Strong solutions to the Cauchy problem of the two-dimensional compressible Navier-Stokes-Smoluchowski equations with vacuum
aa r X i v : . [ m a t h . A P ] A ug Strong solutions to the Cauchy problem of the two-dimensionalcompressible Navier-Stokes-Smoluchowski equations with vacuum ∗ Yang Liu † School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Abstract.
This paper studies the local existence of strong solutions to the Cauchy problemof the 2D fluid-particle interaction model with vacuum as far field density. Notice that the tech-nique used by Ding et al. [14] for the corresponding 3D local well-posedness of strong solutionsfails treating the 2D case, because the L p -norm ( p >
2) of the velocity u cannot be controlledin terms only of √ ρu and ∇ u here. In the present paper, we will use the framework of weightedapproximation estimates introduced in [J. Li, Z. Liang, On classical solutions to the Cauchyproblem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum,J. Math. Pures Appl. (2014) 640–671] for Navier-Stokes equations to obtain the local existenceof strong solutions provided the initial density and density of particles in the mixture do notdecay very slowly at infinity. In particular, the initial density can have a compact support. Thispaper extends Fang et al.’s result [16] and Ding et al.’s result [14], in which, the existence isobtained when the space dimension N = 1 and N = 3 respectively. Keywords : strong solutions, Cauchy problem, compressible Navier-Stokes-Smoluchowskiequaitons, vacuum, two-dimensional space
MSC (2010) : 35Q35, 46E35, 76N10.
In this paper, we consider a fluid-particle interaction model called as Navier-Stokes-Smoluchowskiequations in [3, 8, 9], which in the whole spatial domain R as follows ρ t + div( ρu ) = 0 , ( ρu ) t + div( ρu ⊗ u ) + ∇ ( p F + η ) = µ ∆ u + ( λ + µ ) ∇ div u − ( η + βρ ) ∇ Φ ,η t + ∇ · ( η ( u − ∇ Φ)) = ∆ η, (1.1)in R × R + , with the far-field behavior( ρ, u, η )( x, t ) → (0 , ,
0) as | x | → ∞ , t > , (1.2) ∗ This work was supported by excellent doctorial dissertation cultivation grant from Dalian University ofTechnology. † Corresponding author. E-mail address: [email protected]. ρ ( x,
0) = ρ ( x ) , ρu ( x,
0) = m , η ( x,
0) = η , x ∈ R . (1.3)Here ρ : R × [0 , ∞ ) → R + is the density of the fluid, u : R × [0 , ∞ ) → R the velocity field,and the density of the particles in the mixture η : (0 , ∞ ) × R → R + is related to the probabilitydistribution function f ( t, x, ξ ) in the macroscopic description through the relation η ( t, x ) = Z R f ( t, x, ξ ) dξ. We also denote by p F the pressure of the fluid, given by p F = p F ( ρ ) = aρ γ , a > , γ > , (1.4)and the time independent external potential Φ = Φ( x ) : R → R + is the effects of gravity andbuoyancy, β is a constant reflecting the differences in how the external force affects the fluid andthe particles, λ and µ are constant viscosity coefficients satisfying the physical condition: µ > , λ + µ ≥ . (1.5)The fluid-particle interaction model arises in a lot of industrial procedures such as the analysisof sedimentation phenomenon which finds its applications in biotechnology, medicine, chemicalengineering, and mineral processes. Such interaction systems are also used in combustion theory,when modeling diesel engines or rocket propulsors, see [6, 7, 24, 25]. The system consists in aVlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled tothe equations for the fluid. Generally speaking, at the microscopic scale, the cloud of particlesis described by its distribution function f ( t, x, ξ ), solution to a Vlasov-Fokker-Planck equation.The fluid, on the other hand, is modeled by macroscopic quantities, namely its density ρ ( x, t ) ≥ u ( x, t )(see [8]). If the fluid is compressible and isentropic, then ( ρ, u ) solvesthe compressible Euler (inviscid case) or Navier-Stokes system (viscous case) of equations. Withthe dynamic viscosity terms taken into consideration, system (1.1) was derived formally byCarrillo and Goudon [9]. They obtained the global existence and asymptotic behavior of theweak solutions to (1.1) following the framework of Lions [22] and Feireisl et al. [17, 18]. Withoutthe dynamic viscosity terms in (1.1) , Carrillo and Goudon [8] gave the flowing regime and thebubbling regime under the two different scaling assumptions and investigated the stability andasymptotic limits finally. In dimension one, Fang et al. [16] proved the global existence anduniqueness of the classical large solution with vacuum. In dimension three, Ballew obtainedthe local in time existence of strong solutions in a bounded domain with the no-flux conditionfor the particle density in [3, 4] and studied Low Mach Number Limits under the confinementhypotheses for the spatial domain and external potential Φ in [5]. Recently, motivated by Kim etal. [10–12] on the Navier-Stokes equations, Ding et al. [14] obtained the local classical solutionsof system (1.1) with vacuum in R .When the density of the fluid η = 0, the system (1.1) becomes Navier-Stokes equationsfor the isentropic compressible fluids. Kim et al. proved some local existence results on strongsolutions in a domain of R in [10,11] and the radially symmetric solutions in an annular domainin [13]. Ding et al. [15] obtained global classical solutions with large initial data with vacuumin a bounded domain or exterior domain Ω of R n ( n ≥ R , Cho and Kim also got the local classical solutions [12], in which the initial density needs2ot be bounded below away from zero. For the case that the initial density is allowed to vanish,Huang et al. [19] obtained the global existence of classical solutions to the Cauchy problem forthe isentropic compressible Navier-Stokes equations in three spatial dimensions with smoothinitial data provided that the initial energy is suitably small. Recently, assumed that the initialdensity do not decay very slowly at infinity, Li and Liang [21] have obtained the local existenceof the classical solutions to the two-dimensional Cauchy problem. After that, Li and Xin [20]extended the result of Li and Liang [21] to the global ones, and also get some decay estimatesof solutions.The aim of this paper is to establish the local existence of strong solutions to the Cauchyproblem (1.1) in dimension two. Notice that the local well-posedness of strong solutions fordimension three case established by Ding et al. [14] is not admitted for the case of dimensiontwo. This is mainly due to that in dimension two we fail to control the L p -norm ( p >
2) of thevelocity u in terms only of √ ρu and ∇ u . Moreover, the coupling of u, η and Φ, and the presenceof ∇ · ( ηu − η ∇ Φ) bring additional difficulties. So, some new ideas and careful estimates arenecessary to deal with the two dimension case. In the present paper, we will use the frameworkof weighted approximation estimates introduced in [21] for Navier-Stokes equations to overcomethese difficulties.
