) ˆ ρ (cid:12)(cid:12)(cid:12) m ρ − v (cid:12)(cid:12)(cid:12) + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x . (4.26)For the terms R and R , it follows from (4.13) and (4.21) that, as ǫ → R → ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ( r − ρ )P ′′ ( r ) E ( r, v ) d x d s , (4.27) R → ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ρr E ( r, v ) · ( v − u ) d x d s . (4.28)We turn to the term R . By (4.10), (4.12) and (4.20), it implies that ǫ ( √ ρ ǫ ) − ∇ ρ ǫ · ( √ ρ ǫ u ǫ ) → L (0 , T ; L ) , (4.29) ǫ ∇ ρ ǫ → L (0 , T ; L γγ +1 ) . (4.30)In view of (4.29) and (4.30), together with the regularities v ∈ C ([0 , T ]; L γγ − ) in (2.16) and ∇ v ∈ L (0 , T ; L ∞ ) in (2.17), we have, as ǫ → R = ǫ ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ( √ ρ ǫ ) − ∇ ρ ǫ · ∇ v · √ ρ ǫ u ǫ d x d s − ǫ ˆ t ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17) ( ∇ ρ ǫ · ∇ ) v · v ] d x d s → . (4.31)Finally, for the terms R , noticing that r ≥ r if 1 < γ ≤ r ≤ r ≤ r if γ > ∇ r ∈ L ∞ (0 , T ; L γγ − ) in (2.16) and (2.17), by H¨older’s inequality, it follows from (4.30) that,as ǫ → R ≤ C k ǫ ∇ ρ ǫ k L (0 ,T ; L γγ +1 ) k∇ r k L (0 ,T ; L γγ − ) → . (4.32)Thus, we complete the proof of Theorem 2.1.5. Proof of Theorem 2.3
The goal of this section is to show that the weak solution of the compressible isentropicNavier-Stokes equations is also a dissipative solution in the sense of Definition 2.1.Let ( ρ, u ) be the weak solution of the problem (1.1)-(1.3) or (1.1), (1.2) and (1.4). Let χ ∈ C ∞ c ((0 , T )) and φ m ∈ C ∞ c (Ω). We can also let ( r, v ) be the smooth functions satisfying(2.16) and (2.17) by the arguments in Section 3. Taking the test function Φ = χφ m v in (2.25)of Definition 2.2 gives0 = ˆ T ˆ ρ u · ∂ t ( χφ m v ) d x d t + ˆ T ˆ ρ u ⊗ u : ∇ ( χφ m v ) d x d t + ˆ T ˆ p( ρ )div ( χφ m v ) d x d t − ˆ T ˆ S ( ∇ u ) : ∇ ( χφ m v ) d x d t = ˆ T ∂ t χ ˆ φ m ρ u · v d x d t + ˆ T χ ˆ φ m ρ u · ∂ t v d x d t + ˆ T χ ˆ φ m ρ u · ∇ v · u d x d t + ˆ T χ ˆ φ m p( ρ )div v d x d t − ˆ T χ ˆ φ m S ( ∇ u ) : ∇ v d x d t + ˆ T χ ˆ ρ u ⊗ u : ( ∇ φ m ⊗ v ) d x d t + ˆ T χ ˆ p( ρ ) ∇ φ m · v d x d t − ˆ T χ ˆ S ( ∇ u ) : ( ∇ φ m ⊗ v ) d x d t . (5.1)Choosing Ψ = | v | χφ m and Ψ = P ′ ( r ) χφ m in (2.24) of Definition 2.2, respectively, onesees that 0 = ˆ T ˆ ρ∂ t (cid:16) | v | χφ m (cid:17) d x d t + ˆ T ˆ ρ u · ∇ (cid:16) | v | χφ m (cid:17) d x d t = ˆ T ∂ t χ ˆ φ m ρ (cid:16) | v | (cid:17) d x d t + ˆ T χ ˆ φ m ρ v · ∂ t v d x d t + ˆ T χ ˆ φ m ρ u · ∇ v · v d x d t + ˆ T χ ˆ | v | ρ u · ∇ φ m d x d t , (5.2)and 0 = ˆ T ˆ ρ∂ t (P ′ ( r ) χφ m ) d x d t + ˆ T ˆ ρ u · ∇ (P ′ ( r ) χφ m ) d x d t = ˆ T ∂ t χ ˆ φ m ρ P ′ ( r ) d x d t + ˆ T χ ˆ φ m ρ∂ t P ′ ( r ) d x d t + ˆ T χ ˆ φ m ρ u · ∇ P ′ ( r ) d x d t + ˆ T χ ˆ ρ P ′ ( r ) u · ∇ φ m d x d t . (5.3)Noticing that the relations P ′ ( r ) r − P( r ) = p( r ) and P ′′ ( r ) r = p ′ ( r ), we have0 = ˆ T ddt ˆ [P ′ ( r ) r − P( r )] χφ m d x d t = ˆ T ddt ˆ p( r ) χφ m d x d t + ˆ T ˆ div (p( r ) v χφ m ) d x d t = ˆ T ∂ t χ ˆ φ m [P ′ ( r ) r − P( r )] d x d t + ˆ T χ ˆ φ m r∂ t P ′ ( r ) d x d t + ˆ T χ ˆ φ m r v · ∇ P ′ ( r ) d x d t + ˆ T χ ˆ φ m p( r )div v d x d t + ˆ T χ ˆ p( r ) v · ∇ φ m d x d t . (5.4)Collecting (5.1), (5.2), (5.3) and (5.4) together, we obtain that − ˆ T ∂ t χ ˆ φ m h − ρ u · v + 12 | v | − P ′ ( r ) ρ + P ′ ( r ) r − P( r ) i d x d t = − ˆ T χ ˆ φ m ρ ( u − v ) · ∇ v · ( u − v ) d x d t + ˆ T χ ˆ φ m ρ ( ∂ t v + v · ∇ v ) · ( v − u ) d x d t + ˆ T χ ˆ φ m ( r − ρ ) ∂ t P ′ ( r ) d x d t + ˆ T χ ˆ φ m ( r v − ρ u ) · ∇ P ′ ( r ) d x d t ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 19 − ˆ T χ ˆ φ m (p( ρ ) − p( r ))div v d x d t + ˆ T χ ˆ φ m S ( ∇ u ) : ∇ v d x d t + R χφ m , (5.5)where R χφ m = − ˆ T ∂ t χ ˆ (1 − φ m ) h ρ | u | + P( ρ ) i d x d t | {z } X − ˆ T χ ˆ ρ u ⊗ u : ( ∇ φ m ⊗ v ) d x d t − ˆ T χ ˆ p( ρ ) v · ∇ φ m d x d t | {z } X + ˆ T χ ˆ S ( ∇ u ) : ( ∇ φ m ⊗ v ) d x d t | {z } X + ˆ T χ ˆ | v | ρ u · ∇ φ m d x d t | {z } X − ˆ T χ ˆ ρ P ′ ( r ) u · ∇ φ m d x d t | {z } X + ˆ T χ ˆ p( r ) v · ∇ φ m d x d t | {z } X . We endow a sequence φ m ∈ C ∞ c (Ω) with0 ≤ φ m ≤ , φ m = 1 for x ∈ Ω , dist( x, ∂ Ω) ≥ m ,φ m → , |∇ φ m | ≤ m for x ∈ Ω . Now, we tackle with the term R χφ m . For the term X in R χφ m , in view of (2.23), andusing the Lebesgue’s dominated convergence theorem, one has, as m → ∞ , X → . (5.6)For the term X , by H¨older’s inequality and (2.23), we get, as m → ∞ , X ≤ ˆ T χ ˆ ( ρ | u | + p( ρ )) |∇ φ m dist( x, ∂ Ω) || v [dist( x, ∂ Ω)] − | d x d t ≤ C ˆ T k v [dist( x, ∂ Ω)] − k L ∞ ˆ { x : dist( x,∂ Ω) ≤ m } ρ | u | + p( ρ ) d x d t ≤ C sup ≤ t ≤ T ˆ { x : dist( x,∂ Ω) ≤ m } ρ | u | + p( ρ ) d x × k v [dist( x, ∂ Ω)] − k L (0 ,T ; L ∞ ) → . (5.7)Here, we have employed the fact that v [dist( x, ∂ Ω)] − ∈ L (0 , T ; L ∞ ). Indeed, by Hardy’sinequality, k v [dist( x, ∂ Ω)] − k L p ≤ C k∇ v k L p ≤ C k∇ v k L ∞ , for 1 < p < ∞ , together with v ∈ L (0 , T ; W , ∞ ) (see Remark 2.1) and Lemma A.3, the conclusion holds.With a similar argument to the term X , by H¨older’s inequality and Hardy’s inequality,along with the regularity of r and v in Theorem 2.3, it infers that, as m → ∞ , X ≤ ˆ T χ ˆ | S ( ∇ u ) ||∇ φ m dist(x , ∂ Ω) || v [dist(x , ∂ Ω)] − | dx dt ≤ C k∇ u k L (0 ,T ; L ) (cid:16) ˆ T ˆ { x : dist( x,∂ Ω) ≤ m } |∇ v | d x d t (cid:17) → , (5.8) X ≤ ˆ T χ ˆ ρ | u ||∇ φ m dist( x, ∂ Ω) | | v | [dist( x, ∂ Ω)] − d x d t ≤ C k ρ u k L ∞ (0 ,T ; L γγ +1 ) ˆ T (cid:16) ˆ { x : dist( x,∂ Ω) ≤ m } | v · ∇ v | γγ − d x (cid:17) γ − γ d t → , (5.9) X ≤ ˆ T χ ˆ ρ | u ||∇ φ m dist( x, ∂ Ω) | P ′ ( r )[dist( x, ∂ Ω)] − d x d t ≤ C k ρ u k L ∞ (0 ,T ; L γγ +1 ) ˆ T (cid:16) ˆ { x : dist( x,∂ Ω) ≤ m } |∇ r | γγ − d x (cid:17) γ − γ d t → , (5.10) X ≤ ˆ T χ ˆ | p( r ) ||∇ φ m dist( x, ∂ Ω) || v [dist( x, ∂ Ω)] − | d x d t ≤ C sup ≤ t ≤ T ˆ { x : dist( x,∂ Ω) ≤ m } p( r ) d x k v [dist( x, ∂ Ω)] − k L (0 ,T ; L ∞ ) → . (5.11)Then, we have, as m → ∞ , R χφ m → . Next, letting m → ∞ , if follows from (5.5) that − ˆ T ∂ t χ ˆ h − ρ u · v + 12 | v | − P ′ ( r ) ρ + P ′ ( r ) r − P( r ) i d x d t = − ˆ T χ ˆ ρ ( u − v ) · ∇ v · ( u − v ) d x d t + ˆ T χ ˆ ρ ( ∂ t v + v · ∇ v ) · ( v − u ) d x d t + ˆ T χ ˆ ( r − ρ ) ∂ t P ′ ( r ) d x d t + ˆ T χ ˆ ( r v − ρ u ) · ∇ P ′ ( r ) d x d t − ˆ T χ ˆ (p( ρ ) − p( r ))div v d x d t + ˆ T χ ˆ S ( ∇ u ) : ∇ v d x d t . Since χ ∈ C ∞ c ((0 , T )) is arbitrary, then it holds that, for a.e. t ∈ (0 , T ),ddt ˆ h − ρ u · v + 12 | v | − P ′ ( r ) ρ + P ′ ( r ) r − P( r ) i d x = − ˆ ρ ( u − v ) · ∇ v · ( u − v ) d x + ˆ ρ ( ∂ t v + v · ∇ v ) · ( v − u ) d x + ˆ ( r − ρ ) ∂ t P ′ ( r ) d x d t + ˆ ( r v − ρ u ) · ∇ P ′ ( r ) d x − ˆ (p( ρ ) − p( r ))div v d x + ˆ S ( ∇ u ) : ∇ v d x . (5.12)Adding ddt ´ ρ | u | + P( ρ ) d x + ´ S ( ∇ u ) : ∇ u d x to both sides of (5.12), by means of thedefinitions E ( r, v ) and E ( r, v ) in (1.5), we have, for a.e. t ∈ (0 , T ),ddt E ws ( ρ, u ; r, v ) + ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x = ddt ˆ ρ | u | + P( ρ ) d x + ˆ S ( ∇ u ) : ∇ u d x ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 21 − ˆ ρ ( u − v ) · D ( v ) · ( u − v ) d x − ˆ (p( ρ ) − p ′ ( r )( ρ − r ) − p( r ))div v d x + ˆ ( r − ρ )P ′′ ( r ) E ( r, v ) d x + ˆ ρr E ( r, v ) · ( v − u ) d x + ˆ ρ − rr div S ( ∇ v ) · ( v − u ) d x , (5.13)where E ws ( ρ, u ; r, v ) = ˆ ρ | u − v | + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x . With the same way as (2.11) to deal with the last term on the right-hand side of (5.13),applying Gr¨onwall’s inequality, and using the energy inequality (2.26) in Definition 2.2, ityields, for 0 < ˜ s < t < T , E ws ( ρ, u ; r, v )( t ) + 12 ˆ t ˜ s ˆ S ( ∇ u − ∇ v ) : ∇ ( u − v ) d x d s ≤ exp (cid:16) ˆ t ˜ s C Λ( v ) d s (cid:17) E ws ( ρ, u ; r, v )(˜ s )+ ˆ t ˜ s ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17)(cid:12)(cid:12) ( r − ρ )P ′′ ( r ) E ( r, v ) (cid:12)(cid:12) d x d s + ˆ t ˜ s ˆ exp (cid:16) ˆ ts C Λ( v ) d τ (cid:17)(cid:12)(cid:12)(cid:12) ρr E ( r, v ) · ( v − u ) (cid:12)(cid:12)(cid:12) d x d s . Using (2.24), (2.25) and (2.23) in Definition 2.2, and making the density arguments, theweak solution ( ρ, u ) belongs to the regularity class ρ ∈ C w ([0 , T ]; L γ ) , ρ u ∈ C w ([0 , T ]; L γγ +1 ) . (5.14)Choosing ˜ s := s n with s n → n → ∞ ), and taking advantage of (2.26) and (5.14), itimplies that E ws ( ρ, u ; r, v )( s n ) ≤ ˆ | m | ρ + P( ρ ) d x − ˆ ρ u · v d x (cid:12)(cid:12)(cid:12) t = s n + ˆ ρ | v | d x (cid:12)(cid:12)(cid:12) t = s n − ˆ ρ P ′ ( r ) d x (cid:12)(cid:12)(cid:12) t = s n + ˆ P ′ ( r ) r − P( r ) d x (cid:12)(cid:12)(cid:12) t = s n → ˆ ρ (cid:12)(cid:12)(cid:12) m ρ − v (cid:12)(cid:12)(cid:12) + P( ρ ) − P ′ ( r )( ρ − r ) − P( r ) d x =: E ws ( ρ , m ; r , v ) , as n → ∞ . Thus, we complete the proof of Theorem 2.3.
Appendix
A.In this appendix, we first collect the classical inequality, some properties of L p space, andtwo weak convergence results. Then, we add two supplementary lemmas (Lemmas A.2 andA.8) and give the proofs of them. These facts are frequently used in the proof of our mainresults. We point out that the Lemma A.8 subjects to the density arguments to deduce (3.7),(3.8) and (3.18) in Section 3. Lemma A.1 ([3], Chapter 9, Poincar´e-Wirtinger’s inequality) . Let Ω ⊂ R n be a boundeddomain with a C boundary ∂ Ω . Then, (cid:13)(cid:13)(cid:13) w − | Ω | ˆ Ω w d x (cid:13)(cid:13)(cid:13) L q (Ω) ≤ C k∇ w k L p (Ω) , ∀ w ∈ W ,p , where ≤ q ≤ pnn − p for n > p . Lemma A.2.
