Well-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data
aa r X i v : . [ m a t h . A P ] N ov WELL-POSEDNESS OF A FULLY-COUPLED NAVIER-STOKES/Q-TENSORSYSTEM WITH INHOMOGENEOUS BOUNDARY DATA
HELMUT ABELS, GEORG DOLZMANN, AND YUNING LIU
Abstract.
We prove short-time well-posedness and existence of global weak solutions of theBeris–Edwards model for nematic liquid crystals in the case of a bounded domain with inho-mogeneous mixed Dirichlet and Neumann boundary conditions. The system consists of theNavier-Stokes equations coupled with an evolution equation for the Q -tensor. The solutionspossess higher regularity in time of order one compared to the class of weak solutions withfinite energy. This regularity is enough to obtain Lipschitz continuity of the non-linear termsin the corresponding function spaces. Therefore the well-posedness is shown with the aid of thecontraction mapping principle using that the linearized system is an isomorphism between theassociated function spaces. Introduction
We study the well-posedness of a model for the instationary flow of a nematic liquid crystaldescribed by a model due to Beris and Edwards, cf. [2]. In this model the orientation and degreeof ordering of the liquid crystal is described by a symmetric, traceless d × d tensor Q . Thisdescription goes back to Landau and DeGennes, cf. [4]. In the case that the tensor is uniaxial,i.e., it has two equal non-zero eigenvalues, it can be represented as Q = s (cid:18) n ⊗ n − d I d (cid:19) , where the scalar order parameter s ∈ [ − ,
1] measures the degree of orientational ordering and n is a unit vector and describes the direction of orientation. The Beris-Edwards models leads to asystem, which couples the incompressible Navier-Stokes equations with a second order parabolicequation for the evolution of the tensor Q . More precisely, we consider ∂ t u + ( u · ∇ ) u + ∇ p = div (cid:0) ν ( Q ) D ( u ) (cid:1) + div (cid:0) σ ( Q, H ) + τ ( Q, H ) (cid:1) , div u = 0 ,∂ t Q + (cid:0) u · ∇ (cid:1) Q − S ( ∇ u, Q ) = Γ H ( Q ) (1.1)in Ω T = Ω × (0 , T ) for a sufficiently smooth bounded domain Ω ⊆ R d , d = 2 , T >
0. Here σ is a skew-symmetric tensor and H , τ , and S are symmetric tensors given by H = λ ∆ Q − aQ + b (cid:0) Q − d tr( Q ) I d (cid:1) − c tr( Q ) Q,σ ( Q, H ) = QH − HQ = Q ∆ Q − ∆ QQ ,τ ( Q, H ) = − λ ∇ Q ⊙ ∇ Q − ξ (cid:0) Q + d I d (cid:1) H − ξH (cid:0) Q + d I d (cid:1) + 2 ξ (cid:0) Q + d I d (cid:1) tr( QH ) ,S ( ∇ u, Q ) = (cid:0) ξD ( u ) + W ( u ) (cid:1)(cid:0) Q + d I d (cid:1) + (cid:0) Q + d I d (cid:1)(cid:0) ξD ( u ) − W ( u ) (cid:1) − ξ (cid:0) Q + d I d (cid:1) tr( Q ∇ u ) , (1.2) Mathematics Subject Classification.
Primary 35Q35; Secondary: 35Q30, 76D03, 76D05 .
Key words and phrases.
Beris-Edwards model, liquid crystals, Navier-Stokes equations, Q-tensor, strong-in-time solutions.The third author gratefully acknowledges partial financial support by the DFG through grant AB285/4-2. where we used the notation D ( u ) = 12 (cid:0) ∇ u + ( ∇ u ) T (cid:1) , W ( u ) = 12 (cid:0) ∇ u − ( ∇ u ) T (cid:1) for the stretch and the vorticity tensor, respectively. Moreover, Γ, λ , a , b , and c are positiveconstants. We note that S ( ∇ u, Q ) is introduced to describe how the flow gradient rotates andstretches the director field.Here H relates to the variational derivative of the free energy functional which uses the one-constant approximation for the Oseen-Frank energy of liquid crystals together with a Landau-DeGennes expression for the bulk energy F ( Q ) = Z Ω (cid:0) λ |∇ Q | + f B ( Q ) (cid:1) d x , (1.3)where the bulk energy f B is given by f B ( Q ) = a Q ) − b Q ) + c Q ) . Hence H = H ( Q ) can be rewritten as H ( Q ) = λ ∆ Q + L, L = − aQ + b (cid:0) Q − d tr( Q ) I d (cid:1) − c tr( Q ) Q, (1.4)where L = − Df B ( Q ) consists of lower-order terms in the equation.We complement this system (1.1)-(1.2) by the initial condition( u, Q ) | t =0 = ( u , Q ) in Ω (1.5)and the Dirichlet-Neumann boundary conditions of mixed type, u = 0 on (0 , T ) × ∂ Ω ,Q = Q D on (0 , T ) × Γ D ,∂ n Q = Q N on (0 , T ) × Γ N , (1.6)where ∂ Ω = Γ D ∪ Γ N and Γ D , Γ N are closed, disjoint subsets of R d and ( Q D , Q N ) will beindependent of t ∈ (0 , T ) in the following.So far there are only a few results on the mathematical analysis of this system. First contri-butions were given by Paicu and Zarnescu. In [13] the authors consider the case ξ = 0, Ω = R d .They prove existence of weak solutions for d = 2 , d = 2. In [12] existence of weak solutions is proved providedthat ξ is sufficiently close to 0 and Ω = R d , d = 2 ,
3. Wilkinson studied in [20] the system(1.1)-(1.2) under periodic boundary condition in the case that f B is replaced by a certain sin-gular potential. The potential guarantees that Q attains only physically reasonable values. Heestablished existence of weak solutions for a general ξ and higher regularity in the case of twospace dimensions and ξ = 0. Finally, Feireisl et al. [5] derived a non-isothermal variant of theBeris-Edwards system and proved existence of weak solutions for this system in the case of asingular potential and for periodic boundary conditions. Recently, Wang et al. establish in [18]a rigorous derivation from Beris-Edwards system to the Ericksen-Leslie system, which is widelyinvestigated in the literature. Here we refer to recent works [8], [9], [19], [22] and the referencestherein for more details.In the present paper we discuss existence of weak solutions in a bounded domain with mixedDirichlet-Neumann boundary conditions as well as well-posedness of the system in a class of so-lutions, which possess higher regularity in time than the class of weak solutions. These solutionsare not necessarily more regular with respect to the space variable. We note that in the casewithout boundary, one could establish higher regularity in space for these solutions by usinge.g. standard difference quotient techniques. But in the present case with boundary conditionswe do not have an appropriate regularity result for the principal part of the linearized system, ERIS–EDWARDS MODEL 3 which is a Stokes system coupled with an elliptic equation for Q through the terms S ( ∇ u, e Q )and div σ ( e Q, H ) for a suitable e Q . These coupling terms cancel in the standard energy argument.However, they give rise to extra boundary integrals, when testing with higher order spacialderivatives of the solution, which cannot be absorbed. Fortunately, for higher order temporalderivatives these boundary terms vanish again. The main novelty in the paper is to use thisobservation together with the fact that one more temporal derivative (compared to the regular-ity class of weak solutions) is enough to prove Lipschitz continuity of the non-linear terms inthe associated function spaces. Therefore we are able to prove existence of unique solutions inthis regularity class for sufficiently short times. Let us note that we expect that our solutionsalso possess the natural higher regularity with respect to the spacial variables. This might be afuture work. But to obtain well-posedness of the system locally in time such a regularity resultis not needed.In order to formulate our main results, we have to introduce some assumptions and notation.In the sequel, we shall assume that Γ = λ = a = b = c = 1 to simplify the notation. But allresults hold true for general values of these constants if c >
0. In the following we assume thatΩ is a bounded domain with C -boundary and ν ∈ C ( R d × d ) , < c ν ( · ) c < ∞ (1.7)for some constants c , c . In the following S denotes the vector space of all symmetric and tracefree d × d matrices. More details on the notation are given in Section 2.1 below. We use thefollowing notion of weak solution. Definition 1.1.
Suppose that
T > , u ∈ L σ (Ω) , Q ∈ H (Ω; S ) , Q D ∈ H (Γ D ; S ) , and Q N ∈ H (Γ N ; S ) . A pair ( u, Q ) with u ∈ BC w ([0 , T ]; L σ (Ω)) ∩ L (0 , T ; H ,σ (Ω)) ,Q ∈ BC w ([0 , T ]; H (Ω; S )) ∩ L (0 , T ; H (Ω; S )) is called a weak solution of the system (1.1) in Ω T with initial conditions (1.5) and boundaryconditions (1.6) if the following holds: (1) For any v ∈ C ([0 , T ]; H ,σ (Ω) ∩ W , ∞ (Ω; R d )) and Ψ ∈ C ([0 , T ]; H (Ω; S )) with v | t = T = Ψ | t = T = 0 , it holds that Z Ω T (cid:0) − u · ∂ t v + ( u · ∇ u ) · v + ν ( Q ) D ( u ) : D ( v ) (cid:1) d( x, t )+ Z Ω T (cid:0)(cid:0) σ + τ (cid:1) ( Q, H ( Q )) (cid:1) : ∇ v d( x, t ) = Z Ω u v | t =0 d x (1.8) and − Z Ω T Q : ∂ t Ψ d( x, t ) + Z Ω T u · ∇ Q : Ψ d( x, t ) − Z Ω T S ( ∇ u, Q ) : Ψ d( x, t )= Z Ω T H ( Q ) : Ψ d( x, t ) + Z Ω Q : Ψ | t =0 d x. (1.9)(2) For almost every t ∈ (0 , T ) the following energy inequality holds: Z Ω | u ( t, x ) | d x + F ( Q ( t, · )) + Z Ω t (cid:0) ν ( Q ( τ, x )) | Du ( τ, x ) | + | H ( Q ( τ, x )) | (cid:1) d( x, τ ) Z Ω | u ( x ) | d x + F ( Q ) . (3) For almost every t ∈ [0 , T ] , Q | Γ D = Q D and ∂Q∂n | Γ N = Q N . ERIS–EDWARDS MODEL 4
Throughout the paper Ω is a bounded domain with C -boundary. Our first result is a resulton global existence of weak solutions in the case of homogeneous Neumann boundary conditionsfor the director field Q . Theorem 1.2 (Existence of weak solutions) . Let
T, Q D , Q N , u be as in Definition 1.1. Thenthe system (1.1) has a global weak solution for any T > . Our second result concerns regularity in time for weak solutions of the system (1.1). Thisresult requires a subtle compatibility condition related to the initial data for Q . As (1.1) is anevolution equation, it can be written in the abstract form ddt ( u, Q ) = E ( u, Q ) (1.10)where E : H ,σ (Ω) × H (Ω) → H − σ (Ω) × L (Ω) is defined by (cid:10) E ( u, Q ) , ( ϕ, Ψ) (cid:11) = − Z Ω ( − u ⊗ u + ν ( Q ) D ( u ) + ( τ + σ )( Q, H )) : ∇ ϕ d x + Z Ω (( u · ∇ ) Q + S + Γ H ) : Ψ d x (1.11)for all ( ϕ, Ψ) ∈ H ,σ (Ω) × H (Ω; S ). Since (1.6) specifies a time-independent Dirichlet boundarycondition, it follows that ∂ t Q | Γ D = 0 and this observation leads to the compatibility conditionthat the trace of the second component on the right-hand side of (1.10) vanishes on Γ D . Con-sequently we define the phase space, Z = (cid:8) ( u, Q ) ∈ H ,σ (Ω) × H (Ω) : E ( u, Q ) ∈ L σ × H D , Q = Q D on Γ D , ∂ n Q = Q N on Γ N (cid:9) , where H D = H D (Ω; S ) := { Q ∈ H (Ω; S ) : Q | Γ D = 0 } . Note that the phase space defined above is non-empty. For instance, if we choose u ∈ H (Ω) ∩ H ,σ (Ω) and Q ∈ H (Ω) satisfying the boundary condition in (1.6) such that ∆ Q | Γ D = 0, then( u , Q ) ∈ Z . Theorem 1.3 (Local existence and uniqueness of solutions with regularity in time) . Suppose that ( Q D , Q N ) ∈ H (Γ D ; S ) × H (Γ N ; S ) and that the initial data satisfy ( u , Q ) ∈ Z . Then there exists some T > such that the system (1.1) together with (1.5) and (1.6) has a unique solution ( u, Q ) with u ∈ H (0 , T ; H − σ (Ω)) ∩ H (0 , T ; H ,σ (Ω)) ,Q ∈ H (0 , T ; L (Ω; S )) ∩ H (0 , T ; H (Ω; S )) . Remark 1.4.
