One New Blowup Criterion for the 2D Full Compressible Navier-Stokes System
aa r X i v : . [ m a t h . A P ] J a n One New Blowup Criterion for the 2D Full CompressibleNavier-Stokes System
Yun Wang ∗ Abstract
We establish a blowup criterion for the two-dimensional (2D) full compressibleNavier-Stokes system. The criterion is given in terms of the divergence of thevelocity field only, and is independent of the temperature. The criterion tells thatonce the strong solution blows up, the L ∞ -norm for the divergence of velocityblows up.Keywords: Full compressible Navier-Stokes system, blowup criterion, vacuum.AMS: 35Q35, 35B65, 76N10 The motion of compressible viscous, heat-conductive, ideal polytropic fluid is gov-erned by the full compressible Navier-Stokes system. Suppose that the domain occupiedby the fluid is Ω. The whole system on (0 , T ) × Ω consists of the following equations ρ t + div( ρu ) = 0 , ( ρu ) t + div( ρu ⊗ u ) − µ ∆ u − ( µ + λ ) ∇ div u + ∇ P = 0 ,c v [( ρθ ) t + div( ρuθ )] − κ ∆ θ + P div u = 2 µ | D ( u ) | + λ (div u ) . (1.1)together with the initial-boundary conditions( ρ, u, θ ) | t =0 = ( ρ , u , θ ) , (1.2) u = 0 , ∂θ∂~n = 0 , on ∂ Ω . (1.3)In this paper, we consider the two-dimensional case, i.e., Ω is a bounded smooth domainin R . ρ, u = (cid:0) u , u (cid:1) tr , θ and P = Rρθ ( R >
0) represent respectively the fluid density,velocity, absolute temperature and pressure. In addition, D ( u ) is the deformationtensor: D ( u ) = 12 ( ∇ u + ( ∇ u ) tr ) . The constant viscosity coefficients µ and λ satisfy the physical restrictions: µ > , µ + λ ≥ . (1.4) ∗ School of Mathematics, Soochow University, 1 Shizi Street, Suzhou, Jiangsu 215006, China( [email protected] ). c v and κ are respectively the heat capacity and the ratio of the heatconductivity coefficient over the heat capacity.There is a huge amout of literature on the existence and large time behavior ofsolutions to (1.1). For the case that the initial density is far away from vacuum, localexistence and uniqueness of classical solutions were proved in [21, 26, 28]. Matsumura-Nishida [25] first obtained the global classical solution when the initial data is closeto a non-vacuum equilibrium in some Sobolev space H s . Later, Hoff [8] constructed aglobal weak solution for discontinuous initial data, with the assumption that the initialdensity is strictly positive.Normally the presence of vacuum state makes the problem more complicated. Werecall some results about the weak solution for this case first. For the global weaksolution to the barotropic case, the major breakthrough is due to Lions [24] and subse-quently improved by Jiang-Zhang [20] and Feireisl [7]. They succeeded in constructinga global weak solution with finite energy when the pressure P = ρ γ , γ > N , where N is the dimension. Recently, Huang-Li [12] obtained the global weak solution to the fullNavier-Stokes system (1.1) provided the initial energy is suitably small. Strong solu-tions have also been under investigation. The first local existence result was derivedby Cho-Kim [4]. For a global classical solution with small energy, refer to [12, 16]. Onthe other hand, Xin [33] first contributed a remarkable blow-up result. He proved thatif the initial density has compact support, any smooth solution to the Cauchy problemof the full Navier-Stokes equations without heat conduction blows up in finite time. Inthis direction, for more recent progress, see [3, 27] and references therein.Taking into consideration both the local existence results and the blowup results,then it is important to study the mechanism of possible blowup and structure of possiblesingularity. For the blowup criterion for the compressible flow, there have been manyresults, [3,5,6,10,11,13–15,18,19,30–32]. It should been mentioned here that Huang-Li-Xin [15] first established a Beale-Kato-Majda type blowup criterion for the baratropiccase. In fact they proved that if T ∗ is the maximal time of existence for local strongsolution, then lim T → T ∗ Z T k∇ u k L ∞ dt = ∞ , (1.5)under the assumption 7 µ > λ when Ω is a three-dimensional domain. Jiang-Ou [19]extended this criterion to the full Navier-Stokes system (1.1) over a periodic domain orunit square domain of R and proved thatlim T → T ∗ Z T k∇ u k L ∞ dt = ∞ . (1.6)Recently, Huang-Li-Wang [13] obtained a Serrin type blow up criterion for (1.1) in R N .Here is the criterion,lim T → T ∗ Z T ( k div u k L ∞ + k u k sL r dt ) = ∞ , s + Nr = 1 , N < r ≤ ∞ . (1.7)which is analogue to the Serrin criterion for the 3D incompressible Navier-Stokes equa-tions. In particular, for N = 2, if one can bound a priorily k u k L (0 ,T ; L ∞ ) -norm or k u k L (0 ,T ; L ) -norm, then (1.7) can be replaced bylim T → T ∗ Z T k div u k L ∞ dt = ∞ . (1.8)If (1.8) is proved, it is an improvement of the work by Jiang-Ou [19] and it reveals someconnection between the compressible and incompressible Navier-Stokes equations, sinceglobal strong solutionwith vacuum has been proved for 2D incompressible case [17]. Thequestion is we can not get the uniform bound of k u k L (0 ,T ; L ∞ ) or k u k L (0 ,T ; L ) from thea priori energy estimate. The aim of our paper is to verify (1.8) and the key trick isthe use of Lemma 2.3 below, one critical Sobolev embedding inequality.The results such as (1.5) or (1.6) or (1.7), notice that they are all in terms of velocityfield only. There is another big class of results which are in terms of density ρ andtemprature θ . For example, Sun-Wang-Zhang [30] obtained the following criterion in3D, lim T → T ∗ sup ≤ t ≤ T {k ρ k L ∞ + k ρ − k L ∞ + k θ k L ∞ } ! = ∞ . (1.9)Fang-Zi-Zhang [6] extended the result to the 2D problem with a refiner form,lim T → T ∗ sup ≤ t ≤ T {k ρ k L ∞ + k θ k L ∞ } = ∞ . (1.10)Before stating our main result, we first explain the notations and conventions usedthroughout this paper. We denote Z f dx = Z Ω f dx. For 1 ≤ p ≤ ∞ and integer k ≥
0, the standard homogeneous and inhomogeneousSobolev spaces are denoted by: ( L p = L p (Ω) , W k,p = W k,p (Ω) , H k = W k, ,W ,p = { u ∈ W ,p | u = 0 on ∂ Ω } , H = W , . Let ˙ f := f t + u · ∇ f denote the material derivative of f .Since we are going to work with the blowup criterion of the strong solutions, we’dlike to recall the result for the existence of the local strong solution. The solution to the3D full Navier-Stokes system with vacuum was obtained by Cho-Kim [4]. The methodthere can be applied to the case in this paper, i.e. the case that Ω is a bounded domainin R . And the corresponding result can be stated as follows(or refer to [6]): Theorem 1.1
Let q ∈ (2 , ∞ ) be a fixed constant. Assume that the initial data satisfy ρ ≥ , ρ ∈ W ,q , u ∈ H ∩ H , θ ∈ H , with the compatibility conditions µ ∆ u + ( λ + µ ) ∇ div u − R ∇ ( ρ θ ) = ρ g , (1.11) κ ∆ θ + µ |∇ u + ( ∇ u ) tr | + λ (div u ) = ρ g , (1.12) for some g , g ∈ L . Then there exist a positive constant T and a unique strongsolution ( ρ, u, θ ) to the system (1.1)-(1.3)such that ρ ≥ , ρ ∈ C ([0 , T ]; W ,q ) , θ ∈ C ([0 , T ]; H ) , (1.13) u ∈ C ([0 , T ]; H ∩ H ) , ( u, θ ) ∈ L ([0 , T ]; W ,q ) , (1.14)( u t , θ t ) ∈ L ([0 , T ]; H ) , ( √ ρu t , √ ρθ t ) ∈ L ∞ ([0 , T ]; L ) . (1.15)Regarding the blowup criterion for the local strong solution, here is our main theo-rem. Theorem 1.2
Suppose the assumptions in Theorem 1.1 are satisfied and ( ρ, u, θ ) isthe strong solution. Let T ∗ be the maximal time of existence for that strong solution.If T ∗ < ∞ , then lim T → T ∗ k div u k L (0 ,T ; L ∞ ) = ∞ . (1.16)A few remarks are in order, Remark 1.1
It is worth noting that the conclusion in Theorem 1.2 is somewhat sur-prising since the criterion (1.16) is independent of the temperature and is the sameas that of barotropic case( [14]). In fact, it seems that the nonlinearity of the highlynonlinear terms | D ( u ) | and (div u ) in the temperature equation is stronger than thatof div( ρu ⊗ u ) in the momentum equations ( [31]), however, (1.16) shows that thenonlinear term |∇ u | can be controlled provided one can control div( ρu ⊗ u ) . Remark 1.2
It is well known that the 2D incompressible homogenenous Navier-Stokessystem has a unique global strong solution if the initial velocity belongs to L or somemore regular space, and recently it is proved in [17] that the 2D incompressible non-homogenous Navier-Stokes system also has a unique global strong solution under somecompatibility conditions, so the result in our paper is reasonable from this point. Theblowup criterion here shows that div u plays an important role in the fluid dynamics. Remark 1.3
The techniques in this paper can be easily adapted to the two dimensionalperiodic case. And the same criterion will be derived.
The rest of the paper is organized as follows: In Section 2, we collect some elementaryfacts and inequalities. The main result, Theorem 1.2, will be proved in Section 3.