Let Ω ⊂ R n be a bounded domain with a C boundary ∂ Ω , and a non-negativefunction ˜ r satisfies < M ≤ ˆ Ω ˜ r d x , ˆ Ω ˜ r γ d x ≤ M , for some positive constants M and M , where it assumes γ ≥ nn +2 for n ≥ . Then, thereexists a constant ˜ c γ := C ( n, γ, Ω , M , M ) such that k w k H (Ω) ≤ ˜ c γ (cid:0) k∇ w k L (Ω) + k√ ˜ rw k L (Ω) (cid:1) . Remark A.1.
Lemmas A.1 and A.2 still hold when
Ω = T n . Lemma A.2 can be directlydeduced by Theorem 11.23 called generalized Korn-Poincar´e inequality in [11] . For reader’sconvenience, we give a brief proof by means of Lemma A.1 as follows.Proof. By Minkowski’s inequality and Poincar´e’s inequality in Lemma A.1, it follows that k w k L (Ω) ≤ (cid:13)(cid:13)(cid:13) w − | Ω | ˆ Ω w d x (cid:13)(cid:13)(cid:13) L (Ω) + (cid:13)(cid:13)(cid:13) | Ω | ˆ Ω w d x (cid:13)(cid:13)(cid:13) L (Ω) ≤ C k∇ w k L (Ω) + | Ω | − k w k L (Ω) . Then, we know k w k H (Ω) ≤ C k∇ w k L (Ω) + C ˆ Ω | w | d x . Using H¨older’s inequality, one has ˆ Ω r d x | Ω | ˆ Ω | w | d x ≤ ˆ Ω r (cid:12)(cid:12)(cid:12) w − | Ω | ˆ w d x (cid:12)(cid:12)(cid:12) d x + ˆ Ω r | w | d x ≤k r k L pp − (Ω) (cid:13)(cid:13)(cid:13) w − | Ω | ˆ Ω w d x (cid:13)(cid:13)(cid:13) L p (Ω) + k√ r k L (Ω) k√ rw k L (Ω) . Taking p = nn − , it sees that pp − = nn +2 . By the Poincar´e-Wirtinger’s inequality stated inLemma A.1 and the restrictions on γ , the conclusion follows. (cid:3) Lemma A.3 ([4], Lemma 3.1 in Chapter 4) . Let Ω ⊂ R n be a bounded domain. If w ∈ L p (Ω) for ≤ p < ∞ , and k w k L p (Ω) ≤ M < ∞ , where the constant M is independent of p . Then, w ∈ L ∞ (Ω) , and as p → ∞ , k w k L p (Ω) → k w k L ∞ (Ω) . Lemma A.4 ([3], Lemma 4.3 in Chapter 4) . Let Ω ⊂ R n be an open set, K ⊂⊂ Ω , K = Ω if Ω = T n or R n . Then, for w ∈ L p (Ω) , ≤ p < ∞ , as ξ → , k w ( x + ξ ) − w ( x ) k L p ( K ) → . Lemma A.5 ([3], Theorem 4.15 in Chapter 4) . Let f ∈ L ( R n ) and g ∈ L p ( R n ) with ≤ p ≤ ∞ . Define ( f ∗ g )( x ) = ˆ R n f ( x − y ) g ( y ) d y . Then, k f ∗ g k L p ( R n ) ≤ k f k L ( R n ) k g k L p ( R n ) . ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 23
The following three lemmas are about mollifier. Let w be a locally integrable function.Recall the definitions of mollifiers η ( x ) in (3.1) and ˜ η ( t ) in (3.2). We use w δx to denote themollification of w with respect to x , and w δt,x to denote the mollification of w with respect toboth x and t , that is, w δx ( t, x ) = ( w ∗ η δ )( t, x ) = ˆ R n w ( t, x − y ) η δ ( y ) d y ,w δt,x ( t, x ) = ( w δ ∗ ˜ η δ )( t, x ) = ˆ R ˆ R n w ( s, x − y ) η δ ( y ) d y ˜ η δ ( t − s ) d s . Lemma A.6.
