Incompressible limit for the compressible flow of liquid crystals
aa r X i v : . [ m a t h . A P ] A ug INCOMPRESSIBLE LIMIT FOR THE COMPRESSIBLE FLOW OFLIQUID CRYSTALS
DEHUA WANG AND CHENG YU
Abstract.
The connection between the compressible flow of liquid crystals with lowMach number and the incompressible flow of liquid crystals is studied in a boundeddomain. In particular, the convergence of weak solutions of the compressible flow ofliquid crystals to the weak solutions of the incompressible flow of liquid crystals is provedwhen the Mach number approaches zero; that is, the incompressible limit is justified forweak solutions in a bounded domain. Introduction
In this paper, we consider the incompressible limit of the following hydrodynamic systemof partial differential equations for the three-dimensional compressible flow of nematicliquid crystals [9, 17, 28]: e ρ t + div( e ρ e u ) = 0 , (1.1a)( e ρ e u ) t + div( e ρ e u ⊗ e u ) + ∇ e P ( e ρ ) = e µ ∆ e u − e λ div (cid:18) ∇ e d ⊙ ∇ e d − (cid:0) |∇ e d | + F ( e d ) (cid:1) I (cid:19) , (1.1b) e d t + e u · ∇ e d = e θ (∆ e d − f ( e d )) , (1.1c)where e ρ ≥ e u ∈ R the velocity, e d ∈ R the direction field for theaveraged macroscopic molecular orientations, and e P = a e ρ γ is the pressure with constants a > γ ≥
1. The positive constants e µ, e λ, e θ denote the viscosity, the competitionbetween kinetic energy and potential energy, and the microscopic elastic relation time forthe molecular orientation field, respectively. The symbol ⊗ denotes the Kronecker tensorproduct, I is the 3 × ∇ e d ⊙ ∇ e d denotes the 3 × ij -th entry is < ∂ x i e d , ∂ x j e d > . Indeed, ∇ e d ⊙ ∇ e d = ( ∇ e d ) ⊤ ∇ e d , where ( ∇ e d ) ⊤ denotes the transpose of the 3 × ∇ e d . The vector-valued smoothfunction f ( e d ) denotes the penalty function and has the following form: f ( e d ) = ∇ e d F ( e d ) , where the scalar function F ( e d ) is the bulk part of the elastic energy. A typical exampleis to choose F ( e d ) as the Ginzburg-Landau penalization thus yielding the penalty function Date : November 9, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Liquid crystals, weak solution, compressible flow, Mach number, incompressiblelimit. f ( e d ) as: F ( e d ) = 14 σ ( | e d | − , f ( e d ) = 12 σ ( | e d | − e d , where σ > M = | e u | q e P ′ ( e ρ ) . Thus, letting M approach to zero, we hope that e ρ , e d keep a typical size 1, e u of the order ε , where ε ∈ (0 ,
1) is a small parameter. We scale e ρ , e u , and e d in the following way: e ρ = ρ ε ( εt, x ) , e u = ε u ε ( εt, x ) , e d = d ε ( εt, x ) , and we take the viscosity coefficients as: e µ = εµ ε , e λ = ε λ ε , e θ = εθ ε , where the normalized coefficients µ ε , λ ε , and θ ε satisfy µ ε → µ, λ ε → λ, θ ε → θ as ε → + , with µ , λ and θ positive constants. Under this scaling, system (1.1) becomes ∂ρ ε ∂t + div( ρ ε u ε ) = 0 , (1.2a) ∂ ( ρ ε u ε ) ∂t + div( ρ ε u ε ⊗ u ε ) + ∇ ε ρ γε = µ ε ∆ u ε − λ ε div (cid:18) ∇ d ε ⊙ ∇ d ε − (cid:0) |∇ d ε | + F ( d ε ) (cid:1) I (cid:19) , (1.2b) ∂ d ε ∂t + u ε · ∇ d ε = θ ε (∆ d ε − f ( d ε )) , (1.2c)where we take a = 1 because the exact value of a does not play a role in our paper. Theexistence of global weak solutions to (1.2) in bounded domains was established in [42, 31].By the initial energy bound (2.10) below, we can assume that the initial datum ρ ε is ofthe order 1 + O ( ε ), so it is reasonable to expect that, as ε → ρ ε → u = 0 , which is the incompressible condition of a fluid, and the first two termsin (1.2b) become u t + div( u ⊗ u ) = u t + ( u · ∇ ) u . The corresponding incompressible equations of liquid crystals are: u t + u · ∇ u + ∇ π = µ ∆ u − λ div( ∇ d ⊙ ∇ d ) , (1.3a) d t + u · ∇ d = θ (∆ d − f ( d )) , (1.3b)div u = 0 . (1.3c)Thus, roughly speaking, it is also reasonable to expect from the mathematical point ofview that the weak solutions to (1.2) converge in suitable functional spaces to the weak NCOMPRESSIBLE LIMIT OF LIQUID CRYSTALS 3 solutions of (1.3) as ε →