Theorems 1.2 and 1.3 are also valid for Γ D = ∅ or Γ N = ∅ . Finally, we note that the idea to use higher regularity in time to obtain a unique solution of(1.1)-(1.6) (in the case ξ = 0) has also been used by Guill´en-Gonz´alez and Rodr´ıguez-Bellido [7].2. Preliminaries
Notation.
For two vectors a , b ∈ R d we set a · b = P di =1 a i b i and a ⊗ b = ab T = ( a i b j ) i,j d and for two matrices A , B ∈ R d we set A : B = P di,j =1 A ij B ij = tr( A T B ). Then( AB ) : C = B : ( A T C ) = A : ( CB T ) for all A, B, C ∈ R d × d (2.1)and we omit the parentheses for simplicity in the following if they are clear from the context.Einstein’s summation convention is applied throughout the paper if repeated indices are written. ERIS–EDWARDS MODEL 5
We define the norm of a matrix A ∈ R d × d by | A | = tr( A T A ) = A : A . For a differentiablematrix-valued function F : Ω → R d × d , we denote by div F = ( ∂ β F αβ ) α d the vector whose α thcomponent is the divergence of the α th row in F . Moreover, if A, B : Ω → R d × d are differentiable,we introduce the contraction ⊙ by ∇ A ⊙ ∇ B = (cid:0) ∂ i A αβ ∂ j B αβ (cid:1) i,j d = (cid:0) ∂ i A : ∂ j B (cid:1) i,j d and Q : ∇ A = ( Q : ∂ j A ) j d . Finally, I d denotes the matrix, which represents the identity on R d .For the following it is convenient to rewrite the definitions of τ and S as τ ( Q, H ) = τ ( Q ) + ξτ ( Q, H ) − ξd H,S ( ∇ u, Q ) = S ( ∇ u, Q ) + ξS ( ∇ u, Q ) + 2 ξd D ( u ) , with τ ( Q ) = −∇ Q ⊙ ∇ Q − d I d tr Q ,τ ( Q, H ) = − QH − HQ + 2( Q + 1 d I d ) tr( QH ) ,S ( ∇ u, Q ) = W ( u ) Q − QW ( u ) ,S ( ∇ u, Q ) = D ( u ) Q + QD ( u ) − Q + 1 d I d ) tr( Q ∇ u ) . (2.2)We note that − div τ ( Q ) = ∆ Q ⊙ ∇ Q + ∇ d tr Q + n X j =1 | ∂ j Q | = H ( Q ) : ∇ Q + ∇ d tr Q + n X j =1 | ∂ j Q | + f B ( Q ) . Therefore Z Ω T τ ( Q ) : ∇ v d( x, t ) = Z Ω T ( H ( Q ) : ∇ Q ) · v d( x, t ) (2.3)for all Q and v as in Definition 1.1.Finally, h x ′ , x i X ′ ,X denotes the duality product of x ′ ∈ X ′ and x ∈ X for a Banach space X and ( ., . ) H denotes the inner product of a Hilbert space H .2.2. Function spaces.
We use standard notation for the Lebesgue and Sobolev spaces L p (Ω)and W k,p (Ω) as well as L p (Ω; M ) and W k,p (Ω; M ) for the corresponding spaces for M -valuedfunctions. Sometimes we omit the domain and the range if they are clear from the context. The L -based Sobolev spaces are denoted by H k (Ω) and H k (Ω; M ). The usual spaces of divergencefree vector fields are introduced by L σ (Ω) = (cid:8) u ∈ L (Ω; R d ) , div u = 0 , γ ( u ) = 0 (cid:9) ,H ,σ (Ω) = (cid:8) u ∈ H (Ω; R d ) , div u = 0 (cid:9) , H − σ (Ω) = ( H ,σ (Ω)) ′ where γ ( u ) = u · n ∈ H − ( ∂ Ω) is defined a generalized trace sense, cf. e.g. [16]. Note that L (Ω; R d ) = L σ (Ω) ⊕ ( L σ (Ω)) ⊥ with ( L σ (Ω)) ⊥ = (cid:8) u ∈ L (Ω; R d ) , u = ∇ q for some q ∈ H (Ω) (cid:9) . ERIS–EDWARDS MODEL 6
The Helmholtz projection, i.e., the orthogonal projection L (Ω; R d ) → L σ (Ω), is denoted by P σ . We refer to [16] for its basic properties. For f ∈ H − (Ω; R d ) we define P σ f ∈ H − σ (Ω) by P σ f = f | H ,σ (Ω) . Moreover, for F ∈ L (Ω; R d × d ) we define div F ∈ H − (Ω; R d ) by h div F, Φ i H − ,H = − Z Ω F : ∇ Φ d x for all Φ ∈ H (Ω; R d ) . Finally, if X is a Banach space and T >
0, then C ([0 , T ]; X ) and BC w ([0 , T ]; X ) denotes thespace of all bounded f : [0 , T ] → X that are continuous or weakly continuous, respectively.2.3. An algebraic identity.
The following cancellation property plays an important role inthe sequel.
Lemma 2.1.
Let Q , Q ∈ L (Ω; S ) and u ∈ H ,σ (Ω) . Then S ( ∇ u, Q ) : Q + (cid:18) σ ( Q , Q ) + ξτ ( Q , Q ) − ξd Q (cid:19) : ∇ u = 0 . (2.4) Proof.
We are going to prove the following two identities σ ( Q , Q ) : ∇ u + S ( ∇ u, Q ) : Q = 0 , τ ( Q , Q ) : ∇ u + S ( ∇ u, Q ) : Q = 0 . These identities together with − ξd Q : ∇ u + ξd D ( u ) : Q = 0 imply the assertion. We use (2.1)and the symmetries to obtain2 S ( ∇ u, Q ) : Q = ∇ uQ : Q − ( ∇ u ) T Q : Q − Q ∇ u : Q + Q ( ∇ u ) T : Q = 2 (cid:0) ∇ u : Q Q − ∇ u : Q Q (cid:1) = − ∇ u : σ ( Q , Q ) . Similarly, using the symmetry of Q and Q , τ ( Q , Q ) : ∇ u + S ( ∇ u, Q ) : Q = (cid:0) − Q Q − Q Q + 2( Q + 1 d I d )( Q : Q ) (cid:1) : ∇ u + (cid:0) D ( u ) Q + Q D ( u ) − Q + 1 d I d )( Q : ∇ u ) (cid:1) : Q = 0 , where we use that Q and ∇ u are trace free. The proof is complete. (cid:3) Orthogonal bases of eigenfunctions.
The idea is to solve the system (1.1) by a Galerkinapproach based on eigenfunctions of the Laplace operator for Q and eigenfunctions of the Stokesoperator for u . The existence of these bases follows from the spectral theorem for compactoperators in Hilbert spaces and standard results for elliptic equations and the stationary Stokessystem. We summarize their properties. Note that we consider the Laplace equation as avector-valued equation with values in S . Lemma 2.2 (Eigenvalue problem of the Laplace operator) . There exists an orthonormal basis ( e n ) n ∈ N ⊂ H (Ω; S ) of L (Ω; S ) and a non-decreasing se-quence of corresponding eigenvalues ( λ n ) n ∈ N ⊂ R + with lim n →∞ λ n = ∞ such that − ∆ e n = λ n e n in Ω ,e n = 0 on Γ D ,∂e n ∂n = 0 on Γ N . A similar result holds for Stokes operator A := − P σ ∆. Lemma 2.3 (Eigenvalue problem of the Stokes operator) . There exists an orthonormal basis ( v n ) n ∈ N ⊂ H ,σ (Ω) ∩ W , ∞ (Ω; R d ) of L σ (Ω; R d ) of eigenfunc-tions and a non-decreasing sequence ( ω n ) n ∈ N ⊂ R + of corresponding eigenvalues with lim n →∞ ω n = ∞ and Av i = ω i v i for all i > . ERIS–EDWARDS MODEL 7
The regularity result v n ∈ W , ∞ (Ω; R d ) follows from the standard regularity theory for theStokes system provided that ∂ Ω ∈ C , cf. e.g. [6].This result allows us to define the fractional Stokes operator A m : D ( A m ) → L σ (Ω; R d ) for m ∈ N , m >
1. The following lemma gives the regularity of functions in D ( A m ) and the proofcan be found in [3, Proposition 4.12]. Lemma 2.4.
Let Ω ⊂ R d be an open and bounded set of class C ℓ with ℓ > . Then D ( A m ) iscontained in H m (Ω; R d ) ∩ H ,σ (Ω) provided m ℓ . We recall a compactness result of Aubin-Lions type, see [15] for the proof.
Lemma 2.5.
Suppose that p , p ∈ (1 , ∞ ) . Assume that X − , X and X are three separableand reflexive Banach spaces such that the inclusion X ֒ → X is compact and the inclusion X ֒ → X − is continuous. If ( u ( m ) ) m ∈ N is a bounded sequence in L p (0 , T ; X ) such that ( ∂ t u ( m ) ) m ∈ N isbounded in L p (0 , T ; X − ) , then there exists a subsequence u ( m ′ ) which converges in L p (0 , T ; X ) .Furthermore, if X = ( X − , X ) [1 / , where ( ., . ) [ θ ] denotes the complex interpolation functor, and p = ∞ , then there exists a subsequence u ( m ′ ) which converges in C ([0 , T ]; X ) . The next interpolation result is stated in the three-dimensional situation which is the mainfocus of this paper.
