Large deviations for nematic liquid crystals driven by pure jump noise
aa r X i v : . [ m a t h . P R ] J un Large deviations for nematic liquid crystals driven by purejump noise
Rangrang Zhang , ∗ Guoli Zhou Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, P R China School of Statistics and Mathematics, Chongqing University, Chongqing, 400044, P R China ( [email protected] [email protected] ) Abstract : In this paper, we establish a large deviation principle for a stochastic evolution equation whichdescribes the system governing the nematic liquid crystals driven by pure jump noise. The proof is basedon the weak convergence approach.
AMS Subject Classification : Primary 60F10 Secondary 60H15.
Keywords : large deviations; weak convergence method; nematic liquid crystals.
As we all know, the obvious states of matter are the solid, the liquid and the gaseous state. The liquidcrystal is an intermediate state of a matter, in between the liquid and the crystalline solid, i.e. it mustpossess some typical properties of a liquid as well as some crystalline properties. The nematic liquidcrystal phase is characterized by long-range orientational order, i.e. the molecules have no positionalorder but tend to align along a preferred direction. Much of the interesting phenomenology of liquidcrystals involves the geometry and dynamics of the preferred axis, which is defined by a vector θ . Thisvector is called a director. Since the sign as well as the magnitude of the director has no physicalsignificance, it is taken to be unity.The concrete description of the physical relevance of liquid crystals can be referred to Chandrasekhar[8], Warner and Terentjev [18], Gennes and Prost [11] and the references therein. In the 1960’s, Ericksen[9] and Leslie [14] demonstrated the hydrodynamic theory of liquid crystals. Moreover, they expandedthe continuum theory which has been widely used by most researchers to design the dynamics of thenematic liquid crystals. Inspired by this theory, the most fundamental form of dynamical system repre-senting the motion of nematic liquid crystals has been procured by Lin and Liu [15].The addition of a stochastic noise to this model is fully natural as it represents external randomperturbations or a lack of knowledge of certain physical parameters. More precisely, we consider the ∗ Corresponding author. O T : = (0 , T ] × O , O ⊂ R d , d = du + [( u · ∇ ) u − µ ∆ u + ∇ p ] dt + ∇ · ( ∇ θ ⊙ ∇ θ ) dt = R X G ( t , u ( t − ) , v ) ˜ N ( dt , dv ) , ∇ · u = , d θ + [( u · ∇ ) θ − γ ∆ θ ( t )] dt + f ( θ ( t )) dt = , (1.1)where ˜ N is the compensated time homogeneous Poisson random measure and G , f are measurable func-tions will be specified later. There are several recent works about the existence and uniqueness of path-wise weak solution of (1.1), i.e. strong in the probabilistic sense and weak in the PDE sense. In [4],Brze´zniak, Hausenblas and Razafimandimby studied the Ginzburg-Landau approximation of the nematicliquid crystals under the influence of fluctuating external forces. In that paper, they proved the existenceand uniqueness of local maximal solution in both 2D and 3D cases using fixed point argument. Also theyhave proved the existence of global strong solution to the problem in the 2D case. Brze´zniak, Manna andPanda [5] studied the nematic liquid crystals driven by pure jump noise in both 2D and 3D cases. Theyproved the global well-posedness of strong solution in the 2D case and established the existence of weakmartingale solution of this model in the 3D case, respectively.The purpose of this paper is to prove a large deviations for the 2D nematic liquid crystals driven by apure jump noise, which provides the exponential decay of small probabilities associated with the corre-sponding stochastic dynamical systems with small noise. The proof of the large deviations will be basedon the weak convergence approach introduced in Budhiraja, Chen and Dupuis [6] and Budhiraja, Dupuisand Maroulas [7]. As an important part of the proof, we need to obtain global well-posedness of the socalled skeleton equation. For the uniqueness, we adopt the method introduced in [5]. For the existence,we first apply the Faedo-Galerkin approximation method to construct a sequence of approximating equa-tions as in [5]. We then show that the family of the solutions of the approximating equations is compactin an appropriate space and that any limit of the approximating solutions gives rise to a solution of theskeleton equation. To complete the proof of the large deviation principle, we also need to study the weakconvergence of the perturbations of the system (1.1) in the random directions of the Cameron-Martinspace of the driving Brownian motions.This paper is organized as follows. The mathematical formulation of nematic liquid crystals flows isin Section 2. In Section 3, we recall a general criterion obtained in Budhiraja, Dupuis and Maroulas [7]and state the main result. Section 4 is devoted to the study of the skeleton equations. The large deviationsis proved in Section 5. Let T > O ⊂ R be a bounded domain with smooth boundary ∂ O . Consider the following two-dimensional stochastic evolution equations in O T : = (0 , T ] × O given by du + [( u · ∇ ) u − µ ∆ u + ∇ p ] dt + ∇ · ( ∇ θ ⊙ ∇ θ ) dt = R X G ( t , u ( t − ) , v ) ˜ N ( dt , dv ) , ∇ · u = , d θ + [( u · ∇ ) θ − γ ∆ θ ( t )] dt + f ( θ ( t )) dt = , (2.2)2here the vector field u = u ( x , t ) denotes the velocity of the fluid, θ = θ ( x , t ) is the director field, p denoting the scalar pressure. ˜ N is the compensated time homogeneous Poisson random measure ona certain locally compact Polish space ( X , B ( X )). G and f are measurable functions, which will bespecified in subsection 2.3. The symbol ∇ θ ⊙ ∇ θ is the 2 × ∇ θ ⊙ ∇ θ ] i , j = X k = ∂ x i θ ( k ) ∂ x j θ ( k ) , i , j = , . Without loss of generality, we assume that µ = γ = . The boundary and initial conditions for (2.2) are u = ∂θ∂ n = ∂ O , ( u (0) , θ (0)) = ( u , θ ) , where n is the outward unit normal vector at each point x of O . Denote by N , R , R + , R d the set of positive integers, real numbers, positive real numbers and d − dimensional real vectors, respectively. For a topology space E , denote the corresponding Borel σ − field by B ( E ). For a metric space X , C ([0 , T ]; X ) stands for the space of continuous functionsfrom [0 , T ] into X and D ([0 , T ]; X ) represents the space of right continuous functions with left limitsfrom [0 , T ] into X . For a metric space Y , denote by M b ( Y ) , C b ( Y ) the space of real valued bounded Y / R − measurable maps and real valued bounded continuous functions, respectively.Now, we follow closely the framework of [5]. For any p ∈ [1 , ∞ ) and k ∈ N , let ( L p ( O ) , | · | L p )and ( W k , p ( O ) , k · k W k , p ) be Lebesgue and Sobolev space of R -valued functions, respectively. For p = W k , = H k . For instance, H ( O ) is the Sobolev space of all u ∈ L ( O ), for which there exist weakderivatives ∂ u ∂ x i ∈ L ( O ) , i = , . It’s well-known that H ( O ) is a Hilbert space with the scalar productgiven by ( u , v ) H : = ( u , v ) L + ( ∇ u , ∇ v ) L , u , v ∈ H ( O ) . Now, define working spaces for the system (2.2) as V : = (cid:8) u ∈ C ∞ c ( O ); ∇ · u = (cid:9) , H : = the closure of V in L ( O ) and V : = the closure of V in H ( O ).In the space H , we equip it with the scalar product and the norm inherited from L ( O ) and denotethem by ( · , · ) H and | · | H , respectively, i.e.,( u , v ) H : = ( u , v ) L , | u | H : = | u | L : = | u | , u , v ∈ H .
3n the space V , we equip it with the scalar product inherited from the Sobolev space H ( O ), i.e.,( u , v ) V : = ( u , v ) L + (( u , v )) , where (( u , v )) : = ( ∇ u , ∇ v ) L = Z O ∂ u ∂ x · ∂ v ∂ x dx + Z O ∂ u ∂ x · ∂ v ∂ x dx , u , v ∈ V . (2.3)The norm of V is defined as k u k V : = | u | H + k u k , where k u k : = |∇ u | .As we are working on a bounded domain, it’s clear that V ֒ → H (cid:27) H ′ ֒ → V ′ , where the embedding is compact continuous. Also, we have the embedding H ֒ → H ֒ → L (cid:27) L ֒ → ( H ) ′ ֒ → ( H ) ′ . Set A u : = (( u , · )) , u ∈ V , (2.4)where (( · , · )) is defined by (2.3). If u ∈ V , then A u ∈ V ′ . By the Cauchy-Schwarz inequality, we deducethat | A u | V ′ ≤ k u k , u ∈ V . (2.5)It’s well-known that A is a positive self-adjoint operator. Let { ̺ i } ∞ i = be the orthonormal basis of H composed of eigenfunctions of the Stokes operator A with corresponding eigenvalues 0 ≤ λ ≤ λ ≤· · · → ∞ ( A ̺ i = λ i ̺ i ). We will use fractional powers of the operator A , denoted by A α , as well as theirdomains D ( A α ) for α ∈ R . Note that D ( A α ) = { u = ∞ X i = u i · ̺ i : ∞ X i = λ α i u i < ∞} . We may endow D ( A α ) with the inner product( u , v ) D ( A α ) = ( A α u , A α v ) H . Hence, ( D ( A α ) , ( · , · ) D ( A α ) ) is a Hilbert space and { λ − α i ̺ i } i ∈ N is a complete orthonormal system of D ( A α ).By Riesz representative theorem, D ( A − α ) is the dual space of D ( A α ).4efine a self-adjoint operator A : H → ( H ) ′ by h A θ, w i : = (( θ, w )) : = Z O ∇ θ · ∇ wdx , θ, w ∈ H . (2.6)Let { ς i } ∞ i = be the orthonormal basis of L composed of eigenfunctions of the Stokes operator A . Wehave k A θ k ( H ) ′ ≤ k θ k H . (2.7)Consider the following trilinear form (see [17]) b ( u , v , w ) = X i , j = Z O u ( i ) ∂ x i v ( j ) w ( j ) dx , u ∈ L p , v ∈ W , q , w ∈ L r , where p , q , r ∈ [1 , ∞ ] satisfying 1 p + q + r ≤ . (2.8)Referring to [1], by the Sobolev embedding Theorem and Hölder inequality, we obtain | b ( u , v , w ) | ≤ C k u k V k v k V k w k V , u , v , w ∈ V , (2.9)for some positive constant C . Thus, b is a continuous on V .Now, define a bilinear map B : V × V → V ′ by h B ( u , v ) , w i : = b ( u , v , w ) = X i , j = Z O u ( i ) ∂ x i v ( j ) w ( j ) dx . Then, referring to [17], it gives
Lemma 2.1.
For any u ∈ V, v ∈ V, w ∈ V, (1) h B ( u , v ) , w i = −h B ( u , w ) , v i , h B ( u , v ) , v i = b ( u , v , v ) = . (2) k B ( u , v ) k V ′ ≤ C | u | k u k | v | k v k , for some positive constnat C . Based on Lemma 2.1, the operator B can be uniquely extended to a bounded linear operator B : H × H → V ′ , and it satisfies the following estimate k B ( u , v ) k V ′ ≤ C | u || v | . (2.10)For the convenience for written, denote B ( u ) : = B ( u , u ). Note that B : V → V ′ is locally Lipschitzcontinuous.Now, define a bilinear mapping ˜ B : H × H → ( H ) ′ as h ˜ B ( u , v ) , w i = b ( u , v , w ) u , v , w ∈ H . We still denote by ˜ B ( · , · ) the restriction of ˜ B ( · , · ) to V × H , which map continuously from V × H into L . According to [17], we have 5 emma 2.2. For u ∈ V , θ ∈ H , we have (1) h ˜ B ( u , θ ) , θ i = b ( u , θ, θ ) = . (2) | ˜ B ( u , θ ) | ≤ C | u | k u k k θ k | ∆ θ | , for some positive constant C. Consider the trilinear form defined by m ( θ , θ , u ) = − X i , j , k = Z O ∂ x i θ ( k )1 ∂ x j θ ( k )2 ∂ x j u ( i ) dx for any θ ∈ W , p , θ ∈ W , q and u ∈ W , r with p , q , r ∈ (1 , ∞ ) satisfying (2.8).Define a bilinear operator M : H × H → V ′ such that for any θ , θ ∈ H h M ( θ , θ ) , u i = m ( θ , θ , u ) , u ∈ V . Then, by Hölder inequality and Sobolev interpolation inequality, we have k M ( θ , θ ) k V ′ ≤ C k θ k | ∆ θ | k θ k | ∆ θ | . (2.11)For simplicity, we denote M ( θ ) : = M ( θ, θ ).Collecting all the above functionals, (2.2) can be written as du ( t ) + [ A u ( t ) + B ( u ( t )) + M ( θ ( t ))] dt = R O G ( t , u ( t ) , v ) ˜ N ( dt , dv ) , d θ ( t ) + [ A θ ( t ) + ˜ B ( u ( t ) , θ ( t )) + f ( θ ( t ))] dt = . (2.12) To obtain the global well-posedness of (2.12), we introduce the following hypotheses stated in [5].
