LARGE DEVIATION PRINCIPLE FOR THE MICROPOLAR, MAGNETO-MICROPOLAR FLUID SYSTEMS

. Micropolar ﬂuid and magneto-micropolar ﬂuid systems are systems of equations with distinctive feature in its applicability and also mathe- matical diﬃculty. The purpose of this work is to follow the approach of [8] and show that another general class of systems of equations, that includes the two- dimensional micropolar and magneto-micropolar ﬂuid systems, is well-posed and satisﬁes the Laplace principle, and consequently the large deviation prin- ciple, with the same rate function.


1.
Introduction. The theory of large deviations is an important direction of research and has been studied by many (e.g. [13,20], Chapter 12 [12], [7]). In particular, the authors in [16] developed an approach to this theory through proving the convergence of solutions to variational problems, based on the fact that the large deviation principle (LDP) in a Polish space is equivalent to Laplace principle (see Theorem 1.2.3 [16]). Subsequently, the authors in [4,5] proved a type of extended contraction principle that consists of a weak convergence and compactness conditions, (see Assumption 4.3 [4], and also Assumption 1 on pg. 1401 [5]) that guarantees the Laplace principle with a rate function and hence the LDP with the same rate function.
Various authors have studied the LDP results for many models (e.g. [6,8,9,28,35]). In particular, the work in [8] covered many models that include the Navier-Stokes equations (NSE), magnetohydrodynamics (MHD) system, Bénard problem, magnetic Bénard problem, Leray α-model and shell models of turbulence; we also refer to its accompanying paper [9] for Wong-Zakai approximation results. However, the micropolar and magneto-micropolar fluid systems, which have also attracted much attention from many researchers for reasons to be described in a subsequent section, do not seem to be covered by the work of [8], precisely due to the singular terms that do not exist in most other models of fluid mechanics. The purpose of this work is to establish the well-posedness and LDP of a new class of systems of equations that includes the two-dimensional (2-d) micropolar and magneto-micropolar fluid systems. The well-posedness result on the micropolar and magneto-micropolar fluid system extends the work of [37] in the three-dimensional (3-d) case, the claim ∇ · u = ∇ · b = 0, ∀ t ∈ [0, T ], u| ∂D = w| ∂D = 0, b · n| ∂D = 0, ∇ × b| ∂D = 0, ∀ t ∈ (0, T ). ( Individual particles of complex fluids may consist of different shape, shrink, expand or even rotate independently of the rotation and movement of the fluid and the NSE cannot take into account of such micro-structural aspect. In order to emphasize the micro-structure of fluids, the theory of microfluids and thereafter micropolar fluids were introduced by Eringen in [17,18]; the micropolar fluid (MPF) system is the MMPF system (1a)-(1c) with b ≡ 0. The authors in [1] proposed coupling it furthermore with a magnetic field to study the motion of incompressible electrically conducting micropolar fluid. We note that from the original MMPF system introduced in [1], we made the appropriate modification in (1a)-(1c) by letting u = (u 1 , u 2 , 0), w = (0, 0, w 3 ), b = (b 1 , b 2 , 0) (see pg. 185 [26]). As the MPF system models some fluid better than the NSE, e.g. fluid consisting of bar-like elements such as liquid crystals, dumbbell molecules and animal blood, this system has caught much attention from researchers such as physicists, engineers and mathematicians (e.g. [21,30,32]). We also wish to emphasize that the MPF system has some similarity with the Boussinesq equations, equivalently the Bénard problem via an introduction of a new function (see pg. 133 [36]): where (e 1 , e 2 ) is an orthonormal basis of R 2 . We note that the linear terms χ∇ × w, χ∇ × u in (1a), (1b) respectively are one derivative more singular than χwe 2 , χu 2 of (3a), (3b) respectively. This difference is so immense that despite the fact that in [22], a global well-posedness result for the Boussinesq equation was obtained for the case of γ = 0 and the dissipativity strength being only half of (µ + χ)∆u (see equation (1.1) [22] for details in terms of a fractional Laplacian), such a result is absent and seems very difficult in the case of the MPF system (see [14,38]). Due to this difference, we are interested in the stochastic analysis of MPF and MMPF systems; moreover, this is precisely the reason why this model was excluded from the general case that was covered by the comprehensive result in [8].