Lemma 2.6 (Interpolation) . There is some
C > such that for all f ∈ H (Ω) the estimates k f k L ∞ (Ω) C k f k H (Ω) k f k H (Ω) , (2.5) and k f k L ∞ (Ω) C k f k L (Ω) k f k H (Ω) (2.6) hold. If, additionally, H ֒ → H ֒ → H ′ is a Gelfand-triple, then k f k C ([0 ,T ]; H ) k f k H (0 ,T ; H ′ ) k f k L (0 ,T ; H ) + k f (0 , · ) k H ) . (2.7)Proofs of these statements can be found in [1, Section 2.1]. Lemma 2.7.
Let X and Y be two Banach spaces such that X ⊂ Y with a continuous injection.If f ∈ L ∞ (0 , T ; X ) is weakly continuous with values in Y , then f is weakly continuous withvalues in X . The proof can be found in [17, pp.263].3.
Existence of weak solutions and proof of Theorem 1.2
This section is devoted to the proof of the existence of global weak solutions via a modifiedGalerkin method introduced in [10]. In view of Lemma 2.2 and Lemma 2.3, we define thefinite-dimensional Banach spaces E n = Span { e , . . . , e n } ⊂ H (Ω; S ) ⊂ L (Ω; S ) ,V n = Span { v , . . . , v n } ⊂ H ,σ (Ω) ∩ W , ∞ (Ω; R d ) ⊂ L σ (Ω) , along with the orthogonal projection operators π n : L (Ω; S ) −→ E n and P n : L σ (Ω) −→ V n . These two orthogonal projection operators are bounded linear operators with norms boundedby one, a fact which will be used in the calculations in this section without being mentioned.To simplify notation we use ( · , · ) Ω to denote the inner product in L (Ω; R N ), N >
1, andin L σ (Ω). Since Q D coincides with some element of H ( ∂ Ω) on Γ D , by standard results on ERIS–EDWARDS MODEL 8 elliptic boundary value problems, there exists an harmonic extension e Q ( x ) ∈ H (Ω; S ) suchthat e Q | Γ D = Q D and ∂ n e Q | Γ N = 0.With this notation in place, we seek approximations of the solutions of the system (1.1) ofthe form u ( n ) ( x, t ) = n X i =1 d i ( t ) v i ( x ) , Q ( n ) ( x, t ) = e Q ( x ) + n X i =1 h i ( t ) e i ( x ) (3.1)such that ( u ( n ) , Q ( n ) ) satisfies the generalized Navier-Stokes equations on V n , i.e.,( ∂ t u ( n ) , v k ) Ω + (( u ( n ) · ∇ ) u ( n ) , v k ) Ω + ( ν ( Q ( n ) ) D ( u ( n ) ) , D ( v k )) Ω (3.2)+ (( π n H ( Q ( n ) )) : ∇ Q ( n ) , v k ) Ω + (cid:16) ( σ + ξτ )( Q ( n ) , π n H ( Q ( n ) )) − ξd π n H ( Q ( n ) ) , ∇ v k (cid:17) Ω = 0in (0 , T ) for all k ∈ { , . . . , n } , the evolution equation for the director field on E n , i.e., (cid:0) ∂ t Q ( n ) , e ℓ (cid:1) Ω + (cid:0) ( u ( n ) · ∇ ) Q ( n ) , e ℓ (cid:1) Ω − (cid:0) S ( ∇ u ( n ) , Q ( n ) ) , e ℓ (cid:1) Ω = (cid:0) H ( Q ( n ) ) , e ℓ (cid:1) Ω (3.3)for all ℓ ∈ { , . . . , n } , and the initial conditions u ( n ) | t =0 = P n u , (3.4) Q ( n ) | t =0 = e Q + π n ( Q − e Q ) (3.5)in Ω. Note that we have replaced the term − div τ ( Q, H ( Q )) in the approximate system by H ( Q ) : ∇ Q because of (2.3). In the following we will use that ∂ t Q ( n ) , ∆ Q ( n ) ∈ E n since e Q isindependent of t and harmonic. Hence π n ∂ t Q ( n ) = ∂ t Q ( n ) and π n ∆ Q ( n ) = ∆ Q ( n ) .By Lemma 2.2 and 2.3, the above system is well defined and can be regarded as a finite-dimensional system of ordinary differential equations which has a solution on a maximal timeinterval [0 , T n ) with T n > n >
1. The following proposition implies that T n = ∞ .Moreover, the following a priori bounds will be essential to pass to the limit n → ∞ in the proofof Theorem 1.2. Proposition 3.1 (Lyapunov functional) . Let n > . Then the system (3.2) – (3.5) has the Lyapunov functional E (cid:0) u ( n ) ( t, · ) , Q ( n ) ( t, · ) (cid:1) = 12 Z Ω (cid:12)(cid:12) u ( n ) ( t, x ) (cid:12)(cid:12) d x + F (cid:0) Q ( n ) ( t, · ) (cid:1) which satisfies ddt E (cid:0) u ( n ) ( t, · ) , Q ( n ) ( t, · ) (cid:1) + Z Ω ν ( Q ( n ) ) (cid:12)(cid:12) D ( u ( n ) ) (cid:12)(cid:12) d x + Z Ω (cid:12)(cid:12) π n H ( Q ( n ) ) (cid:12)(cid:12) d x = 0 (3.6) for all t ∈ [0 , T n ) . Consequently T n = ∞ for any n ∈ N .Proof. We multiply (3.2) by d k ( t ), integrate in space, and sum over k = 1 , . . . , n . This isequivalent to replacing v k by u ( n ) ( t ) and an integration by parts leads to12 ddt Z Ω | u ( n ) | d x + Z Ω ν ( Q ( n ) ) | D ( u ( n ) ) | d x + (cid:0) π n H ( Q ( n ) ) : ∇ Q ( n ) , u ( n ) (cid:1) Ω + (cid:16) ( σ + ξτ ) (cid:0) Q ( n ) , π n H ( Q ( n ) ) (cid:1) − ξd π n H ( Q ( n ) ) , ∇ u ( n ) (cid:17) Ω = 0 (3.7)in [0 , T n ). Note that by (3.3) the boundary conditions for Q ( n ) are time-independent and there-fore ∂ t Q ( n ) : ∂ n Q ( n ) = 0 on ∂ Ω . This fact shows in combination with the chain rule and an integration by parts that (cid:0) ∂ t Q ( n ) , π n H ( Q ( n ) ) (cid:1) Ω = (cid:0) ∂ t Q ( n ) , H ( Q ( n ) ) (cid:1) Ω = − ddt F ( Q ( n ) ) . ERIS–EDWARDS MODEL 9
Consequently we may replace e ℓ in (3.3) by π n H ( Q ( n ) ) and an integration by parts leads to ddt F ( Q ( n ) ) − (cid:0) ( u ( n ) · ∇ ) Q ( n ) , π n H ( Q ( n ) ) (cid:1) Ω + (cid:0) S ( ∇ u ( n ) , Q ( n ) ) , π n H ( Q ( n ) ) (cid:1) Ω + Z Ω (cid:12)(cid:12) π n H ( Q ( n ) ) (cid:12)(cid:12) d x = 0 (3.8)in [0 , T n ). Moreover, recall that by the algebraic identity (2.4) (cid:0) S ( ∇ u ( n ) , Q ( n ) ) , π n H ( Q ( n ) ) (cid:1) Ω + (cid:0) ( σ + ξτ ) (cid:0) Q ( n ) , π n H ( Q ( n ) ) (cid:1) − ξd π n H ( Q ( n ) ) , ∇ u ( n ) (cid:1) Ω = 0and this identity implies the assertion of the proposition together with (3.7) and (3.8).Finally, (3.6) implies that the norm of the solution ( u ( n ) , Q ( n ) ) cannot blow up in finite time.Hence the characterization of the maximal existence time for solutions of ordinary differentialequations yields T n = ∞ . (cid:3) To construct the solution of the system (1.1) as a weak limit of approximations we needstronger a priori estimates concerning regularity in space and time. The following results holdfor all
T >
0. Note that, unless otherwise indicated, all constants are generic constant whichmay depend on Ω, T , ξ , ν and its derivatives, and other parameters of the system (1.1) but areindependent of t ∈ [0 , T ] and the index n in the approximating system (3.2)–(3.5). Proposition 3.2 (Regularity in space) . Let n ∈ N and let ( u ( n ) , Q ( n ) ) be the solution of thesystem (3.2) – (3.5) . Then u ( n ) ∈ L (0 , T ; H ,σ (Ω)) ∩ L ∞ (0 , T ; L σ (Ω)) , Q ( n ) ∈ L (0 , T ; H (Ω)) and we have the a priori estimates k u ( n ) k L (0 ,T ; H ,σ (Ω)) ∩ L ∞ (0 ,T ; L σ (Ω)) + k Q ( n ) k L (0 ,T ; H (Ω)) C ( E ) where the constant C ( E ) is independent of n but depends on E = E ( u , Q ) .Proof. Proposition 3.1 implies that the ODE system (3.2)–(3.5) has a solution for all times
T > t ∈ [0 ,T ] F ( Q ( n ) ( t, · )) + k π n H ( Q ( n ) ) k L (Ω T ) C ( E ) , (3.9a) k D ( u ( n ) ) k L (Ω T ) + k u ( n ) k L ∞ (0 ,T ; L σ (Ω)) C ( E ) (3.9b)with a constant C independent of n . Korn’s inequality implies the bound on u ( n ) of the propo-sition.To prove the bound on Q ( n ) , we need to improve the bound for π n H ( Q ( n ) ) to a uniform boundfor H ( Q ( n ) ). Since k π n k L ( L ) π n H ( Q ( n ) ) = ∆ Q ( n ) + π n L ( Q ( n ) )these bounds can be obtained from ∆ Q ( n ) and L ( Q ( n ) ), respectively. We obtain from (1.3), andYoung’s inequality for almost all t ∈ [0 , T ] that Z Ω (cid:16) |∇ Q ( n ) ( x, t ) | + | Q ( n ) ( x, t ) | (cid:17) d x C (cid:16) F ( Q ( n ) ( t, · )) + 1 (cid:17) . As a result k Q ( n ) k L ∞ (0 ,T ; H (Ω)) C . (3.10)The definition of H in (1.4) and (3.9a) imply k ∆ Q ( n ) k L (Ω T ) C + k π n L ( Q ( n ) ) k L (Ω T ) . (3.11) ERIS–EDWARDS MODEL 10