0, and the hydrostatic pressure π in (1.3a) is the “limit” of1 ε ( ρ γε − − λ ε |∇ d ε | − λ ε F ( d ε )in (1.2b). This paper is devoted to the rigorous justification of the convergence of theabove incompressible limit (i.e., the low Mach number limit) for global weak solutions ofthe compressible equations of liquid crystals in bounded smooth domains. We remark thatthe existence of global weak solutions to the incompressible flow of liquid crystals (1.3)was established in Lin-Liu [29].When the direction field d does not appear, (1.2) reduces to the compressible Navier-Stokes equations. Lions-Masmoudi [34] investigated the incompressible limits of the com-pressible isentropic Navier-Stokes equations in the whole space and periodic domains usingthe group method generated by the wave operator, a method introduced in earlier works[16, 39] which requires certain smoothness of solutions. The study in bounded smoothdomains with the no-slip boundary condition on the velocity is much harder than that inthe whole space or periodic domains, because in bounded domains, there are extra dif-ficulties arising from the appearance of the boundary layers, and the subtle interactionsbetween dissipative effects and wave propagation near the boundary, and hence requiresa different approach. Desjardins-Grenier-Lions-Masmoudi in [6] relied on spectral analy-sis and Duhamel’s principle to treat these difficulties to find the limit of global solutionsin a bounded domain. These results have been extended by others; see for examples[2, 10, 5, 35, 41]. We also remark that in Hoff [20] some convergence results were provedfor well-prepared data as long as the solution of incompressible limit is suitably smooth.For the case of nonisentropic flows, see [13, 14] for some recent developments. Recently,Hu-Wang [22] studied the convergence of weak solutions of the compressible magnetohy-drodynamic equations to the weak solutions of the incompressible magnetohydrodynamicequations when the Mach number goes to zero in a periodic domain, the whole space,or a bounded domain; and Jiang-Ju-Li [23] studied the incompressible MHD limit in theinviscid case as long as the strong solution of the incompressible inviscid MHD exists withperiodic boundary conditions. For other related studies on the incompressible limits ofviscous and inviscid flows, see [1, 4, 11, 19, 21, 24, 25, 27, 36, 37, 38, 40] and the referencesin [13]. Finally, we remark that the incompressible flow can also be derived from thevanishing Debye length type limit of a compressible flow with a Poisson damping; see forexamples [7, 8].In this paper, we shall establish the incompressible limit of (1.2) in a sufficiently smoothbounded domain Ω ⊂ R . As mentioned earlier, this limit problem in a bounded smoothdomain has more difficulties and requires a different approach due to the appearanceof the boundary layers and the subtle interactions between dissipative effects and wavepropagation near the boundary. Comparing with those works on the compressible Navier-Stokes equations, we will encounter extra difficulties in studying the compressible liquidcrystals. More precisely, besides the difficulties from compressible Navier-Stokes equations,the appearance of the direction field and the coupling effect between the hodrodynamicequations and the direction field should also be taken into account with new estimates.We will overcome all these difficulties by adapting the spectral analysis of the semigroupgenerated by the dissipative wave operator, Duhamel’s principle, and the weak convergencemethod to establish the convergence of the global weak solutions of compressible flow of DEHUA WANG AND CHENG YU liquid crystals (1.2) to the weak solutions of the incompressible flow of liquid crystals (1.3)as ε goes to zero in a bounded domain.We organize the rest of the paper as follows. In Section 2, we will provide some prelim-inaries and state our main result. In Section 3, we will prove in four steps the convergenceof the incompressible limit in a bounded domain.2. Preliminaries and Main Results