In order to present our results, we now recall the standard notations for fluid mechanics mathematical literature (e.g. [10,36,37]). Firstly, to emphasize the significance of a constant on certain parameters, we write A a,b B to imply the ex- For the MPF and the MMPF systems, we may denote and the inner products of Furthermore, for the MPF and the MMPF systems we may define Moreover, we may denote where e.g. B 1 (u, u) = B 1 (u) and analogously B(y, y) = B(y).
In this paper, we study a general class of equations (see (10)) that includes the MPF and the MMPF systems. We now state the general conditions on an operator A and B : The operator B is a bilinear continuous mapping such that 3. There exists a Banach space H such that the following properties hold: H , equivalently (see Remark 2.1 (1) pg. 383 [8])

KAZUO YAMAZAKI
Finally, for the MPF and the MMPF systems, we may denote We emphasize already that although the authors in [8] considered a wide class of different systems, they required that R is a linear bounded operator in H and hence does not apply for the stochastic MPF and the MMPF systems of our consideration.
Concerning the external forces f u , f w , f b in (1a), (1b), (1c) respectively, we let Q be a linear positive operator in H such that it is in the trace class and hence compact (see [12]).
The embedding i : H 0 → H is Hilbert-Schmidt and hence compact; moreover, i · i * = Q with i * being the adjoint operator of i. We denote by L Q L Q (H 0 , H) the space of all linear operators S : with S * being the adjoint operator of S, and {e i } i an orthonormal basis of H. We let W be the H-valued Wiener process on (Ω, F, (F t ) t≥0 , P) with a covariance operator Q so that W is Gaussian, has time independent increments, where {β i } i are mutually independent standard Wiener processes, and {q i } i satisfy q i e i Qe i for all i. Moreover, (F t ) t≥0 is the Brownian filtration, the smallest right-continuous complete filtration with respect to which {W (t)} t≥0 is adapted. With these notations we may reconsider from (1a), (1b), (1c), the MPF system with y = (u, w, 0) and the MMPF system with y = (u, w, b) as where σ : [0, T ] × V → L Q (H 0 , H) is the noise intensity. For the general class of equations which we will study, let us state the condition on σ: endowed with the weak topology that is defined by the metric of d 1 (y 1 , y 2 ) is an orthonormal basis for the space L 2 ([0, T ]; H 0 ), becomes a Polish space. We furthermore define Finally, we let M > 0, h ∈ A M , ζ ∈ H and write where |·| L(H0,H) is the operator norm of linear operators from H 0 to H. 2.R ∈ C([0, T ] × V ; H) and there exist constants R i , i = 0, 1 such that for all y j ∈ V, j = 1, 2, We define the Polish space X C( We may now state our first result on the well-posedness of a class of systems of equations, that includes the MPF and the MMPF systems (1a), (1b), (1c). It holds that for all M > 0 and T > 0, there exists  (7) satisfy the Condition (C1) with H = L 4 (D) (see Lemma 6.2 pg. 70 [25] and also [27]), and also the Condition (C3) because R defined in (4) satisfies the role ofR in Condition (C3)(2). 2. The well-posedness result of stochastic equations in fluid mechanics has been investigated by many mathematicians; for brevity we only mention the most relevant work on the stochastic NSE here with no intention to be complete (e.g. [2,19,29]). We also note that for the LDP result, we only need the well-posedness result in the case 0 < L 2 < 2,σ = σ. As in the case of [8], we chose to state Theorem 2.2 in all possible cases for completeness.
Next, we prepare to state our result on the LDP for a class of systems of equations, that includes the systems (7). We let > 0 and y solve By Theorem 2.2, we see that for any [23] an also [31] for details). Now we let B(X) denote the Borel σ-field generated by X and recall some definitions relevant to LDP theory. Firstly, with the standard convention that an infimum over an empty set is +∞, we recall the definition of LDP (cf. [13]). Now for any h ∈ L 2 ([0, T ]; H 0 ), we let y h be the solution to a corresponding control equation We need the following additional assumption of the Hölder regularity on the noise intensity σ(·, y) in time t: We now state our LDP result.
Remark 2. Most importantly, the difference between Theorem 2.4 and Theorem 3.2 of [8] is the Condition (C3) (2). By Poincare's inequality we see that anyR that satisfies also satisfies the Condition (C3)(2) with a modified constant; thus, our work recovers the result of [8] as well.