Since L ( Q ( n ) ) contains at most cubic terms in Q , we infer from Sobolev’s inequality and (3.10)that k L ( Q ( n ) ) k L (Ω T ) C and the combination of this bound with (3.11) leads to k ∆ Q ( n ) k L (Ω T ) C (3.12)and thus k H ( Q ( n ) ) k L (Ω T ) C . (3.13)Note that the leading part of H ( Q ( n ) ) is ∆ Q ( n ) and therefore we obtain an H -estimate for Q ( n ) .In fact, for all t ∈ [0 , T ], k Q ( n ) ( · , t ) k H (Ω) C (cid:16) k ∆ Q ( n ) ( · , t ) k L (Ω) + k Q ( n ) ( · , t ) k L (Ω) (cid:17) . (3.14)This estimate combined with (3.12) gives the second assertion in the proposition. The proof isnow complete. (cid:3) Proposition 3.3 (Regularity in time) . Let ( u ( n ) , Q ( n ) ) be the solution of the system (3.2) – (3.5) for some n ∈ N . Then we have the a priori estimate k ∂ t u ( n ) k L (0 ,T ; D ( A / ) ′ ) + k ∂ t Q ( n ) k L (0 ,T ; H − (Ω)) C ( E ) , where C ( E ) is independent of n .Proof. We begin with the estimate for ∂ t u ( n ) . By Lemma 2.4, we have the embedding D ( A ) ֒ → L ∞ (Ω). Thus (cid:12)(cid:12)(cid:0) ( u ( n ) · ∇ ) u ( n ) ( t ) , v k (cid:1) Ω (cid:12)(cid:12) k u ( n ) ( t ) k L (Ω) k∇ u ( n ) ( t ) k L (Ω) k v k k L ∞ (Ω) C k u ( n ) ( t ) k L (Ω) k∇ u ( n ) ( t ) k L (Ω) k v k k D ( A ) , (3.15)for all t ∈ [0 , T ] as well as (cid:12)(cid:12)(cid:0) ( π n H ( Q ( n ) )) : ∇ Q ( n ) ( t ) , v k (cid:1) Ω (cid:12)(cid:12) C k H ( Q ( n ) ( t )) k L (Ω) k∇ Q ( n ) ( t ) k L (Ω) k v k k D ( A ) , (3.16)and by (1.7) (cid:12)(cid:12)(cid:0) ν ( Q ( n ) ) D ( u ( n ) )( t ) , D ( v k ) (cid:1) Ω (cid:12)(cid:12) C k∇ u ( n ) ( t ) k L (Ω) k v k ( t ) k D ( A / ) . (3.17)Moreover, Lemma 2.4 implies the embedding D ( A ) ֒ → W , ∞ (Ω) and hence (cid:12)(cid:12)(cid:0) σ (cid:0) Q ( n ) , π n H ( Q ( n ) ) (cid:1) ( t ) − ξd (cid:0) π n H ( Q ( n ) ) (cid:1) ( t ) , ∇ v k (cid:1) Ω (cid:12)(cid:12) C (cid:0) k Q ( n ) ( t ) k L (Ω) (cid:1) k H ( Q ( n ) ( t )) k L (Ω) k v k k D ( A / ) and (cid:12)(cid:12)(cid:0) ξτ (cid:0) Q ( n ) , π n H ( Q ( n ) ) (cid:1) ( t ) , ∇ v k (cid:1)(cid:12)(cid:12) C k Q ( n ) ( t ) k L (Ω) (cid:0) k Q ( n ) ( t ) k L (Ω) + 1 (cid:1) k H ( Q ( n ) ( t )) k L (Ω) k v k k D ( A / ) . (3.18)The combination of (3.15)-(3.18) together with (3.2) yields for all k ∈ { , . . . , n } and almost all t ∈ [0 , T ] that (cid:12)(cid:12)(cid:0) ∂ t u ( n ) ( t ) , v k (cid:1)(cid:12)(cid:12) Cb n ( t ) k v k ( t ) k D ( A / ) , (3.19)where b n ( t ) is defined by b n ( t ) = (cid:0) k u ( n ) ( t ) k L (Ω) (cid:1) k∇ u ( n ) ( t ) k L (Ω) + k H ( Q ( n ) )( t ) k L (Ω) k∇ Q ( n ) ( t ) k L (Ω) + (cid:0) k Q ( n ) ( t ) k L (Ω) (cid:1) k H ( Q ( n ) )( t ) k L (Ω) . In view of (cid:0) ∂ t u ( n ) ( t ) , v (cid:1) Ω = 0 for v ⊥ V n one obtains (cid:12)(cid:12)(cid:0) ∂ t u ( n ) ( t ) , v (cid:1) Ω (cid:12)(cid:12) Cb n ( t ) k v k D ( A / ) for all v ∈ D ( A / ) and for almost all t ∈ [0 , T ] . (3.20) ERIS–EDWARDS MODEL 11
Thus (cid:13)(cid:13) ∂ t u ( n ) ( t ) (cid:13)(cid:13) D ( A / ) ′ Cb n ( t ) for almost all ∈ [0 , T ] . (3.21)The above estimate implies the a priori bound for ∂ t u since b n ( t ) is in L (0 , T ) due to (3.9) and(3.10).Now we turn to the estimate for ∂ t Q ( n ) . By H¨older’s inequality, Sobolev’s embedding H (Ω) ֒ → L (Ω) and (3.10) we find for almost all t ∈ [0 , T ] that (cid:12)(cid:12)(cid:0) ( u ( n ) · ∇ ) Q ( n ) ( t ) , e ℓ (cid:1) Ω (cid:12)(cid:12) k u ( n ) ( t ) k L (Ω) k∇ Q ( n ) ( t ) k L (Ω) k e ℓ k L (Ω) C k u ( n ) ( t ) k H (Ω) k∇ Q ( n ) ( t ) k L (Ω) k e ℓ k H (Ω) , and (cid:12)(cid:12)(cid:0) S ( u ( n ) , Q ( n ) )( t ) , e ℓ (cid:1) Ω (cid:12)(cid:12) C k∇ u ( n ) ( t ) k L (Ω) (cid:0) k Q ( n ) ( t ) k L (Ω) + k Q ( n ) ( t ) k L (Ω) (cid:1) k e ℓ k L (Ω) e C k∇ u ( n ) ( t ) k L (Ω) (cid:0) k∇ Q ( n ) ( t ) k L (Ω) + 1 (cid:1) k e ℓ k H (Ω) , as well as (cid:12)(cid:12)(cid:0) H ( Q ( n ) ( t )) , e ℓ (cid:1) Ω (cid:12)(cid:12) k H ( Q ( n ) ( t )) k L (Ω) k e ℓ k L (Ω) . These estimates imply together with (3.3) that for almost every t ∈ [0 , T ], (cid:12)(cid:12)(cid:0) ∂ t Q ( n ) ( t ) , e (cid:1) Ω (cid:12)(cid:12) Cy n ( t ) k e k H (Ω) , ∀ e ∈ E n , (3.22)where y n ( t ) is defined by y n ( t ) = k u ( n ) k H (Ω) (cid:0) k∇ Q ( n ) ( t ) k L (Ω) + 1 (cid:1) + k H ( Q ( n ) ( t )) k L (Ω) . (3.23)By the orthogonality of the eigenvectors, ( ∂ t Q ( n ) , e ) = 0 for all e ⊥ E n and consequently (cid:12)(cid:12)(cid:0) ∂ t Q ( n ) ( t ) , e (cid:1) Ω (cid:12)(cid:12) Cy n ( t ) k e k H (Ω) for all e ∈ H (Ω; S ) and for almost all t ∈ [0 , T ] , (3.24)which leads to k ∂ t Q ( n ) ( t ) k H − (Ω) Cy n ( t ) . (3.25)The assertion of the proposition follows now since y n ( t ) is integrable in L (0 , T ) in view of theestimates in Proposition 3.2 and (3.23), (3.10) and (3.13). (cid:3) After these preparations we are in a position to give the proof of Theorem 1.2.
Proof of Theorem 1.2.
We divide the proof into several steps.
Step 1: Compactness and construction of weak limits.
We conclude from Proposition 3.2 and 3.3the following bounds on the solutions ( u ( n ) , Q ( n ) ) of the Galerkin approximation k u ( n ) k L (0 ,T ; H ,σ ) ∩ L ∞ (0 ,T ; L σ ) + k ∂ t u ( n ) k L (0 ,T ; D ( A / ) ′ ) + k ∂ t Q ( n ) k L (0 ,T ; H − ) + k∇ Q ( n ) k L ∞ (0 ,T ; L ) + k ∆ Q ( n ) k L (Ω T ) + k H ( Q ( n ) ) k L (Ω T ) C , (3.26)where the constant C is independent of n . Moreover, k Q ( n ) k L (0 ,T ; H ) C (Ω) . By the weak compactness of reflexive Banach spaces and the weak compactness of the dualspaces of separable spaces we can extract a subsequence of ( u ( n ) , Q ( n ) ), which we denote againby ( u ( n ) , Q ( n ) ), such that the weak convergences u ( n ) ⇀ n →∞ u in L (0 , T ; H ,σ (Ω)) ,Q ( n ) ⇀ n →∞ Q in L (0 , T ; H (Ω; S )) , ∆ Q ( n ) ⇀ n →∞ ∆ Q in L (Ω T ; S ) , (3.27) ERIS–EDWARDS MODEL 12 and the weak-*-convergences Q ( n ) ∗ ⇀ n →∞ Q in L ∞ (0 , T ; H (Ω; S )) ,u ( n ) ∗ ⇀ n →∞ u in L ∞ (0 , T ; L σ (Ω)) (3.28)hold. Moreover, for fixed ǫ > . Q ( n ) → n →∞ Q in L (0 , T ; H − ǫ (Ω)) ∩ L p (Ω T ) , ∀ p ∈ (1 , ,u ( n ) → n →∞ u in L (0 , T ; L σ (Ω)) , (3.29)and Q ( n ) → n →∞ Q in C ([0 , T ]; L (Ω)) ,u ( n ) → n →∞ u in C ([0 , T ]; H − σ (Ω)) (3.30)hold. The estimates (3.30), (3.28) and Lemma 2.7 imply that u ∈ BC w ([0 , T ]; L σ (Ω)) , Q ∈ BC w ([0 , T ]; H (Ω)) . (3.31)In order to pass to the limit we assert that the subsequence satisfies additionally ν ( Q ( n ) ) D ( u ( n ) ) ⇀ n →∞ ν ( Q ) D ( u ) in L (Ω T ; R d ) ,H ( Q ( n ) ) ⇀ n →∞ H ( Q ) in L (Ω T ; S ) . (3.32)In fact, for all ϕ ∈ L (Ω T ) we infer from Lebesgue’s dominated convergence theorem, (3.29) andthe properties of the viscosity coefficient that ϕν ( Q ( n ) ) converges strongly to ϕν ( Q ) in L (Ω T )and the conclusion follows from the weak convergence of D ( u ( n ) ) to D ( u ) in L (Ω T ).For the second assertion, note that by the strong convergence of π n to the identity mapon L (Ω; S ), we deduce from the third convergence in (3.27) that π n ∆ Q ( n ) ⇀ ∆ Q . Since H ( Q ) = ∆ Q + L ( Q ), where L ( Q ) is a polynomial of degree less than or equal to three in Q ,cf. (1.4), we only need to show that L ( Q ( n ) ) ⇀ n →∞ L ( Q ) weakly in L (Ω T ). However, since( L ( Q ( n ) )) n ∈ N is bounded in L ∞ (0 , T ; L (Ω)) due to H (Ω) ֒ → L (Ω) and the L ∞ (0 , T ; H (Ω))-bound for Q ( n ) , ( L ( Q ( n ) )) n ∈ N possesses a weak limit in L (Ω T ) (for a suitable subsequence).This weak limit coincides with L ( Q ) since L ( Q ( n ) ) → n →∞ L ( Q ) in L (Ω T ) because of (3.29)with p = 3. Step 2: Derivation of the equation for u . We replace v k in (3.2) by v ∈ C ([0 , T ]; W , ∞ (Ω)) ofthe form v ( t ) = N X k =1 d k ( t ) v k (3.33)and obtain the following equation which holds pointwise for t ∈ [0 , T ]: (cid:0) ∂ t u ( n ) , v (cid:1) Ω + (cid:0) ( u ( n ) · ∇ ) u ( n ) , v (cid:1) Ω + (cid:0) ν ( Q ( n ) ) D ( u ( n ) ) , D ( v ) (cid:1) Ω + (cid:0) ( π n H ( Q ( n ) )) : ∇ Q ( n ) , v (cid:1) Ω + (cid:0) ( σ + ξτ )( Q ( n ) , π n H ( Q ( n ) )) − ξd π n H ( Q ( n ) ) , ∇ v (cid:1) Ω = 0 . ERIS–EDWARDS MODEL 13