Hypothesis H0 (A) ˜ N is a compensated time homogeneous Poisson random measure on a locally com-pact space ( X , B ( X )) over a probability space ( Ω , F , P ) with a σ − finite intensity measure ϑ . (B) Let G : [0 , T ] × H × X → H is a measurable function and there exists a constant L such that Z X | G ( t , u , v ) − G ( t , u , v ) | H ϑ ( dv ) ≤ L | u − u | , u , u ∈ H , t ∈ [0 , T ] . (2.13)and for each p ≥
1, there exists a constant C p such that Z X | G ( t , u , v ) | p H ϑ ( dv ) ≤ C p (1 + | u | p ) , u ∈ H , t ∈ [0 , T ] . (2.14) (C) For N ∈ N , numbers b j , j = , · · · , N with b j >
0, we define a function ˜ f : [0 , ∞ ) → R by˜ f ( r ) = N X j = b j r j , for any r ∈ R + . f : R → R by f ( θ ) = ˜ f ( | θ | ) θ. (2.15)Let F : R → R be a Frechét di ff erentiable map such that for any θ ∈ R and g ∈ R F ′ ( θ )[ g ] = f ( θ ) · g . (2.16)Under Hypothesis H0 (C) , referring to Appendix D in [5], we have
Lemma 2.3.
For any κ > and κ > , there exists C ( κ ) > , C ( κ ) > , C ( κ ) > such that |h f ( θ ) − f ( θ ) , θ − θ i| ≤ κ |∇ θ − ∇ θ | + C ( κ ) | θ − θ | β ( θ , θ ) , (2.17) and |h f ( θ ) − f ( θ ) , ∆ ( θ − θ ) i|≤ κ | ∆ θ − ∆ θ | + [ C ( κ ) |∇ ( θ − θ ) | + C ( κ ) | θ − θ | ] β ( θ , θ ) , (2.18) where β ( θ , θ ) : = C (1 + | θ | NL N + + | θ | NL N + ) . (2.19) Moreover, for any θ ∈ H , it gives | f ( θ ) | ≤ C (1 + | θ | qL q ) , q = N + . (2.20)Now, we recall the definition of a strong solution to (2.12) in [5]. Definition 2.1.
The system (2.12) has a strong solution if for every stochastic basis ( Ω , F , {F t } t ≥ , P ) anda time homogeneous Poisson random measure ˜ N on ( X , B ( X )) over the stochastic basis with intensitymeasure ϑ , there exist progressively measurable process u : [0 , T ] × Ω → H with P − a.e.u ( · , ω ) ∈ D ([0 , T ]; H ) ∩ L ([0 , T ]; V ) (2.21) and progressively measurable process θ : [0 , T ] × Ω → H with P − a.e. θ ( · , ω ) ∈ C ([0 , T ]; H ) ∩ L ([0 , T ]; H ) (2.22) such that for all t ∈ [0 , T ] and χ ∈ V , the following identity holds P − a.e. ( u ( t ) , χ ) + Z t h A u ( s ) , χ i ds + Z t h B ( u ( s )) , χ i ds + Z t h M ( θ ( s )) , χ i ds = ( u , χ ) + Z t Z X ( G ( s , u ( s ) , v ) , χ ) ˜ N ( dsdv ) , (2.23) and for all υ ∈ H , the following identity holds P − a.e. ( θ ( t ) , υ ) + Z t h A θ ( s ) , υ i ds + Z t h ˜ B ( u ( s ) , θ ( s )) , υ i ds + Z t h f ( θ ( s )) , υ i ds = ( θ , υ ) . (2.24)7ccording to [5], we have Theorem 2.1.
Let the initial value ( u , θ ) ∈ H × H . Under Hypothesis H0 , the system (2.12) has astrong solution ( u , θ ) in the sense of Definition 2.1. Also, the solution satisfies the following estimates E sup t ∈ [0 , T ] | u ( t ) | H + Z T k u ( t ) k V dt < ∞ , (2.25) and E sup t ∈ [0 , T ] k θ ( t ) k H + Z T k θ ( t ) k H dt < ∞ . (2.26) In this section, we will recall a general criterion for a large deviation principle introduced by Budhiraja,Dupuis and Maroulas in [7]. To this end, we closely follow the framework and notations in Budhiraja,Chen and Dupuis [6] and Budhiraja, Dupuis and Maroulas [7].Let { X ε } be a family of random variables defined on a probability space ( Ω , F , P ) taking values insome Polish space E . The large deviation principle is concerned with exponential decay of P ( X ε ∈ · ), as ε → Definition 3.1. (Rate Function) A function I : E → [0 , ∞ ] is called a rate function if for each M < ∞ ,the level set { x ∈ E : I ( x ) ≤ M } is a compact subset of E . For O ∈ B ( E ) , we define I ( O ) : = inf x ∈ O I ( x ) . Definition 3.2. (Large Deviation Principle) The sequence { X ε } is said to satisfy a large deviation prin-ciple with rate function I if the following two conditions hold. (a) Large deviation upper bound. For each closed subset F of E , lim sup ε → ε log P ( X ε ∈ F ) ≤ − I ( F ) , (b) Large deviation lower bound. For each open subset G of E , lim inf ε → ε log P ( X ε ∈ G ) ≥ − I ( G ) . The following notations will be used. Let X be a locally compact Polish space. Set C c ( X ) be the spaceof continuous functions with compact supports. Denote M FC ( X ) : = n measure ϑ on ( X , B ( X )) such that ϑ ( K ) < ∞ for every compact K in X o . Endow M FC ( X ) with the weakest topology such that for every f ∈ C c ( X ), the function ϑ → h f , ϑ i = R X f ( u ) d ϑ ( u ) , ϑ ∈ M FC ( X ) is continuous. This topology can be metrized such that M FC ( X ) is a Polishspace (see [7]).Let T >
0, set X T = [0 , T ] × X . Fix a measure ϑ ∈ M FC ( X ) and let ϑ T = λ T ⊗ ϑ , where λ T isLebesgue measure on [0 , T ]. We recall the definition of Poisson random measure from [12] that8 efinition 3.3. We call measure n a Poisson random measure on X T with intensity measure ϑ T is a M FC ( X ) − valued random variable such that (1) for each B ∈ B ( X T ) with ϑ T ( B ) < ∞ , n ( B ) is a Poisson distribution with mean ϑ T ( B ) , (2) for disjoint B , · · · , B k ∈ B ( X T ) , n ( B ) , · · · , n ( B k ) are mutually independent random variables. Denote by P the measure induced by n on ( M FC ( X T ) , B ( M FC ( X T ))). Let M = M FC ( X T ). P is theunique probability measure on ( M , B ( M )), under which the canonical map N : M → M , N ( m ) : = m is aPoisson random measure with intensity measure ϑ T . In this paper, we also consider probability P θ , for θ >
0, under which N is a Poisson random measure with intensity θϑ T . The corresponding expectationoperators will be denoted by E and E θ , respectively.Set Y = X × [0 , ∞ ) , Y T = [0 , T ] × Y . Similarly, let ¯ M = M FC ( Y T ) and let ¯ P be the unique probability measure on ( ¯ M , B ( ¯ M )) under whichthe canonical mapping ¯ N : ¯ M → ¯ M , ¯ N ( m ) : = m is a Poisson random measure with intensity measure¯ ϑ T = λ T ⊗ ϑ ⊗ λ ∞ , with λ ∞ being Lebesgue measure on [0 , ∞ ). The expectation operator will be denotedby ¯ E . Let F t : = σ { ¯ N ((0 , s ] × O ) : 0 ≤ s ≤ t , O ∈ B ( Y ) } , and denote by ¯ F t the completion under ¯ P . Let¯ P be the predictable σ − field on [0 , T ] × ¯ M with the filtration { ¯ F t : 0 ≤ t ≤ T } on ( ¯ M , B ( ¯ M ))and ¯ A be the class of all ( ¯ P ⊗ B ( X )) / ( B [0 , ∞ )) − measurable maps ϕ : X T × ¯ M → [0 , ∞ ) . For ϕ ∈ ¯ A , define a counting process N ϕ on X T by N ϕ ((0 , t ] × U ) = Z (0 , t ] × U Z (0 , ∞ ) I [0 ,ϕ ( s , x )] ( r ) ¯ N ( dsdxdr ) , t ∈ [0 , T ] , U ∈ B ( X ) . (3.27) N ϕ is the controlled random measure with ϕ selecting the intensity for the points at location x andtime s , in a possibly random but nonanticipating way. If ϕ ( s , x , ¯ m ) ≡ θ ∈ (0 , ∞ ). We write N ϕ = N θ . Notethat N θ has the same distribution with respect to ¯ P as N has with respect to P θ . Define l : [0 , ∞ ) → [0 , ∞ )by l ( r ) = r log r − r + , r ∈ [0 , ∞ ) . For any ϕ ∈ ¯ A , the quantity L T ( ϕ ) = Z X T l ( ϕ ( t , x , w )) ϑ T ( dtdx ) (3.28)is well-defined as a [0 , ∞ ] − valued random variable.9 .2 A general criterion In order to state a general criteria for large deviation principle (LDP) obtained by Budhiraja et al. in [7],we introduce the following notations. Define S M = { g : X T → [0 , ∞ ) : L T ( g ) ≤ M } , S = ∪ M ≥ S M . A function g ∈ S M can be identified with a measure ϑ gT ∈ M , which is defined by ϑ gT ( O ) = Z O g ( s , x ) ϑ T ( dsdx ) , O ∈ B ( X T ) . This identification induces a topology on S M under which S M is a compact space (see the Appendixof [6]). Throughout this paper, we always use this topology on S M . Let U M = { ϕ ∈ ¯ A : ϕ ( ω ) ∈ S M , ¯ P − a . e .ω } , where ¯ A is defined in subsection 3.1.Let {G ε } ε> be a family of measurable maps from ¯ M to U , where ¯ M is introduced in subsection3.1 and U is a Polish space. Let Z ε = G ε ( ε N ε − ). Now, we list the following su ffi cient conditions forestablishing LDP for the family { Z ε } ε> . Condition A
There exists a measurable map G : ¯ M → U such that the following hold. (i) For every M < ∞ , let g n , g ∈ S M be such that g n → g as n → ∞ . Then, G ( ϑ g n T ) → G ( ϑ gT ) in U . (ii) For every M < ∞ , let { ϕ ε : ε > } ⊂ U M be such that ϕ ε converges in distribution to ϕ as ε → G ε ( ε N ε − ϕ ε ) converges to G ( ϑ ϕ T ) in distribution.The following result is due to Budhiraja et al. in [7]. Theorem 3.1.
Suppose the above
Condition A holds. Then Z ε satisfies a large deviation principle on U with the good rate function I given byI ( f ) = inf { g ∈ S : f = G ( ϑ gT ) } n L T ( g ) o , ∀ f ∈ U . (3.29) By convention, I ( ∅ ) = ∞ . In order to obtain LDP for (2.12), we need additional conditions on the coe ffi cients. Here, we adopt thesame conditions as [19] and state some preliminary results from Budhiraja et al. [6].Let G : [0 , T ] × H × X → H be a measurable mapping. Set | G ( t , v ) | , H : = sup u ∈ H | G ( t , u , v ) | H + | u | H , ( t , v ) ∈ [0 , T ] × X , | G ( t , v ) | , H : = sup u , u ∈ H , u , u | G ( t , u , v ) − G ( t , u , v ) | H | u − u | H , ( t , v ) ∈ [0 , T ] × X , ypothesis H1 For i = ,
1, there exists δ i > E ∈ B ([0 , T ] × X ) satisfying ϑ T ( E ) < ∞ ,the following holds Z E e δ i | G ( s , v ) | i , H ϑ ( dv ) ds < ∞ . Now, we state the following Lemmas established by [6] and [19].