Definition 1.1
If all derivatives involved in (1.1) for ( ρ, u, η ) are regular distributions, andequations (1.1) hold almost everywhere in R × (0 , T ) , then ( ρ, u, η ) is called a strong solutionto (1.1) . In this section, for 1 ≤ r ≤ ∞ , we denote the standard Lebesgue and Sobolev spaces asfollows: L r = L r ( R ) , W s,r = W s,r ( R ) , H s = W s, . (1.6)Denote ¯ x , (e + | x | ) / log σ (e + | x | ) , with σ > B N , { x ∈ R | | x | < N } . The main result of this paper is stated as the followingtheorem: Theorem 1.1
Suppose that the initial data ( ρ , u , η ) satisfy ρ ≥ , ¯ x a ρ ∈ L ∩ H ∩ W ,q , ∇ u ∈ L , ∇ η ∈ L , ¯ x a η ∈ L , Φ ∈ H , √ ρ u ∈ L , with q > and a > . Then there exist T , N > such that the problem (1.1) – (1.3) has a uniquestrong solution ( ρ, u, η ) on R × (0 , T ] satisfying ρ ∈ C ([0 , T ]; L ∩ H ∩ W ,q ) , ¯ x a ρ ∈ L ∞ (0 , T ; L ∩ H ∩ W ,q ) , √ ρu, ∇ u, ¯ x − u, √ t √ ρu t ∈ L ∞ (0 , T ; L ) , ∇ u ∈ L (0 , T ; H ) ∩ L q +1 q (0 , T ; W ,q ) , √ t ∇ u ∈ L (0 , T ; W ,q ) ,η, ∇ η, ¯ x a η, √ tη t ∈ L ∞ (0 , T ; L ) , ∇ η ∈ L (0 , T ; H ) , √ t ∇ u ∈ L (0 , T , W ,q ) , √ ρu t , ¯ x a ∇ η, √ t ∇ u t , √ t ∇ η t , √ t ¯ x − u t ∈ L ( R × (0 , T )) , (1.7) and inf ≤ t ≤ T Z B N ρ ( x, t ) dx ≥ Z R ρ ( x, t ) dx. (1.8)3he rest of the paper is organized as follows: In Section 2, we recall some elementary factsand inequalities used in the sequel. Sections 3 deals with an approximation problem (2.2) on B R to derive uniform estimates for the unique strong solution with respect to R . Finally, theproof of Theorem 1.1 will be given in Section 4. Firstly, the follow local existence theory on bounded ball B R , { x ∈ R : | x | < R } , where theinitial density is strictly away from vacuum, can be shown by arguments as in [14]. Lemma 2.1
For any given
R > and B R = { x ∈ R || x | < R } , assume that ( ρ , u , η ) satisfies ( ρ , u , η ) ∈ H ( B R ) , Φ ∈ H ( B R ) , inf x ∈ B R ρ > . (2.1) Then there exist a small time T R > and a unique classical solution ( ρ, u, η ) to the followinginitial-boundary-value problem ρ t + div( ρu ) = 0 , ( ρu ) t + div( ρu ⊗ u ) + ∇ ( p F + η ) = µ ∆ u + ( λ + µ ) ∇ div u − ( η + βρ ) ∇ Φ − R − u,η t + ∇ · ( η ( u − ∇ Φ)) = ∆ η,u = 0 , ( ∇ η + η ∇ Φ) · n = 0 , x ∈ ∂B R , t > , ( ρ, u, η )( x,
0) = ( ρ , u , η )( x ) , x ∈ B R , (2.2) on B R × (0 , T R ] such that ρ ∈ C ([0 , T R ]; H ) , ρ t ∈ L ∞ (0 , T R ; H ) , √ ρu t ∈ L ∞ (0 , T R ; L ) , ( u, η ) ∈ C ([0 , T R ]; H ) ∩ L (0 , T R ; H ) , ( u t , η t ) ∈ L ∞ (0 , T R ; H ) ∩ L (0 , T R ; H ) , ( √ tu, √ tη ) ∈ L ∞ (0 , T R ; H ) , ( √ tu t , √ tη t ) ∈ L ∞ (0 , T R ; H ) , ( √ tu tt , √ tη tt ) ∈ L (0 , T R ; H ) , √ t √ ρu tt ∈ (0 , T R ; L ) , ( tu t , tη t ) ∈ L ∞ (0 , T R ; H ) , ( tu tt , tη tt ) ∈ L ∞ (0 , T R ; H ) ∩ L (0 , T R ; H ) ,t √ ρu ttt ∈ L ∞ (0 , T R ; L ) , ( t u tt , t η tt ) ∈ L ∞ (0 , T R ; H ) ,t √ ρu ttt ∈ L ∞ (0 , T R ; L ) , ( t u ttt , t η ttt ) ∈ L (0 , T R ; H ) , (2.3)where we denote L = L ( B R ) and H k = H k ( B R ) for positive integer k .Next, for either Ω = R or Ω = B R with R ≥
1, the following weighted L p -bounds forelements of the Hilbert space ˜ D , (Ω) , { v ∈ H (Ω) |∇ v ∈ L (Ω) } can be found in [23, theoremB.1]. Lemma 2.2
For m ∈ [2 , ∞ ) and θ ∈ (1 + m/ , ∞ ) , there exists a positive constant C suchthat for either Ω = R or Ω = B R with R ≥ and for any v ∈ ˜ D , (Ω) , (cid:18)Z Ω | v | m e + | x | (log( e + | x | )) − θ dx (cid:19) m ≤ C k v k L ( B ) + C k∇ v k L (Ω) . (2.4) Lemma 2.3 (Lemma 2.4 in [21])
Let ¯ x and σ be as in Theorem 1.1 with Ω = R or Ω = B R ,and ρ ∈ L (Ω) ∩ L γ (Ω) with γ > be a non-negative function satisfying Z B N ρdx ≥ M , Z Ω ρ γ dx ≤ M , ith M , M > , and B N ⊂ Ω ( N ≥ ). Then for every v ∈ ˜ D , (Ω) , there is C = C ( M , M , N , γ, σ ) > such that k v ¯ x − k L (Ω) ≤ C k√ ρv k L (Ω) + C k∇ v k L (Ω) . (2.5) Moreover, for ε > and σ > there is C = C ( ε, η, M , M , N , γ, σ ) > such that every v ∈ ˜ D , (Ω) satisfies k v ¯ x − σ k L ε ˜ σ (Ω) ≤ C k√ ρv k L (Ω) + C k∇ v k L (Ω) , (2.6) with ˜ σ = min { , σ } . Next, the following L p -bound for elliptic systems, whose proof is similar to that of [10, lemma12], is a direct consequence of the combination of a well-known elliptic theory due to Agmon-Douglis-Nirenberg [1, 2] with a standard scaling procedure. Lemma 2.4
For p > and k ≥ , there exists a positive constant C depending only on p and k such that k∇ k +2 v k L p ( B R ) ≤ C k ∆ v k W ,p ( B R ) , (2.7) for every v ∈ W k +2 ,p ( B R ) satisfying either v · n = 0 , rot v = 0 , on ∂B R , or v = 0 , on ∂B R . Throughout this section and the next, for p ∈ [1 , ∞ ] and k ≥
0, we denote Z f dx = Z B R f dx, L p = L p ( B R ) , W k,p = W k,p ( B R ) , H k = W k, . Moreover, for
R > N ≥
4, assume that ( ρ , u , η ) satisfies, in addition to (2.1), that12 ≤ Z B N ρ ( x ) dx ≤ Z B R ρ ( x ) dx ≤ . (3.1)Lemma 2.1 thus yields that there exists some T R > ρ, u, η ) on B R × [0 , T R ] satisfying (2.3).For ¯ x, σ , a and q as in theorem 1.1, the main aim of this section is to derive the followingkey a priori estimate on ψ defined by ψ ( t ) , k√ ρu k L + k∇ u k L + k∇ η k L + k ¯ x a η k L + k ¯ x a ρ k L ∩ H ∩ W ,q + R − k u k L . (3.2)5 roposition 3.1 Assume that ( ρ , u , η ) satisfies (2.1) and (3.1) . Then there exist T , M > , both depending only on µ, γ, q, a, η , N , and E , such that sup ≤ t ≤ T ψ ( t ) + Z T ( k∇ u k q +1 q L q + t k∇ u k L q + k∇ u k L + k∇ η k L ) dt ≤ M. (3.3) where E , k√ ρ u k L + k∇ u k L + k∇ η k L + k ¯ x a ρ k L ∩ H ∩ W ,q + k ¯ x a η k L , To prove proposition 3.1, whose proof will be postponed to the end of this section, we beginwith the following standard energy estimate for ( ρ, u, η ). Lemma 3.1
Let ( ρ, u, η ) be a smooth solution to the initial-boundary value problem (2.2) . Thenthere exists T = T ( N , E ) > such that for all t ∈ (0 , T ]sup ≤ s ≤ t Z h ρ | u | + aγ − ρ γ + η ln η + ( βρ + η )Φ i dx + Z t Z h |∇ u | + | ∇√ η + √ η ∇ Φ | i dxds ≤ C, (3.4) and moreover, sup ≤ s ≤ t Z η dx + Z t Z |∇ η | dxds ≤ C. (3.5) where and throughout the paper, denote by C generic positive constants depending only on thefixed constants µ, λ, γ, β, a, q, σ , N , E , and k Φ k H ( R ) . Proof.