If we choose d k ( t ) such that v | t = T = 0 and integrate this equation in time, then an integrationby parts for the first term yields Z T (cid:0) − (cid:0) u ( n ) , ∂ t v (cid:1) Ω + (cid:0) ( u ( n ) · ∇ ) u ( n ) , v (cid:1) Ω + (cid:0) ν ( Q ( n ) ) D ( u ( n ) ) , D ( v ) (cid:1) Ω (cid:1) d t + Z T (cid:2)(cid:0) ( σ + ξτ )( Q ( n ) , π n H ( Q ( n ) )) − ξd π n H ( Q ( n ) ) , ∇ v (cid:1) Ω (cid:3) d t + Z T (cid:0) ( π n H ( Q ( n ) )) : ∇ Q ( n ) , v (cid:1) Ω d t = (cid:0) u ( n ) , v (cid:1) Ω (cid:12)(cid:12)(cid:12) t =0 . (3.34)By the convergences (3.27), (3.32) and (3.29), one can pass to the limit n → ∞ in the firstintegral in (3.34). It remains to show Z Ω T ( σ + ξτ )( Q ( n ) , π n H ( Q ( n ) )) : ∇ v d( x, t ) → Z Ω T ( σ + ξτ ) ( Q, H ( Q )) : ∇ v d( x, t ) , Z Ω T ( π n H ( Q ( n ) )) : ∇ Q ( n ) · v d( x, t ) → Z Ω T H ( Q ) : ∇ Q · v d( x, t )as n → ∞ . To prove the second assertion, we use (3.27), (3.29) and (3.32) to obtain ∇ Q ( n ) · v → n →∞ ∇ Q · v in L (Ω T ) ,π n H ( Q ( n ) ) ⇀ n →∞ H ( Q ) in L (Ω T ) . Using the strong convergence of ( Q ( n ) ) n ∈ N in L (Ω T ) and the weak convergence of H ( Q ( n ) ) in L (Ω T ) one can easily prove the first assertion since all terms in τ ( Q, H ) and σ ( Q, H ) are linearwith respect to H and at most quadratic with respect to Q . Hence we conclude Z Ω T ( − u · ∂ t v + ( u · ∇ ) u · v + ν ( Q ) D ( u ) : D ( v ) + H ( Q ) : ∇ Q · v ) d( x, t )+ Z Ω T (cid:18)(cid:0) σ + ξτ (cid:1) ( Q, H ( Q )) − ξd H ( Q ) (cid:19) : ∇ v d( x, t ) = Z Ω u ( x ) · v (0 , x ) d x for any v of the form (3.33) with v ( T, · ) = 0. By a density argument, the above equationalso holds for any v ∈ C ([0 , T ); V (Ω)). This equation together with (2.3) implies the weakformulation (1.8). Step 3: Derivation of the equation for Q . We replace e ℓ in (3.3) by Ψ ∈ C ([0 , T ]; H (Ω; S ))of the form Ψ( t ) = P Nℓ =1 d ℓ ( t ) e ℓ , integrate in time on [0 , T ] and integrate by parts in the firstterm. This yields − Z Ω T Q ( n ) : ∂ t Ψ d( x, t ) + Z Ω T ( u ( n ) · ∇ ) Q ( n ) : Ψ d( x, t ) − Z Ω T S ( ∇ u ( n ) , Q ( n ) ) : Ψ d( x, t )= Z Ω T H ( Q ( n ) ) : Ψ d( x, t ) + (cid:0) Q ( n ) , Ψ (cid:1) Ω (cid:12)(cid:12)(cid:12) t =0 . Employing (3.27) and (3.29) we conclude Z Ω T S ( ∇ u ( n ) , Q ( n ) ) : Ψ d( x, t ) → n →∞ Z Ω T S ( ∇ u, Q ) : Ψ d( x, t ) . Hence we can pass to the limit in the equation above. Through a density argument we obtainthe weak formulation (1.9). Finally, the boundary conditions (1.6) (for almost every t ) followfrom the fact that ( u ( n ) , Q ( n ) ) satisfy these boundary conditions and the (weak) continuity ofthe Dirichlet and Neumann trace operators on H (Ω), H (Ω), respectively. ERIS–EDWARDS MODEL 14 Regularity in time and proof of Theorem 1.3
The proof of a unique local solution with additional regularity in time is obtained by Banach’sfixed-point theorem. In Section 4.1 we define the function spaces and the operators to which wewill apply the fixed-point theorem, in Section 4.2 we prove that the linear operator L : X → Y defined in (4.4) is bounded, onto and one-to-one, in Section 4.3 we verify that the nonlinearoperator N in (4.6) is locally Lipschitz continuous with small Lipschitz constant for T sufficientlysmall, and in Section 4.4 we give the proof of Theorem 1.3. In this section we assume that( u , Q ) ∈ Z . As usual, we formulate the first equation in (1.1) weakly by testing with divergencefree vector fields. Then we obtain ∂ t u − P σ div( ν ( Q ) D ( u )) = P σ div (cid:0) τ ( Q, H ( Q )) + σ ( Q, H ( Q )) − u ⊗ u (cid:1) , (4.1) ∂ t Q − ∆ Q = − ( u · ∇ ) Q + S ( ∇ u, Q ) + L ( Q ) , (4.2)where P σ : H − (Ω; R d ) → H − σ (Ω) and div : L (Ω; R d × d ) → H − (Ω; R d ) are defined as in Sec-tion 2.1.4.1. Function spaces and operators.
The idea is to rewrite the nonlinear system (1.1) as anoperator equation between suitable Banach spaces. We begin with the definition of the linearand the nonlinear operator in this fixed-point formulation and use these definitions together withthe regularity in time asserted in Theorem 1.3 as motivation for the definition of the functionspaces for the domain and the range of the operators. We linearize the system about the constanttrajectory Q of the Q -tensor. Then the principal part of the linear system is given by S and L , where S ( Q ) (cid:18) uQ (cid:19) = (cid:18) P σ div (cid:2) ν ( Q ) D ( u ) + ( σ + ξτ )( Q , ∆ Q ) − ξd ∆ Q (cid:3) ∆ Q + S ( ∇ u, Q ) (cid:19) , (4.3)and L ( Q ) (cid:18) uQ (cid:19) = ddt (cid:18) uQ (cid:19) − S ( Q ) (cid:18) uQ (cid:19) , (4.4)respectively.As a result, we can consider all the terms in (4.1) as a functional over H ,σ (Ω) and once weobtain a solution to (4.1), we can disregard the P σ in (4.1) by adding a pressure term ∇ p dueto standard results. The nonlinear operator N in the reformulation of the system of partialdifferential equations as the operator equation L ( Q )( u, Q ) = N ( Q )( u, Q ) is given by N ( Q ) (cid:18) uQ (cid:19) = (cid:18) P σ div [( ν ( Q ) − ν ( Q )) D ( u ) + τ ( Q ) − u ⊗ u ] − ( u · ∇ ) Q − L ( Q ) (cid:19) + (cid:18) P σ div (cid:2) ( σ + ξτ )( Q, ∆ Q ) − ( σ + ξτ )( Q , ∆ Q ) + ( σ + ξτ )( Q, L ( Q )) − ξd L ( Q ) (cid:3) S ( ∇ u, Q ) − S ( ∇ u, Q ) + ξS ( ∇ u, Q ) − ξS ( ∇ u, Q ) (cid:19) . It is also useful to pass to a formulation with homogeneous initial and boundary conditions.Note that (1.1) together with the initial and boundary conditions can be formulated by theoperator equation L ( Q ) (cid:18) u h + u Q h + Q (cid:19) = N ( Q ) (cid:18) u h + u Q h + Q (cid:19) , where ( u h , Q h ) satisfies the corresponding homogeneous initial and boundary conditions. Bythe definition of the linear operator L in (4.4), the above identity is equivalent to L ( Q ) (cid:18) u h Q h (cid:19) = N ( Q ) (cid:18) u h + u Q h + Q (cid:19) + S ( Q ) (cid:18) u Q (cid:19) (4.5)and the right-hand side defines a nonlinear operator N ( Q ) (cid:18) u h Q h (cid:19) = N ( Q ) (cid:18) u h + u Q h + Q (cid:19) + S ( Q ) (cid:18) u Q (cid:19) . (4.6) ERIS–EDWARDS MODEL 15
We now turn to the definition of functions spaces X and Y such that L , N : X → Y with L an isomorphism. Motivated by the idea to construct solutions which are twice differentiablein time and the precise assertions in Theorem 1.3, we define the function space for the range ofthe operators by Y u = H (0 , T ; H − σ (Ω)) , Y Q = H (0 , T ; L (Ω; S )) . In particular, we need to prove regularity of solutions of the linear equation L ( Q )( u h , Q h ) =( f, g ) with right-hand side ( f, g ) ∈ Y subject to homogeneous initial data. The general lineartheory requires a compatibility condition which is taken care of by the definition of Y as Y = (cid:8) ( f, g ) ∈ Y u × Y Q : ( f, g ) | t =0 ∈ L σ (Ω) × H D (Ω) (cid:9) . (4.7)These spaces are equipped with the usual norms in product spaces and for spaces of functionsof one variable with values in a Banach space together with the correct norm of the initial data.More precisely, the norm of Y is given by k ( f, g ) k Y = (cid:16) k ( f, g ) k Y u × Y Q + k ( f, g ) | t =0 k L σ (Ω) × H (Ω) (cid:17) . (4.8)Note that the second part of the norm is not controlled by trace theorems applied to Y u × Y Q .The domains of the operators are given by the Banach spaces X u = H (0 , T ; H − σ (Ω)) , X u = H (0 , T ; H ,σ (Ω)) , X u = X u ∩ X u ,X Q = H (0 , T ; L (Ω; S )) , X Q = H (0 , T ; H (Ω; S )) , X Q = X Q ∩ X Q together with the norms k u k X u = (cid:16) k u k X u + k u k X u + k u | t =0 k H ,σ (Ω) + k u t | t =0 k L (Ω) (cid:17) , k Q k X Q = (cid:16) k Q k X Q + k Q k X Q + k Q | t =0 k H (Ω) + k ∂ t Q | t =0 k H (Ω) (cid:17) . (4.9)Note that the last two terms in the norms are important to obtain in the subsequent constantsthat are uniformly bounded as T →
0, cf. e.g. (2.7). The corresponding subspaces related tothe homogeneous initial and boundary conditions in the formulation of the problem are definedby X = { ( u, Q ) ∈ X u × X Q : T ( Q ) = (0 , , ( u, Q ) | t =0 = (0 , } . (4.10)Here the trace operator T ( Q ) is given by T ( Q ) = (cid:0) Q | (0 ,T ) × Γ D , ∂ n Q | (0 ,T ) × Γ N (cid:1) , (4.11)and X is equipped with the product norm k ( u, Q ) k X = k ( u, Q ) k X u × X Q . Together with these norms the space X and Y are closed subspaces of the spaces X u × X Q and Y u × Y Q , respectively.One can check the compatibility condition in Z that the right-hand side of (4.5) belongs to Y if ( u h , Q h ) ∈ X , cf. the proof of Proposition 4.3 (i) below.4.2. Existence and uniqueness for the linear system.