Lemma 3.1.
Under
Hypothesis H0 and
Hypothesis H1 , (i) For i = , and every M ∈ N ,C Mi , : = sup g ∈ S M Z X T | G ( s , v ) | i , H | g ( s , v ) − | ϑ ( dv ) ds < ∞ , (3.30) C Mi , : = sup g ∈ S M Z X T | G ( s , v ) | i , H | g ( s , v ) + | ϑ ( dv ) ds < ∞ . (3.31) (ii) For every η > , there exists δ > such that for any A ⊂ [0 , T ] satisfying λ T ( A ) < δ sup g ∈ S M Z A Z X | G ( s , v ) | i , H | g ( s , v ) − | ϑ ( dv ) ds ≤ η. (3.32) Lemma 3.2. (1)
For any g ∈ S , if sup t ∈ [0 , T ] | Y ( t ) | H < ∞ , then Z X G ( · , Y ( · ) , v )( g ( · , v ) − ϑ ( dv ) ∈ L ([0 , T ]; H ) . (2) If the family of mappings { Y n : [0 , T ] → H , n ≥ } satisfying sup n sup t ∈ [0 , T ] | Y n ( t ) | H < ∞ , then ˜ C M : = sup g ∈ S M sup n Z T | Z X G ( t , Y n ( t ) , v )( g ( t , v ) − ϑ ( dv ) | H ds < ∞ . Lemma 3.3.
Let h : [0 , T ] × X → R be a measurable function such that Z X T | h ( s , v ) | ϑ ( dv ) ds < ∞ , and for all δ ∈ (0 , ∞ ) and E ∈ B ([0 , T ] × X ) satisfying ϑ T ( E ) < ∞ , Z E exp( δ | h ( s , v ) | ) ϑ ( dv ) ds < ∞ . Then, we have (1)
Fix M ∈ N . Let g n , g ∈ S M be such that g n → g as n → ∞ . Then lim n →∞ Z X T h ( s , v )( g n ( s , v ) − ϑ ( dv ) ds = Z X T h ( s , v )( g ( s , v ) − ϑ ( dv ) ds . (2) Fix M ∈ N . Given ε > , there exists a compact set K ε ⊂ X , such that sup g ∈ S M Z T Z K c ε | g ( s , v ) − | ϑ ( dv ) ds ≤ ε. For every compact set K ⊂ X , lim M →∞ sup g ∈ S M Z T Z K | h ( s , v ) | I { h ≥ M } g ( s , v ) ϑ ( dv ) ds = . In this paper, we consider the following nematic liquid crystals driven by small multiplicative Lévynoise: du ε ( t ) + [ A u ε ( t ) + B ( u ε ( t )) + M ( θ ε ( t ))] dt = ε R X G ( t , u ε ( t ) , v ) ˜ N ε − ( dtdv ) , d θ ε ( t ) + [ A θ ε ( t ) + ˜ B ( u ε ( t ) , θ ε ( t )) + f ( θ ε ( t ))] dt = . (3.33)By Theorem 2.1, under Hypothesis H0 , there exists a unique strong solution of (2.12) in D ([0 , T ]; H ) × C ([0 , T ]; H ). Therefore, there exists a Borel-measurable mapping: G ε : ¯ M → D ([0 , T ]; H ) × C ([0 , T ]; H )such that ( u ε ( · ) , θ ε ( · )) = G ε ( ε N ε − ).For g ∈ S , consider the following skeleton equations du g ( t ) + [ A u g ( t ) + B ( u g ( t )) + M ( θ g ( t ))] dt = R X G ( t , u g ( t ) , v )( g ( t , v ) − ϑ ( dv ) dt , d θ g ( t ) + [ A θ g ( t ) + ˜ B ( u g ( t ) , θ g ( t )) + f ( θ g ( t ))] dt = . (3.34)The solution ( u g , θ g ) defines a mapping G : ¯ M → D ([0 , T ]; H ) × C ([0 , T ]; H ) such that ( u g ( · ) , θ g ( · )) = G ( ϑ gT ).In this paper, our main result is Theorem 3.2.
Let ( u , θ ) ∈ H × H . Under Hypothesis H0 and
Hypothesis H1 , ( u ε , θ ε ) satisfies a largedeviation principle on D ([0 , T ]; H ) × C ([0 , T ]; H ) with the good rate function I defined by (3.29) withrespect to the uniform convergence.Proof. According to Theorem 2.1, we need to prove (i) and (ii) in
Condition A . The verification of (i)will be established by Proposition 5.1, (ii) will be proved by Theorem 5.2. (cid:3)
In this section, we will show that the skeleton equation (3.34) admits a unique solution for every g ∈ S .Let K be a Banach space with norm k · k K . Given p > , α ∈ (0 , W α, p ([0 , T ]; K ) bethe Sobolev space of all u ∈ L p ([0 , T ]; K ) such that Z T Z T k u ( t ) − u ( s ) k pK | t − s | + α p dtds < ∞ , endowed with the norm k u k pW α, p ([0 , T ]; K ) = Z T k u ( t ) k pK dt + Z T Z T k u ( t ) − u ( s ) k pK | t − s | + α p dtds . The following results can be found in [10]. 12 emma 4.1.
Let B ⊂ B ⊂ B be Banach spaces, B and B reflexive, with compact embedding B ⊂ B.Let p ∈ (1 , ∞ ) and α ∈ (0 , be given. Let X be the spaceX = L p ([0 , T ]; B ) ∩ W α, p ([0 , T ]; B ) , endowed with the natural norm. Then the embedding of X in L p ([0 , T ]; B ) is compact. Lemma 4.2.
For V and H are two Hilbert spaces (V ′ is the dual space of V) with V ⊂⊂ H = H ′ ⊂ V ′ ,where V ⊂⊂ H denotes V is compactly embedded in H. If u ∈ L ([0 , T ]; V ) , dudt ∈ L ([0 , T ]; V ′ ) , thenu ∈ C ([0 , T ]; H ) . For the skeleton equation (3.34), we have
Theorem 4.1.
Given ( u , θ ) ∈ H × H and g ∈ S . Assume
Hypothesis H0 and
Hypothesis H1 hold,then there exists a unique solution ( u g , θ g ) such thatu g ∈ C ([0 , T ]; H ) ∩ L ([0 , T ]; V ) , θ g ∈ C ([0 , T ]; H ) ∩ L ([0 , T ]; H ) , and u g ( t ) = u − Z t A u g ( s ) ds − Z t B ( u g ( s )) ds − Z t M ( θ g ( s )) ds + Z t Z X G ( s , u g ( s ) , v )( g ( s , v ) − ϑ ( dv ) ds , (4.35) θ g ( t ) = θ − Z t A θ g ( s ) ds − Z t ˜ B ( u g ( s ) , θ g ( s )) ds − Z t f ( θ g ( s )) ds . (4.36) Moreover, for any M ∈ N , there exists C ( p , M ) > such that sup g ∈ S M sup s ∈ [0 , T ] | θ g ( s ) | pL + Z T | θ g ( s ) | p − L ( k θ g ( s ) k + | θ g ( s ) | N + L N + ) ds ≤ C ( p , M ) , (4.37) and sup g ∈ S M sup s ∈ [0 , T ] ( Ψ ( θ g ( s )) + | u g ( s ) | ) p + (cid:16) Z T ( k u g ( s ) k + | ∆ θ g ( s ) − f ( θ g ( s )) | ) ds (cid:17) p ≤ C ( p , M ) , (4.38) where Ψ ( θ g ( s )) : = k θ g ( s )) k + R O G ( | θ g ( s ) | ) dx.Proof. (Existence) We apply the Faedo-Galerkin approximation method to deduce the existence of so-lution of (2.12). Let Φ n : R → [0 ,
1] be a smooth function such that Φ n ( t ) =
1, if | t | ≤ n , Φ n ( t ) =
0, if | t | > n +
1. Define χ n : H → H as χ n ( u ) = Φ n ( | u | H ) u , and χ n : L → L as χ n ( θ ) = Φ n ( | θ | L ) θ . Define thefollowing finite dimensional spaces for any n ∈ N , H n : = Span { ̺ , · · · , ̺ n } , L n : = Span { ς , · · · , ς n } . P n the projection from H onto H n , and ˜ P n the projection from L onto L n . Let B n ( u ) : = χ n ( u ) B ( u ) , u ∈ H n , M n ( θ ) : = χ n ( θ ) M ( θ ) , θ ∈ L n , ˜ B n ( θ ) : = χ n ( u ) ˜ B ( u , θ ) , u ∈ H n , θ ∈ L n . Based on the above mappings, consider the following Faedo-Galerkin approximations: ( u n ( t ) , θ n ( t )) ∈ H n × L n , which is the solution of du n ( t ) + A u n ( t ) dt + P n B n ( u n ( t )) dt + P n M n ( θ n ( t )) dt = P n R X G ( t , u n ( t ) , v )( g ( t , v ) − ϑ ( dv ) dt , (4.39) d θ n ( t ) + A θ n ( t ) dt + ˜ P n ˜ B n ( u n ( t ) , θ n ( t )) dt + ˜ P n f n ( θ n ( t )) dt = , (4.40)with the initial condition ( u n (0) , θ n (0)) = ( P n u , ˜ P n θ ).Since B n , M n , ˜ B n are all globally Lipschitz continuous, the existence of solutions to (4.39)-(4.40) canbe obtained using similar method as [2].Now, for the solution ( u n ( t ) , θ n ( t )) of (4.39)-(4.40), we aim to show that, for any p ≥ n sup s ∈ [0 , T ] | θ n ( s ) | pL + Z T | θ n ( s ) | p − L ( k θ n ( s ) k + | θ n ( s ) | N + L N + ) ds ≤ C ( p , M ) , (4.41)and sup n sup s ∈ [0 , T ] ( Ψ ( θ n ( s )) + | u n ( s ) | ) p + (cid:16) Z T ( k u n ( s ) k + | ∆ θ n ( s ) − f ( θ n ( s )) | ) ds (cid:17) p ≤ C ( p , M ) , (4.42)where Ψ ( θ n ( s )) : = k θ n ( s )) k + R O F ( | θ n ( s ) | ) dx and F is defined by (2.16).Firstly, we prove (4.41). For p ≥