First, multiplying (2.2) by u , integrating the resulting equation over B R and usingEq.(2.2) , we have ddt Z h ρ | u | + aγ − ρ γ i dx + Z h µ |∇ u | + ( µ + λ ) | div u | + u · ∇ η + ( βρ + η ) ∇ Φ · u + R − | u | i dx = 0 , (3.6)where we have used the fact Z ρ γ ∇ · udx = Z ρ γ − ρ ∇ · udx = − Z ( ρ t + u · ∇ ρ ) ρ γ − = − ddt Z ρ γ γ dx + Z ρ γ γ ∇ · u, so that − a Z ρ γ ∇ · u = ddt Z aρ γ γ − . Using (2.2) and (2.2) , we have Z ( η + βρ ) ∇ Φ · udx = − Z div( ηu ) dx − Z β div( ρu )Φ dx = ddt Z βρφdx + Z h η t − ∇ · ( η ∇ φ ) − ∆ η i Φ dx ddt Z ( η + βρ )Φ dx + Z ( η ∇ Φ + ∇ η ) ∇ Φ dx. (3.7)Multiplying (2.2) by log η , integrating the resulting equation over B R , and using the boundarycondition (2.2) , one deduces that Z η t log ηdx − Z h ηu − η ∇ Φ − ∇ η i ∇ ηη dx = ddt Z η log ηdx − Z h u · ∇ η − ∇ Φ · ∇ η − |∇ η | η i dx = 0 . (3.8)Substituting (3.7) and (3.8) into (3.6), we immediately complete the proof of (3.4).Next, multiplying (2.2) by η , integrating the resulting equation over B R , using boundarycondition (2.2) , we have12 ddt Z η dx + Z |∇ η | dx = Z η ( u − ∇ Φ) ∇ ηdx = Z ηu ∇ ηdx − Z η ∇ Φ ∇ ηdx = − Z div uη dx − Z η ∇ Φ ∇ ηdx ≤ Z |∇ η | dx + C Z | η | |∇ u | dx + C Z η |∇ Φ | dx ≤ Z |∇ η | dx + C Z η |∇ Φ | dx + C k∇ u k L k η k L ≤ Z |∇ η | dx + C Z η |∇ Φ | dx + C k∇ u k L k η k L k∇ η k L ≤ Z |∇ η | dx + C Z η dx + C k∇ u k L k η k L . (3.9)According to energy inequality (3.4), we have R t R |∇ u | dxds ≤ C . Thus, we can use Gronwall’sinequality to deduce that sup ≤ s ≤ t Z η dx + Z t Z |∇ η | dxds ≤ C. (3.10) Lemma 3.2
Under the conditions of Proposition 3.1, let ( ρ, u, η ) be a smooth solution tothe initial-boundary value problem (2.1) - (2.2) . Then there exists T = T ( N , E ) > and α = α ( γ, q ) > such that for all t ∈ (0 , T ]sup ≤ s ≤ t k ¯ x a η k L + Z t k ¯ x a ∇ η k L ds ≤ C, (3.11)sup ≤ s ≤ t ( k∇ u k L + k∇ η k L ) + Z t ( k√ ρu t k L + k η t k L + k ∆ η k L ) ds ≤ C Z t ψ α ds + C. (3.12) Proof.
First, we always assume that t ≤ T . The conservation of ρ with (2.2) yields thatthere exists T > ≤ t ≤ T Z B N ρdx ≥ , (3.13)7hat is (3.8) in [21]. Furthermore, corresponding to (3.10) obtained in [21], we have by (3.4),(3.13), and Lemma 2.3 that k ρ η u k L ε ˜ σ + k u ¯ x − η k L ε ˜ σ ≤ C ( ε, σ ) ψ σ , t ∈ (0 , T ] (3.14)with ˜ σ = min { , σ } .Next, to obtain (3.11), multiplying (2.2) by η ¯ x a and integrating by parts yield12 (cid:16) Z | η | ¯ x a dx (cid:17) t + Z |∇ η | ¯ x a dx = 12 Z η ∆¯ x a dx − Z ∇ · ( ηu − η ∇ Φ) η ¯ x a dx = 12 Z η ∆¯ x a dx − Z ηu · ∇ η ¯ x a dx − Z η ( ∇ · u )¯ x a dx + Z η ∇ η · ∇ Φ¯ x a dx + Z η ∆Φ¯ x a dx ≤ C Z | η | ∆¯ x a dx + C Z | η | |∇ u | ¯ x a dx + C Z | η | | u | · ∇ ¯ x a dx + C Z | η | | ∆Φ | ¯ x a dx + C Z | η | |∇ Φ |∇ ¯ x a dx , X i =1 I i . (3.15)Direct calculations yield that I ≤ C Z | η | ¯ x a ¯ x − log σ ) ( e + | x | ) dx ≤ C Z | η | ¯ x a dx, (3.16) I ≤ C Z |∇ u || η | ¯ x a dx ≤ C k∇ u k L k η ¯ x a k L ≤ C k∇ u k L k η ¯ x a k L ( k∇ η ¯ x a k L + k η ∇ ¯ x a k L ) ≤ C ( k∇ u k L + 1) k η ¯ x a k L + 14 k∇ η ¯ x a k L , (3.17) I ≤ C Z ¯ x a | η | ¯ x − | u | ¯ x − log σ ( e + | x | ) dx ≤ C k η ¯ x a k L k η ¯ x a k L k u ¯ x − k L ≤ C k η ¯ x a k L + C k η ¯ x a k L ( k√ ρu k L + k∇ u k L ) ≤ C (1 + k∇ u k L ) k η ¯ x a k L + 14 k∇ η ¯ x a k L , (3.18) I + I ≤ C Z | η | ¯ x a dx + C Z | η | ¯ x a ¯ x − log σ ( e + | x | ) dx ≤ C Z | η | ¯ x a dx. (3.19)Putting (3.17)-(3.19) into (3.15), after using Gronwall’s inequality and (3.4), we havesup ≤ s ≤ t k ¯ x a η k L + Z t k ¯ x a ∇ η k L dx ≤ C exp n C Z t (1 + k∇ u k L ) ds o ≤ C. (3.20)8ext, to prove (3.12), multiplying Eqs. (2.2) by u t and integration by parts yield12 ddt Z h (2 µ + λ )(div u ) + µω + R − | u | i dx + Z ρ | u t | dx ≤ C Z ρ | u | |∇ u | dx + 2 Z ( p F + η )div u t dx − Z ( βρ + η ) ∇ Φ · u t dx, (3.21)where ω , rot u is defined in the following (3.40).We estimate each term on the right-hand side of (3.21) as follows:First, the Gagliardo-Nirenberg inequality implies that for all p ∈ (2 , + ∞ ), k∇ u k L p ≤ C ( p ) k∇ u k /pL k∇ u k − /pH ≤ C ( p ) ψ + C ( p ) ψ k∇ u k − /pL , (3.22)which together with (3.14) yields that for σ > σ = min { , σ } , Z ρ σ | u | |∇ u | dx ≤ C k ρ σ/ u k L / ˜ σ k∇ u k L / (4 − ˜ σ ) ≤ C ( σ ) ψ σ (1 + k∇ u k ˜ σ/ L ) ≤ C ( ε, σ ) ψ α ( σ ) + εψ − k∇ u k L . (3.23)Next, noticing that p F satisfies p F t + div( p F u ) + ( γ − p F div u = 0 . (3.24)we deduce from (2.2) and the Sobolev