The key point in the proof of thelocal existence of solutions with additional regularity in time is the verification of global solv-ability of the linear system and of its regularity properties. This is achieved based on resultson abstract parabolic evolution equations which we recall for the convenience of the reader.Suppose that V and H are two separable Hilbert spaces such that the embedding V ֒ → H isinjective, continuous, and dense. Fix T ∈ (0 , ∞ ). Suppose that for all t ∈ [0 , T ] a bilinear form a ( t ; · , · ) : V × V → R is given which satisfies for all φ , ψ ∈ V the following assumptions:(a) a ( · ; φ, ψ ) is measurable on [0 , T ]; ERIS–EDWARDS MODEL 16 (b) there exists a constant c >
0, independent of t, φ and ψ , with (cid:12)(cid:12) a ( t ; φ, ψ ) (cid:12)(cid:12) c k φ k V k ψ k V for all t ∈ [0 , T ];(c) there exist k , α > t and φ , with a ( t ; φ, φ ) + k k φ k H > α k φ k V for all t ∈ [0 , T ] ;(d) a ( · ; φ, ψ ) is differentiable, a ( · ; φ, ψ ) is continuous in [0 , T ] and ∂ t a ( t ; φ, ψ ) is measurablewith | ∂ jt a ( t ; φ, ψ ) | c k φ k V k ψ k V for j = 0, 1 with c independent of t . Theorem 4.1.
Suppose that (a)–(c) hold. Then there exists a representation operator L ( t ) : V → V ′ with a ( t ; φ, ψ ) = h L ( t ) φ, ψ i V ′ , V , which is continuous and linear for fixed t . Moreover,for all f ∈ L ((0 , T ); V ′ ) and y ∈ H , there exists a unique solution y ∈ (cid:8) v : [0 , T ] → V with v ∈ L (0 , T ; V ) , ∂ t v ∈ L (0 , T ; V ′ ) (cid:9) which solves the equation ∂ t y + L ( t ) y = f in V ′ for a.e. t ∈ (0 , T ) , subject to the initial condition y (0) = y . Finally, assume additionally that (d) holds and that y ∈ V . Then L : H ((0 , T ); V ) → H ((0 , T ); V ′ ) is continuous and for all f ∈ H ((0 , T ); V ′ ) which satisfy the compatibility condition f (0) − L (0) y ∈ H the solution y satisfies y ∈ H ((0 , T ); V ) and ∂ tt y ∈ L (cid:0) (0 , T ); V ′ (cid:1) . The proof of this theorem can be found in [21, Lemma 26.1 and Theorem 27.2].The following result establishes the invertibility of the linear operator equation. Note thatwe are seeking a solution of the linear equation in X , i.e., a solution with homogeneous initialand boundary conditions. Proposition 4.2 (Homogeneous linear system) . Let T ∈ (0 , . Then L : X → Y , and forevery ( f, g ) ∈ Y , the operator equation L ( Q )( u, Q ) = ( f, g ) has a unique solution ( u, Q ) ∈ X satisfying kL − ( Q )( f, g ) k X = k ( u, Q ) k X C L k ( f, g ) k Y (4.12) where C L is independent of T ∈ (0 , and ( f, g ) ∈ Y . In particular L ( Q ) : X → Y isinvertible and L − ( Q ) is a bounded linear operator with norm independent of T ∈ (0 , .Proof. The idea is to apply Theorem 4.1 and we carry out this program in the subsequent steps.
Step 1: Function spaces.
Since the regularity is in time, we only need to incorporate theregularity in space into the spaces V and H . We define the Hilbert spaces H = H × H = L σ (Ω) × H D (Ω; S ) , V = V × V = H ,σ (Ω) × (cid:8) Q ∈ H (Ω; S ) ∩ H D (Ω; S ) , ∂ n Q | Γ N = 0 (cid:9) and equip them with the inner products(( u, Q ) , ( v, P )) H = ( u, v ) L (Ω) + ( Q, P ) H (Ω) for all ( u, Q ) , ( v, P ) ∈ H , (( u, Q ) , ( v, P )) V = ( u, v ) H (Ω) + ( Q, P ) H (Ω) for all ( u, Q ) , ( v, P ) ∈ V . Here the inner product in the Sobolev spaces H k , k >
1, is the usual inner product in thesespaces. The spaces H and V are Hilbert spaces. ERIS–EDWARDS MODEL 17
Recall that V ֒ → H ∼ = H ′ ֒ → V ′ , where H is identified with H ′ via the Riesz isomorphism P ( ∇ P, ∇· ) L (Ω) + ( P, · ) L (Ω) . This implies that for all P, Φ ∈ V h P, Φ i V ′ , V = ( P, Φ) H = Z Ω ∇ P : ∇ Φ d x + Z Ω P : Φ d x = Z Ω P : ( I − ∆)Φ d x. (4.13) Step 2: Operators and bilinear forms.
The most subtle point is the correct definition of thebilinear form a since the natural bilinear form associated with S given by (cid:10) L ( v, P ) , ( ϕ, Φ) (cid:11) V ′ , V = Z Ω ν ( Q ) D ( v ) : D ( ϕ ) d x + Z Ω (( σ + ξτ )( Q , ∆ P ) − ξd ∆ P ) : ∇ ϕ d x − Z Ω (∆ P + S ( ∇ v, Q )) : Φ d x does not lead to a bilinear form which is coercive on V . This is achieved by taking advantage ofthe cancellation property in (2.4) and by defining a bilinear form which is independent of timeby (cid:10)e L ( v, P ) , ( ϕ, Φ) (cid:11) V ′ , V = Z Ω ν ( Q ) D ( v ) : D ( ϕ ) d x + Z Ω (( σ + ξτ )( Q , ∆ P ) − ξd ∆ P ) : ∇ ϕ d x − Z Ω (∆ P + S ( ∇ v, Q )) : ( I − ∆)Φ d x for all ( v, P ) , ( ϕ, Φ) ∈ V . The additional term in the equation has to be compensated for onthe right-hand side of the linear system and therefore we associate to ( f, g ) ∈ Y the element( F, G ) ∈ L (cid:0) (0 , T ); V ′ (cid:1) by h ( F ( t ) , G ( t )) , ( φ, Φ) i V ′ , V = Z Ω (cid:0) f ( t ) · φ + g ( t ) : ( I − ∆)Φ (cid:1) d x for all ( φ, Φ) ∈ V and almost all t ∈ (0 , T ). We now assert that the solution of the abstractevolution equation (cid:10) ( ∂ t u, ∂ t Q ) , ( φ, Φ) (cid:11) V ′ , V + (cid:10)e L ( u, Q ) , ( ϕ, Φ) (cid:11) V ′ , V = h ( F, G ) , ( φ, Φ) i V ′ , V (4.14)for all ( ϕ, Φ) ∈ V subject to the initial condition ( u (0) , Q (0)) = (0 , ∈ H is indeed a weaksolution of the linear evolution equation. The choice of ( φ, ∈ V implies the correct equationfor u . To identify the equation for Q , choose (0 , Φ) ∈ V as test function and obtain by (4.13) Z Ω g ( x, t ) : ( I − ∆)Φ( x ) d x = h ∂ t Q, Φ i V ′ , V − Z Ω (∆ Q + S ( ∇ u, Q )) : ( I − ∆)Φ d x = Z Ω ( ∂ t Q − ∆ Q − S ( ∇ u, Q )) : ( I − ∆)Φ d x . Since ( I − ∆) : V → L (Ω; S ) is bijective, cf. e.g. [11, Theorems 4.10 and 4.18], we conclude ∂ t Q − ∆ Q − S ( ∇ u, Q ) = g a.e. in Ω × (0 , T ) . Step 3: Existence of time-regular solutions.
The existence of time-regular solutions followsfrom Theorem 4.1 once we have verified the assumptions (a)–(d) on the bilinear form and theregularity assumptions on the right-hand side. Since a does not depend on time, (a) and (d)are immediate. By Sobolev’s embedding theorem, H ֒ → C , hence Q ∈ L ∞ and (b) followsfrom (1.7) and H¨older’s inequality. Moreover, in view of the cancellation property (2.4), Korn’sinequality and Young’s inequality, (cid:10) L ( v, P ) , ( v, P ) (cid:11) V ′ , V = Z Ω ν ( Q ) D ( v ) : D ( v ) d x + Z Ω | ∆ P | d x − Z Ω (∆ P + S ( ∇ v, Q )) : P d x > c k ( v, P ) k V − C k ( v, P ) k H ERIS–EDWARDS MODEL 18 for all ( v, P ) ∈ V , t ∈ [0 , T ], and suitable constants c , C >
0. Therefore (c) is satisfied. Finallywe obtain by the definition of Y that ( f, g ) ∈ Y is equivalent to ( F, G ) ∈ H (0 , T ; V ′ ) and( F (0) , G (0)) ∈ H ′ ∼ = H . Hence there exists a unique solution ( u, Q ) ∈ H (0 , T ; V ′ ) ∩ H (0 , T ; V )of the abstract evolution equation and therefore for the linear equation. Step 4: L is a bounded isomorphism. The only regularity statement which does not follow fromthe regularity of the solution in Step 3 is the assertion Q ∈ H ( L ). Note that the right-handside in the equation for Q belongs to H ( L ). Therefore ∂ t Q ∈ H ( L ) and Q ∈ H ( L ).Altogether, we have proven that L ( Q ) : X → Y is an isomorphism. The boundedness ofthe operator norm of L ( Q ) − : Y → X uniformly in 0 < T u, Q ) T andintegration in time, we derivesup t T k ( u ( t ) , Q ( t )) k H + c Z T k ( u ( t ) , Q ( t )) k V d t C k ( F, G ) k L (0 ,T ; V ′ ) with constants c , C independent of T >
0. Moreover, if we differentiate (4.14) with respect to t and take the duality product with ( ∂ t u, ∂ t Q ) T , then we discoversup t T k ( ∂ t u ( t ) , ∂ t Q ( t )) k H + c Z T k ( ∂ t u ( t ) , ∂ t Q ( t )) k V d t C (cid:0) k ( ∂ t F, ∂ t G ) k L (0 ,T ; V ′ ) + k ( ∂ t u (0) , ∂ t Q (0)) k H (cid:1) . By the previous estimate, Young’s inequality, and (4.14) for t = 0 we concludesup t T k ( ∂ t u ( t ) , ∂ t Q ( t )) k H + c Z T k ( ∂ t u ( t ) , ∂ t Q ( t )) k V d t C (cid:0) k ( F, G ) k H (0 ,T ; V ′ ) + k ( F (0) , G (0)) k H (cid:1) = C k ( F, G ) k Y for all 0 < T
1. Finally, second order time derivatives of (4.14) imply the same estimate for( ∂ t u, ∂ t Q ) ∈ L (0 , T ; V ′ ). The foregoing estimates can be summarize by k ( u, Q ) k X C k ( F, G ) k Y for all T ∈ (0 , (cid:3) Local Lipschitz continuity of the nonlinear operator.