2, let ψ ( · ) be the mapping defined by ψ ( θ ( t )) : = p | θ ( t ) | p , θ ∈ L . (4.43)The first Fréchet derivative is ψ ′ ( θ )[ h ] = | θ ( t ) | p − h θ, h i . (4.44)Based on (4.40), (4.43) and (4.44), we deduce that d ψ ( θ n ( t )) = −| θ n ( t ) | p − h A θ n ( t ) + ˜ B n ( u n ( t ) , θ n ( t )) + f n ( θ n ( t )) , θ n ( t ) i dt . Due to (4.43) and Lemma 2.2, we arrive at | θ n ( t ) | p + Z t | θ n ( s ) | p − k θ n ( s ) k ds + Z t | θ n ( s ) | p − h f n ( θ n ( s )) , θ n ( s ) i ds ≤ | θ | p . (4.45)Referring to equations (5.12) and (5.13) in [5], it gives that h f ( θ ) , θ i ≥ Z O | θ ( x ) | N + dx − C Z O | θ ( x ) | dx . (4.46)14utting (4.46) into (4.45), we deduce that | θ n ( t ) | p + Z t | θ n ( s ) | p − k θ n ( s ) k ds + Z t | θ n ( s ) | p − ( | θ n ( s ) | N + L N + − C | θ n ( s ) | ) ds ≤ | θ | p , which implies that | θ n ( t ) | p + Z t | θ n ( s ) | p − k θ n ( s ) k ds + Z t | θ n ( s ) | p − | θ n ( s ) | N + L N + ds ≤ | θ | p + C Z t | θ n ( s ) | p ds . (4.47)Applying Gronwall’s inequality, we reachsup t ∈ [0 , T ] | θ n ( t ) | p ≤ | θ | p e CT . (4.48)Combining (4.47) and (4.48), we conclude that Z T | θ n ( s ) | p − k θ n ( s ) k ds + Z T | θ n ( s ) | p − | θ n ( s ) | N + L N + ds ≤ | θ | p + C Z T | θ n ( s ) | p ds ≤ | θ | p (1 + CT e CT ) . Thus, we complete the result (4.41).For (4.42), we firstly define a stopping time τ Rn : = inf { t ≥ | u n ( t ) | ≥ R or | θ n ( t ) | L ≥ R or k θ n ( t ) k H ≥ R } ∧ T . (4.49)From (4.41), we deduce that τ Rn ↑ T , P − a . s . , as R ↑ ∞ .For any t ∈ [0 , τ Rn ], define a mapping φ ( · ) as φ ( u ( t )) = | u ( t ) | , u ∈ H . Then, we have d φ ( u n ( t )) = −h A u n ( t ) + B n ( u n ( t )) + M n ( θ n ( t )) , u n ( t ) i dt + h P n Z X G ( t , u n ( t ) , v )( g ( t , v ) − ϑ ( dv ) , u n ( t ) i dt . Using Lemma 2.1, we obtain d φ ( u n ( t )) + k u n ( t ) k dt = −h M n ( θ n ( t )) , u n ( t ) i dt + h P n Z X G ( t , u n ( t ) , v )( g ( t , v ) − ϑ ( dv ) , u n ( t ) i dt . Let Ψ ( · ) be the mapping defined by Ψ ( θ ) = k θ k + Z O F ( | θ | ) dx . Ψ ′ ( θ )[ g ] = h∇ θ, ∇ g i + h f ( θ ) , g i = h− ∆ θ + f ( θ ) , g i . Then, we have d Ψ ( θ n ( t )) = Ψ ′ ( θ n )[ d θ n ( t )] = h− ∆ θ n + f n ( θ n ) , d θ n ( t ) i = h− ∆ θ n + f n ( θ n ) , − A θ n ( t ) − f n θ n ( t ) − ˜ B n ( u n ( t ) , θ n ( t )) i dt = −| ∆ θ n − f n ( θ n ) | dt − h ˜ B n ( u n ( t ) , θ n ( t )) , − ∆ θ n + f n ( θ n ) i dt . Referring to (5.28)-(5.29) in [5], it gives h ˜ B n ( u n ( t ) , θ n ( t )) , − ∆ θ n + f n ( θ n ) i = −h M n ( θ n ) , u n i . (4.50)This implies d Ψ ( θ n ( t )) + | ∆ θ n − f n ( θ n ) | dt = h M n ( θ n ) , u n i dt . (4.51)Adding (4.50) and (4.51), we get d [ Ψ ( θ n ( t )) + φ ( u n ( t ))] + ( k u n ( t ) k + | ∆ θ n − f n ( θ n ) | ) dt = h P n Z X G ( t , u n ( t ) , v )( g ( t , v ) − ϑ ( dv ) , u n ( t ) i dt . (4.52)Since h P n Z X G ( t , u n ( t ) , v )( g ( t , v ) − ϑ ( dv ) , u n ( t ) i≤ Z X | G ( t , u n ( t ) , v ) || g ( t , v ) − || u n ( t ) | ϑ ( dv ) ≤ Z X | G ( t , u n ( t ) , v ) | + | u n ( t ) | | g ( t , v ) − | (1 + | u n ( s ) | ) | u n ( t ) | ϑ ( dv ) ≤ Z X | G ( t , v ) | , H | g ( t , v ) − | (1 + | u n ( t ) | ) ϑ ( dv ) ≤ Z X | G ( t , v ) | , H | g ( t , v ) − | ϑ ( dv ) + Z X | G ( t , v ) | , H | g ( t , v ) − || u n ( t ) | ϑ ( dv ) , we conclude that [ Ψ ( θ n ( t )) + | u n ( t ) | ] + Z t ( k u n ( s ) k + | ∆ θ n − f n ( θ n ) | ) ds ≤ Ψ ( θ ) + | u | + Z t Z X | G ( s , v ) | , H | g ( s , v ) − | ϑ ( dv ) ds + Z t | u n ( s ) | Z X | G ( s , v ) | , H | g ( s , v ) − | ϑ ( dv ) ds . (4.53)16pplying Gronwall inequality to (4.53), we havesup t ∈ [0 ,τ Rn ] [ Ψ ( θ n ( t )) + | u n ( t ) | ] + Z τ Rn ( k u n ( s ) k + | ∆ θ n − f n ( θ n ) | ) ds ≤ h Ψ ( θ ) + | u | + Z τ Rn Z X | G ( s , v ) | , H | g ( s , v ) − | ϑ ( dv ) ds i × exp n Z τ Rn Z X | G ( s , v ) | , H | g ( s , v ) − | ϑ ( dv ) ds o ≤ h Ψ ( θ ) + | u | + Z T Z X | G ( s , v ) | , H | g ( s , v ) − | ϑ ( dv ) ds i × exp n Z T Z X | G ( s , v ) | , H | g ( s , v ) − | ϑ ( dv ) ds o . (4.54)Utilizing (3.30), we deduce thatsup t ∈ [0 ,τ Rn ] [ Ψ ( θ n ( t )) + | u n ( t ) | ] + Z τ Rn ( k u n ( s ) k + | ∆ θ n − f n ( θ n ) | ) ds ≤ [ Ψ ( θ ) + | u | + C M , ] T e C M , T . (4.55)As the constant in the right hand side of (4.55) is independent of R and n , passing to the limit as R → ∞ ,we obtain sup s ∈ [0 , T ] [ Ψ ( θ n ( t )) + | u n ( t ) | ] + Z T ( k u n ( s ) k + | ∆ θ n − f n ( θ n ) | ) ds ≤ C ( p , T ) . (4.56)Moreover, with the help of (4.56), we can obtain an estimate for ∆ θ n and k θ n ( t ) k H using similarmethod as Proposition 5.6 in [5]. Concretely, for any p ≥
1, there exists a positive constant C independentof n such that | Z T | ∆ θ n ( s ) | ds | p ≤ C ( p ) , sup t ∈ [0 , T ] k θ n ( t ) k pH ≤ C ( p ) . (4.57)In the following, we want to prove that for α ∈ (0 , ), there exists C ( α ) , L ( α ) > n ≥ k u n k W α, ([0 , T ]; V ′ ) ≤ C ( α ) . (4.58)sup n ≥ k θ n k W α, ([0 , T ];( H ) ′ ) ≤ L ( α ) . (4.59)Firstly, u n ( t ) can be written as u n ( t ) = P n u − Z t A u n ( s ) ds − Z t B n ( u n ( s )) ds − Z t M n ( θ n ( s )) ds + Z t Z X G ( s , u n ( s ) , v )( g ( s , v ) − ϑ ( dv ) ds : = I n + I n ( t ) + I n ( t ) + I n ( t ) + I n ( t ) . | I n | ≤ C . Since k A u n k V ′ ≤ k u n k , for t > s , we have k I n ( t ) − I n ( s ) k V ′ = k Z ts A u n ( r ) dr k V ′ ≤ C ( t − s ) Z ts k A u n ( r ) k V ′ dr ≤ C ( t − s ) Z ts k u n ( r ) k dr . Hence, by (4.42), we have for α ∈ (0 , ), k I n k W α, ([0 , T ]; V ′ ) ≤ Z T k I n ( t ) k V ′ dt + Z T Z T k I n ( t ) − I n ( s ) k V ′ | t − s | + α dsdt ≤ C ( α ) . Moreover, using (2.10), for t > s , we get k I n ( t ) − I n ( s ) k V ′ = k Z ts B n ( u n ( r )) dr k V ′ ≤ C ( t − s ) Z ts k B n ( u n ( r )) k V ′ dr ≤ C ( t − s ) Z ts | u n ( r ) | dr ≤ C ( t − s ) sup t ∈ [0 , T ] | u n ( t ) | Z ts k u n ( r ) k dr , thus, by (4.42), for α ∈ (0 , ), we have k I n k W α, ([0 , T ]; V ′ ) ≤ C ( α ) . Utilizing (2.11), for t > s , we deduce that k I n ( t ) − I n ( s ) k V ′ = k Z ts M n ( θ n ( r )) dr k V ′ ≤ Z ts k M n ( θ n ( r )) k V ′ dr ! ≤ Z ts k θ n ( r ) k| ∆ θ n ( r ) | dr ! ≤ C ( t − s ) sup t ∈ [0 , T ] k θ n ( r ) k Z ts | ∆ θ n ( r ) | dr , hence, by (4.42) and (4.57), for α ∈ (0 , ), we have k I n k W α, ([0 , T ]; V ′ ) ≤ C ( α ) . I n , we have | I n ( t ) − I n ( s ) | H = (cid:12)(cid:12)(cid:12)(cid:12) Z ts P n Z X G ( r , u n ( r ) , v )( g ( r , v ) − ϑ ( dv ) dr (cid:12)(cid:12)(cid:12)(cid:12) H ≤ Z ts Z X | G ( r , u n ( r ) , v ) || g ( r , v ) − | ϑ ( dv ) dr ! ≤ Z ts Z X | G ( r , v ) | , H | g ( r , v ) − | (1 + | u n ( r ) | ) ϑ ( dv ) dr ! ≤ (1 + sup t ∈ [0 , T ] | u n ( r ) | ) Z ts Z X | G ( r , v ) | , H | g ( r , v ) − | ϑ ( dv ) dr ! ≤ (1 + sup t ∈ [0 , T ] | u n ( r ) | ) Z T Z X | G ( r , v ) | , H | g ( r , v ) − | ϑ ( dv ) dr × Z ts Z X | G ( r , v ) | , H | g ( r , v ) − | ϑ ( dv ) dr , with the help of (3.30) and (4.42), for α ∈ (0 , ), we get k I n k W α, ([0 , T ]; H ) ≤ C ( α ) . Based on the above estimates, we complete the proof of (4.58). The proof of (4.59) is similar to (4.58). θ n ( t ) can be written as θ n ( t ) = P n θ − Z t A θ n ( s ) ds − Z t ˜ B n ( u n ( s ) , θ n ( s )) ds − Z t f n ( θ n ( s )) ds : = J n + J n ( t ) + J n ( t ) + J n ( t ) . It’s easy to know k J n k H ≤ L . For t > s , we have k J n ( t ) − J n ( s ) k H ) ′ = k Z ts A θ n ( r ) dr k H ) ′ ≤ C Z ts k A θ n ( r ) k H ) ′ dr ≤ C Z ts | θ n ( r ) | dr ≤ C ( t − s ) sup t ∈ [0 , T ] | θ n ( t ) | , thus, by (4.41), for α ∈ (0 , ), we have k J n k W α, ([0 , T ];( H ) ′ ) ≤ L ( α ) . t > s , we get k J n ( t ) − J n ( s ) k H ) ′ = k Z ts ˜ B n ( u n ( r ) , θ n ( r )) dr k H ) ′ ≤ C Z ts | ˜ B n ( u n ( r ) , θ n ( r )) | dr ! ≤ C Z ts | u n ( r ) | k u n ( r ) k k θ n ( r ) k | ∆ θ n ( r ) | dr ! ≤ C ( t − s ) sup t ∈ [0 , T ] | u n ( t ) | Z ts k u n ( r ) k dr ! + C ( t − s ) sup t ∈ [0 , T ] k θ n ( t ) k Z ts | ∆ θ n ( r ) | dr ! , hence, using (4.41)-(4.42) and (4.57), for α ∈ (0 , ), it gives that k J n k W α, ([0 , T ];( H ) ′ ) ≤ L ( α ) . Utilizing (2.20), for t > s , we deduce that k J n ( t ) − J n ( s ) k H ) ′ = k Z ts f n ( θ n ( r )) dr k H ) ′ ≤ C ( Z ts | f n ( θ n ( r )) | dr ) ≤ C ( t − s )( Z ts | f n ( θ n ( r )) | dr ) ≤ C ( t − s )[ Z ts (1 + | θ n ( r ) | N + L N + ) dr ] ≤ C ( t − s )[ Z ts (1 + k θ n ( r ) k N + H ) dr ] ≤ C ( t − s ) + C ( t − s ) Z ts k θ n ( r ) k N + H dr ≤ C ( t − s ) + C ( t − s ) sup t ∈ [0 , T ] k θ n ( t ) k N + H , where for any N ∈ N + , H ( O ) ֒ → L N + ( O ) is used. By (4.57), for α ∈ (0 , ), we deduce that k J n k W α, ([0 , T ];( H ) ′ ) ≤ L ( α ) . Therefore, collecting all the above estimates, it gives (4.59).Based on (4.58)-(4.59), applying Lemma 4.1, we conclude that u n is compact in L ([0 , T ]; H ) ∩ C ([0 , T ]; V ′ ) and θ n is compact in L ([0 , T ]; H ) ∩ C ([0 , T ]; ( H ) ′ ). Moreover, using (4.41)-(4.42), wededuce that there exists (ˆ u , ˆ θ ) and a subsequence still denoted by ( u n , θ n ) such that ˆ u ∈ L ([0 , T ]; H ) ∩ C ([0 , T ]; ( V ) ′ ) ∩ L ∞ ([0 , T ]; H ) ∩ L ([0 , T ]; V ), ˆ θ ∈ L ([0 , T ]; H ) ∩ C ([0 , T ]; ( H ) ′ ) ∩ L ∞ ([0 , T ]; H ) ∩ L ([0 , T ]; H ),20 . u n → ˆ u weakly star in L ∞ ([0 , T ]; H ), u n → ˆ u strongly in L ([0 , T ]; H ), u n → ˆ u weakly in L ([0 , T ]; V ), u n → ˆ u strongly in C ([0 , T ]; ( V ) ′ ). θ n → ˆ θ weakly star in L ∞ ([0 , T ]; H ), θ n → ˆ θ strongly in L ([0 , T ]; H ), θ n → ˆ θ weakly in L ([0 , T ]; H ), θ n → ˆ θ strongly in C ([0 , T ]; ( H ) ′ ).Next, we need to show (ˆ u , ˆ θ ) is the unique solution of (4.35)-(4.36). We will use the same method as[19].Let ψ be a continuously di ff erential function defined on [0 , T ] with ψ ( T ) =