inequality that2 Z p F div u t dx = 2 ddt Z p F div udx − Z p F u · ∇ div udx + 2( γ − Z p F (div u ) dx ≤ ddt Z p F div udx + εψ − k∇ u k L + C ( ε ) ψ α . (3.25)Moreover, we have − Z ( ∇ η + η ∇ Φ) · u t dx − Z βρ ∇ Φ · u t dx = ddt Z ( ∇ η + η ∇ Φ) · udx − Z ( ∇ η + η ∇ Φ) t · udx − Z βρ ∇ Φ · u t dx = ddt Z ( η ∇ · u − η ∇ Φ · u ) dx − Z η t ∇ · udx + Z η t ∇ Φ · udx − Z βρ ∇ Φ · u t dx = ddt Z ( η ∇ · u − η ∇ Φ · u ) dx − Z η t ∇ · udx + Z (∆ η − ∇ · ( ηu − η ∇ Φ)) ∇ Φ · udx − Z βρ ∇ Φ · u t dx = ddt Z ( η ∇ · u − η ∇ Φ · u ) dx − Z η t ∇ · udx − Z ∇ u · ∇ η · ∇ Φ dx − Z u · ∇ η · ∆Φ dx + Z ηu · ∇ u · ∇ Φ dx − Z βρ ∇ Φ · u t dx Z ηu · ∆Φ dx − Z ηu · ∇ Φ · ∆Φ dx − Z η ∇ u |∇ Φ | dx, , ddt J + X i =1 J i . (3.26)Direct calculations yield that J ≤ C Z | η t ||∇ u | dx ≤ k η t k L + C Z |∇ u | dx, (3.27) J ≤ C k∇ Φ k L ∞ Z |∇ u ||∇ η | dx ≤ C k∇ η k L + C k∇ u k L , (3.28) J ≤ C Z | u | ¯ x − a |∇ η | ¯ x a ∆Φ dx ≤ C k ∆Φ k L k u ¯ x − a k L k∇ η ¯ x a k L ≤ C k ¯ x a ∇ η k L + C (1 + k∇ u k L ) , (3.29) J ≤ C Z | η | ¯ x a | u | ¯ x − a |∇ u ||∇ Φ | dx ≤ C k∇ Φ k L ∞ k η ¯ x a k L k u ¯ x − a k L k∇ u k L ≤ C k η ¯ x a k L + C (1 + k∇ u k L ) k∇ u k L k∇ u k L ≤ Cψ α + εψ − k∇ u k L , (3.30) J ≤ C k∇ Φ k L ∞ Z √ ρ √ ρu t dx ≤ Z ρ | u t | dx + C k∇ Φ k L ∞ Z ρ dx ≤ Z ρ | u t | dx + C, (3.31) J ≤ C Z | η | ¯ x a | u | ¯ x − a | ∆Φ | dx ≤ C k ∆Φ k L ∞ k η ¯ x a k L k u ¯ x − a k L ≤ C k ∆Φ k L ∞ k η ¯ x a k L ( k√ ρu k L + k∇ u k L ) ≤ Cψ α , (3.32) J ≤ C k ∆Φ k L ∞ Z | η | ¯ x a | u | ¯ x − a |∇ Φ | dx ≤ C k η ¯ x a k L k u ¯ x − a k L k∇ Φ k L ≤ C k ¯ x a ∇ η k L + Cψ α , (3.33) J ≤ C k∇ Φ k L ∞ Z | η ||∇ u | dx ≤ C k∇ Φ k L ∞ k η k L k∇ u k L ≤ Cψ α + C. (3.34)Substituting (3.27)-(3.40) into (3.26), and combining (3.26) and (3.21) lead to12 ddt Z h (2 µ + λ )(div u ) + µω + R − | u | i dx + Z ρ | u t | dx ddt B ( t ) + εψ − k∇ u k L + C k ¯ x a ∇ η k L + Cψ α , (3.35)where B ( t ) = − Z p F div udx + Z ( η ∇ · u − η ∇ Φ u ) dx ≤ µ k∇ u k L + C k p F k L + C k η k L + C k∇ Φ k L k ¯ x a η k L k u ¯ x − a k L ≤ µ k∇ u k L + C k p F k L + C, (3.36)owing to (3.4), (3.5), (3.20) and (3.14).Moreover, multiplying the equation (2.2) by η t and integrating the result equation withrespect to x over B R , we have12 ddt k∇ η k L + k η t k L = Z ∇ · ( η ( u − ∇ Φ)) · η t dx ≤ k η t k L + C Z | u | |∇ η | dx + C Z | η | | div u | dx + C Z |∇ η | |∇ Φ | dx + C Z | η | | ∆Φ | dx ≤ k η t k L + C Z | u | ¯ x − a |∇ η | ¯ x a |∇ η | dx + C k η k L k∇ u k L + C Z |∇ η | dx + C ≤ k η t k L + C k ¯ x − a u k L k ¯ x a ∇ η k L k∇ η k L + C k η k L k∇ η k L k∇ u k L k∇ u k H + C Z |∇ η | dx + C ≤ k η t k L + εψ − k∇ u k L + C k ¯ x a ∇ η k L + Cψ α k∇ η k L + C ( ε ) ψ α ≤ k η t k L + 12 k∇ η k L + εψ − k∇ u k L + C k ¯ x a ∇ η k L + C ( ε ) ψ α . (3.37)From (2.2) , taking it by L -norm, using Gagliardo-Nirenberg inequality, we get k ∆ η k L ≤ C k η t k L + C k∇ · ( ηu − η ∇ Φ) k L ≤ C k η t k L + C ( k u ∇ η k L + k η ∇ u k L + k∇ Φ k L ∞ k∇ η k L + k ∆Φ k L ∞ k η k L ) ≤ C k η t k L + C k ¯ x − a u k L k ¯ x a ∇ η k L k∇ η k L + C k η k L k∇ η k L k∇ u k L k∇ u k H + C k∇ η k L + C ≤ C k η t k L + 12 k∇ η k L + εψ − k∇ u k L + C k ¯ x a ∇ η k L + C ( ε ) ψ α (3.38)Finally, to estimate the last term on the right-hand side of (3.37), (3.38), and (3.35), denoting ∇ ⊥ , ( ∂ , − ∂ ), we rewrite the momentum equation (2.2) as R − u + ρ ˙ u = ∇ F + µ ∇ ⊥ ω − ( η + βρ ) ∇ Φ , (3.39)where ˙ f , f t + u · ∇ f, F , (2 µ + λ )div u − p F − η, ω , ∇ ⊥ · u (3.40)11re the material derivative of f , the effective viscous flux and the vorticity respectively. Thus,(3.39) implies that ω satisfies (cid:26) µ ∆ ω = ∇ ⊥ (cid:0) ρ ˙ u + ( η + βρ ) ∇ Φ + R − u (cid:1) , in B R ,ω = 0 , on ∂B R . (3.41)Applying the standard L p -estimate to (3.41) yields that, for p ∈ (1 , ∞ ), k∇ ω k L p ≤ C ( ρ )( k ρ ˙ u k L p + k ( η + βρ ) ∇ Φ k L p + R − k u k L p ) , which together with (3.39) gives k∇ F k L p + k∇ ω k L p ≤ C ( ρ )( k ρ ˙ u k L p + k ( η + βρ ) ∇ Φ k L p + R − k u k L p ) , (3.42)It follows from (2.7) and (3.42) that for p ∈ [2 , q ], k∇ u k L p ≤ C k∇ ω k L p + C k∇ div u k L p ≤ C ( k∇ ω k L p + k (2 µ + λ ) ∇ div u k L p ) ≤ C ( k ρ ˙ u k L p + k ( η + βρ ) ∇ Φ k L p + k∇ p F k L p + k∇ η k L p + R − k u k L p ) , (3.43)which together with (3.5), (3.22) and (3.23) leads to k∇ u k L ≤ C p ψ k√ ρu t k L + C k ρu · ∇ u k L + Cψ α ≤ C p ψ k√ ρu t k L + Cψ α + 12 k∇ u k L . (3.44)Putting (3.44) into (3.37), (3.38), and (3.35), integrating the resulting inequality over (0 , t ) andchoosing ε suitably small yield R − k u k L + k∇ u k L + k∇ η k L + Z t ( k√ ρu t k L + k η t k L + k∇ η k L ) ds ≤ C + C k p F k L + C Z t ψ α ds + C Z t k ¯ x a ∇ η k L ds ≤ C + C k p F k L + C Z t ψ α ds ≤ C + C Z t ψ α ds, (3.45)where we have used (3.15) and the following estimate k p F k L ≤ k p F ( ρ ) k L + C Z t k p F k / L k p F k / L ∞ k∇ u k L ds ≤ C + C Z t ψ α ds (3.46)due to (3.24). The proof of lemma 3.2 is completed. Lemma 3.3
Let ( ρ, u, η ) and T be as in Lemma 3.2. Then, for all t ∈ (0 , T ] , sup ≤ s ≤ t s k ¯ x a ∇ η k L + Z t s k ¯ x a ∆ η k L ds ≤ C exp n Z t ψ α ds o , (3.47)sup ≤ s ≤ t s k√ ρu t k L + Z t s (cid:0) k∇ u t k L + R − | u t | (cid:1) ds ≤ C exp n C Z t ψ α ds o . (3.48)12 roof. Differentiating (2.2) with respect to t gives ρu tt + ρu · ∇ u t − µ ∇ ⊥ ω t − ∇ ((2 µ + λ )div u t ) + R − u t = − ρ t ( u t + u · ∇ u ) − ρu t · ∇ u − ∇ ( p F t + η t ) − ( η t + βρ t ) ∇ Φ . (3.49)Multiplying (3.49) by u t and integrating the resulting equation over B R , we obtain after using(2.2) that 12 ddt Z ρ | u t | dx + Z (cid:0) (2 µ + λ )(div u t ) + µω t + R − | u t | (cid:1) dx = − Z ρu · ∇ u t · u t dx − Z ρu · ∇ ( u · ∇ u · u t ) dx − Z ρu t · ∇ u · u t dx + Z ( p F t + η t )div u t dx − Z ( η t + βρ t ) ∇ Φ · u t dx , Ψ( t ) + Z η t div u t dx − Z ( η t + βρ t ) ∇ Φ · u t dx. (3.50)By the arguments (3.27)–(3.31) for the proof of Lemma 3.3 in [21], it follows from (3.4), (3.5)and (3.14) for ε ∈ (0 ,
1) thatΨ( t ) ≤ ε k∇ u t k L + C ( ε ) ψ α ( k∇ u k L + k ρ / u t k L + 1) . (3.51)On the other hand, Z η t div u t dx − Z ( η t + βρ t ) ∇ Φ u t dx = Z (∆ η − ∇ · ( η ( u − ∇ Φ))) · div u t dx − Z (∆ η − ∇ · ( η ( u − ∇ Φ)) − β div( ρu )) ∇ Φ · u t dx ≤ C Z | ∆ η ||∇ u t | dx + C Z | u ||∇ η ||∇ u t | dx + C Z η |∇ u ||∇ u t | dx + C Z |∇ η ||∇ u t ||∇ Φ | dx + C Z η |∇ u t || ∆Φ | dx + C Z |∇ η || ∆Φ || u t | dx + C Z η | u || ∆Φ || u t | dx + C Z η | u ||∇ Φ ||∇ u t | dx + C Z η |∇ Φ || ∆Φ || u t | dx + C Z η |∇ Φ | |∇ u t | dx + C Z |∇ ρ || u ||∇ Φ || u t | dx + C Z ρ |∇ u ||∇ Φ || u t | dx , X i =1 R i . (3.52)Using Gagliardo-Nirenberg and H¨older’s inequalities, we get R ≤ C k ∆ η k L k∇ u t k L ≤ k∇ u t k L + C k η t k L + εψ − k∇ u k L + C k ¯ x a ∇ η k L + C ( ε ) ψ α , (3.53) R ≤ C Z | u | ¯ x − a |∇ η | ¯ x a |∇ u t | dx ≤ C k u ¯ x − a k L k ¯ x a ∇ η k L k∇ u t k L k∇ u t k L + C k u ¯ x − a k L k ¯ x a ∇ η k L k ¯ x a ∇ η k L ≤ k∇ u t k L + 18 k ¯ x a ∇ η k L + Cψ α k ¯ x a ∇ η k L , (3.54) R ≤ C k∇ u t k L k η k L k∇ u k L ≤ C k∇ u t k L k∇ η k L k η k L k∇ u k L k∇ u k L ≤ k∇ u t k L + εψ − k∇ u k L + C ( ε ) ψ α , (3.55) R ≤ C k∇ Φ k L ∞ k∇ η k L k∇ u t k L ≤ k∇ u t k L + C ( ε ) ψ α , (3.56) R ≤ C k ∆Φ k L ∞ k η k L k∇ u t k L ≤ k∇ u t k L + C, (3.57) R ≤ C Z ¯ x a |∇ η | ¯ x − a | u t || ∆Φ | dx ≤ C k ¯ x a ∇ η k L k u t ¯ x − a k L k ∆Φ k L ≤ C k ¯ x a ∇ η k L k ¯ x a ∆ η k L ( k√ ρu t k L + k∇ u t k L ) ≤ k∇ u t k L + C k ¯ x a ∇ η k L k ¯ x a ∆ η k L + C k√ ρu t k L , (3.58) R ≤ C Z ¯ x a η ¯ x − a | u || ∆Φ | ¯ x − a | u t | dx ≤ C k ∆Φ k L ∞ k ¯ x a η k L k u ¯ x − a k L k u t ¯ x − a k L ≤ C k ¯ x a η k L ( k√ ρu k L + k∇ u k L )( k√ ρu t k L + k∇ u t k L ) ≤ k∇ u t k L + Cψ α k√ ρu t k L + Cψ α , (3.59) R ≤ C Z ¯ x a η ¯ x − a | u ||∇ Φ ||∇ u t | dx ≤ C k∇ Φ k L ∞ k ¯ x a η k L k u ¯ x − a k L k∇ u t k L ≤ C k ¯ x a η k L k ¯ x a ∇ η k / L ( k√ ρu k L + k∇ u k L ) k∇ u t k L ≤ k∇ u t k L + C k ¯ x a ∇ η k L + Cψ α , (3.60) R ≤ C Z ¯ x a η |∇ Φ || ∆Φ | ¯ x − a | u t | dx ≤ C k ∆Φ k L ∞ k∇ Φ k L k ¯ x a η k L k u t ¯ x − a k L ≤ C k ¯ x a η k L k ¯ x a ∇ η k L ( k√ ρu t k L + k∇ u t k L ) ≤ k∇ u t k L + C k ¯ x a ∇ η k L + Cψ α k√ ρu t k L + Cψ α ,R ≤ C k∇ Φ k L ∞ k η k L k∇ u t k L ≤ k∇ u t k L + C, (3.61) R ≤ C Z ¯ x a |∇ ρ | ¯ x − a | u ||∇ Φ | ¯ x − a | u t | dx ≤ C k∇ Φ k L ∞ k ¯ x a ∇ ρ k L k u ¯ x − a k L k u t ¯ x − a k L ≤ Cψ α ( k√ ρu k L + k∇ u k L )( k√ ρu t k L + k∇ u t k L )14 k∇ u t k L + Cψ α k√ ρu t k L + Cψ α , (3.62) R ≤ C Z ρ |∇ Φ || u t ||∇ u | dx ≤ C k ρ k L ∞ k∇ Φ k L ∞ k√ ρu t k L k∇ u k L ≤ Cψ α k√ ρu t k L + Cψ α . (3.63)Substituting (3.53)-(3.63) into (3.50), and we get12 ddt Z ρ | u t | dx + Z (cid:0) (2 µ + λ )(div u t ) + µω t + R − | u t | (cid:1) dx ≤ Cψ α (1 + k√ ρu t k L + k∇ u k L ) + Cψ α k ¯ x a ∇ η k L + 12 k ¯ x a ∆ η k L ≤ k ¯ x a ∆ η k L + Cψ α k√ ρu t k L + Cψ α k ¯ x a ∇ η k L + C k η t k L + Cψ α , (3.64)where in the last inequality