We consider the incompressible limit in a smooth bounded domain Ω ⊂ R . To stateprecisely our main result, we need to introduce a geometrical condition on Ω (cf. [6]). Letus consider the following overdetermined problem: − ∆ ϕ = λϕ in Ω , ∂ϕ∂ν = 0 on ∂ Ω , and ϕ is constant on ∂ Ω . (2.1)A solution to (2.1) is said to be trivial if λ = 0 and ϕ is a constant. We say that Ω satisfiesthe assumption (H) if all solutions of (2.1) are trivial. In the two-dimensional case, it wasproved that every bounded simply connected open set with Lipschitz boundary satisfies(H). We refer the readers to [6] for more information about assumption (H).Let us recall the definition of Leray’s projectors: P onto the space of divergence-freevector fields and Q onto the space of gradients, defined by u = P u + Q u , with div( P u ) = 0 , curl( Q u ) = 0 , (2.2)for u ∈ L . Indeed, in view of the results in [15], we know that the operators P and Q arelinear bounded operators in W s,p for s ≥ < p < ∞ in any bounded domain withsmooth boundary.We consider a sequence of weak solutions { ( ρ ε , u ε , d ε ) } ε> to (1.2) in a smooth boundeddomain Ω ⊂ R with the following boundary condition: u ε | ∂ Ω = 0 , d ε | ∂ Ω = d ε , (2.3)and initial condition: ρ ε | t =0 = ρ ε , ρ ε u ε | t =0 = m ε , d ε | t =0 = d ε , (2.4)satisfying ρ ε ≥ , ρ ε ∈ L γ (Ω) , (2.5) m ε ∈ L γγ +1 (Ω) , m ε = 0 if ρ ε = 0 , (2.6) ρ ε | u ε | ∈ L (Ω) , d ε ∈ H (Ω) , (2.7) (cid:0) ρ ε (cid:1) u ε converges weakly in L to some u as ε →
0, (2.8) d ε converges weakly in L to some d as ε →
0, (2.9)and Z Ω (cid:18) ρ ε | u ε | + 12 λ ε |∇ d ε | + λ ε F ( d ε ) (cid:19) dx + 1 ε ( γ − Z Ω (cid:0) ( ρ ε ) γ − γρ ε + ( γ − (cid:1) dx ≤ C, (2.10) NCOMPRESSIBLE LIMIT OF LIQUID CRYSTALS 5 for some constant
C >
0. We remark that (2.10) implies, roughly speaking, that ρ ε is oforder 1 + O ( ε ) since ( ρ ε ) γ − γρ ε + ( γ −
1) = ( ρ ε ) γ − − γ (cid:0) ρ ε − (cid:1) , and ρ γ is a convex function for γ >
1. As proved in [42, 31], for any fixed ε >
0, thereexists a global weak solution ( ρ ε , u ε , d ε ) to the compressible flow of liquid crystals (1.2)satisfying ρ ε ∈ L ∞ ([0 , T ]; L γ (Ω)) , √ ρ ε u ε ∈ L ∞ ([0 , T ]; L (Ω)) , u ε ∈ L ([0 , T ]; H (Ω)) , d ε ∈ L ([0 , T ]; H (Ω)) ∩ L ∞ ([0 , T ]; H (Ω)) , for any given T >
0; and in addition, ρ ε u ε ∈ C ([0 , T ]; L γγ +1 (Ω)) ,ρ ε ∈ C ([0 , T ]; L ploc (Ω)) , if 1 ≤ p < γ ; as well as E ε ( t ) + µ ε Z T Z Ω |∇ u ε | dxdt + λ ε θ ε Z T Z Ω | ∆ d ε − f ( d ε ) | dxdt ≤ E ε (0) , (2.11)for t ∈ [0 , T ] a.e., where E ε := Z Ω (cid:18) ρ ε | u ε | + 1 ε ( γ − ρ γε + 12 λ ε |∇ d ε | + λ ε F ( d ε ) (cid:19) dx, and E ε (0) := Z Ω (cid:18) ρ ε | u ε | + 1 ε ( γ −
1) ( ρ ε ) γ + 12 λ ε |∇ d ε | + λ ε F ( d ε ) (cid:19) dx. We now recall the existence result of global weak solutions to the incompressible flowof liquid crystals in [29]:
Proposition 2.1.
For u ∈ L (Ω) and d ∈ H (Ω) with d | ∂ Ω ∈ H / ( ∂ Ω) , system (1.3) with the following initial and boundary conditions: u | t =0 = u ( x ) with div u = 0 , d | t =0 = d ( x ) , and u | ∂ Ω = 0 , d | ∂ Ω = d ( x ) has a global weak solution ( u , d ) such that u ∈ L (0 , T ; H (Ω)) ∩ L ∞ (0 , T ; L (Ω)) , d ∈ L (0 , T ; H (Ω)) ∩ L ∞ (0 , T ; H (Ω)) , and Z Ω (cid:0) | u | + λ |∇ d | + 2 λF ( d ) (cid:1) dx + Z T Z Ω (cid:0) µ |∇ u | + λθ | ∆ d − f ( d ) | (cid:1) dxdt ≤ Z Ω (cid:0) | u | + λ |∇ d | + 2 λF ( d ) (cid:1) dx for all T ∈ (0 , ∞ ) . DEHUA WANG AND CHENG YU
Remark 2.1.
The global weak solutions obtained in Proposition 3.1 are the weak solutionsin Leray’s sense. The uniqueness of such solutions can be reached in two-dimensionalspaces, see for example [29]. For more details on the existence and regularity of weaksolutions to the incompressible flow of liquid crystals, we refer the readers to [29, 30, 18].Our main result reads as follows:
Theorem 2.1.
Assume that { ( ρ ε , u ε , d ε ) } ε> is a sequence of weak solutions to the com-pressible flow of liquid crystals (1.2) in a smooth bounded domain Ω ⊂ R with the initialand boundary conditions (2.3) - (2.10) and γ > . Then, for any given
T > , as ε → , { ( ρ ε , u ε , d ε ) } converges to a weak solution ( u , d ) of the incompressible flow of liquid crys-tals (1.3) with the initial data: u | t =0 = P u , d | t =0 = d and the boundary condition: u | ∂ Ω = 0 , d | ∂ Ω = d . More precisely, as ε → ,ρ ε converges to in C ([0 , T ]; L γ (Ω)); u ε converges to u weakly in L ((0 , T ) × Ω) and strongly if Ω satisfies condition ( H ); d ε converges to d strongly in L ([0 , T ]; H (Ω)) and weakly in L ([0 , T ]; H (Ω)) . Proof of Theorem 2.1
In this section we prove Theorem 2.1 in four steps.3.1.