In this section, we recall some known facts and elementary inequalities that will beused later.The first proposition is for the Lam´e system, which comes from the momentumequation (1 . . Assume that Ω ⊂ R is a bounded smooth domain. Suppose U ∈ H is a weak solution to the Lam´e system, ( µ ∆ U + ( µ + λ ) ∇ div U = F, in Ω ,U ( x ) = 0 , on ∂ Ω . (2.1)In what follows, we denote L U = µ ∆ U + ( µ + λ ) ∇ div U . Owing to the uniqueness ofsolution, we denote U = L − F .The system is an elliptic system under the assumption (1.4), hence some regularityestimates can be derived. For a proof, refer to [30]. Proposition 2.1
Let q ∈ (1 , ∞ ) . Then there exists some constant C depending onlyon λ, µ, p and Ω such that • if F ∈ L p , then k U k W ,p ≤ C k F k L p ; (2.2) • if F ∈ W − ,p (i.e., F = div f with f = ( f ij ) × , f ij ∈ L p ), then k U k W ,p ≤ C k f k L p . (2.3) Moreover, for the endpoint case, if f ij ∈ L ∞ ∩ L , then ∇ U ∈ BM O (Ω) and thereexists some constant C depending only on λ, µ, Ω such that k∇ U k BMO (Ω) ≤ C ( k f k L ∞ + k f k L ) . (2.4) Here k g k BMO (Ω) := k g k L + [ g ] BMO (Ω) , with [ g ] BMO (Ω) := sup x ∈ Ω , r ∈ (0 ,d ) | Ω r ( x ) | Z Ω r ( x ) | g ( y ) − g Ω r ( x ) | dy,g Ω r ( x ) = 1 | Ω r ( x ) | Z Ω r ( x ) g ( y ) dy, where Ω r ( x ) = B r ( x ) ∩ Ω and | Ω r ( x ) | denotes the Lebesgue measure of Ω r ( x ) . Two logarithmic Sobolev inequalities will be presented, which originate from thework owing to Brezis-Gallouet [1] and Brezis-Wainger [2]. The first one, together withProposition 2.1, will give the estimate of ∇ ρ . For its proof, see also [30]. Lemma 2.2
Let Ω be a bounded Lipschitz domain in R and f ∈ W ,p with p ∈ (2 , ∞ ) ,there exists a constant C depending only on p such that k f k L ∞ ≤ C (cid:0) k f k BMO (Ω) ln( e + k f k W ,p ) (cid:1) . (2.5)The second inequality is in terms of both space integral and time integral. The proofcan be found in [17] or refer to [23] for a similar proof. It plays an important role inthe proof of Lemma 3.3. Lemma 2.3
Let Ω be a smooth domain in R , and f ∈ L (0 , T ; H ∩ W ,p ) , with p > .Then there exists a constant C depending only on p such that k f k L (0 ,T ; L ∞ ) ≤ C (cid:16) k f k L (0 ,T ; H ) ln( e + k f k L (0 ,T ; W ,p ) ) (cid:17) . (2.6) Let ( ρ, u, θ ) be a strong solution described in Theorem 1.2. Suppose that (1.16) isfalse, i.e., lim T → T ∗ k div u k L (0 ,T ; L ∞ ) ≤ M < + ∞ , (3.1)which together with (1.1) yields immediately the following upper bound of the den-sity(see [14, Lemma 3.4]). Lemma 3.1
Assume that (3.1) holds. Then it is true that for ≤ T < T ∗ , sup ≤ t ≤ T k ρ k L ∞ ≤ C, (3.2) where and in what follows, C, C , C and C denote generic constants depending onlyon M , µ, λ, R, κ, c v , T ∗ , and the initial data. The next estimate is similar to an energy estimate.
Lemma 3.2
Under the assumption (3.1) , it holds that for ≤ T < T ∗ , sup ≤ t ≤ T Z (cid:0) ρθ + ρ | u | (cid:1) dx + Z T k∇ u k L dt ≤ C. (3.3) Proof.
Applying standard maximum principle to (1.1) together with θ ≥ R × [0 ,T ] θ ( x, t ) ≥ . (3.4)It follows from (1.1) that the specific energy E , c v θ + | u | satisfies( ρE ) t + div( ρEu + P u ) = ∆( κθ + 12 µ | u | ) + µ div( u · ∇ u ) + λ div( u div u ) . (3.5)Integrating (3.5) over Ω × [0 , T ] directly yieldssup ≤ t ≤ T Z (cid:0) ρθ + ρ | u | (cid:1) dx ≤ C. (3.6)Multiplying (1 . by u and integrating the resulting equation over Ω, we obtain afterusing (3.4) and (3.6) that12 ddt Z ρ | u | dx + µ k∇ u k L + ( µ + λ ) k div u k L ≤ C k div u k L ∞ Z ρθ dx ≤ C k div u k L ∞ , which together with (3.1) and (3.6) gives (3.3). It completes the proof of Lemma 3.2.For a slightly higher order estimate, we derive the bound for the L ∞ (0 , T ; L )-normof ∇ u, which will play a key role in obtaining more high order estimates. Lemma 3.3
Under the condition (3.1) , it holds that for ≤ T < T ∗ , sup ≤ t ≤ T Z (cid:0) ρθ + |∇ u | (cid:1) dx + Z T Z (cid:0) |∇ θ | + θ |∇ u | + ρ | ˙ u | (cid:1) dxdt ≤ C. (3.7)Before the proof of Lemma 3.3, we present an auxiliary lemma, which controls L p -norm of θ by k∇ θ k L . And it will help the proof of Lemma 3.3. Lemma 3.4
Under the condition (3.1) , it holds on [0 , T ∗ ) that for every p ∈ [1 , ∞ ) , k θ k L p ≤ C + C ( p ) k∇ θ k L . (3.8) Proof.