Let w ∈ W k,p with ≤ p < ∞ , then, for | α | ≤ k , D α w δx = ( D α w ) δx . Remark A.2.
This lemma is one conclusion during the proof of Theorem 1 in Section 5.3of [8] , which means that the α th order partial derivative of the smooth function w δx is themollification of the α th order weak partial derivative of w . Similarly, the regularizing in t hasthe same property. We directly write it as follows, if ∂ t w ∈ L q (0 , T ; L p ) , ≤ p, q < ∞ , then ∂ t w δt,x = ( ∂ t w ) δt,x . Lemma A.7 ([3], Theorem 4.22 in Chapter 4) . Let Ω ⊂ R n be an open set, K ⊂⊂ Ω , K = Ω if Ω = T n or R n . Then, for w ∈ L p (Ω) , ≤ p < ∞ , k w δx − w k L p ( K ) → as δ → . Lemma A.8.
Let Ω ⊂ R n be a bounded domain, K ⊂⊂ Ω , K = Ω if Ω = T n , and ( t , t ) ⊂⊂ (0 , T ) with any T ∈ (0 , ∞ ) . Then1. For ∂ t w ∈ L (0 , T ; L (Ω)) , ∇ w ∈ L ∞ (0 , T ; L (Ω)) , we have k w δt,x − w k L ∞ ( t ,t ; L ∞ ( K )) → as δ → .
2. For ∂ t w ∈ L (0 , T ; L (Ω)) , w ∈ L ∞ (0 , T ; L p (Ω)) with ≤ p < ∞ , we have k w δt,x − w k L ∞ ( t ,t ; L p ( K )) → as δ → . Proof.
1. By the definition of mollifiers in (3.1) and (3.2), and with some direct computations,it has w δt,x ( t, x ) − w ( t, x ) = ˆ R ˆ R n [ w ( t − s, x − y ) − w ( t, x )] η δ ( y ) d y ˜ η δ ( s ) d s = ˆ R ˆ R n [ w ( t − s, x − y ) − w ( t, x − y )] η δ ( y ) d y ˜ η δ ( s ) d s + ˆ R n [ w ( t, x − y ) − w ( t, x )] η δ ( y ) d y = ˆ R ˆ R n [ w ( t − δs, x − δy ) − w ( t, x − δy )] η ( y ) d y ˜ η ( s ) d s | {z } y := y ′ = δy , s := s ′ = δs + ˆ R n [ w ( t, x − δy ) − w ( t, x )] η ( y ) d y | {z } y := y ′ = δy = ˆ R ˆ R n ˆ dd τ w ( t − τ δs, x − δy ) d τ η ( y ) d y ˜ η ( s ) d s + ˆ R n ˆ dd τ w ( t, x − τ δy ) d τ η ( y ) d y = ˆ ˆ R ˆ R n ∂ t w ( t − τ δs, x − δy ) × ( − δs ) η ( y ) d y ˜ η ( s ) d s d τ | {z } Fubini’s theorem + ˆ ˆ R n ∇ w ( t, x − τ δy ) · ( − δy ) η ( y ) d y d τ | {z } Fubini’s theorem =: I + II . (A.1)For the first part I , noticing that supp η ( y ) = { y ∈ R n : | y | ≤ } and supp ˜ η ( s ) = { s ∈ R : | s | ≤ } , and by H¨older’s inequality, we have I ≤ Cδ ˆ ˆ { s : | s |≤ } ˆ { y : | y |≤ } | ∂ t w ( t − τ δs, x − δy ) | d y d s d τ = Cδ ˆ ˆ { s : | s |≤ τδ } ˆ { y : | y |≤ δ } | ∂ t w ( t − s, x − y ) | d y d s | {z } s := s ′ = τδs , y := y ′ = δy d τ ≤ Cδ ˆ ˆ [0 ,T ] ∪ ([0 ,T ] −{ s : | s |≤ δ } ) ˆ Ω ∪ (Ω −{ y : | y |≤ δ } ) | ∂ t w ( t, x ) | d x d t d τ ≤ Cδ k ∂ t w k L (0 ,T ; L (Ω)) , (A.2)where Ω − { y : | y | ≤ δ } = { x − y : x ∈ Ω , y ∈ R n , | y | ≤ δ } and [0 , T ] − { s : | s | ≤ δ } = { t − s : t ∈ [0 , T ] , s ∈ R , | s | ≤ } .Similarly, for the second part II , it holds II ≤ Cδ ˆ ˆ { y : | y |≤ } |∇ w ( t, x − τ δy ) | d y d τ ≤ Cδ ˆ ˆ Ω ∪ (Ω −{ y : | y |≤ δ } |∇ w ( t, x ) | d x d τ ≤ Cδ k∇ w ( t, · ) k L (Ω) . Therefore, k w δt,x − w k L ∞ ( t ,t ; L ∞ ( K )) ≤ Cδ k ∂ t w k L (0 ,T ; L (Ω)) + Cδ k∇ w k L ∞ (0 ,T ; L (Ω)) → δ → w δt,x ( t, x ) − w ( t, x ) = ˆ R ˆ R n [ w ( t − s, x − y ) − w ( t, x )] η δ ( y ) d y ˜ η δ ( s ) d s = ˆ ˆ R ˆ R n ∂ t w ( t − τ δs, x − δy ) × ( − δs ) η ( y ) d y ˜ η ( s ) d s d τ + ˆ R n [ w ( t, x − δy ) − w ( t, x )] η ( y ) d y ≤ Cδ k ∂ t w k L (0 ,T ; L (Ω)) + C ˆ { y : | y |≤ } | w ( t, x − δy ) − w ( t, x ) | d y . ISSIPATIVE SOLUTIONS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS 25
Using Minkowski’s integral inequality, then we arrive at k w δt,x ( t, x ) − w ( t, x ) k L ∞ ( t ,t ; L p ( K )) ≤ Cδ k ∂ t w k L (0 ,T ; L (Ω)) + C sup ≤ t ≤ T ˆ { y : | y |≤ } k w ( t, x − δy ) − w ( t, x ) k L p ( K ) d y . By Lemma A.4, it gives k w δt,x − w k L ∞ ( t ,t ; L p ( K )) → δ → . (cid:3) Finally, we recall two important lemmas to deal with the product of two weak convergencesequences.
Lemma A.9 ([16], Lemma C.1) . Let X be a reflexive Banach space, Y be a Banach space, X ֒ → Y , Y ′ is separable and dense in X ′ . Assume a sequence { f n } satisfies f n ∈ L ∞ (0 , T ; X ) and ∂ t f n ∈ L p (0 , T ; Y ) with < p ≤ ∞ . Then, f n is relatively compact in C w ([0 , T ]; X ) . Lemma A.10 ([17], Lemma 5.1) . Assume g n ⇀ g weakly in L p (0 , T ; L p ) , h n ⇀ h weakly in L q (0 , T ; L q ) , p + p = q + q = 1 , ≤ p , p ≤ ∞ , q > . In addition, ∂ t g n is uniformlybounded in L (0 , T ; W − k, ) for some k ≥ , and k h n ( x, t ) − h n ( x + ξ, t ) k L q (0 ,T ; L q ) → as | ξ | → for any n . Then, g n h n → gh in D ′ ((0 , T ) × Ω) . Acknowledgements:
Liang Guo is supported by NSFC (Grant No. 11671193). Fucai Li issupported in part by NSFC (Grant Nos.11971234 and 11671193) and a project funded by thepriority academic program development of Jiangsu higher education institutions. Cheng Yuis partially supported by Collaboration Grants for Mathematicians from Simons Foundation.
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Email address : [email protected] (Fucai Li) Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China
Email address : [email protected] (Cheng Yu) Department of Mathematics, University of Florida, Gainesville, FL 32611, UnitedStates of America
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