A priori estimates and consequences.
We first recall the spectral analysis of thesemigroup [6] generated by the dissipative wave operator. Let { λ k, } k ∈ N ( λ k, >
0) bethe nondecreasing sequence of eigenvalues and { Φ k, } k ∈ N in L (Ω) be the eigenvectors withzero mean value of the Laplace operator satisfying the homogeneous Neumann boundarycondition: − ∆Φ k, = λ k, Φ k, in Ω , ∂ Φ k, ∂ν = 0 on ∂ Ω , where ν is the unit outer normal of Ω. By the Gram-Schmidt orthogonalization method,it is possible to assume that { Φ k, } k ∈ N is an orthonormal basis of L (Ω) and that up to aslight modification, if λ k, = λ l, , and k = l , then Z ∂ Ω ∇ Φ k, · ∇ Φ l, ds = 0 . From (2.11) and the conservation of mass, we have for almost all t ≥ Z Ω (cid:18) ρ ε | u ε | + 1 ε ( γ −
1) ( ρ γε − γρ ε + γ −
1) + λ ε |∇ ε d ε | + λF ( d ε ) (cid:19) dx + µ ε Z T Z Ω |∇ u ε | dxdt + λ ε θ ε Z T Z Ω | ∆ d ε − f ( d ε ) | dxdt ≤ Z Ω (cid:18) ρ ε | u ε | + 1 ε ( γ −
1) (( ρ ε ) γ − γρ ε + γ −
1) + λ ε |∇ ε d ε | + λF ( d ε ) (cid:19) dx ≤ C. (3.1)By (3.1), we have the following properties: √ ρ ε u ε is bounded in L ∞ ([0 , T ]; L (Ω)) , (3.2) NCOMPRESSIBLE LIMIT OF LIQUID CRYSTALS 7 d ε is bounded in L ∞ ([0 , T ]; H (Ω)) , (3.3)1 ε ( γ −
1) ( ρ γε − γρ ε + γ −
1) is bounded in L ∞ ([0 , T ]; L (Ω)); (3.4)and ∇ u ε is bounded in L ([0 , T ]; L (Ω)) , (3.5)∆ d ε − f ( d ε ) is bounded in L ([0 , T ]; L (Ω)) (3.6)for all T > c > x ≥ x γ − − γ ( x − ≥ c | x − | if γ ≥ ,x γ − − γ ( x − ≥ c | x − | if γ < x ≤ R,x γ − − γ ( x − ≥ c | x − | γ if γ < x ≥ R, where R ∈ (0 , ∞ ) . Thus from (3.4), we have Z Ω (cid:18) ε | ρ ε − | χ | ρ ε − |≤ / + 1 ε | ρ ε − | γ χ | ρ ε − |≥ / (cid:19) dx ≤ C, (3.8)where χ is the characteristics function and C denotes a generic positive constant hereafter.By (3.8), one hassup t ≥ k ρ ε − k L γ (Ω) ≤ Cε κ/γ , and sup t ≥ k ρ ε − k L κ (Ω) ≤ Cε, where κ = min { , γ } , which implies that ρ ε → C ([0 , T ]; L γ (Ω)) as ε → . (3.9)Now we split the velocity u ε = u ε + u ε with u ε = u ε χ | ρ ε − |≤ , u ε = u ε χ | ρ ε − | > , which means that sup t ≥ Z Ω | u ε | dx ≤ t ≥ Z Ω ρ ε | u ε | dx ≤ C (3.10)and k u ε k L (Ω) ≤ Z Ω | ρ ε − || u ε | dx ≤ Cε k u ε k L κ/ ( κ − (Ω) ≤ Cε k∇ u ε k L (Ω) , (3.11)where we used some embedding inequality. By (3.10) and (3.11), it is easy to see that u ε is bounded in L ∞ ([0 , T ]; L (Ω)) , and u ε ε − / is bounded in L ([0 , T ]; L (Ω)) . This implies that u ε is bounded in L ((0 , T ) × Ω) for all
T > . By (1.2a) and (3.9), one deduces that,div u ε → L ([0 , T ]; L (Ω))for all T > , where we used a fact Z T Z Ω | div u ε | dxdt ≤ C Z T Z Ω |∇ u ε | dxdt ≤ C. DEHUA WANG AND CHENG YU
By (3.6), smoothness of f , and the standard elliptic theory, we have ∇ d ε ∈ L ([0 , T ]; L (Ω)) . On the other hand, multiplying d ε on the both sides of (1.2c), then applying maximalprinciple, we have d ε ∈ L ∞ ([0 , T ] × Ω) . Thus, d ε ∈ L ([0 , T ]; H (Ω)) . Similarly to [42], using the Gagliardo-Nirenberg inequality, k∇ d ε k L (Ω) ≤ c k ∆ d ε k L (Ω) k d ε k L ∞ (Ω) + c k d ε k L ∞ (Ω) , one has ∇ d ε ∈ L ((0 , T ) × Ω) . (3.12)Summing up the above estimates, we can assume that, up to a subsequence if necessary, u ε → u weakly in L ([0 , T ]; H (Ω) , div u ε → L ([0 , T ]; L (Ω)) ,ρ ε → C ([0 , T ]; L γ (Ω)) , ∆ d ε − f ( d ε ) → ∆ d − f ( d ) weakly in L ([0 , T ]; L (Ω)) , and d ε → d weakly in L ∞ (0 , T ; H (Ω)) ∩ L ([0 , T ]; H (Ω)) . To show the strong convergence of d ε , we rely on the following Aubin-Lions compactnesslemma (see [33]): Lemma 3.1.