Denote by ¯ θ = | Ω | R θ dx , the average of θ , | ¯ θ | Z ρ dx ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z ρθdx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ρ (cid:0) θ − ¯ θ (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C k∇ θ k L , (3.9)which together with Poincar´e’s inequality implies k θ k L ≤ C + C k∇ θ k L . (3.10)And consequently, (3.8) holds. Proof of Lemma 3.3.
First, multiplying (1 . by θ and integrating the resultingequation over Ω lead to ddt Z ρθ dx + 2 κ k∇ θ k L ≤ C k div u k L ∞ Z ρθ dx + C Z θ |∇ u | dx. (3.11)To make the estimate close, one needs to bound the term R θ |∇ u | dx in (3.11). Toachieve that, we borrowed the idea from [13], multiplying (1.1) by uθ and integratingthe resulting equation over Ω. Then µ Z |∇ u | θ dx + ( µ + λ ) Z | div u | θ dx = − Z ρ ˙ u · uθ dx − µ Z u · ∇ u · ∇ θ dx − ( µ + λ ) Z div u ( u · ∇ θ ) dx − Z ∇ P · uθ dx. (3.12)We estimate the terms on the righthand of (3.12). By the Young’s inequality, (cid:12)(cid:12)(cid:12)(cid:12)Z ρ ˙ u · uθ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ η Z ρ | ˙ u | dx + C η Z ρθ | u | dx, (3.13)and (cid:12)(cid:12)(cid:12)(cid:12) µ Z u · ∇ u · ∇ θ dx + ( µ + λ ) Z div u · ( u · ∇ θ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ k∇ θ k L + C ǫ Z | u | |∇ u | dx, (3.14)where η, ǫ are small positive constants to be determined later. Using integration byparts, (cid:12)(cid:12)(cid:12)(cid:12)Z ∇ P uθ dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z P θ div u dx + Z P u · ∇ θ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ k∇ θ k L + C k div u k L ∞ Z ρθ dx + C ǫ Z ρ θ | u | dx. (3.15)Combining the estimates (3.11)-(3.15), we obtain after choosing ǫ suitably small that ddt Z ρθ dx + Z (cid:0) θ |∇ u | + κ |∇ θ | (cid:1) dx ≤ Cη Z ρ | ˙ u | dx + C k div u k L ∞ Z ρθ dx + C η Z (cid:0) ρθ | u | + | u | |∇ u | (cid:1) dx. (3.16)Note that there are some terms such as R ρ | ˙ u | dx whose estimates are not clear.These terms look less frightening than R θ |∇ u | dx , if one is familiar with the techniquesfor regularity estimates of compressible Navier-Stokes equation. One standard way is tomultiply (1 . by u t and integrate the resulting equation over Ω. Then by the Young’sinequality, we obtain that12 ddt Z (cid:2) µ |∇ u | + ( µ + λ )(div u ) (cid:3) dx + Z ρ | ˙ u | dx = Z ρ ˙ u · ( u · ∇ ) u dx + Z P div u t dx ≤ Z ρ | ˙ u | dx + C Z | u | |∇ u | dx + ddt Z P div u dx − Z P t div u dx. (3.17)To deal with the term R P t div u dx , we employ some technique which is a combinationof those from [30] and [32]. First, we split u into two parts, v and w . Let v = L − ∇ P and w = u − v . (In what follows, v and w always denote L − ∇ P and u − v . ) As notedin [30], div w acts as the effective viscous flux for the bounded domain case. Now Z P t div u dx = Z P t div v dx + Z P t div w dx. Herein Z P t div v dx = − Z ∇ P t v dx = − Z ( L v ) t v dx = 12 ddt Z | ( −L ) / v | dx, (3.18)and according to (3.5), Z P t div w dx = Rc v (cid:20)Z ( ρE ) t div w dx − Z