0. Recall { ̺ j } j ≥ is anorthonormal eigenfunction of H , which can be viewed as an orthonormal eigenfunction of V . Multiplying(4.39) by ψ ( t ) ̺ j and using integration by parts, we obtain − Z T h u n ( t ) , ψ ′ ( t ) ̺ j i dt + Z T h u n ( t ) , ψ ( t ) A ̺ j i dt = h P n u , ψ (0) ̺ j i − Z T h P n B n ( u n ( t )) , ψ ( t ) ̺ j i dt − Z T h P n M n ( θ n ( t )) , ψ ( t ) ̺ j i dt + Z T h P n Z X G ( t , u n ( t ) , v )( g ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt . For every n > sup m ∈ N + sup t ∈ [0 , T ] | u m ( t ) | H ∨ sup m ∈ N + sup t ∈ [0 , T ] | θ m ( t ) | L ∨ j , we have − Z T h u n ( t ) , ψ ′ ( t ) ̺ j i dt + Z T h u n ( t ) , ψ ( t ) A ̺ j i dt = h u , ψ (0) ̺ j i − Z T h B ( u n ( t )) , ψ ( t ) ̺ j i dt − Z T h M ( θ n ( t )) , ψ ( t ) ̺ j i dt + Z T h Z X G ( t , u n ( t ) , v )( g ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt . Denote the above equality by symbols J ( T ) + J ( T ) = J + J ( T ) + J ( T ) + J ( T ) . Since u n → ˆ u strongly in C ([0 , T ]; V ′ ), we have J ( T ) → − Z T h ˆ u ( t ) , ψ ′ ( t ) ̺ j i dt . With the aid of Cauchy-Schwarz inequality and u n → ˆ u strongly in L ([0 , T ]; H ), we get J ( T ) → Z T h ˆ u ( t ) , ψ ( t ) A ̺ j i dt .
21y the triangle inequality and (2.10), we have (cid:12)(cid:12)(cid:12)(cid:12) Z T h B ( u n ( t )) , ψ ( t ) ̺ j i dt − Z T h B (ˆ u ( t )) , ψ ( t ) ̺ j i dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T |h B ( u n ( t ) − ˆ u ( t ) , u n ( t )) , ψ ( t ) ̺ j i| dt + Z T |h B (ˆ u ( t ) , u n ( t ) − ˆ u ( t )) , ψ ( t ) ̺ j i| dt ≤ Z T | u n ( t ) − ˆ u ( t ) | ( | u n ( t ) | + | u ( t ) | ) k ψ ( t ) ̺ j k V dt ≤ Z T | u n ( t ) − ˆ u ( t ) | dt ) sup t ∈ [0 , T ] ( | u n ( t ) | + | u ( t ) | ) | ψ ( t ) | , hence, J ( T ) → Z T h B (ˆ u ( t )) , ψ ( t ) ̺ j i dt . Using (2.11), we get | Z T h M ( θ n ( t )) , ψ ( t ) ̺ j i dt − Z T h M (ˆ θ ( t )) , ψ ( t ) ̺ j i dt |≤ Z T |h M ( θ n ( t ) − ˆ θ ( t ) , θ n ( t )) , ψ ( t ) ̺ j i| dt + Z T |h M (ˆ θ ( t ) , θ n ( t ) − ˆ θ ( t )) , ψ ( t ) ̺ j i| dt ≤ Z T ( k θ n ( t ) − ˆ θ ( t ) k | ∆ ( θ n ( t ) − ˆ θ ( t )) | k θ n ( t ) k | ∆ θ n ( t ) | ) k ψ ( t ) ̺ j k V dt + Z T ( k θ n ( t ) − ˆ θ ( t ) k | ∆ ( θ n ( t ) − ˆ θ ( t )) | k ˆ θ ( t ) k | ∆ ˆ θ ( t ) | ) k ψ ( t ) ̺ j k V dt ≤ sup t ∈ [0 , T ] | ψ ( t ) | ( Z T k θ n − ˆ θ k ds ) [( Z T | ∆ θ n | ds ) + ( Z T | ∆ ˆ θ | ds ) ][ T sup t ∈ [0 , T ] k θ n k ( Z T | ∆ θ n | ds ) ] + sup t ∈ [0 , T ] | ψ ( t ) | ( Z T k θ n − ˆ θ k ds ) [( Z T | ∆ θ n | ds ) + ( Z T | ∆ ˆ θ | ds ) ][ T sup t ∈ [0 , T ] k ˆ θ k ( Z T | ∆ ˆ θ | ds ) ] . With the help of (4.42) and θ n → ˆ θ strongly in L ([0 , T ]; H ), we conclude that J ( T ) → Z T h M (ˆ θ ( t )) , ψ ( t ) ̺ j i dt . The proof of J ( T ) → R T h R X G ( t , ˆ u ( t ) , v )( g ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt is the same as (4.25) in [19], weomit it. Based on the above steps, we conclude that for any j ≥ − Z T h ˆ u ( t ) , ψ ′ ( t ) ̺ j i dt + Z T h ˆ u ( t ) , ψ ( t ) A ̺ j i dt = h u , ψ (0) ̺ j i − Z T h B (ˆ u ( t )) , ψ ( t ) ̺ j i dt − Z T h M (ˆ θ ( t )) , ψ ( t ) ̺ j i dt + Z T h Z X G ( t , ˆ u ( t ) , v )( g ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt . (4.60)Actually, (4.60) holds for any ζ , which is a finite linear combination of ̺ j . That is22 Z T h ˆ u ( t ) , ψ ′ ( t ) ζ i dt + Z T h ˆ u ( t ) , ψ ( t ) A ζ i dt = h u , ψ (0) ζ i − Z T h B (ˆ u ( t )) , ψ ( t ) ζ i dt − Z T h M (ˆ θ ( t )) , ψ ( t ) ζ i dt + Z T h Z X G ( t , ˆ u ( t ) , v )( g ( t , v ) − ϑ ( dv ) , ψ ( t ) ζ i dt . (4.61)Since V is dense in H , we get d ˆ u ( t ) + A ˆ u ( t ) dt + B (ˆ u ( t )) dt + M (ˆ θ ( t )) dt = Z X G ( t , ˆ u ( t ) , v )( g ( t , v ) − ϑ ( dv ) dt (4.62)holds as an equality in distribution in L ([0 , T ]; H ′ ).Finally, it remains to prove ˆ u (0) = u . Multiplying (4.62) with the same ψ ( t ) as above and integratingwith respect to t . By integration by parts, we have − Z T h ˆ u ( t ) , ψ ′ ( t ) ζ i dt + Z T h ˆ u ( t ) , ψ ( t ) A ζ i dt = h ˆ u , ψ (0) ζ i − Z T h B (ˆ u ( t )) , ψ ( t ) ζ i dt − Z T h M (ˆ θ ( t )) , ψ ( t ) ζ i dt + Z T h Z X G ( t , ˆ u ( t ) , v )( g ( t , v ) − ϑ ( dv ) , ψ ( t ) ζ i dt . (4.63)By comparison with (4.61), it gives h u − ˆ u , ψ (0) ζ i = , ∀ ζ ∈ V . Choosing ψ such that ψ (0) ,
0, then(ˆ u (0) − u , ζ ) = , ∀ ζ ∈ V . Since V is dense in H , we have ˆ u (0) = u .Using the same method as above, we can obtain the following equality holds − Z T h ˆ θ ( t ) , ψ ′ ( t ) ς j i dt + Z T h ˆ θ ( t ) , ψ ( t ) A ς j i dt = h ˆ θ , ψ (0) ς j i − Z T h ˜ B (ˆ u ( t ) , ˆ θ ( t )) , ψ ( t ) ς j i dt − Z T h f n (ˆ θ ( t )) , ψ ( t ) ς j i dt . Therefore, (ˆ u , ˆ θ ) satisfies (4.35)-(4.36).(Continuity) According to Lemma 4.2, we need to show d ˆ udt ∈ L ([0 , T ]; V ′ ) , d ˆ θ dt ∈ L ([0 , T ]; ( H ) ′ ) . The proof is similar to the proof process of (4.58)-(4.59), we omit it. Thus, we obtainˆ u ∈ C ([0 , T ]; H ) , ˆ θ ∈ C ([0 , T ]; H ) . u , θ ) , ( u , θ ) are two solutions of (4.35)-(4.36). Let u = u − u , θ = θ − θ ,then ( u , θ ) = (0 , du ( t ) + A u ( t ) dt + ( B ( u ( t ) , u ( t )) + B ( u ( t ) , u ( t ))) dt = − ( M ( θ ( t ) , θ ( t )) + M ( θ ( t ) , θ ( t ))) dt + Z X ( G ( t , u ( t ) , v ) − G ( t , u ( t ) , v ))( g ( t , v ) − ϑ ( dv ) dt , d θ ( t ) + A θ ( t ) dt + ( ˜ B ( u ( t ) , θ ( t )) + ˜ B ( u ( t ) , θ ( t ))) dt = − ( f ( θ ( t )) − f ( θ ( t ))) dt . Define Υ ( t ) : = exp ( − Z t ( ξ ( s ) + ξ ( s ) + ξ ( s )) ds ) , ∀ t > , where ξ ( s ) = c ( κ ) | u | k u k + C ( κ ) k θ k + C ( κ , κ ) k θ k | ∆ θ | ,ξ ( s ) = c ( κ ) + c ( κ ) β ( θ , θ ) ,ξ ( s ) = c ( κ , κ ) k θ k | ∆ θ | + c ( κ , κ ) k θ k | ∆ θ | + c ( κ ) | u | k u k + c ( κ ) β ( θ , θ ) , with β ( θ , θ ) = C (1 + | θ | NL N + + | θ | NL N + ) . Using Lemma 2.2, we get d [ Υ ( t ) | θ ( t ) | ] = − Υ ( t )[ k θ ( t ) k + h ˜ B ( u ( t ) , θ ( t )) + f ( θ ( t )) − f ( θ ( t )) , θ ( t ) i ] dt + Υ ′ ( t ) | θ ( t ) | dt , (4.64)and d [ Υ ( t ) k θ ( t ) k ] = Υ ( t )[ −| ∆ θ ( t ) | + h ˜ B ( u ( t ) , θ ( t )) + ˜ B ( u ( t ) , θ ( t )) + f ( θ ( t )) − f ( θ ( t )) , ∆ θ ( t ) i ] dt +Υ ′ ( t ) k θ ( t ) k dt . (4.65)By Lemma 2.1, we obtain d [ Υ ( t ) | u ( t ) | ] = − Υ ( t )[ k u ( t ) k + h B ( u ( t ) , u ( t )) + M ( θ ( t ) , θ ( t )) + M ( θ ( t ) , θ ( t )) , u ( t ) i ] dt + Υ ( t ) h Z X ( G ( t , u ( t ) , v ) − G ( t , u ( t ) , v ))( g ( t , v ) − ϑ ( dv ) , u ( t ) i dt + Υ ′ ( t ) | u ( t ) | dt . (4.66)Referring to (9.13) in [5], it gives |h B ( u , u ) , u i| ≤ k k u k + C ( κ ) | u | k u k | u | , |h M ( θ , θ ) , u i| ≤ κ k u k + κ | ∆ θ | + C ( κ , κ ) k θ k | ∆ θ | k θ k , |h M ( θ, θ ) , u i| ≤ κ k u k + κ | ∆ θ | + C ( κ , κ ) k θ k | ∆ θ | k θ k , |h ˜ B ( u , θ ) , ∆ θ i| ≤ κ | ∆ θ | + C ( κ ) | u | k u k k θ k , |h ˜ B ( u , θ ) , θ i| ≤ κ | ∆ θ | + C ( κ ) | u | k θ k , |h ˜ B ( u , θ ) , ∆ θ i| ≤ κ | ∆ θ | + κ k u k + C ( κ , κ ) | u | k θ k | ∆ θ | . d [ Υ ( t )( | u ( t ) | + | θ ( t ) | + k θ ( t ) k )] + Υ ( t )[ k u ( t ) k + k θ ( t ) k + | ∆ θ ( t ) | ] dt ≤ Υ ( t )[ ξ ( t ) | u ( t ) | + ξ ( t ) | θ ( t ) | + ξ ( t ) k θ ( t ) k ] dt + Υ ( t )[ L k u ( t ) k + L k θ ( t ) k + L | ∆ θ ( t ) | ] dt + Υ ( t ) | u ( t ) | Z X | G ( t , v ) | , H | g ( t , v ) − | ϑ ( dv ) dt +Υ ′ ( t )[ | u ( t ) | + | θ ( t ) | + k θ ( t ) k ] dt , where L = κ + κ + κ + κ , L = κ , L = κ + κ + κ + κ + κ + κ . Choosing κ = κ = κ = κ = , κ = and κ = κ = κ = κ = κ = κ = , we have d [ Υ ( t )( | u ( t ) | + | θ ( t ) | + k θ ( t ) k )] + Υ ( t )[ k u ( t ) k + k θ ( t ) k + | ∆ θ ( t ) | ] dt ≤ Υ ( t )[ ξ ( t ) | u ( t ) | + ξ ( t ) | θ ( t ) | + ξ ( t ) k θ ( t ) k ] dt + Υ ( t ) | u ( t ) | Z X | G ( t , v ) | , H | g ( t , v ) − | ϑ ( dv ) dt +Υ ′ ( t )[ | u ( t ) | + | θ ( t ) | + k θ ( t ) k ] dt . By the choice of Υ ( t ), we deduce that2 Υ ( t )[ ξ ( t ) | u ( t ) | + ξ ( t ) | θ ( t ) | + ξ ( t ) k θ ( t ) k ] + Υ ′ ( t )[ | u ( t ) | + | θ ( t ) | + k θ ( t ) k ] ≤ . Hence, we conclude that d [ Υ ( t )( | u ( t ) | + | θ ( t ) | + k θ ( t ) k )] + Υ ( t )[ k u ( t ) k + k θ ( t ) k + | ∆ θ ( t ) | ] dt ≤ Υ ( t )( | u ( t ) | + | θ ( t ) | + k θ ( t ) k )] Z X | G ( t , v ) | , H | g ( t , v ) − | ϑ ( dv ) dt . Applying Gronwall inequality to the above inequality and using (3.30), we obtain the uniqueness. Up tonow, we complete the proof of Theorem 4.1. (cid:3)
This section is devoted to the proof of the main result. According to Theorem 2.1, we need to prove (i)and (ii) in
Condition A .Firstly, we prove (i) in
Condition A . For g ∈ S , from Theorem 4.1, we can define G ( ϑ gT ) = ( u g , θ g ) . Proposition 5.1.
For any M ∈ N + , and { g n } n ≥ ⊂ S M , g ∈ S M satisfying g n → g as n → ∞ . Then G ( ϑ g n T ) → G ( ϑ gT ) in C ([0 , T ]; H ) × C ([0 , T ]; H ) . roof. Recall that G ( ϑ g n T ) = ( u g n , θ g n ). For simplicity, denote ( u n , θ n ) = ( u g n , θ g n ).Using similar method as Theorem 4.1 and by Lemma 3.1, we can prove thatsup n sup s ∈ [0 , T ] | θ n ( s ) | pL + Z T | θ n ( s ) | p − L ( k θ n ( s ) k + | θ n ( s ) | N + L N + ) ds ≤ C ( p , M ) , (5.67)for any p ≥ n sup s ∈ [0 , T ] ( Ψ ( θ n ( s )) + | u n ( s ) | ) p + (cid:16) Z T ( k u n ( s ) k + | ∆ θ n ( s ) − f ( θ n ( s )) | ) ds (cid:17) p ≤ C ( p , M ) , (5.68)where Ψ ( θ n ( s )) : = k θ n ( s )) k + R O F ( | θ n ( s ) | ) dx and C ( p , M ) is independent of n . Moreover, for α ∈ (0 , ), there exist C ( α ) , L ( α ) such thatsup n ≥ k u n k W α, ([0 , T ]; V ′ ) ≤ C ( α ) , sup n ≥ k θ n k W α, ([0 , T ];( H ) ′ ) ≤ L ( α ) . Hence, we deduce from Lemma 4.1 that there exists an element ( u , θ ) and a subsequence still denoted by( u n , θ n ) such that u ∈ L ([0 , T ]; H ) ∩ C ([0 , T ]; ( V ) ′ ) ∩ L ∞ ([0 , T ]; H ) ∩ L ([0 , T ]; V ), θ ∈ L ([0 , T ]; H ) ∩ C ([0 , T ]; ( H ) ′ ) ∩ L ∞ ([0 , T ]; H ) ∩ L ([0 , T ]; H ), u n → u weakly star in L ∞ ([0 , T ]; H ), u n → u strongly in L ([0 , T ]; H ), u n → u weakly in L ([0 , T ]; V ), u n → u strongly in C ([0 , T ]; ( V ) ′ ). θ n → θ weakly star in L ∞ ([0 , T ]; H ), θ n → θ strongly in L ([0 , T ]; H ), θ n → θ weakly in L ([0 , T ]; H ), θ n → θ strongly in C ([0 , T ]; ( H ) ′ ).We will prove ( u , θ ) = ( u g , θ g ).Let ψ be a continuously di ff erential function defined on [0 , T ] with ψ ( T ) =
0. Multiply-ing u n ( t ) by ψ ( t ) ̺ j and using integration by parts, for every n > (sup m ∈ N + sup t ∈ [0 , T ] | u m ( t ) | H ) ∨ (sup m ∈ N + sup t ∈ [0 , T ] | θ m ( t ) | L ) ∨ j , we obtain − Z T h u n ( t ) , ψ ′ ( t ) ̺ j i dt + Z T h u n ( t ) , ψ ( t ) A ̺ j i dt = h u , ψ (0) ̺ j i − Z T h B ( u n ( t )) , ψ ( t ) ̺ j i dt − Z T h M ( θ n ( t )) , ψ ( t ) ̺ j i dt + Z T h Z X G ( t , u n ( t ) , v )( g n ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt . − Z T h u n ( t ) , ψ ′ ( t ) ̺ j i dt + Z T h u n ( t ) , ψ ( t ) A ̺ j i dt −h u , ψ (0) ̺ j i + Z T h B ( u n ( t )) , ψ ( t ) ̺ j i dt + Z T h M ( θ n ( t )) , ψ ( t ) ̺ j i dt → − Z T h u ( t ) , ψ ′ ( t ) ̺ j i dt + Z T h u ( t ) , ψ ( t ) A ̺ j i dt −h u , ψ (0) ̺ j i + Z T h B ( u ( t )) , ψ ( t ) ̺ j i dt + Z T h M ( θ ( t )) , ψ ( t ) ̺ j i dt . For the remain term R T h R X G ( t , u n ( t ) , v )( g n ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt , referring to Proposition 4.1. in [20],it gives that Z T h Z X G ( t , u n ( t ) , v )( g n ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt → Z T h Z X G ( t , u ( t ) , v )( g ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt . Therefore, we conclude that − Z T h u ( t ) , ψ ′ ( t ) ̺ j i dt + Z T h u ( t ) , ψ ( t ) A ̺ j i dt = h u , ψ (0) ̺ j i − Z T h B ( u ( t )) , ψ ( t ) ̺ j i dt − Z T h M ( θ ( t )) , ψ ( t ) ̺ j i dt + Z T h Z X G ( t , u ( t ) , v )( g ( t , v ) − ϑ ( dv ) , ψ ( t ) ̺ j i dt . Similarly, we can obtain − Z T h θ ( t ) , ψ ′ ( t ) ς j i dt + Z T h θ ( t ) , ψ ( t ) A ς j i dt = h θ , ψ (0) ς j i − Z T h ˜ B ( u ( t ) , θ ( t )) , ψ ( t ) ς j i dt − Z T h f ( θ ( t )) , ψ ( t ) ς j i dt . Thus, we get ( u , θ ) = ( u g , θ g ).Next, we prove ( u n , θ n ) → ( u , θ ) in C ([0 , T ]; H ) × C ([0 , T ]; H ). Let w n = u n − u , r n = θ n − θ . Then dw n + A w n dt + [ B ( w n , u n ) + B ( u , w n )] dt = − [ M ( r n , θ n ) + M ( θ, r n )] dt + Z X (cid:16) G ( t , u n ( t ) , v )( g n ( t , v ) − − G ( t , u ( t ) , v )( g ( t , v ) − (cid:17) ϑ ( dv ) dt , and dr n + A r n dt + [ ˜ B ( w n , θ n ) + ˜ B ( u , r n )] dt + ( f ( θ n ) − f ( θ )) dt = . I m ′ of(4.54) in [20], we obtain sup t ∈ [0 , T ] ( | w n ( t ) | + | r n ( t ) | + k r n ( t ) k ) = , which implies the desired result. (cid:3) Recall G ε ( ε N ε − ) = ( u ε ( · ) , θ ε ( · )). Let ϕ ε ∈ U M and ϑ ε = ϕ ε . The following lemma was proved byBudhiraja et al. [7]. Lemma 5.1. E ε t ( ϑ ε ) : = exp n Z (0 , t ) × X × [0 ,ε − ] log( ϑ ε ( s , x )) ¯ N ( dsdxdr ) + Z (0 , t ) × X × [0 ,ε − ] ( − ϑ ε ( s , x ) +
1) ¯ ϑ T ( dsdxdr ) o , is an { ¯ F t }− martingale. Then Q ε t ( G ) = Z G E ε t ( ϑ ε ) d ¯ P , for G ∈ B ( ¯ M ) defines a probability measure on ¯ M . Since ε N ε − ϕ ε under Q ε T has the same law as that of ε N ε − under ¯ P , it follows that there exists a uniquesolution to the following controlled stochastic evolution equations (˜ u ε , ˜ θ ε ):˜ u ε ( t ) = u − Z t A ˜ u ε ( s ) ds − Z t B (˜ u ε ( s )) ds − Z t M (˜ θ ε ( s )) ds + Z t Z X G ( s , ˜ u ε ( s ) , v )( ε N ε − ϕ ε ( dsdv ) − ϑ ( dv ) ds ) = u − Z t A ˜ u ε ( s ) ds − Z t B (˜ u ε ( s )) ds − Z t M (˜ θ ε ( s )) ds + Z t Z X G ( s , ˜ u ε ( s ) , v )( ϕ ε ( s , v ) − ϑ ( dv ) ds ) + Z t Z X ε G ( s , ˜ u ε ( s ) , v )( N ε − ϕ ε ( dsdv ) − ε − ϕ ε ( s , v ) ϑ ( dv ) ds ) , (5.69)˜ θ ε ( t ) = θ − Z t A ˜ θ ε ( s ) ds − Z t ˜ B (˜ u ε ( s ) , ˜ θ ε ( s )) ds − Z t f (˜ θ ε ( s )) ds , (5.70)and we have G ε ( ε N ε − ϕ ε ) = (˜ u ε , ˜ θ ε ) . (5.71)Before proving (ii) in Condition A , we make a priori estimates of (˜ u ε , ˜ θ ε ).28 emma 5.2. There exists ε > such that for any p ≥ , sup <ε<ε h E sup t ∈ [0 , T ] | ˜ θ ε ( t ) | p H + E Z T | ˜ θ ε ( s ) | p − L ( k ˜ θ ε ( s ) k + | ˜ θ ε ( s ) | N + L N + ) ds i ≤ C ( p , M ) , (5.72) and sup <ε<ε E sup s ∈ [0 , T ] ( Ψ (˜ θ ε ( s )) + | ˜ u ε ( s ) | ) p + E (cid:16) Z T ( k ˜ u ε ( s ) k + | ∆ ˜ θ ε ( s ) − f (˜ θ ε ( s )) | ) ds (cid:17) p ≤ C ( p , M ) , (5.73) where Ψ (˜ θ ε ( s )) : = k ˜ θ ε ( s )) k + R O F ( | ˜ θ ε ( s ) | ) dx . Moreover, for α ∈ (0 , ) , there exists constantsL ( α ) , R ( α ) > such that sup <ε<ε E k ˜ u ε k W α, ([0 , T ]; V ′ ) ≤ L ( α ) , sup <ε<ε E k ˜ θ ε k W α, ([0 , T ];( H ) ′ ) ≤ R ( α ) . (5.74) Proof.