Let X , X and X be three Banach spaces with X ⊆ X ⊆ X and X and X be reflexive. Suppose that X is compactly embedded in X and that X is continuouslyembedded in X . For < p, q < ∞ , let W = (cid:26) u ∈ L p ([0 , T ]; X ) : dudt ∈ L q ([0 , T ]; X ) (cid:27) . Then the embedding of W into L p ([0 , T ]; X ) is also compact. From (1.2c), it is easy to see that k ∂ t d ε k L (Ω) ≤ k θ ε (∆ d ε − f ( d ε )) k L (Ω) + k u ε · ∇ d ε k L (Ω) ≤ C k u ε k L (Ω) + C k∇ d ε k L (Ω) + C k ∆ d ε − f ( d ε ) k L (Ω) , ≤ C k∇ u ε k L (Ω) + C k∇ d ε k L (Ω) + C k ∆ d ε − f ( d ε ) k L (Ω) . (3.13)This, combined with (3.5), (3.6), (3.12) and embedding theorem, implies that k ∂ t d ε k L ([0 ,T ]; L (Ω)) ≤ C. Since H ⊂ H ⊂ L and the injection H ֒ → H is compact, we apply Lemma 3.1 todeduce that the sequence { d ε } is precompact in L (0 , T ; H (Ω)) . By taking a subsequenceif necessary, we can assume that, d ε → d weakly in L (0 , T ; H (Ω)) , NCOMPRESSIBLE LIMIT OF LIQUID CRYSTALS 9 and d ε → d strongly in L (0 , T ; H (Ω)) . Therefore, by a standard argument, we deduce that the limit u and d satisfy equation(1.3c) in the sense of distributions. By strong convergence of d ε and smoothness of F , wededuce the convergence of the nonlinear term ∇ d ε ⊙ ∇ d ε − ( 12 |∇ d ε | + F ( d ε )) I → ∇ d ⊙ ∇ d − ( 12 |∇ d | + F ( d )) I as ε → π ε = 1 ε ρ γε − λ ε |∇ d ε | − λ ε F ( d ε ) , then we rewrite equation (1.2b) as ∂ ( ρ ε u ε ) ∂t + div( ρ ε u ε ⊗ u ε ) + ∇ π ε = µ ε ∆ u ε − λ ε div( ∇ d ε ⊙ ∇ d ε ) . (3.14)We project equation (3.14) onto divergence-free vector fields: ∂ t P ( ρ ε u ε ) + P (div( ρ ε u ε ⊗ u ε )) − µ ε ∆ P u ε = − P ( λ ε div( ∇ d ε ⊙ ∇ d ε )) , (3.15)where P is defined by (2.2). Then (3.15) yields a bound on ∂ t P ( ρ ε u ε ) in L ([0 , T ]; H − (Ω)) + L ([0 , T ]; W − , (Ω)) + L ([0 , T ]; H − (Ω)) , and hence in L ([0 , T ]; W − , (Ω)) . In addition, P ( ρ ε u ε ) is bounded in L ∞ ([0 , T ]; L γγ +1 (Ω)) ∩ L ([0 , T ]; L r (Ω)) with 1 r = 1 γ + 16 . To continue our proof, we need the following lemma (cf. Lemma 5.1 in [32]).
Lemma 3.2.
Let the functions g n , h n converge weakly to the functions g , h , respectively,in L p (0 , T ; L p (Ω)) , L q (0 , T ; L q (Ω)) , where ≤ p , p ≤ ∞ and p + 1 q = 1 p + 1 q = 1 . Assume, in addition, that ∂g n ∂t is bounded in L (0 , T ; W − m, (Ω)) for some m ≥ independent of n, and k h n − h n ( t, · + ζ ) k L q (0 ,T ; L q ) → as | ζ | → uniformly in n. Then g n h n converges to gh in the sense of distributions in (0 , T ) × Ω . Applying Lemma 3.2, we deduce that P ( ρ ε u ε ) · P ( u ε ) converges to | u | in the sense ofdistributions. It is easy to see that P u ε → u = P u in the sense of distributions becausethe weak convergence of P u ε to u = P u in L ((0 , T ); L (Ω)) and Z T Z Ω ( | P u ε | − P ( ρ ε u ε ) · P u ε ) dxdt ≤ C k ρ ε − k C ([0 ,T ]; L γ (Ω)) k u ε k L ([0 ,T ]; L s (Ω)) , with s = γγ − < γ > . The convergence of Q u ε . To prove our main result, it remains to show the conver-gence of the gradient part of the velocity Q u ε . The argument for the convergence of Q u ε in our paper follows the same line in [6] and [22], except the argument for the directionfield d ε . For the convenience of readers and the completeness of argument, we provide thedetails here.First, we introduce the spectral problem associated with the viscous wave operator L ε in terms of eigenvalues and eigenvectors of the invisicid wave operator L . In the sequel,we write the density fluctuation as ϕ ε = ρ ε − ε , and φ = (Φ , m ) ⊤ . We define the wave operators L and L ε in D ′ (Ω) × D ′ (Ω) as follows: L (cid:18) Φ m (cid:19) = (cid:18) div m ∇ Φ (cid:19) (3.16)and L ε (cid:18) Φ m (cid:19) = L (cid:18) Φ m (cid:19) + ε (cid:18) µ ε ∆ m (cid:19) . (3.17)The eigenvalues and eigenvectors of L read as follows: φ ± k, = Φ k, m ± k, = ± ∇ Φ k, iλ k, ! , (3.18)and Lφ ± k, = ± iλ k, φ ± k, in Ω , m ± k, · ν = 0 on ∂ Ω . And we need the following lemma [6]:
Lemma 3.3.