12 ( ρ | u | ) t div w dx (cid:21) = Rc v (cid:26)Z ( ρEu + P u − κ ∇ θ − µ ∇ u · u − µu · ∇ u − λu div u ) · ∇ div w dx + 12 Z div( ρu ) | u | div w dx − Z ρu t · u div w dx (cid:27) = Rc v (cid:26)Z (cid:18) ( c ν + R ) ρθu + 12 ρ | u | u − κ ∇ θ − µ ∇ u · u − µu · ∇ u − λu div u (cid:19) · ∇ div w dx − Z ρ | u | u · ∇ div w dx − Z ρ ˙ u · u div w dx (cid:27) = Rc v (cid:26)Z [( c v + R ) ρθu − κ ∇ θ − µ ∇ u · u − µu · ∇ u − λu div u ] · ∇ div w dx − Z ρ ˙ u · u div w dx (cid:27) . (3.19)By virtue of Proposition 2.1, we have k∇ v k L ≤ C k ρθ k L , (3.20)and k∇ w k L ≤ C k ρ ˙ u k L . (3.21)Making use of these two inequalities (3.20) and (3.21), we obtain that − Z P t div w dx ≤ C ( k√ ρθ k L k u k L ∞ + k∇ θ k L + k∇ u k L k u k L ∞ ) k∇ div w k L + C k√ ρ ˙ u k L k u k L ∞ k div w k L ≤ C δ k√ ρθ k L k u k L ∞ + C δ k∇ θ k L + C δ k∇ u k L k u k L ∞ + δ k√ ρ ˙ u k L . (3.22)Substituting (3.18) and (3.22) into (3.17), we obtain after choosing δ suitably smallthat ddt Z (cid:16) µ |∇ u | + ( µ + λ )(div u ) − P div u + | ( −L ) / v | (cid:17) dx + Z ρ | ˙ u | dx ≤ C k√ ρθ k L k u k L ∞ + C κ k∇ θ k L + C k∇ u k L k u k L ∞ . (3.23)Choose some constant C ≥ C + 1 suitably large such that µ |∇ u | − P div u + C ρθ ≥ µ |∇ u | + ρθ . Adding (3.16) multiplied by C to (3.23), we have after choosing η suitably small that ddt Z (cid:16) µ |∇ u | + ( µ + λ )(div u ) − P div u + C ρθ + | ( −L ) / v | (cid:17) dx + 12 Z ρ | ˙ u | dx + µ Z θ |∇ u | dx + κ Z |∇ θ | dx ≤ C k div u k L ∞ Z ρθ dx + C k u k L ∞ (cid:18)Z ρθ dx + k∇ u k L (cid:19) . (3.24)LetΨ( t ) = e + sup τ ∈ [0 ,t ] (cid:18) k∇ u ( τ ) k L + Z ρθ ( τ ) dx (cid:19) + Z t Z (cid:0) ρ | ˙ u | + θ |∇ u | + |∇ θ | (cid:1) dxdτ. By virtue of Gownwall’s inequality, for every 0 ≤ s ≤ T < T ∗ ,Ψ( T ) ≤ C Ψ( s ) exp (cid:26) C Z Ts k u ( τ ) k L ∞ dτ (cid:27) . (3.25)Now it is time to get a good control of k u k L ( s,T ; L ∞ ) . Making use of Lemma 2.3, wecan get that k u k L ( s,T ; L ∞ ) ≤ C (cid:16) k u k L ( s,T ; H ) ln (cid:0) e + k u k L ( s,T ; W , ) (cid:1)(cid:17) . (3.26)By Proposition 2.1 and Lemma 3.4, k u k W , ≤ C k w k W , + k v k W , ≤ C k ρ ˙ u k L + C k P k L + C k u k L ≤ C k ρ ˙ u k L + C k∇ θ k L + C k∇ u k L + C, k u k L ( s,T ; W , ) ≤ C k ρ ˙ u k L ( s,T ; L ) + C k∇ θ k L ( s,T ; L ) + C k∇ u k L ( s,T ; L ) + C ≤ C Ψ( T ) . (3.27)Substituting (3.27) to (3.26), k u k L ( s,T ; L ∞ ) ≤ C (cid:16) k u k L ( s,T ; H ) ln ( C Ψ( T )) (cid:17) . (3.28)Taking this inequality (3.28) back to (3.25), then we getΨ( T ) ≤ C Ψ( s )( C Ψ( T )) C k u k L s,T ; H . Recalling the energy like estimate (3.3), we choose some s which is close enough to T ∗ such that lim T → T ∗ − C k u k L ( s,T ; H ) ≤ , then Ψ( T ) ≤ C Ψ( s ) < ∞ , (3.29)which completes the proof for Lemma 3.3.Lemma 3.3 tells that lim T → T ∗ − k∇ u k L ∞ (0 ,T ; L ) < ∞ , which implies that lim T → T ∗ − k u k L (0 ,T ; L ) < ∞ . According Huang-Li-Wang [13]’s criterion (1.7), we can claim here that the strongsolution can be extended. For readers’ convenience, we give the complete proof. Theremaining proof consists of higher order estimates of the solutions which are needed toguarantee the extension of local strong solution to be a global one under the conditions(3.1) and (3.7). Compared to [13], there are some slight changes, since we consider thebounded case, instead of the whole space one.
Lemma 3.5
Under the condition (3.1) , it holds that for ≤ T < T ∗ , sup ≤ t ≤ T Z (cid:0) |∇ θ | + ρ | ˙ u | (cid:1) dx + Z T Z (cid:16) ρ ˙ θ + |∇ ˙ u | (cid:17) dxdt ≤ C. (3.30) Proof . First, applying ˙ u j [ ∂ t + div( u · )] to (1 . j and integrating the resulting equationover Ω, we obtain after integration by parts that12 ddt Z ρ | ˙ u | dx = − Z ˙ u j [ ∂ j P t + div( u∂ j P )] dx + µ Z ˙ u j [∆ u jt + div( u ∆ u j )] dx +( µ + λ ) Z ˙ u j [ ∂ j div u t + div( u∂ j div u )] dx = X i =1 N i . (3.31)1We get after integration by parts and using the equation (1 .
1) that N = − Z ˙ u j [ ∂ j P t + div( ∂ j P u )] dx = R Z ∂ j ˙ u j (cid:16) ρ ˙ θ − ρu · ∇ θ − θu · ∇ ρ − θρ div u (cid:17) dx + Z ∂ k ˙ u j ∂ j P u k dx = R Z ∂ j ˙ u j · ρ ˙ θ dx − Z P ∂ k ˙ u j ∂ j u k dx ≤ µ k∇ ˙ u k L + C k ρ ˙ θ k L + C Z ρ θ |∇ u | dx ≤ µ k∇ ˙ u k L + C k ρ ˙ θ k L + C k θ k L + C k∇ u k L . (3.32)Integration by parts leads to N = µ Z ˙ u j [ △ u jt + div( u △ u j )] dx = − µ Z (cid:16) ∂ i ˙ u j ∂ i u jt + △ u j u · ∇ ˙ u j (cid:17) dx = − µ Z (cid:16) |∇ ˙ u | − ∂ i ˙ u j u k ∂ k ∂ i u j − ∂ i ˙ u j ∂ i u k ∂ k u j + △ u j u · ∇ ˙ u j (cid:17) dx = − µ Z (cid:16) |∇ ˙ u | + ∂ i ˙ u j ∂ i u j div u − ∂ i ˙ u j ∂ i u k ∂ k u j − ∂ i u j ∂ i u k ∂ k ˙ u j (cid:17) dx ≤ − µ Z |∇ ˙ u | dx + C Z |∇ u | dx. (3.33)Similarly, we have N ≤ −
78 ( µ + λ ) k div ˙ u k L + C Z |∇ u | dx. (3.34)Substituting (3.32)-(3.34) into (3.31) implies ddt Z ρ | ˙ u | dx + µ k∇ ˙ u k L ≤ C Z ρ ˙ θ dx + C k θ k L + C Z |∇ u | dx ≤ C Z ρ ˙ θ dx + C k∇ θ k L + C k√ ρ ˙ u k L + C, (3.35)where for the last inequality we have used the fact, k∇ u k L ≤ k∇ v k L + k∇ w k L ≤ C k ρθ k L + C k ρ ˙ u k L / ≤ C k∇ θ k L + C k√ ρ ˙ u k L + C , (3.36)owing to Proposition 2.1 and Lemma 3.4.Next, multiplying (1 . by ˙ θ and integrating the resulting equation over Ω yieldthat c v Z ρ | ˙ θ | dx + κ ddt Z |∇ θ | dx = κ Z ∆ θ · ( u · ∇ θ ) dx + λ Z (div u ) ˙ θ dx + 2 µ Z | D ( u ) | ˙ θ dx − R Z ρθ div u ˙ θ dx , X i =1 I i . (3.37)2We estimate each I i ( i = 1 , · · · ,
4) as follows:First, it follows from Sobolev embedding theory that for any ǫ ∈ (0 , , Z θ |∇ u | dx ≤ C k θ k L ∞ k∇ u k L ≤ ǫ k∇ θ k L + C ǫ k∇ θ k L + C ǫ , (3.38)which together with the standard W , -estimate of (1 . gives k θ k H ≤ C Z ρ ˙ θ dx + C Z ρ θ |∇ u | dx + C Z |∇ u | dx + C k θ k L ≤ C Z ρ ˙ θ dx + Cǫ k∇ θ k L + C ǫ k∇ θ k L + C Z |∇ u | dx + C ǫ . Hence, choosing some ǫ small enough, we have k θ k H ≤ C Z ρ ˙ θ dx + C k∇ θ k L + C Z |∇ u | dx + C. (3.39)Consequently, by Gagliardo-Nirenberg inequality, | I | ≤ C Z |∇ θ ||∇ θ || u | dx ≤ C k∇ θ k L k∇ θ k L k u k L ≤ C k∇ θ k L (cid:0) k∇ θ k L + k∇ θ k L (cid:1) / k∇ θ k / L ≤ δ k∇ θ k L + C δ k∇ θ k L ≤ Cδ Z ρ ˙ θ dx + C δ k∇ θ k L + C δ Z |∇ u | dx + C δ . (3.40)Next, integration by parts yields that, for any η, δ ∈ (0 , ,I = λ Z (div u ) θ t dx + λ Z (div u ) u · ∇ θ dx = λ (cid:18)Z (div u ) θ dx (cid:19) t − λ Z θ div u div( ˙ u − u · ∇ u ) dx + λ Z (div u ) u · ∇ θ dx = λ (cid:18)Z (div u ) θ dx (cid:19) t − λ Z θ div u div ˙ u dx + 2 λ Z θ div u div( u · ∇ u ) dx + λ Z (div u ) u · ∇ θ dx = λ (cid:18)Z (div u ) θdx (cid:19) t − λ Z θ div u div ˙ u dx + 2 λ Z θ div u∂ i u j ∂ j u i dx + λ Z u · ∇ (cid:0) θ (div u ) (cid:1) dx ≤ λ (cid:18)Z (div u ) θ dx (cid:19) t + C k θ |∇ u |k L (cid:0) k∇ ˙ u k L + k∇ u k L (cid:1) ≤ λ (cid:18)Z (div u ) θ dx (cid:19) t + η k∇ ˙ u k L + Cδ Z ρ ˙ θ dx + C δ,η k∇ θ k L + C δ k∇ u k L + C δ,η , (3.41)3where in the last inequality, we have used (3.38) and (3.39).Then, similar to (3.41), we have that, for any η, δ ∈ (0 , ,I ≤ µ (cid:18)Z | D ( u ) | θ dx (cid:19) t + η