Remark 3. We remark that the work within this manuscript focuses on the 2-d case. In the 3-d case, the LDP result will face difficulty because the uniqueness of the global weak solution is unknown; this problem is actually analogous to the deterministic case. While LDP result on the local unique strong solution may be pursued, it seems to require a different approach from what is considered in this manuscript. Moreover, even in the 2-d case, it is not clear to the author if the well-posedness and LDP results may be extended to the stochastic density-dependent or nonhomogeneous fluid equations. In particular, we mention the work of [3] which proved the local existence of a unique strong solution to the 3-d deterministic nonhomogeneous MPF system, and [11,33] which established the global existence of a weak solution to the 3-d stochastic nonhomogeneous NSE. The work of [40] proved the global existence of a weak solution to the stochastic 3-d nonhomogeneous MHD system and following the proof therein immediately deduces an analogous results for the 3-d stochastic nonhomogeneous MPF and MMPF systems. These works in [11,33,40] all may be extended to the 2-d case with no difficulty; however, the uniqueness of the global weak solution remains unknown due to technical difficulty and therefore, whether or not the LDP result may be obtained for such 2-d stochastic density-dependent or nonhomogeneous fluid equations remain unknown.
In subsequent sections we prove Theorem 2.2 and Theorem 2.4. We elaborate in proof where the difference with the work of [8] is significant, in particular when estimates involveR or Condition (C3) (2). Moreover, we intentionally be brief when our proof can be similarly done as in the work of [8] while give details when it was missing in [8]; some parts of our proofs distinctively differ, e.g. choice of r(t) in (47).
3. Proof of Theorem 2.2. We define a V -valued mapping from [0, T ] × V by F (t, y) −Ay − B(y) −R(t, y) and note its properties: for any φ, ψ ∈ V, η > 0, there exist constants R 1 , c η > 0 such that This is same as the equation (A.1) in the Appendix of [8] except the termR in our case, which we compute carefully. We may write by Hölder's inequality, Condition (C3)(2) and Young's inequality. Thus, in sum of (18), (19), (20) applied to (17), we obtain (16). Now we let {φ j } j be an orthonormal basis of H such that by denseness we assume φ j ∈ D(A) for any j ≥ 1, define H n span{φ 1 , . . . , φ n } ⊂ D(A) and denote P n : H → H n as the orthogonal projection from H onto H n , and σ n P n σ,σ n P nσ . Thus, |σ n (y)| 2 L Q ≤ |σ(y)| 2 L Q by (5) and because P n is a contraction on H. For h ∈ A M , and v ∈ H n , we consider the following approximating system of (10): where (6)). It follows that for v ∈ H n , the map y → Ay +Ry, v u ∈ H n is globally Lipschitz uniformly in t because by Hölder's inequality, the Condition (C3)(2) and because v ∈ H n ⊂ D(A). By analogous computations using Conditions (C1)(2), (C1)(3c) and (C1)(3a), the map y → B(y), v , v ∈ H n may be shown to be locally Lipschitz. Moreover, similarly to (19), we can also compute for any η > 0, by Conditions (C1)(2), (C1)(3c), (C1)(3b) and Young's inequalities. Furthermore, there exists a constant c = c(n) such that v V ≤ c(n)|v| for all v ∈ H n . Now since v ∈ H n = span{φ 1 , . . . , φ n }, we may substitute φ j , j ∈ {1, . . . , n} for v in (21). By hypothesis of Theorem 2.2, the Condition (C2) holds, and either σ =σ andR satisfies Condition (C3)(2) or (C3) holds. It follows that y ∈ H n → (σ n (y)h(t), φ j ) 1≤j≤n is globally Lipschitz from H n to n × n matrices and y ∈ H n → (σ n (y)h(t), φ j ) 1≤j≤n is globally Lipschitz from H n to R n uniformly in t; this is because for all y i ∈ H n , i = 1, 2, if the Condition (C2)(2) holds, then If σ =σ, then σ n = P n σ = P nσ =σ n so that the computation in (23) shows that