Combing Theorem 4.1 in our paper and Proposition 5.4-5.5 proved by Brze´zniak et al. in [5], wecan obtain the estimates of (5.72) and (5.73).Let ϕ ε ∈ U M , we have˜ u ε ( t ) = u − Z t A ˜ u ε ( s ) ds − Z t B (˜ u ε ( s )) ds − Z t M (˜ θ ε ( s )) ds + Z t Z X G ( s , ˜ u ε ( s ) , v )( ϕ ε ( s , v ) − ϑ ( dv ) ds ) + Z t Z X ε G ( s , ˜ u ε ( s ) , v )( N ε − ϕ ε ( dsdv ) − ε − ϕ ε ( s , v ) ϑ ( dv ) ds ): = J ε + J ε + J ε + J ε + J ε + J ε , ˜ θ ε ( t ) = θ − Z t A ˜ θ ε ( s ) ds − Z t ˜ B (˜ u ε ( s ) , ˜ θ ε ( s )) ds − Z t f (˜ θ ε ( s )) ds : = K ε + K ε + K ε + K ε . Clearly, sup <ε<ε E | J ε | H ≤ L , sup <ε<ε E | K ε | H ≤ R . By the same method as in the proof of (4.58) and (4.59), we getsup <ε<ε E k J ε k W α, ([0 , T ]; V ′ ) ≤ CT sup <ε<ε E Z T k ˜ u ε ( r ) k dr ≤ L ( α ) , sup <ε<ε E k K ε k W α, ([0 , T ];( H ) ′ ) ≤ CT sup <ε<ε E sup t ∈ [0 , T ] | ˜ θ ε ( t ) | ≤ R ( α ) , <ε<ε E k J ε k W α, ([0 , T ]; V ′ ) ≤ CT sup <ε<ε (cid:16) E sup t ∈ [0 , T ] | ˜ u ε ( t ) | (cid:17) (cid:16) E Z ts k ˜ u ε ( r ) k dr (cid:17) ≤ L ( α ) , sup <ε<ε E k K ε k W α, ([0 , T ];( H ) ′ ) ≤ CT sup <ε<ε (cid:16) E sup t ∈ [0 , T ] | ˜ u ε ( t ) | (cid:17) (cid:16) E Z ts k ˜ u ε ( r ) k dr (cid:17) + CT sup <ε<ε (cid:16) E sup t ∈ [0 , T ] k ˜ θ ε ( t ) k (cid:17) (cid:16) E Z ts | ∆ ˜ θ ε ( r ) | dr (cid:17) ≤ R ( α ) . Moreover, we getsup <ε<ε E k J ε k W α, ([0 , T ]; V ′ ) ≤ CT sup <ε<ε (cid:16) E sup t ∈ [0 , T ] k ˜ θ ε ( t ) k (cid:17) (cid:16) E Z ts | ∆ ˜ θ ε ( r ) | dr (cid:17) ≤ L ( α ) , sup <ε<ε E k K ε k W α, ([0 , T ];( H ) ′ ) ≤ CT + CT sup <ε<ε E sup t ∈ [0 , T ] k ˜ θ ε ( t ) k N + H ≤ R ( α ) , where (5.72)-(5.73) are used.For the remain two terms J ε and J ε , referring to Lemma 4.2 in [20], we havesup <ε<ε E k J ε k W , ([0 , T ]; H ) ≤ L , sup <ε<ε E k J ε k W , ([0 , T ]; H ) ≤ L . Based on all the above estimates, we complete the proof. (cid:3)
To prove (ii) in
Condition A , we need to obtain the tightness of { (˜ u ε , ˜ θ ε ) } <ε<ε in D ([0 , T ]; D ( A − α )) × C ([0 , T ]; H − ), for some α > { (˜ u ε , ˜ θ ε ) } <ε<ε . The proof can be foundin [13] and [3]. Lemma 5.3.
Let E be a separable Hilbert space with the inner product ( · , · ) . For an orthonormal basis { ξ k } k ∈ N in E, define the function r N : E → R + byr N ( x ) = X k ≥ N + ( x , ξ k ) , N ∈ N . Let E be a total and closed under addition subset of E. Then a sequence { X ε } ε ∈ (0 , of stochastic processwith trajectories in D ([0 , T ] , E ) i ff the following Condition B holds: { X ε } ε ∈ (0 , is E − weakly tight, that is, for every h ∈ E , { ( X ε , h ) } ε ∈ (0 , is tight in D ([0 , T ]; R ) , For every η > , lim N →∞ lim ε → P (cid:16) r N ( X ε ( s ) > η ) f or some s ∈ [0 , T ] (cid:17) = . (5.75)30onsider a sequence { τ ε , δ ε } satisfying the following Condition C : (1) For each ε , τ ε ia a stopping time with respect to the natural σ − fildes, and takes only finitely manyvalues. (2) The constant δ ε ∈ [0 , T ] satisfying δ ε → ε → { Y ε } ε ∈ (0 , be a sequence of random elements of D ([0 , T ]; R ). For f ∈ D ([0 , T ]; R ), let J ( f ) denotethe maximum of the jump | f ( t ) − f ( t − ) | . We introduce the following Condition D on { Y ε } : (I) For each sequence { τ ε , δ ε } satisfying Condition C , Y ε ( τ ε + δ ε ) − Y ε ( τ ε ) → ε → Lemma 5.4.
Assume { Y ε } ε ∈ (0 , satisfies Condition D , and either { Y ε (0) } and J ( Y ε ) are tight on the lineor { Y ε ( t ) } is tight on the line for each t ∈ [0 , T ] , then { Y ε } is tight in D ([0 , T ]; R ) . Let (˜ u ε , ˜ θ ε ) be defined by (5.71). We have Lemma 5.5. { (˜ u ε , ˜ θ ε ) } <ε<ε is tight in D ([0 , T ]; D ( A − α )) × C ([0 , T ]; H − ) , for some α > .Proof. With the help of (5.74) and ˜ θ ε ∈ C ([0 , T ]; H ), we deduce that ˜ θ ε is tight in C ([0 , T ]; H − ). Now,we prove { ˜ u ε } <ε<ε is tight in D ([0 , T ]; D ( A − α )). Note that { λ α i ̺ i } i ∈ N is a complete orthonormal systemof D ( A − α ). Sincelim N →∞ lim ε → E sup t ∈ [0 , T ] r N (˜ u ε ( s )) = lim N →∞ lim ε → E sup t ∈ [0 , T ] ∞ X i = N + (˜ u ε ( s ) , λ α i ̺ i ) D ( A − α ) = lim N →∞ lim ε → E sup t ∈ [0 , T ] ∞ X i = N + ( A − α ˜ u ε ( s ) , ̺ i ) H = lim N →∞ lim ε → E sup t ∈ [0 , T ] ∞ X i = N + (˜ u ε ( s ) , ̺ i ) H λ α i ≤ lim N →∞ lim ε → E sup t ∈ [0 , T ] | ˜ u ε ( t ) | H λ α N + = , which implies (5.75) holds with E = D ( A − α ).Choosing E = D ( A α ). We now prove { ˜ u ε , < ε < ε } is E − weakly tight. Let h ∈ D ( A α ), and { τ ε , δ ε } satisfies Condition C . It’s easy to see { (˜ u ε ( t ) , h ) E , < ε < ε } is tight on the real line for each t ∈ [0 , T ].We now prove that { (˜ u ε ( t ) , h ) E , < ε < ε } satisfies (D) . From (5.69)-(5.70), we have˜ u ε ( τ ε + δ ε ) − ˜ u ε ( τ ε ) = − Z τ ε + δ ε τ ε A ˜ u ε ( s ) ds − Z τ ε + δ ε τ ε B (˜ u ε ( s )) ds − Z τ ε + δ ε τ ε M (˜ θ ε ( s )) ds + Z τ ε + δ ε τ ε Z X G ( s , ˜ u ε ( s ) , v )( ϕ ε ( s , v ) − ϑ ( dv ) ds ) + Z τ ε + δ ε τ ε Z X ε G ( s , ˜ u ε ( s ) , v )( N ε − ϕ ε ( dsdv ) − ε − ϕ ε ( s , v ) ϑ ( dv ) ds ): = K ε + K ε + K ε + K ε + K ε . ε → E | ( K ε , h ) E | =
0. For K ε , using (5.73), we havelim ε → E | ( K ε , h ) E | ≤ k h k D ( A ) lim ε → δ ε E [ sup t ∈ [0 , T ] | ˜ u ε ( t ) | H ] = . By (2.10), we get lim ε → E | ( K ε , h ) E | ≤ k h k V lim ε → E Z τ ε + δ ε τ ε k B (˜ u ε ( t )) k V ′ dt ≤ k h k V lim ε → E Z τ ε + δ ε τ ε | ˜ u ε ( t ) | dt ≤ k h k V lim ε → δ ε E sup t ∈ [0 , T ] | ˜ u ε ( t ) | = . With the help of (2.11) and (5.73), we deduce thatlim ε → E | ( K ε , h ) E | ≤ k h k V lim ε → E Z τ ε + δ ε τ ε k M (˜ θ ε ( t )) k V ′ dt ≤ k h k V lim ε → E Z τ ε + δ ε τ ε k ˜ θ ε ( t ) k| ∆ ˜ θ ε ( t ) | dt ≤ k h k V lim ε → E [ sup t ∈ [0 , T ] k ˜ θ ε ( t ) k Z τ ε + δ ε τ ε | ∆ ˜ θ ε ( t ) | dt ] ≤ k h k V lim ε → δ ε (cid:16) E sup t ∈ [0 , T ] k ˜ θ ε ( t ) k (cid:17) (cid:16) E Z τ ε + δ ε τ ε | ∆ ˜ θ ε ( t ) | dt (cid:17) = . For K ε , referring to (4.82) in [20], we getlim ε → E | ( K ε , h ) E | = . Hence, we conclude the desired result. (cid:3)
Fix the solution (˜ u ε , ˜ θ ε ) of (5.69)-(5.70), consider the following equation: d ˜ ξ ε ( t ) = − A ˜ ξ ε ( t ) dt + ε Z X G ( t , ˜ u ε ( t − ) , v )( N ε − ϕ ε ( dtdv ) − ε − ϕ ε ( t , v ) ϑ ( dv ) dt ) , (5.76)with ˜ ξ ε (0) =
0. Referring to Proposition 3.1 in [16], there exists a unique solution ˜ ξ ε ( t ) , t ≥ ξ ε ∈ D ([0 , T ]; H ) ∩ L ([0 , T ]; V ) , (5.77)and there exists constant C and ˜ ε < ε such that for any 0 < ε < ˜ ε , E sup t ∈ [0 , T ] | ˜ ξ ε | H + E Z T k ˜ ξ ε k V dt ≤ √ ε C . (5.78)Now, we are ready to prove (ii) in Condition A . Recall G ε ( ε N ε − ϕ ε ) = (˜ u ε , ˜ θ ε ) is defined by (5.71).32 heorem 5.2. Fix M ∈ N , and let { ϕ ε , ε < ε } ⊂ U M , ϕ ∈ U M be such that ϕ ε converges in distributionto ϕ as ε → . Then G ε ( ε N ε − ϕ ε ) converges in distribution to G ( ϑ ϕ T ) , in D ([0 , T ]; H ) × C ([0 , T ]; H ) .Proof. Note that G ε ( ε N ε − ϕ ε ) = (˜ u ε , ˜ θ ε ). From Lemma 5.2 and Lemma 5.5, we know that ˜ u ε is tight in D ([0 , T ]; D ( A − α )) ∩ L ([0 , T ]; H ), for α > ˜ θ ε is tight in C ([0 , T ]; ( H ) ′ ) ∩ L ([0 , T ]; H ), lim ε → E h sup t ∈ [0 , T ] | ˜ ξ ε ( t ) | H + R T k ˜ ξ ε ( t ) k V dt i = , where ˜ ξ ε is defined in (5.76). Let Ξ = (cid:16) D ([0 , T ]; D ( A − α )) ∩ L ([0 , T ]; H ) (cid:17) × (cid:16) C ([0 , T ]; ( H ) ′ ) ∩ L ([0 , T ]; H ) (cid:17) . Set Π = ( Ξ , U M , D ([0 , T ]; H ) ∩ L ([0 , T ]; V )) . Let ((˜ u , ˜ θ ) , ϕ,