we have used (3.44).Next, we should estimate k ¯ x a ∆ η k L . Indeed, multiplying (2.2) by ∆ η ¯ x a , integrating theresult equation by parts yields that12 (cid:16) Z |∇ η | ¯ x a dx (cid:17) t + Z | ∆ η | ¯ x a dx = − Z η t · ∇ η ∇ ¯ x a dx + Z ∇ · ( ηu − η ∇ Φ) · ∆ η ¯ x a dx = − Z (∆ η − ∇ · ( ηu − η ∇ Φ)) · ∇ η ∇ ¯ x a dx + Z ∇ · ( ηu − η ∇ Φ) · ∆ η ¯ x a dx ≤ C Z |∇ η || ∆ η ||∇ ¯ x a | dx + C Z |∇ η | | u ||∇ ¯ x a | dx + C Z η |∇ η ||∇ u ||∇ ¯ x a | dx + C Z |∇ η | |∇ Φ ||∇ ¯ x a | dx + C Z η |∇ η || ∆Φ ||∇ ¯ x a | dx + C Z |∇ u ||∇ η | ¯ x a dx + C Z η |∇ u || ∆ η | ¯ x a dx + C Z |∇ η ||∇ Φ || ∆ η | ¯ x a dx + C Z η | ∆Φ || ∆ η | ¯ x a dx , X i =1 S i . (3.65)Using the Gagliardo-Nirenberg inequality, (3.15), ( ?? ), (3.4) and (3.5), we get S ≤ C Z ¯ x a |∇ η | ¯ x a | ∆ η | ¯ x − log σ ( e + | x | ) dx ≤ ε k ¯ x a ∆ η k L + C k ¯ x a ∇ η k L , (3.66) S ≤ C Z |∇ η | a − a ¯ x a − |∇ η | a | u | ¯ x − ¯ x − log σ ( e + | x | ) dx ≤ C k ¯ x a − |∇ η | a − a k L a a − k u ¯ x − k L a k|∇ η | a k L a ≤ Cψ α k ¯ x a ∇ η k L + ε k ¯ x a ∆ η k L , (3.67) S ≤ C Z ¯ x a η |∇ u | ¯ x a |∇ η | ¯ x − log σ ( e + | x | ) dx C k ¯ x a η k L k∇ u k L k ¯ x a ∇ η k L ≤ C k ¯ x a η k L + C k∇ u k L + C k ¯ x a ∇ η k L ≤ C k ¯ x a η k L ( k ¯ x a ∇ η k L + k ¯ x a η k L ) + C k∇ u k L + C k ¯ x a ∇ η k L ≤ Cψ α ( k ¯ x a ∇ η k L + k∇ u k L ) , (3.68) S ≤ C Z |∇ η | |∇ Φ | ¯ x a ¯ x − log σ ( e + | x | ) dx ≤ C k∇ Φ k L ∞ k ¯ x a ∇ η k L , (3.69) S ≤ C Z ¯ x a η | ∆Φ | ¯ x a |∇ η | ¯ x − log σ ( e + | x | ) dx ≤ C k ∆Φ k L ∞ k ¯ x a η k L + C k ¯ x a ∇ η k L (3.70) S ≤ C k∇ u k L k ¯ x a ∇ η k L ≤ C k∇ u k L k ¯ x a ∇ η k L ( k ¯ x a ∇ η k L + k ¯ x a ∆ η k L ) ≤ ε k ¯ x a ∆ η k L + C ( ε ) ψ α k ¯ x a ∇ η k L , (3.71) S ≤ C k ¯ x a ∆ η k L k ¯ x a η k L k∇ u k L ≤ ε k ¯ x a ∆ η k L + C k ¯ x a η k L k ¯ x a ∇ η k L k∇ u k L k∇ u k H ≤ ε k ¯ x a ∆ η k L + Cψ − k∇ u k L + Cψ α k ¯ x a ∇ η k L , (3.72) S + S ≤ C ( k∇ Φ k L ∞ k ¯ x a ∇ η k L + k ∆Φ k L ∞ k ¯ x a η k L ) k ¯ x a ∆ η k L ≤ ε k ¯ x a ∆ η k L + C k ¯ x a ∇ η k L + Cψ α . (3.73)Substituting (3.66)-(3.73) into (3.65) and choosing ε suitably small lead to12 ddt k ¯ x a ∇ η k L + k ¯ x a ∆ η k L ≤ ε k ¯ x a ∆ η k L + Cψ α k ¯ x a ∇ η k L + Cψ − k∇ u k L + Cψ α . (3.74)Thus, multiplied (3.74) by s , together with Gronwall’s inequality, we getsup ≤ s ≤ t s k ¯ x a ∇ η k L + Z t s k ¯ x a ∆ η k L ds ≤ C exp n Z t ψ α ds o , (3.75)due to (3.44) and (3.12).Now, multiplying (3.64) by t , we obtain (3.48) using Gronwall’s inequality and (3.75). Theproof of Lemma 3.3 is completed. Lemma 3.4
Let ( ρ, u, η ) and T be as in Lemma 3.2. Then, for all t ∈ (0 , T ] , sup ≤ s ≤ t k ¯ x a ρ k L ∩ H ∩ W ,q ≤ exp n C exp n Z t ψ α ds oo . (3.76) Proof.
Notice that following the framework of Lemma 3.4 in [21] for proving an estimatesimilar to (3.76), it suffices to verify the following estimate: Z t (cid:0) k∇ u k q +1 q L ∩ L q + s k∇ u k L ∩ L q (cid:1) ds ≤ C exp n C Z t ψ α ds o . (3.77)16n fact, on the one hand, it follows from (3.44), (3.48) and (3.38) that Z t (cid:0) k∇ u k L + s k∇ u k L (cid:1) ds ≤ C Z t ( ψ α + k√ ρu t k L ) ds + C sup ≤ s ≤ t (cid:0) s k√ ρu t k L (cid:1) Z t ψds ≤ C exp n C Z t ψ α ds o . (3.78)On the other hand, choosing p = q in (3.43), using the Gagliardo-Nirenberg inequality gives k∇ u k L q ≤ C ( k ρ ˙ u k L p + k ( η + βρ ) ∇ Φ k L p + k∇ p F k L p + k∇ η k L p ) ≤ C ( k ρu t k L q + k ρu k L q k∇ u k L q + k η k L p + ψ α + k∇ η k L p ) ≤ C k ρu t k q − q − L k ρu t k q − qq − L + Cψ α (1 + k∇ u k − q L + k∇ η k − q L ) ≤ Cψ α ( k√ ρu t k q − q − L k∇ u t k q − qq − L + k√ ρu t k L ) + Cψ α (1 + k∇ u k − q L + k∇ η k − q L ) . (3.79)This combined with (3.78) (3.48), (3.4), and (3.5) Z t k∇ u k q +1 q L q ds ≤ C Z t ψ α s − q +12 q ( s k√ ρu t | L ) q − q ( q − ( s k∇ u t k L ) ( q − q +1)2( q − ds + C Z t ψ α k√ ρu t k q +1 q L ds + C Z t (cid:16) k∇ u k q − q L + k∇ η k q − q L (cid:17) ds ≤ C sup ≤ s ≤ t ( s k√ ρu t k L ) q − q ( q − Z t ψ α s − q +12 q ( s k∇ u t k L ) ( q − q +1)2( q − ds + C Z t (cid:16) ψ α + k√ ρu t k L + k∇ u k L + k∇ η k L (cid:17) ds ≤ C exp n C Z t ψ α ds oh Z t (cid:16) ψ α + s − q q − q − q q − q + s k∇ u t k L (cid:17) ds i ≤ C exp n C Z t ψ α ds o , (3.80)and that Z t s k∇ u k L q ds ≤ C exp n C Z t ψ α ds o . (3.81)One thus obtains (3.77) from (3.78)- and completes the proof of lemma 3.4.Now, proposition 3.1 is a direct consequence of lemmas 3.1-3.4. Proof of proposition 3.1.