In this section we analyze thenonlinear terms. The fundamental properties of the nonlinear operator are summarized in thefollowing proposition.
Proposition 4.3.
Fix < T , R > , ( u , Q ) ∈ Z , let N ( Q ) be the nonlinear operatordefined in (4.6) , and recall that B X (0 , R ) = { ( v, P ) ∈ X , k ( v, P ) k X R } . Then the followingassertions are true for all ( u hi , Q hi ) ∈ B X (0 , R ) , i = 1 , : (i) N ( Q ) maps X to Y . (ii) Local Lipschitz continuity: There exists a constant C N ( T, R, Q , u ) > such that kN ( Q )( u h , Q h ) − N ( Q )( u h , Q h ) k Y C N ( T, R, Q , u ) k ( u h − u h , Q h − Q h ) k X . (4.15)(iii) Local boundedness: There exists a constant C R ( u , Q ) > independent of T and R such that kN ( Q )( u h , Q h ) k Y C N ( T, R, Q , u ) k ( u h , Q h ) k X + kE ( u , Q ) k Y . (4.16)(iv) For
R > fixed we have lim T → C N ( T, R, Q , u ) = 0 . ERIS–EDWARDS MODEL 19
Proof.
We divide the proof into several steps. In order to simplify the presentation, the de-pendence of the generic constant C on Ω, ξ , ν and ( u , Q ) will be neglected. Moreover, wewill skip the time interval (0 , T ) and domain Ω in the vector-valued functions spaces for betterreadability, e.g. we denote L p ( L q ) = L p (0 , T ; L q (Ω)). For any function F : R k → R ℓ with k , ℓ ∈ N , and any points a , a ∈ R k we define J F ( a ) K = F ( a ) − F ( a ) . Note that by the definitions of P σ : H − (Ω; R d ) → H − σ (Ω) and div : L (Ω; R d × d ) → H − (Ω; R d )we can estimate the H ( H − σ ) norm of the divergence of the difference of the fields in the firstcomponent in N in Y u by their H ( L )-norm. Therefore all estimates in the proof of theLipschitz continuity involve the H ( L )-norm and will be accomplished based on the fact thatmost of the expressions are bilinear or trilinear. Proof of (i): The range of N lies in Y . Fix ( u h , Q h ) ∈ X . The compatibility condition forthe initial conditions of elements in Y in (4.7) follows from ( u , Q ) ∈ Z , (4.10) and (4.6) sincefor all ( u h , Q h ) ∈ Z , N ( Q ) (cid:18) u h Q h (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = N ( Q ) (cid:18) u h + u Q h + Q (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t =0 + S ( Q ) (cid:18) u Q (cid:19) = N ( Q ) (cid:18) u Q (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) t =0 + S ( Q ) (cid:18) u Q (cid:19) = E (cid:18) u Q (cid:19) ∈ L σ (Ω) × H D (Ω) . Moreover, N ( Q )( u h , Q h ) ∈ Y u × Y Q follows by inspection of all terms in the definition of N in the same way as in the proof of (ii). Proof of (ii): Local Lipschitz continuity of N . Let X T = { ( u, Q ) ∈ X u × X Q : T ( Q ) = ( Q D , Q N ) , ( u, Q ) | t =0 = ( u , Q ) } . We define ( u i , Q i ) = ( u hi + u , Q hi + Q ) ∈ X T and ( u, Q ) = ( u − u , Q − Q ) ∈ X , where weidentify as usual a function independent of t with its extension to (0 , T ) as a constant function.By definition, J N ( Q )( u h , Q h ) K = J N ( Q )( u, Q ) K and since N involves only spatial derivatives, we infer N ( Q )( u , Q ) (cid:12)(cid:12) t =0 − N ( Q )( u , Q ) (cid:12)(cid:12) t =0 = 0 . Hence (4.15) is equivalent to kN ( Q )( u , Q ) − N ( Q )( u , Q ) k Y u × Y Q C N ( T, R, Q , u ) k ( u, Q ) k X (4.17)for all ( u i , Q i ) ∈ X T such that k ( u i , Q i ) − ( u , Q ) k X R . (4.18)The proof of the Lipschitz continuity requires additional estimates and is therefore divided intoseveral steps in which we estimate the differences between the various terms in the operators.
Step 1: Uniform bounds.
We have for i = 1, 2 uniform bounds in space–time, k Q i − Q k L ∞ (Ω T ) + k ∂ t Q i − ∂ t Q k L ( L ∞ ) CT R , k Q k L ∞ (Ω T ) + k ∂ t Q k L ( L ∞ ) CT k Q k X Q , (4.19) ERIS–EDWARDS MODEL 20 as well as uniform bounds in time for higher-order norms in space, k Q i k L ∞ ( H k ) C (cid:0) R + k Q k H k (cid:1) , k ∈ { , , } , k u i k L ∞ ( H k ) C (cid:0) R + k u k H k (cid:1) , k ∈ { , } , k u k L ∞ ( H ) + k Q k L ∞ ( H ) C (cid:0) k u k X u + k Q k X Q (cid:1) . (4.20)Note carefully, that the constants are independent of T . More precisely, by the interpolationresult (2.5), k Q i − Q k L ∞ (Ω T ) C k Q i − Q k L ∞ ( H ) k Q i − Q k L ∞ ( H ) . (4.21)We apply (2.7) with H = H = H k (Ω), 0 k Q i − Q , observe that the term related tothe initial conditions vanishes since Q i | t =0 = Q and obtain for i = 1 , k Q i − Q k L ∞ ( H ) k Q i − Q k L ∞ ( H ) C k Q i − Q k L ( H ) k Q i − Q k H ( H ) k Q i − Q k L ( H ) k Q i − Q k H ( H ) CT k Q i − Q k L ∞ ( H ) k Q i − Q k H ( H ) k Q i − Q k H ( H ) CT k Q i − Q k H ( H ) CT R , (4.22)where we used (4.18) in the last step. The uniform bound for Q i − Q in (4.19) follows imme-diately. To estimate the time derivatives of Q i − Q , we use H¨older’s inequality and (2.6) andfind for i = 1 , k ∂ t Q i − ∂ t Q k L ( L ∞ ) C k ∂ t Q i − ∂ t Q k L ( L ) k ∂ t Q i − ∂ t Q k L ( H ) CT k ∂ t Q i − ∂ t Q k L ∞ ( L ) k ∂ t Q i − ∂ t Q k L ( H ) CT (cid:0) k ∂ t Q i − ∂ t Q k H ( L ) + k ( ∂ t Q i − ∂ t Q ) | t =0 k L (Ω) (cid:1) k ∂ t Q i − ∂ t Q k L ( H ) CT k Q i − Q k X Q CT R . (4.23)In the last step, we used the definition (4.9) of the norm in X Q . The estimates for Q are analogousand the proof of (4.19) is complete. To verify (4.20) we employ the triangle inequality, (2.7) and(4.18) and find for i = 1 , k = 0 , , k Q i k L ∞ ( H k ) k Q i − Q k L ∞ ( H k ) + k Q k L ∞ ( H k ) C (cid:0) k Q i − Q k H ( H k ) + k ( Q i − Q ) | t =0 k H k (cid:1) + k Q k H k C (cid:0) R + k Q k H k (cid:1) (4.24)and k Q k L ∞ ( H k ) C ( k Q k H ( H k ) + k Q | t =0 k H k ) = C k Q k X Q . (4.25)The estimates for u and u i are similar and therefore (4.20) has been established. Step 2: Estimates for differences of viscosities.
Note that by the fundamental theorem ofcalculus, ν ( Q ) − ν ( Q ) = Z ddτ ν (cid:0) τ Q + (1 − τ ) Q (cid:1) d τ = Z ( ∇ ν ) (cid:0) τ Q + (1 − τ ) Q (cid:1) : Q d τ , and by (4.19) and for i = 1, 2, k ν ( Q ) − ν ( Q ) k L ∞ (Ω T ) C ( R, ν, Q ) k Q k L ∞ (Ω T ) C ( R, ν, Q ) T k Q k X Q , k ν ( Q i ) − ν ( Q ) k L ∞ (Ω T ) C ( R, ν, Q ) k Q i − Q k L ∞ (Ω T ) C ( R, ν, Q ) T . (4.26) ERIS–EDWARDS MODEL 21
If one differentiates the integral representation, then one finds ∂ t (cid:0) ν ( Q ) − ν ( Q ) (cid:1) = Z (cid:8) ( ∇ ν ) (cid:0) τ Q + (1 − τ ) Q (cid:1) [ Q, τ ∂ t Q + (1 − τ ) ∂ t Q ] + ( ∇ ν ) (cid:0) τ Q + (1 − τ ) Q (cid:1) : Q t (cid:9) d τ . We deduce from (1.7), (4.19), (4.20) and the foregoing formula that for i = 1, 2 and a.e. in Ω T , | ∂ t ( ν ( Q )) − ∂ t ( ν ( Q ))) | C ( R ) (cid:0) | Q | + | ∂ t Q | (cid:1) , | ∂ t ( ν ( Q i )) − ∂ t ( ν ( Q )) | C ( R ) (cid:0) | Q i − Q | + | ∂ t Q i − ∂ t Q | (cid:1) . (4.27)Finally note that k ∂ t Q i k L ∞ (Ω T ) C k Q i k X . Step 3: Estimates for differences of viscous stress tensor.