Let Ω be a C bounded domain of R d and let k ≥ , N ≥ . Then, thereexist approximate eigenvalues iλ ± k,ε,N and eigenvectors φ ± k,ε,N = (Φ ± k,ε,N , m ± k,ε,N ) ⊤ of L ε such that L ε φ ± k,ε,N = iλ ± k,ε,N φ ± k,ε,N + R ± k,ε,N , with iλ ± k,ε,N = ± iλ k, + iλ ± k, √ ε + O ( ε ) , where Re ( iλ ± k, ) ≤ , and for all p ∈ [1 , ∞ ] , we have, (cid:12)(cid:12)(cid:12) R ± k,ε,N (cid:12)(cid:12)(cid:12) L p (Ω) ≤ C p ( √ ε ) N +1 /p and (cid:12)(cid:12)(cid:12) φ k,ε,N − φ ± k, (cid:12)(cid:12)(cid:12) L p (Ω) ≤ C p ( √ ε ) /p . Remark 3.1.
The key idea is to construct an approximation scheme of L ε in terms of φ ± k, . We refer the readers to [6] for more details and the proof. From Lemma 3.3 and itsproof, we have, for any integers k , iλ ± k, = − ± i s µ ε λ k, Z ∂ Ω |∇ Φ k, | dx, which satisfies Re ( iλ ± k, ) ≤ . (3.19) NCOMPRESSIBLE LIMIT OF LIQUID CRYSTALS 11
We observe that the first order term iλ ± k, clearly yields an instantaneous damping ofacoustic waves, as soon as Re ( iλ ± k, ) < . Thus, we let I ⊂ N to be a collection of the all eigenvectors Φ k, of the Laplace operator such that Re ( iλ ± k, ) < . Denote J = N \ I, that is to say, when k ∈ J , we have Re ( iλ ± k, ) = 0due to (3.19). This implies λ k, = 0 . In the case that λ k, = 0, m ± k, must vanish on ∂ Ωand therefore not only m ± k, · ν = 0 but also m ± k, = 0 on ∂ Ω . Thus, no significant boundarylayer is created, and there is no enhanced dissipation of energy in these layers.Remark that { ∇ Φ k, λ k, } k ∈ N is an orthonormal basis of L (Ω) functions with zero meanvalue on Ω. We write Q u ε = X k ∈ N (cid:18) Q u ε , ∇ Φ k, λ k, (cid:19) ∇ Φ k, λ k, , where ( f, g ) = Z Ω f ( x ) g ( x ) dx. We split Q u ε into two parts Q u ε and Q u ε , defined by Q u ε = X k ∈ I (cid:18) Q u ε , ∇ Φ k, λ k, (cid:19) ∇ Φ k, λ k, , and Q u ε = X k ∈ J (cid:18) Q u ε , ∇ Φ k, λ k, (cid:19) ∇ Φ k, λ k, , which, respectively, correspond to damped terms and nondamped terms. What remainsis to show that Q u ε → L ([0 , T ] × Ω) , and curl div( Q m ε ⊗ Q u ε ) → J = ∅ , which is equivalent to saying that div( Q m ε ⊗ Q u ε )converges to a gradient in the sense of distributions.Let us observe that in view of the bound on u ε in L (0 , T ; H (Ω)) , the problem reducesto a finite number of modes. Remark that the eigenvalues { λ k, } k ≥ is a nondecreasingsequence, we have X k>N Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Q i u ε , ∇ Φ k, λ k, (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Cλ N +1 |∇ u ε | L ((0 ,T ) × Ω) , i = 1 or 2 . Letting N → ∞ , then λ N → ∞ , which implies that ( Q u ε , m ± k, ) → L (0 , T ) for any k > N. So we need to show that ( Q u ε , m ± k, ) converges to 0 strongly in L (0 , T ) for anyfixed k and study the interaction of a finite number of terms in div( Q u ε ⊗ Q u ε ) . Recalling that ϕ ε = ρ ε − ε , we have Q u ε = Q m ε − εQ ( ϕ ε u ε )and ε | ( Q ( ϕ ε u ε ) , ∇ Φ k, ) | = ε (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ϕ ε u ε · ∇ Φ k, dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε k ϕ ε k L γ (Ω) k u ε k L γγ − (Ω) k∇ Φ k, k L ∞ (Ω) , (3.20)which tends to zero in L (0 , T ) due to γ > . So we move to study ( Q m ε , m ± k, ) . We write β ± k,ε = ( φ ε ( t ) , φ ± k, )with φ ε ( t ) = (cid:18) ϕ ε m ε (cid:19) . (3.21)It is easy to see 2( Q m ε , m ± k, ) = β ± k,ε − β ∓ k,ε . Applying Lemma 3.3 with N = 2 and the H¨older inequality, one obtains | ( φ ε ( t ) , φ ± k, ) − φ ± k,ε, | ≤ k φ ε k L (Ω) k φ ± k, − φ ± k,ε, k L (Ω) ≤ Cε α (cid:18) k ϕ ε k L ∞ ([0 ,T ]; L κ (Ω)) + k m ε k L ∞ ([0 ,T ]; L γγ +1 (Ω)) (cid:19) (3.22)where α = min { − κ , − γ } . It remains to show that b ± k,ε ( t ) = ( φ ε ( t ) , φ ± k,ε, ) converges to zero strongly in L ([0 , T ])when k ∈ I and check the oscillations when k ∈ J. Using L ∗ ε to denote the adjoint operator of L ε with respect to ( · , · ), we have ∂ t φ ε − L ∗ ε φ ε ε = (cid:18) g ε (cid:19) (3.23)where g ε = − div( m ε ⊗ m ε ) − ∇ π ε − λ ε div( ∇ d ε ⊙ ∇ d ε ) . Taking the scalar product of (3.23) with φ ± k,ε, , one obtains( ∂ t φ ε , φ ± k,ε, ) − (cid:18) L ∗ ε φ ε ε , φ ± k,ε, (cid:19) = c ± k,ε ( t ) (3.24)where c ± k,ε ( t ) = ( g ε , m ± k,ε, ) + ε − ( φ ε , R ± k,ε, ) . Letting b ± k,ε ( t ) = ( φ ε , φ ± k,ε, ) and observing that (cid:18) L ∗ ε φ ε ε , φ ± k,ε, (cid:19) = 1 ε iλ ± k,ε ( φ ε , φ ± k,ε, ) + 1 ε R ± k,ε, , NCOMPRESSIBLE LIMIT OF LIQUID CRYSTALS 13 we rewrite (3.24) as ∂ t b ± k,ε − ε iλ ± k,ε, b ± k,ε = c ± k,ε . (3.25)We can view (3.25) as an ordinary differential equation, which satisfies the initial value: b ± k,ε | t =0 = b ± k,ε (0) . (3.26)3.3. The case k ∈ I . By the basic theory of ordinary differential equation, the solutionto (3.25)-(3.26) is given by b ± k,ε ( t ) = b ± k,ε (0) e iλ ± k,ε, tε + Z t c ± k,ε ( s ) e iλ ± k,ε, t − s ) ε ds. (3.27)Next, we need to estimate b ± k,ε ( t ). Note that iλ ± k,ε, = ± iλ k, + iλ ± k, √ ε + o ( ε ) , which is helpful in estimating the first part of (3.27): k b ± k,ε (0) e iλ k,ε, tε k L (0 ,T ) ≤ C k b ± k,ε (0) e Re ( iλ ± k, ) t √ ε k L (0 ,T ) ≤ Cε , (3.28)where we used Lemma 3.3.To estimate the second term in (3.27), observe that for any a ∈ L q (0 , T ) and 1 ≤ p, q ≤∞ such that p + q = 1 , we have (cid:12)(cid:12)(cid:12)(cid:12)Z T e iλ k,ε, t − sε a ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T (cid:12)(cid:12)(cid:12)(cid:12) e Re ( iλ ± k, ) t − s √ ε (cid:12)(cid:12)(cid:12)(cid:12) | a ( s ) | ds ≤ C | a | L q (0 ,T ) √ ε p . (3.29)To prove b ± k,ε ( t ) → L ([0 , T ]) , it remains to show that c ± k,ε is bounded in L q (0 , T ) for some q > . Remark that | c ± k,ε | ≤ c + c + c + c , where c = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ( m ε ⊗ m ε )( t ) · ∇ m ± k,ε, dx (cid:12)(cid:12)(cid:12)(cid:12) and c , c , c will be defined as follows. Using m ε = εϕ ε u ε + u ε , u ε = u ε + u ε , and integration by parts, we have c ( t ) ≤ ε Z Ω | ϕ ε u ε ⊗ u ε · ∇ m ± k,ε, | dx + C Z Ω | u ε · ∇ u ε · m ± k,ε, | dx ≤ k m ± k,ε, k L ∞ (Ω) k u ε + u ε k L (Ω) k∇ u ε k L (Ω) + ε k ϕ ε k L ∞ ([0 ,T ]; L κ (Ω)) k ( u ε ) k L κ/κ − (Ω) kk∇ m ± k,ε, k L ∞ (Ω) ≤ C k u ε k L ∞ ([0 ,T ]; L (Ω)) · k∇ u ε k L (Ω) + Cε / k∇ u ε k L (Ω) + Cε / k∇ u ε k L (Ω) . (3.30)For the second term c = Z Ω ( π ε div m ± k,ε, ) dx, recalling that div m ± k,ε, = iλ ± k,ε, Φ k,ε, + R ± k,ε, , we have c ( t ) ≤ k π ε k L ∞ ([0 ,T ]; L (Ω)) ( k Φ ± k,ε, k L ∞ (Ω) + k R ± k,ε, k L ∞ (Ω) ) ≤ C. (3.31)The third term can be estimated as: c ( t ) = ε − | ( φ ε , R ± k,ε, ) |≤ ε − k R ± k,ε, k L κ/ ( κ − (Ω) k φ ε k L ∞ ([0 ,T ]; L κ (Ω)) ≤ Cε / − / κ . (3.32)And the last term is c ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω div( ∇ d ε ⊙ ∇ d ε ) m ± k,ε, dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z Ω |∇ d ε | |∇ m ± k,ε, | dx ≤ C k∇ m ± k,ε, k L ∞ (Ω) k∇ d ε k L (Ω) ≤ C k∇ d ε k L (Ω) , (3.33)where we used ∇ d ε ∈ L (Ω) . From (3.30)-(3.33), it is easy to see that the term c ± k,ε isbounded, and consequently, b ± k,ε → L (0 , T ) . To complete our proof, we need to consider further the case k ∈ J .3.4. The case k ∈ J . As in Remark 3.1, when k ∈ J , we have λ ± k, = 0 . This impliesthat, together with (3.27), e ± iλ k, t/ε b ± k,ε is bounded in L (0 , T ) , and ∂ t ( e ± iλ k, t/ε b ± k,ε ) is bounded in √ εL (0 , T ) + L p (0 , T ) for some p > . It follows that, up to a subsequence if necessary, b ± k,ε converges strongly in L (0 , T ) tosome element b ± k,osc . Since ρ ε → C ([0 , T ]; L γ (Ω)) and b ± k,ε is uniformly bounded in L (0 , T ) , one obtains that ρ ε b ± k,ε b ± l,ε ∇ Φ k, λ k, ⊗ ∇ Φ l, λ l, − b ± k,ε b ± l,ε ∇ Φ k, λ k, ⊗ ∇ Φ l, λ l, → b ± k,ε b ± l,ε ∇ Φ k, λ k, ⊗ ∇ Φ l, λ l, for all k, l ∈ J. NCOMPRESSIBLE LIMIT OF LIQUID CRYSTALS 15
On the other hand, b ± k,ε b ± l,ε ∇ Φ k, λ k, ⊗ ∇ Φ l, λ l, = e i ( λ k, − λ l, ) t/ε e − iλ k, t/ε b k, e iλ l, t/ε b l, ∇ Φ k, λ k, ⊗ ∇ Φ l, λ l, → e i ( λ k, − λ l, ) t/ε b ± k,osc b ± l,osc ∇ Φ k, λ k, ⊗ ∇ Φ l, λ l, as ε → e ± λ k, t/ε b ± k,ε in L (0 , T ) when k ∈ J. If λ k, = λ l, , we can writediv( ∇ Φ k, ⊗ ∇ Φ l, + ∇ Φ l, ⊗ ∇ Φ k, ) = − λ k, ∇ (Φ k, Φ l, ) + ∇ ( ∇ Φ k, · ∇ Φ l, ) , which is a gradient, and thus disappears in the pressure term. It remains to consider thecase λ k, = λ l, . In this case, we have a fact that b ± k,osc ( t ) b ± l,osc ( t ) ∈ L ([0 , T ])for all k, l ∈ J due to b ± k,osc ( t ) ∈ L ([0 , T ]) for all k ∈ J. This implies, together with the Riemann-Lebesgue Lemma, Z Ω e i ( λ k, − λ l, ) t/ε b ± k,osc ( t ) b ± l,osc ( t ) dt → ε → . Thus, we conclude that e i ( λ k, − λ l, ) t/ε b ± k,osc b ± l,osc ∇ Φ k, λ k, ⊗ ∇ Φ l, λ l, → ε → k, l ≤ N, div X k,l ∈ J ρ ε b ± k,ε b ± l,ε ∇ Φ k, λ k, ⊗ ∇ Φ l, λ l, converges to a gradient in the sense of distributions, which means thatdiv ( ρ ε Q u ε ⊗ Q u ε )converges to a gradient in the sense of distributions.The proof of Theorem 2.1 is now complete. Acknowledgments
D. Wang’s research was supported in part by the National Science Foundation underGrant DMS-0906160 and by the Office of Naval Research under Grant N00014-07-1-0668.C. Yu’s research was supported in part by the National Science Foundation under GrantDMS-0906160.
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Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260.
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