It follows from (3.76), (3.4), (3.5), and (3.11) and that ψ ( t ) ≤ exp n C exp n C Z t ψ α ds oo . Standard arguments thus yield that for M , e Ce and T , min { T , ( CM α ) − } ,sup ≤ t ≤ T ψ ( t ) ≤ M, which together with (3.44), (3.77) and (3.12). The proof of Proposition 3.1 is thus completed.17 emma 3.5 Let ( ρ, u, η ) be a smooth solution to the initial-boundary-value problem (2.2) ,and T is obtained in proposition (3.1) , then we have sup ≤ s ≤ T (cid:0) s k η t k L + s k ∆ η k L (cid:1) + Z T s k∇ η t k L ds ≤ C. (3.82) Proof.
Differentiating (2.2) with respect to t shows η tt + ∇ · ( η t u + ηu t − η t ∇ Φ) − ∆ η t = 0 , (3.83)Multiplying (3.83) by η t and then integrating equation over B R , integrating by parts, we have12 ddt Z | η t | dx + Z |∇ η t | dx = Z ( η t u + ηu t − η t ∇ Φ) · ∇ η t dx ≤ C Z | η t | |∇ u | dx + C Z η | u || ∆Φ ||∇ η t | dx + C Z η | u t ||∇ η t | dx + C Z | η t ||∇ Φ ||∇ η t | dx , X i =1 K i . (3.84)Using the H¨older’s inequality, Gagliardo-Nirenberg inequality, we have K ≤ C k η t k L k∇ u k L ≤ C k η t k L k∇ η t k L k∇ u k L ≤ ε k∇ η t k L + C ( ε ) k η t k L , (3.85) K ≤ Z η ¯ x a | u | ¯ x a |∇ Φ ||∇ η t | dx ≤ C k∇ Φ k L ∞ k∇ η t k L k ¯ x a η k L k u ¯ x − a k L ≤ ε k∇ η t k L + C k ¯ x a ∇ η k L , (3.86) K ≤ C k∇ η t k L k η | u t |k L ≤ ε k∇ η t k L + C Z η ¯ x a η | u t | ¯ x − a dx ≤ ε k∇ η t k L + C k η k L k u t ¯ x − a k L k ¯ x a η k L ≤ ε k∇ η t k L + C ( ε ) k∇ u t k L + C ( ε ) k ρ u t k L , (3.87) K ≤ C k∇ Φ k L ∞ k η t k L k∇ η t k L ≤ ε k∇ η t k L + C ( ε ) k η t k L . (3.88)Now, putting (3.85) and (3.88) into (3.84), and multiplying the resulting inequality by s , wehave after choosing ε suitably small that ddt (cid:0) s k η t k L (cid:1) + s k∇ η t k L ≤ C ( s k η t k L ) + C ( k η t k L + s k ¯ x a ∇ η k L + s k∇ u t k L + s k ρ u t k L ) , (3.89)which together with Gronwall’s inequality and (3.38) yields thatsup ≤ s ≤ T (cid:0) s k η t k L + s k ∆ η k L (cid:1) + Z T s k∇ η t k L ds ≤ C. (3.90)The proof of lemma 3.5 is completed. 18 Proofs of theorems 1.1
Let ( ρ , u , η ) be as in Theorem 1.1. For simplicity, assume that Z R ρ dx = 1 , which implies that there exists a positive constant N such that Z B N ρ dx ≥ Z R ρ dx = 34 . (4.1)We construct ρ R = ˆ ρ R + R − e −| x | where 0 ≤ ˆ ρ R ∈ C ∞ ( R ) satisfies that Z B N ˆ ρ R dx ≥ , (4.2)and that ¯ x a ˆ ρ R → ¯ x a ρ , in L ( R ) ∩ H ( R ) ∩ W ,q ( R ) as R → ∞ . (4.3)Notice that ¯ x a η ∈ L ( R ) and ¯ x a ∇ η ∈ L ( R ), choosing η R ∈ C ∞ ( B R ) such that¯ x a η R → ¯ x a η , ∇ η R → ∇ η in L ( R ) , as R → ∞ . (4.4)Since ∇ u ∈ L ( R ), choosing v Ri ∈ C ∞ ( B R )( i = 1 ,
2) such thatlim R →∞ k v Ri − ∂ i u k L ( R ) = 0 , i = 1 , , (4.5)and let smooth u R uniquely solve ( − ∆ u R + R − u R = − ρ R u R + q ρ R h R − ∂ i v Ri in B R ,u R = 0 on ∂B R , (4.6)where h R = ( √ ρ w R ) ∗ j /R with the standard mollifying kernel j δ , δ >
0. Extend u R to R bydefining 0 outside B R , and denote w R , u R ϕ R . By the same arguments as those for the proofof Theorem 1.1 in [21], we obtained thatlim R →∞ (cid:16) k∇ ( w R − u ) k L ( R ) + k q ρ R w R − √ ρ u k L ( R ) (cid:17) = 0 , (4.7)where 0 ≤ ϕ R ≤ , ϕ R ( x ) = 1 , if | x | ≤ R/ , |∇ k ϕ R | ≤ CR − k ( k = 1 , . (4.8)Then, in terms of lemma 2.1, the initial-boundary value problem (2.2) with the initial data( ρ R , u R , η R ) has a classical solution ( ρ R , u R , η R ) on B R × [0 , T R ]. Moreover, proposition 3.1show that exists a T independent of R such that (3.3) and (3.82) hold for ( ρ R , u R , η R ). By(3.3), (3.11), (4.4), (4.7), and (3.82), after taking a subsequence, ( ρ R , u R , η R ) locally and weakly(in the corresponding spaces) converges to a strong solution ( ρ, u, η ) of (1.1)-(1.5) on R × (0 , T ]satisfying (1.7) and (1.8). The proof of the existence part of theorem 1.1 is completed.19ext prove the uniqueness of the strong solutions. Take two strong solutions ( ρ i , u i , η i )( i =1 ,
2) sharing the same initial data with (1.7) and (1.8), and let ¯ ρ = ρ − ρ , ¯ u = u − u , ¯ η = η − η .Then, ¯ ρ t + ( u · ∇ )¯ ρ + ¯ u · ∇ ρ + ¯ ρ div u + ρ div ¯ u = 0 ,ρ ¯ u t + ρ u · ∇ ¯ u + ∇ ( p F ( ρ ) − p F ( ρ )) + ∇ ( η − η )= µ ∆¯ u + ( µ + λ ) ∇ div ¯ u − ¯ ρ ( u t + u · ∇ u ) − ρ ¯ u · ∇ u − (¯ η + β ¯ ρ ) ∇ Φ , ¯ η t + ∇ · (¯ ηu − ¯ η ∇ Φ) + ∇ · ( η ¯ u ) − ∆¯ η = 0 . (4.9)for ( x, t ) ∈ R × (0 , T ] with¯ ρ ( x,
0) = ¯ u ( x,
0) = ¯ η ( x,