k∇ ˙ u k L + Cδ Z ρ ˙ θ dx + C δ,η k∇ θ k L + C δ k∇ u k L + C δ,η . (3.42)Finally, it follows from (3.38) and (3.39) that | I | ≤ δ Z ρ ˙ θ dx + C δ Z θ |∇ u | dx ≤ Cδ Z ρ ˙ θ dx + C δ k∇ θ k L + C δ . (3.43)Substituting (3.40)-(3.43) into (3.37), we obtain after choosing δ suitably small that,for any η ∈ (0 , ,ddt Z (cid:16) κ |∇ θ | − θ (cid:2) λ (div u ) + 2 µ | D ( u ) | (cid:3)(cid:17) dx + c v Z ρ ˙ θ dx ≤ Cη k∇ ˙ u k L + C η k∇ u k L + C η k∇ θ k L + C η ≤ Cη k∇ ˙ u k L + C η k√ ρ ˙ u k L + C η k∇ θ k L + C η , (3.44)where the last inequality is owing to (3.36).Noticing that Z θ (cid:2) λ (div u ) + 2 µ | D ( u ) | (cid:3) dx ≤ C k θ k L k∇ u k L / ≤ C ( k∇ θ k L + 1) · k∇ u k / L k∇ u k / L ≤ C ( k∇ θ k L + 1) (cid:16) k√ ρ ˙ u k / L + k ρθ k / L + 1 (cid:17) ≤ κ k∇ θ k L + η / k√ ρ ˙ u k L + C η , (3.45)so adding (3.35) multiplied by 2 η / to (3.44), we obtain (3.30) after choosing η suitablysmall and using Gronwall’s inequality. Thus we complete the proof of Lemma 3.5.As a corollary, we can bound k θ k L and k∇ u k L . Corollary 3.6
Under the condition (3.1) , it holds that for ≤ T < T ∗ , sup ≤ t ≤ T ( k θ k L + k∇ u k L ) ≤ C. (3.46) Proof . By virtue of Lemma 3.4 and Lemma 3.5, k θ k L ≤ C k∇ θ k L + C ≤ C. Consequently, according to (3.36) and Lemma 3.5, k∇ u k L ≤ C k ρ ˙ u k L + C k ρθ k L + C k∇ u k L + C ≤ C. Next, we will derive the desired estimates for ˙ θ . In fact, we have4 Lemma 3.7
Under the condition (3.1) , it holds that for ≤ T < T ∗ , sup ≤ t ≤ T Z ρ ˙ θ dx + Z T k∇ ˙ θ k L dt ≤ C. (3.47) Proof . Applying the operator ∂ t + div( u · ) to (1.1) , and using (1.1) , we get c v ρ (cid:16) ∂ t ˙ θ + u · ∇ ˙ θ (cid:17) = κ ∆ ˙ θ + κ (div u ∆ θ − ∂ i ( ∂ i u · ∇ θ ) − ∂ i u · ∇ ∂ i θ ) − Rρ ˙ θ div u − Rρθ div ˙ u + (cid:0) λ (div u ) + 2 µ | D ( u ) | (cid:1) div u + 2 λ (cid:16) div ˙ u − ∂ k u l ∂ l u k (cid:17) div u + µ ( ∂ i u j + ∂ j u i ) (cid:16) ∂ i ˙ u j + ∂ j ˙ u i − ∂ i u k ∂ k u j − ∂ j u k ∂ k u i (cid:17) . (3.48)Multiplying (3.48) by ˙ θ, we obtain after integration by parts and Corollary 3.6 that c v (cid:18)Z ρ | ˙ θ | dx (cid:19) t + κ k∇ ˙ θ k L ≤ C Z |∇ u | (cid:16) |∇ θ || ˙ θ | + |∇ θ ||∇ ˙ θ | (cid:17) dx + C Z |∇ u | | ˙ θ ||∇ u | dx + C Z ρ | ˙ θ | |∇ u | dx + C Z ρθ |∇ ˙ u || ˙ θ | dx + C Z |∇ u ||∇ ˙ u || ˙ θ | dx ≤ C k∇ u k L k∇ θ k L k ˙ θ k L + C k∇ u k L k∇ θ k L k∇ ˙ θ k L + C k∇ u k L k ˙ θ k L + C k∇ u k L k√ ρ ˙ θ k L k ˙ θ k L + C k√ ρ ˙ θ k L k∇ ˙ u k L k θ k L ∞ + C k∇ u k L k∇ ˙ u k L k ˙ θ k L ≤ κ k∇ ˙ θ k L + C k∇ θ k L k ˙ θ k L + C k ˙ θ k L + C k√ ρ ˙ θ k L k ˙ θ k L + C k√ ρ ˙ θ k L k∇ ˙ u k L k θ k L ∞ + C k∇ ˙ u k L k ˙ θ k L + C. (3.49)It follows from (3.39) and Lemma 3.5 that k∇ θ k L ≤ C k√ ρ ˙ θ k L + C k∇ θ k L + C k∇ u k L + C ≤ C k√ ρ ˙ θ k L + C. (3.50)For the estimate for k ˙ θ k L , we will follow the method used in Lemma 3.4. Let ¯˙ θ = | Ω | R ˙ θ dx , ¯˙ θ Z ρ dx ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z ρ (cid:16) ˙ θ − ¯˙ θ (cid:17) dx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ρ ˙ θ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ ˙ θ k L + C k√ ρ ˙ θ k L , (3.51)which together with Poincar´e’s inequality leads to k ˙ θ k L ≤ C k∇ ˙ θ k L + C (cid:12)(cid:12)(cid:12) ¯˙ θ (cid:12)(cid:12)(cid:12) ≤ C k∇ ˙ θ k L + C k√ ρ ˙ θ k L . (3.52)And k θ k L ∞ can be estimated as follows, k θ k L ∞ ≤ C k∇ θ k L + C k θ k L ≤ C k√ ρ ˙ θ k L + C. (3.53)5Substituting (3.50)-(3.53) to (3.49), we arrive at c v (cid:18)Z ρ | ˙ θ | dx (cid:19) t + κ k∇ ˙ θ k L ≤ C Z ρ | ˙ θ | dx + C k√ ρ ˙ θ k L k∇ ˙ θ k L + C k∇ ˙ θ k L + C k√ ρ ˙ θ k L + C k∇ ˙ u k L Z ρ | ˙ θ | dx + C k∇ ˙ u k L k√ ρ ˙ θ k L + C k∇ ˙ u k L + 14 k∇ ˙ θ k L + C, (3.54)which together with the Gronwall’s inequality completes the proof for Lemma 3.7.As a corollary, the bounds for k θ k H and k θ k L ∞ can be derived. Corollary 3.8
Under the condition (3.1) , it holds that for ≤ T < T ∗ , sup ≤ t ≤ T ( k θ k H + k θ k L ∞ ) ≤ C. (3.55) Proof . First, it follows from (3.39), Lemma 3.5, Corollary 3.6 and Lemma 3.7 that k∇ θ k L ≤ C. (3.56)Hence, k θ k L ∞ ≤ C k θ k H ≤ C. (3.57)Up to now, we have get the bounds for k ρ k L ∞ and k θ k L ∞ , which imply other nec-essary high order estimates for the extension of the strong solution, according to thetheorem proved in [6]. We sketch the proof for completeness. Corollary 3.9
Under the condition (3.1) , it holds that for ≤ T < T ∗ , sup ≤ t ≤ T k w k H + Z T (cid:0) k∇ w k L p + k∇ w k L ∞ (cid:1) dt ≤ C, p ∈ (2 , ∞ ) . (3.58) Proof . By virtue of Proposition 2.1 and Lemma 3.5, k w k H ≤ C k ρ ˙ u k L ≤ C, and by Sobolev embedding inequality, k∇ w k L ∞ ≤ C k∇ w k W ,p ≤ C k ρ ˙ u k L p ≤ C k∇ ˙ u k L , which implies (3.58).The next lemma is used to bound the density gradient and k u k H . Lemma 3.10
Under the condition (3.1) , it holds that for ≤ T < T ∗ , sup ≤ t ≤ T ( k ρ k W ,q + k u k H ) ≤ C. (3.59)6 Proof . For 2 ≤ p ≤ q , |∇ ρ | p satisfies the following equation( |∇ 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 . Hence, ddt k∇ ρ k L p ≤ C (1 + k∇ u k L ∞ ) k∇ ρ k L p + C k∇ u k L p ≤ C (1 + k∇ v k L ∞ + k∇ w k ∞ ) k∇ ρ k L p + C (cid:0) k∇ v k L p + k∇ w k L p (cid:1) ≤ C (1 + k∇ v k L ∞ + k∇ w k ∞ ) k∇ ρ k L p + C k∇ w k L p + C, (3.60)where for the last inequality we used the fact k∇ v k L p ≤ C k∇ ( ρθ ) k L p ≤ C k∇ ρ k L p k θ k L ∞ + C k∇ θ k L p k ρ k L ∞ ≤ C k∇ ρ k L p + C. (3.61)To bound k∇ v k L ∞ , we make use of the endpoint case of Proposition 2.1, Lemma 2.2and (3.61), k∇ v k L ∞ ≤ C (cid:0) k∇ v k BMO (Ω) ln( e + k∇ v k W ,p ) (cid:1) ≤ C (1 + ( k P k L ∞ + k P k L ) ln( e + k∇ v k W ,p )) ≤ C (1 + ln( e + k∇ ρ k L p )) . (3.62)Substituting (3.62) into (3.60), we get that ddt ( e + k∇ ρ k L p ) ≤ C (1 + k∇ w k L ∞ ) k∇ ρ k L p + C ln( e + k∇ ρ k L p ) k∇ ρ k L p + C k∇ w k L p + C, (3.63)which together with Gronwall’s inequality and Corollary 3.9 gives thatsup
The author would like to thank Professor Jing Li and Dr.Xiangdi Huang for introducing this topic and all the helpful discussions. The author ispartially supported by NSF of China under Grant 11241004.
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