We verify the estimate (cid:13)(cid:13) q P σ div (cid:0) ( ν ( Q ) − ν ( Q )) D ( u ) (cid:1) y (cid:13)(cid:13) H ( H − σ ) C ( ν, R ) T k ( u, Q ) k X u × X Q . (4.28)To this end we rewrite this expression as (cid:13)(cid:13) P σ div (cid:2) ( ν ( Q ) − ν ( Q )) D ( u ) − ( ν ( Q ) − ν ( Q )) D ( u ) (cid:3)(cid:13)(cid:13) H ( H − σ ) (cid:13)(cid:13) div (cid:2) ( ν ( Q ) − ν ( Q )) D ( u ) + ( ν ( Q ) − ν ( Q )) D ( u ) (cid:3)(cid:13)(cid:13) H ( H − ) k ( ν ( Q ) − ν ( Q )) D ( u ) k H ( L ) + k ( ν ( Q ) − ν ( Q )) D ( u ) k H ( L ) , Expressing these norms as L ( L )-norms of the functions and their first order derivative in timeleads to four higher-order and two lower-order terms. For the higher-order terms we find by(4.27) and (4.19) k ∂ t ( ν ( Q ) − ν ( Q )) D ( u ) k L (Ω T ) C ( R ) k ( | Q | + | ∂ t Q | ) | D ( u ) |k L (Ω T ) C ( R ) k Q k H ( L ∞ ) k D ( u ) k L ∞ ( L ) C ( R ) T k Q k X , and analogously k ∂ t ( ν ( Q i ) − ν ( Q )) Du k L (Ω T ) C ( R ) k ( | Q i − Q | + | ∂ t ( Q i − Q ) | | Du |k L (Ω T ) C ( R ) k Q i − Q k H ( L ∞ ) k Du k L ∞ ( L ) C ( R ) T k u k X u . We obtain for the remaining two higher-order terms by (4.19), (4.20) and (4.26) k ( ν ( Q ) − ν ( Q )) ∂ t D ( u ) k L (Ω T ) C ( R ) k Q k L ∞ (Ω T ) k ∂ t D ( u ) k L (Ω T ) C ( R ) k Q k L ∞ (Ω T ) (cid:0) k u − u k X u + k ∂ t D ( u ) k L (Ω T ) (cid:1) C ( R ) T k Q k X Q and k ( ν ( Q ) − ν ( Q )) ∂ t D ( u ) k L (Ω T ) C ( R ) T k ∂ t D ( u ) k L (Ω T ) C ( R ) T k u k X u . To estimate the lower-order terms in the H ( L )-norm we use (4.26) and obtain k ( ν ( Q ) − ν ( Q )) D ( u ) k L ( L ) + k ( ν ( Q ) − ν ( Q )) D ( u ) k L ( L ) C ( R ) T (cid:0) k Q k X Q k D ( u ) k L ( L ) + k Q − Q k X Q k D ( u ) k L ( L ) (cid:1) C ( R ) T k ( u, Q ) k X u × X Q . The combination of the foregoing estimates implies the assertion of this step.
ERIS–EDWARDS MODEL 22
Step 4: Fundamental estimates for bilinear forms.
Suppose that P , P are time dependenttensor fields with initial value P , that P = P − P , and that B : R d × d × R d × d → R d × d is abilinear form with constant coefficients. Then k J B ( Q − Q , P ) K k H ( L ) CT R (cid:0) k P k L ∞ ( L ) + k P k H ( L ) ) (cid:1) + CT k Q k X Q (cid:0) k P k L ∞ ( L ) + k P k H ( L ) (cid:1) where we assume that all norms on the right-hand side are finite. In particular, k J B ( Q − Q , P ) K k H ( L ) C ( R ) T k ( u, Q ) k X for P ∈ (cid:8) ∇ u, D ( u ) , W ( u ) , ∆ Q (cid:9) . In fact, by the triangle inequality and the product rule forbilinear forms, k B ( Q − Q , P ) − B ( Q − Q , P ) k H ( L ) k B ( Q − Q , P ) k H ( L ) + k B ( Q, P ) k H ( L ) k B ( ∂ t Q − ∂ t Q , P ) k L ( L ) + k B ( Q − Q , ∂ t P ) k L ( L ) + k B ( Q − Q , P ) k L ( L ) + k B ( ∂ t Q, P ) k L ( L ) + k B ( Q, ∂ t P ) k L ( L ) + k B ( Q, P ) k L ( L ) C (cid:8) k ∂ t Q − ∂ t Q k L ( L ∞ ) k P k L ∞ ( L ) + k Q − Q k L ∞ (Ω T ) k P k H ( L ) + k ∂ t Q k L ( L ∞ ) k P k L ∞ ( L ) + k Q k L ∞ (Ω T ) k P k H ( L ) (cid:9) . The assertion follows from (4.19) and (4.20).
Step 5: Fundamental estimates for trilinear forms.
Suppose that P , P are time dependenttensor fields with initial values P and that E : R d × d × R d × d × R d × d → R d × d is a trilinear formwith constant coefficients. Then k J E ( Q, Q, P ) − E ( Q , Q , P ) K k H ( L ) CT R (cid:0) k P k L ∞ ( L ) + k P k H ( L ) ) (cid:1) + C ( R, k P k L ) T k Q k X Q (cid:0) k P k H ( L ) + k P k L (cid:1) where we assume that all norms on the right-hand side are finite. In particular, k J E ( Q, Q, P ) − E ( Q , Q , P ) K k H ( L ) C ( R ) T k ( u, Q ) k X for P ∈ (cid:8) ∇ u, D ( u ) , W ( u ) , ∆ Q (cid:9) . To see this, note that E ( Q , Q , P ) − E ( Q , Q , P ) − E ( Q , Q , P ) + E ( Q , Q , P )= E ( Q , Q , P ) − E ( Q , Q , P ) + E ( Q , Q , P ) − E ( Q , Q , P )= E ( Q − Q , Q , P ) + E ( Q , Q − Q , P ) + E ( Q − Q , Q , P ) + E ( Q , Q − Q , P ) . We need to estimate this sum in the H ( L )-norm. By H¨older’s inequality and the product rulefor trilinear forms, each term leads to four terms that need to be estimated in L ( L ). For thefirst term we find k E ( Q − Q , Q , P ) k H ( L ) k E ( ∂ t Q − ∂ t Q , Q , P ) + E ( Q − Q , ∂ t Q , P ) + E ( Q − Q , Q , ∂ t P ) k L ( L ) + k E ( Q − Q , Q , P ) k L ( L ) C (cid:8) k ∂ t Q − ∂ t Q k L ( L ∞ ) k Q k L ∞ (Ω T ) k P k L ∞ ( L ) + k Q − Q k L ∞ (Ω T ) k ∂ t Q k L ( L ∞ ) k P k L ∞ ( L ) + k Q − Q k L ∞ (Ω T ) k Q k L ∞ (Ω T ) k P k H ( L ) (cid:9) . ERIS–EDWARDS MODEL 23
The second term can be estimated the same way. For the third term one finds k E ( Q − Q , Q , P ) k H ( L ) k E ( ∂ t Q − ∂ t Q , Q , P ) + E ( Q − Q , ∂ t Q , P ) + E ( Q − Q , Q , ∂ t P ) k L ( L ) + k E ( Q − Q , Q , P ) k L ( L ) C (cid:8) k ∂ t Q − ∂ t Q k L ( L ∞ ) k Q k L ∞ (Ω T ) k P k L ∞ ( L ) + k Q − Q k L ∞ (Ω T ) k ∂ t Q k L ( L ∞ ) k P k L ∞ ( L ) + k Q − Q k L ∞ (Ω T ) k Q k L ∞ (Ω T ) k P k H ( L ) (cid:9) . The fourth term can be estimated as before.
Step 6: Estimates for additional stress tensors in the fluid equation.
We have (cid:13)(cid:13) J P σ div (cid:8) σ ( Q − Q , ∆ Q ) + ξ (cid:0) τ ( Q, ∆ Q ) − τ ( Q , ∆ Q ) (cid:1)(cid:9) K (cid:13)(cid:13) H ( H − σ ) C ( R ) T k Q k X Q . This follows for σ ( Q − Q , ∆ Q ) and the bilinear part in τ ( Q, ∆ Q ) − τ ( Q , ∆ Q ) from Step 4and for the trilinear part Q tr( Q ∆ Q ) − Q tr( Q ∆ Q ) from Step 5. Step 7: The coupling term in the evolution of the tensor field.
We have (cid:13)(cid:13) q S ( ∇ u, Q ) + ξS ( ∇ u, Q ) y (cid:13)(cid:13) H ( H − σ ) C ( R ) T k ( u, Q ) k X u × X Q . This follows for the bilinear part in S ( ∇ u, Q ) + ξS ( ∇ u, Q ) from Step 4 and for the trilinearpart Q tr( Q ∇ u ) − Q tr( Q ∇ u ) from Step 5. Step 8: Additional lower-order terms.
The terms u ⊗ u , ( u · ∇ ) Q + L ( Q ) and J ( Q ) = P σ div (cid:2) τ ( Q ) + σ ( Q, L ( Q )) + ξτ ( Q, L ( Q )) − ξd L ( Q ) (cid:3) are of lower-order and lead to the estimates kJ ( Q ) − J ( Q ) k Y u T C ( R ) k Q k X Q , k div (cid:0) u ⊗ u − u ⊗ u (cid:1) k Y u T C ( R ) k u k X u , k ( u · ∇ ) Q + L ( Q ) − ( u · ∇ ) Q − L ( Q ) k Y Q T C ( R ) k ( u, Q ) k X u × X Q . These estimates can be done the same way as in Step 4 and 5.
Proof of (iii): Boundedness of N . If suffices to show that (4.16) is a consequence of (4.15). In fact, the choice of ( u , Q ) = 0 in(4.15) implies kN ( Q )( u , Q ) − N ( Q )(0 , k Y C N ( T, R ) k ( u , Q ) k Y (4.29)and the assertion follows by the triangle inequality since N ( Q )(0 ,
0) = E ( u , Q ), see the proofof (i). Proof of (iv): Asymptotic behaviour of the constant.
This assertion follows from the scaling ofthe constants in step (ii) in T . (cid:3) Proof of Theorem 1.3.
By (4.5) and (4.6), the proof of Theorem 1.3 can be reduced tothe statement that the nonlinear mapping L ( Q ) := L − ( Q ) N ( Q ) : X → X (4.30)has a unique fixed-point. By (4.12) and (4.15) we find for all ( u hi , Q hi ) ∈ B X (0 , R ) that (cid:13)(cid:13) L − ( Q ) N ( Q )( u h , Q h ) − L − ( Q ) N ( Q )( u h , Q h ) (cid:13)(cid:13) X C L kN ( Q )( u h , Q h ) − N ( Q )( u h , Q h ) k Y C L C N ( T, R, Q ) k ( u h − u h , Q h − Q h ) k X . ERIS–EDWARDS MODEL 24
Therefore L ( Q ) is a contraction mapping for T ≪
1. A similar argument shows that L maps B X (0 , R ) into itself. In fact, by by (4.16) (cid:13)(cid:13) L ( Q )( u h , Q h ) (cid:13)(cid:13) X C L (cid:13)(cid:13) N ( Q )( u h , Q h ) (cid:13)(cid:13) Y C L (cid:0) C N ( T, R, Q ) k ( u h , Q h ) k X + kE ( u , Q ) k Y (cid:1) and this estimate allows us to fix R ≫ T ≪ (cid:13)(cid:13) L ( Q )( u h , Q h ) (cid:13)(cid:13) X C L C N ( T, R ) k ( u h , Q h ) k X + R R .
We conclude from Banach’s fixed-point theorem that L possess a unique fixed-point ( u h , Q h ) ∈ X and this fixed-point is a solution of the system (1.1) subject to (1.5) and (1.6).The argument implies the uniqueness as well. Suppose that there was another solution(ˆ u h , ˆ Q h ) in B X (0 , R ) with R > R . Choose ˆ T T and repeat the above argument to showthe uniqueness of fixed-points of L , which implies ( u h , Q h ) = (ˆ u h , ˆ Q h ) on (0 , ˆ T ) × Ω. Then theuniqueness follows by the continuity argument. (cid:3)
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