y ∈ H n → (σ n (y)h(t), φ j ) 1≤j≤n is also globally Lipschitz from H n to R n . On the other hand, if σ =σ, then using the hypothesis that the Condition (C3) holds and in particular (12), it follows that y ∈ H n → (σ n (y)h(t), φ j ) 1≤j≤n is also globally Lipschitz from H n to R n . Thus, by existence and uniqueness theorem for stochastic ordinary differential equations (e.g. [24]), there exists a unique solution y n,h = n j=1 (y n,h , φ j )φ j and a stopping time τ n,h ≤ T such that (21) holds for all t < τ n,h . To prove the next proposition that deduces that T n,h = T , we rely on the following Gronwall's inequality type result from [8]: , I(t) and φ(t) be non-negative processes and Z(t) be a non-negative integrable random variable. Suppose that I(t) is non-decreasing in t and there exist non-negative constants C, α, β, γ, δ such that instead of (11). Then for any p ≥ 1, there exists a constant  Proof. We let y n,h (t) be the approximate solution to (21) and set the stopping time for any N ∈ R + , τ N inf{t : |y n,h (t)| ≥ N } ∧ T . Now we know y n,h exists at least locally on [0, t], t < τ n,h where τ n,h is a stopping time such that τ n,h ≤ T and lim t τn, |y n,h (t)| = ∞. We also set π n : H 0 → H 0 the projection operator such that π n y n i=1 (y, e i )e i for {e i } i , an orthonormal basis of H. Applying Ito's formula on (21) with f (t, x) = x 2 and again with f (t, x) = x p gives d|y n,h | 2p + 2p|y n,h | 2(p−1) y n,h 2 V dt = − 2p(R(y n,h (t)), y n,h )|y n,h | 2(p−1) dt + 2p(σ n (y n,h (t))dW n (t), y n,h (t))|y n,h | 2(p−1) + 2p(σ n (y n,h (t))h(t), y n,h )|y n,h | 2(p−1) dt + p|σ n (y n,h (t))π n | 2 L Q |y n,h | 2(p−1) dt + 2p(p − 1)|π n σ * n (y n,h (t))y n,h (t)| 2 0 |y n,h | 2(p−2) dt.
Step 3. The purpose of this Step is to show that ds ⊗ dP a.e. on Ω T , S h (s) = σ (y h (s)), F h (s) = F (y h (s)),S h (s) =σ(y h (s))h(s). We let v ∈ X where X is defined in (38). By hypothesis of Theorem 2.2, the constant L 2 of the Condition (C2) satisfies L 2 < 2 so that we can find ∈ (0, 2 − L 2 ) and then choose Then for any t ∈ [0, T ], we set where c η is the constant from (16); we emphasize that our choice of r(t) here differs from that of equation (A.17) on pg. 413 [8]. It is clear that 0 ≤ r(t) < ∞ P-a.s. because v ∈ X implies by (38) that v ∈ L 4 (Ω T ; H) and by hypothesis of Theorem 2.2, h ∈ A M so that by (8), (9), P-a.s., h ∈ L 2 ([0, T ]; H 0 ). Hence r ∈ L 1 (Ω; L ∞ (0, T )), e −r ∈ L ∞ (Ω T ) and Now using the fact that P n ζ → ζ in H as n → ∞, while by (50) with v = y n,h , using Ito's formula on (21)
Step 5. We now show the uniqueness of y h ∈ X = C([0, T ]; H) ∩ L 2 ([0, T ]; V ). We let v ∈ X be another solution to (61) and set where Ψ(s) e −a s and Now for any η > 0 we estimate in particular by Condition (C3)(2) and Young's inequality to deduce from (66) where we took a = 2c η 2 and in particular computed that Ψ(s) ≤ e −a s
In order to apply Lemma 3.1, we denote s. as h ∈ A M by hypothesis of Theorem 2.2, and 2βe C = 1. Finally, 2δe C ≤ α if L 2 > 0 is sufficiently small. Thus, under the hypothesis that L 2 > 0 is sufficiently small, we can apply Lemma 3.
By hypothesis of Proposition 2, h → h as → 0 in distribution (as A M -valued random elements). As A M is a Polish space, by Skorokhod's Representation theorem (see [34], also [12]), there exists (h ,h,W ) such that 1. the joint distribution of (h ,W ) is the same as that of (h , W ), 2. the distribution ofh coincides with that of h, 3.h →h P-a.s. in the weak topology of S M so that P-a.s. for all t ∈ [0, T ], t 0h (s)ds − t 0h (s)ds → 0 weakly in H 0 .
This will imply by arbitrariness of {y n } in K M , that every sequence in K M has a convergent subsequence and hence K M is compact. We let Y n y n − y. From (95), where in particular we may compute − 2 R (s, y n (s)) −R(s, y(s)), Y n (s) ≤ 2|R(s, y n (s)) −R(s, y(s))||Y n (s)|