0) = 0 , x ∈ R . (4.10)Firstly, multiply (4.9) by 2¯ ρ ¯ x r and integrate by parts. Similar to the inequality (5.32)in [21], we get that k ¯ ρ ¯ x r k L ≤ C Z t ( k∇ ¯ u k L + k√ ρ ¯ u k L ) ds, t ∈ (0 , T ] , (4.11)where r ∈ (1 , ˜ a ) with ˜ a = min { a, } .Secondly, multiplying (4.9) by ¯ u and integrating by parts lead to12 ddt Z ρ | ¯ u | dx + Z (2 µ + λ ) | div¯ u | + µ | ω | dx = − Z ¯ ρ ( u t + u · ∇ u ) · ¯ udx − Z ρ ¯ u · ∇ u · ¯ udx + Z ( p F ( ρ ) − p F ( ρ ))div¯ udx + Z ¯ η div¯ udx + Z (¯ η + β ¯ ρ ) ∇ Φ¯ udx ≤ C k∇ u k L ∞ Z ρ | ¯ u | dx + C Z | ¯ ρ || ¯ u | ( | u t | + | u ||∇ u | ) dx + C k p F ( ρ ) − p F ( ρ ) k L k div¯ u k L + C k ¯ η k L k div¯ u k L + C Z ¯ η ¯ x r | ¯ u | ¯ x − r |∇ Φ | dx + C Z ¯ ρ ¯ x r | ¯ u | ¯ x − r |∇ Φ | dx , C k∇ u k L ∞ Z ρ | ¯ u | dx + X i =1 Q i . (4.12)Just like (4.11), it has been obtained via (5.33) and (5.36) in [21] that Q + Q ≤ C ( ε )(1 + t k∇ u t k L + t k∇ u k L q ) Z t ( k∇ ¯ u k L + k√ ρ ¯ u k L ) ds + ε ( k√ ρ ¯ u k L + k∇ ¯ u k L ) . (4.13)With the Cauchy inequality and (3.3), (3.14), and (3.82), we have X i =3 Q i ≤ ε k∇ ¯ u k L + C ( ε ) k ¯ η k L + C k∇ Φ k L k ¯ x r ¯ η k L k ¯ u ¯ x − r k L + C k∇ Φ k L k ¯ x r ¯ ρ k L k ¯ u ¯ x − r k L ε k∇ ¯ u k L + C ( ε ) k ¯ η k L + C k ¯ x r ¯ η k L ( k√ ρ ¯ u k L + k∇ ¯ u k L )+ C ( k√ ρ ¯ u k L + k∇ ¯ u k L ) Z t ( k∇ ¯ u k L + k√ ρ ¯ u k L ) ds ≤ ε ( k∇ ¯ u k L + k√ ρ ¯ u k L ) + C ( ε ) k ¯ η k L + C ( ε ) k ¯ x r ¯ η k L + Z t ( k∇ ¯ u k L + k√ ρ ¯ u k L ) ds. (4.14)It remains to estimate the k ¯ η k L and k ¯ x r ¯ η k L . In fact, multiplying (1.1) by ¯ η and integrating12 ddt Z | ¯ η | dx + k∇ ¯ η k L = − Z ¯ ηu · ∇ ¯ ηdx − Z ¯ η div u dx + Z ¯ η ∇ ¯ η · ∇ Φ dx + Z ¯ η ∆Φ dx − Z ¯ η ¯ u · ∇ η dx − Z η div¯ u ¯ ηdx , X i =1 II i . (4.15)Using the H¨older’s inequality, Gagliardo-Nirenberg inequality, we have II ≤ C Z ¯ x r ¯ η ¯ x − r | u ||∇ ¯ η | dx ≤ C k∇ ¯ η k L k ¯ x r ¯ η k L k u ¯ x − r k L ≤ ε k∇ ¯ η k L + C ( ε ) k ¯ x r ¯ η k L + δ k ¯ x r ∇ ¯ η k L , (4.16) X i =2 II i ≤ C k∇ u k L k ¯ η k L + C k ∆Φ k L ∞ k η k L + k∇ Φ k L ∞ k ¯ η k L k∇ ¯ η k L ≤ ε k∇ ¯ η k L + C ( ε ) k ¯ η k L + C, (4.17) II ≤ C Z ¯ x r ¯ η ¯ x − r | ¯ u ||∇ η | dx ≤ C k∇ η k L k ¯ u ¯ x − r k L k ¯ x r ¯ η k L ≤ ε ( k√ ρ ¯ u k L + k∇ ¯ u k L ) + C ( ε ) k ¯ x r ¯ η k L + ε k ¯ x r ∇ ¯ η k L , (4.18) II ≤ C k∇ ¯ u k L k η k L k ¯ η k L ≤ ε k∇ ¯ u k L + δ k∇ ¯ η k L + C ( ε, δ ) k ¯ η k L . (4.19)Moreover, multiplying (1.1) by ¯ x r ¯ η and integrating by parts yield12 ddt Z | ¯ η | ¯ x r dx + Z |∇ ¯ η | ¯ x r dx = Z ¯ η ∆¯ x r dx − Z u · ∇ ¯ η ¯ η ¯ x r dx − Z ¯ η div u ¯ η ¯ x r dx + Z ∇ ¯ η ∇ · Φ¯ η ¯ x r dx + Z ¯ η ¯ x r ∆Φ dx + Z ¯ uη ∇ ¯ η ¯ x r dx Z ¯ uη ¯ η ∇ ¯ x r dx , X i =1 III i . (4.20)For the term III i ( i = 1 , · · · ,
7) on the right hand side of (4.20), we get that
III ≤ C Z | ¯ η | ¯ x r ¯ x − log σ ) ( e + | x | ) dx ≤ C Z | ¯ η | ¯ x r dx, (4.21) III + III = − Z ¯ η div u ¯ x r dx − Z ¯ η u ∇ ¯ x r dx ≤ C k∇ u k L k ¯ x a ¯ η k L + C k ¯ x r ¯ η k L k ¯ x r ¯ η k L k u ¯ x − k L ≤ ε k ¯ x r ∇ ¯ η k L + C ( ε ) k ¯ x r ¯ η k L + C, (4.22) III + III ≤ k∇ Φ k L ∞ k ¯ x r ¯ η k L k ¯ x r ∇ ¯ η k L + C k ∆Φ k L ∞ k ¯ x r ¯ η k L ≤ ε k ¯ x r ∇ ¯ η k L + C ( ε ) k ¯ x r ¯ η k L + C, (4.23) III ≤ Z ¯ u ¯ x − b η ¯ x b + r ∇ ¯ η ¯ x r dx ≤ k ¯ x r ∇ ¯ η k L k ¯ u ¯ x − b k L k ¯ x a η k L ≤ ε k ¯ x r ∇ ¯ η k L + ε ( k√ ρ ¯ u k L + k∇ ¯ u k L ) + C, (4.24) III ≤ C Z ¯ u ¯ x − η ¯ x r ¯ η ¯ x r ¯ x − log σ ( e + | x | ) dx ≤ C k ¯ x r ¯ η k L k ¯ u ¯ x − k L k ¯ x r η k L ≤ ε ( k√ ρ ¯ u k L + k∇ ¯ u k L ) + C ( ε ) k ¯ x r ¯ η k L , (4.25)where b + r < ˜ a .Denoting G ( t ) , k√ ρ ¯ u k L + k ¯ η k L + k ¯ x r ¯ η k L + Z t ( k∇ ¯ u k L + k√ ρ ¯ u k L ) ds, (4.26)with all these estimates (4.13)-(4.25), choosing ε, δ suitably small lead to G ′ ( t ) ≤ C (1 + k∇ u k L ∞ + t k∇ u t k L + t k∇ u k L q ) G ( t ) , (4.27)which together with Gronwall’s inequality, and (1.7) yields G ( t ) = 0. Hence, ¯ u ( x, t ) = 0 and¯ η ( x, t ) = 0 for almost everywhere ( x, t ) ∈ R × (0 , T ). Then, one can deduce from (4.11) that¯ ρ = 0 for almost everywhere ( x, t ) ∈ R × (0 , T ). The proof of theorem 1.1 is completed. References [1] S. Agmon, A. Douglis, L, Nirenberg, Estimates near the boundary for solutions of elliptic partialdifferential equations satisfying general boundary conditions: I Commun. Pure Appl. Math. 12(1959)623C727.[2] S. Agmon, A. Douglis, L, Nirenberg, Estimates near the boundary for solutions of elliptic partial dif-ferential equations satisfying general boundary conditions: II Commun. Pure Appl. Math. 17(1964)35C92.[3] J. Ballew, Mathematical Topics in Fluid-Particle Interaction. PHD dissertation, University of Mary-land, USA, 2014.
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