0) be any limit of the tight family { ((˜ u ε , ˜ θ ε ) , ϕ ε , ˜ ξ ε ) , ε ∈ (0 , ˜ ε ) } . We will show that (˜ u , ˜ θ ) hasthe same law as G ( ϑ ϕ T ) and (˜ u ε , ˜ θ ε ) converges in distribution to (˜ u , ˜ θ ) in D ([0 , T ]; H ) × C ([0 , T ]; H ).By the Skorokhod representative theorem, there exists a stochastic basis ( Ω , F , {F t } t ∈ [0 , T ] , P )and, on this basis, Π − valued random variables ((˜ u , ˜ θ ) , ϕ , , ((˜ u ε , ˜ θ ε ) , ϕ ε , ˜ ξ ε ) such that ((˜ u ε , ˜ θ ε ) , ϕ ε , ˜ ξ ε )(resp. ((˜ u , ˜ θ ) , ϕ , u ε , ˜ θ ε ) , ϕ ε , ˜ ξ ε ) (resp. ((˜ u , ˜ θ ) , ϕ, u ε , ˜ θ ε ) , ϕ ε , ˜ ξ ε ) → ((˜ u , ˜ θ ) , ϕ ,
0) in Π , P − a.s.From the equations satisfied by ((˜ u ε , ˜ θ ε ) , ϕ ε , ˜ ξ ε ), we see that ((˜ u ε , ˜ θ ε ) , ϕ ε , ˜ ξ ε ) satisfies the followingintegral equations:˜ u ε ( t ) − ˜ ξ ε ( t ) = u − Z t A (˜ u ε ( s ) − ˜ ξ ε ( s )) ds − Z t B (˜ u ε ( s )) ds − Z t M (˜ u ε ( s ) , ˜ θ ε ( s )) ds + Z t Z X G ( s , ˜ u ε ( s ) , v )( ϕ ε ( s , v ) − ϑ ( dv ) ds , ˜ θ ε ( t ) = θ − Z t A ˜ θ ε ( s ) ds − Z t ˜ B (˜ u ε ( s ) , ˜ θ ε ( s )) ds − Z t f (˜ θ ε ( s )) ds . Define
Σ = ( C ([0 , T ]; H ) ∩ L ([0 , T ]; V )) × ( C ([0 , T ]; H ) ∩ L ([0 , T ]; H )), we have P (cid:16) (˜ u ε − ˜ ξ ε , ˜ θ ε ) ∈ Σ (cid:17) = ¯ P (cid:16) (˜ u ε − ˜ ξ ε , ˜ θ ε ) ∈ Σ (cid:17) = . Let Ω be the subset of Ω such that ((˜ u ε , ˜ θ ε ) , ϕ ε , ˜ ξ ε ) → ((˜ u , ˜ θ ) , ϕ ,
0) in Π , then P ( Ω ) =
1. Now, wehave to show that, for any fixed ω ∈ Ω ,sup t ∈ [0 , T ] | ˜ u ε ( ω , t ) − ˜ u ( ω , t ) | H → and sup t ∈ [0 , T ] k ˜ θ ε ( ω , t ) − ˜ θ ( ω , t ) k H → , as ε → . (5.79)33et p ε ( t ) = ˜ u ε ( t ) − ˜ ξ ε ( t ) and q ε ( t ) = ˜ θ ε ( t ). Then, ( p ε ( ω , t ) , q ε ( ω , t )) satisfies p ε ( t ) = u − Z t A p ε ( s ) ds − Z t B ( p ε ( s ) + ˜ ξ ε ( s )) ds − Z t M ( p ε ( s ) + ˜ ξ ε ( s ) , q ε ( s )) ds + Z t Z X G ( s , p ε ( s ) + ˜ ξ ε ( s ) , v )( ϕ ε ( s , v ) − ϑ ( dv ) ds , q ε ( t ) = θ − Z t A q ε ( s ) ds − Z t ˜ B ( p ε ( s ) + ˜ ξ ε ( s ) , q ε ( s )) ds − Z t f ( q ε ( s )) ds . Since lim ε → [ sup t ∈ [0 , T ] | ˜ ξ ε ( ω , t ) | H + Z T k ˜ ξ ε ( ω , t ) k V dt ] = , we have lim ε → sup t ∈ [0 , T ] | ˜ u ε ( ω , t ) − ˆ u ( ω , t ) | H + lim ε → sup t ∈ [0 , T ] k ˜ θ ε ( t ) − ˆ θ ( ω , t ) k H ≤ lim ε → sup t ∈ [0 , T ] [ | p ε ( ω , t ) − ˆ u ( ω , t ) | H + | ˜ ξ ε ( ω , t ) | H ] + lim ε → sup t ∈ [0 , T ] k q ε ( ω , t ) − ˆ θ ( ω , t ) k H = lim ε → sup t ∈ [0 , T ] | p ε ( ω , t ) − ˆ u ( ω , t ) | H + lim ε → sup t ∈ [0 , T ] k q ε ( ω , t ) − ˆ θ ( ω , t ) k H . (5.80)By the similar method as Theorem 4.1, we can obtain thatlim ε → sup t ∈ [0 , T ] | p ε ( ω , t ) − ˆ u ( ω , t ) | H = , and lim ε → sup t ∈ [0 , T ] k q ε ( ω , t ) − ˆ θ ( ω , t ) k H = , (5.81)where ˆ u ( t ) = u − Z t A ˆ u ( s ) ds − Z t B (ˆ u ( s )) ds − Z t M (ˆ θ ( s )) ds + Z t Z X G ( s , ˆ u ( s ) , v )( ϕ ( s , v ) − ϑ ( dv ) ds , ˆ θ ( t ) = θ − Z t A ˆ θ ( s ) ds − Z t ˜ B (ˆ u ( s ) , ˆ θ ( s )) ds − Z t f (ˆ θ ( s )) ds . Hence, combining (5.80) and (5.81), we obtainlim ε → sup t ∈ [0 , T ] | ˜ u ε ( ω , t ) − ˆ u ( ω , t ) | H = , and lim ε → sup t ∈ [0 , T ] k ˜ θ ε ( t ) − ˆ θ ( ω , t ) k H = , (5.82)which imply that (˜ u , ˜ θ ) = (ˆ u , ˆ θ ) = G ( ϑ ϕ ), and (˜ u , ˜ θ ) has the same law as G ( ϑ ϕ ). Since (˜ u ε , ˜ θ ε ) = (˜ u ε , ˜ θ ε ) in law, we deduce from (5.82) that (˜ u ε , ˜ θ ε ) converges to G ( ϑ ϕ ). We complete the proof. (cid:3) Acknowledgements
This work is supported by National Natural Science Foundation of China (GrantNo. 11401057), Natural Science Foundation Project of CQ (Grant No. cstc2016jcyjA0326), Funda-mental Research Funds for the Central Universities(Grant No. 106112015CDJXY100005) and ChinaScholarship Council (Grant No.:201506055003). 34 eferences [1] R.A. Adams:
Sobolev Space . New York: Academic Press, 1975.[2] S. Albeverio, Z. Brze´zniak, J. Wu:
Existence of global solutions and invariant measures for stochas-tic di ff erential equations driven by Poisson type noise with non-Lipschitz coe ffi cients. J. Math. Anal.Appl. 371, no. 1, 309-322 (2010).[3] D. Aldous:
Stopping times and tightness.
Ann. Probab. 6 335-340 (1978).[4] Z. Brze´zniak, E. Hausenblas, P. Razafimandimby:
Stochastic Nonparabolic dissipative systems mod-elling the flow of liquid crystals: strong solution . RIMS Kokyuroku Proceeding of RIMS Symposiumon Mathematical Analysis of Incompressible Flow: 41-72 (2014).[5] Z. Brze´zniak, U. Manna, A. A. Panda:
Existence of weak martingale solution of nematic liquidcrystals driven by pure jump noise. arXiv:1706.05056 (2017).[6] A. Budhiraja, J. Chen, P. Dupuis:
Large deviations for stochastic partial di ff erential equations drivenby a Poisson random measure. Stochastic Process. Appl. 123, no. 2, 523-560 (2013).[7] A. Budhiraja, P. Dupuis and V. Maroulas :
Variational representations for continuous time processes .Ann. Inst. Henri Poincaré Probab. Stat. 47, no.3, 725-747 (2011).[8] S. Chandrasekhar:
Liquid Crystals.
Cambridge University Press (1992).[9] J.L. Ericksen:
Conservation laws for liquid crystals.
Trans. Soc. Rheology, 5: 23-34 (1961).[10] F. Flandoli, D. Gatarek:
Martingale and stationary solutions for stochastic Navier-Stokes equa-tions . Probab. Theory Related Fields 102, no. 3, 367-391 (1995).[11] P.G. Gennes, J. Prost:
The Physics of Liquid Crystals.
Clarendon Press, Oxford (1993).[12] N. Ikeda, Nobuyuki, S. Watanabe:
Stochastic di ff erential equations and di ff usion processes. Secondedition. North-Holland Mathematical Library, 24.[13] A. Jakubowski:
On the Skorokhod topology.
Ann. Inst. H. Poincaré Probab. Statist. 22, no. 3, 263-285 (1986).[14] F.M. Leslie:
Some constitutive equations for liquid crystals.
Arch. Rational Mech. Anal. 28(04):265-283 (1968).[15] F. Lin, C. Liu:
Nonparabolic dissipative systems modelling the flow of liquid crystals.
Communi-cations on Pure and Applied Mathematics, Vol. XLVIII: 501-537 (1995).[16] M. Röckner, T. Zhang:
Stochastic evolution equations of jump type: existence, uniqueness andlarge deviation principles.
Potential Anal. 26, no. 3, 255-279 (2007).3517] R. Temam:
Navier-Stokes equations and nonlinear functional analysis . Second edition. CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and AppliedMathematics (SIAM), Philadelphia, PA, 1995.[18] M. Warner, E. Terentjev:
Liquid Crystal Elastomers.
International Series of Monographs onPhysics, Oxford University Press (2003).[19] X. Yang, J. Zhai, T. Zhang:
Large deviations for SPDEs of jump type.
Stoch. Dyn. 15, no. 4,1550026, 30 pp (2015).[20] J. Zhai, T. Zhang: