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Invariant Measures for a Stochastic Electroconvection Model

We consider a stochastic electroconvection model describing the nonlinear evolution of a surface charge density in a two-dimensional fluid with additive stochastic forcing. We prove the existence and uniqueness of solutions and we show that the corresponding Markov semigroup is weak Feller. We also prove the existence of invariant measures for the Markov transition kernels associated with the model.


INTRODUCTION
We consider a stochastic electroconvection model describing the evolution of a surface charge density interacting with a two-dimensional fluid. The surface charge density q evolves according to the stochastic partial differential equation dq + ∇ ⋅ Jdt =gdW. (1) The current density J is given by and Φ is a potential due to applied voltage restricted to the surface whereas Λ −1 q is the potential due to the surface charge density q restricted to the surface. Here Λ denotes the square root of the two-dimensional periodic Laplacian, and Λ −1 denotes its inverse. The fluid velocity u obeys a stochastic forced Navier-Stokes equation given by du + u ⋅ ∇udt − ∆udt + ∇pdt = qEdt + f dt + gdW, (4) and the divergence-free condition ∇ ⋅ u = 0, (5) where f are body forces and p is the fluid pressure. The potential Φ is assumed to be time independent and smooth whereas the body forces f are assumed to be time independent and divergencefree. We denote by W (t, w) = (W 1 , ..., W n ) a collection of standard independent Brownian motions. The stochastic noise processes gdW andgdW are given by andg dW = n l=1g l (x)dW l (t, w), where g = (g 1 , ..., g n ) andg = (g 1 , ...,g n ) are time-independent and the components of g are divergence-free. The system of equations (1)- (7) is posed on the two-dimensional torus T 2 = [0, 2π] 2 with periodic boundary conditions.
In [1], we considered the two-dimensional periodic deterministic electroconvection model (1)- (7), where the equations are not forced by noise, and established the existence and uniqueness of global regular solutions, provided that the initial data is sufficiently regular. We addressed the long-time behavior of solutions and proved the existence of a finite-dimensional global attractor. In [2], global existence of regular solutions of the deterministic model (1)- (7) with homogeneous Dirichlet boundary conditions was established in the absence of body forces in the fluid (f = 0).
In this paper, we study the stochastic model described by (1)- (7) in the presence of noises forcing the equations satisfied by the charge density q and the velocity u. We show that the stochastic system (1)- (7) has unique global solutions when the initial deterministic charge density is at least L 4 (T 2 ) regular and the initial deterministic velocity is at least H 1 (T 2 ) regular. The existence of solutions is obtained by taking a mollification of (1)- (7), establishing uniform bounds for the mollified solutions, and using the Banach Alaoglu theorem in order to obtain weak convergence. The identification of the drift in the case of the stochastic electroconvection model (1)-(7) is highly challenging. The reason is that the nonlinearity qRq is not weakly continuous in the spaces we have control in. The remedy is a coercive estimate (112) and use of ideas from [9] where the authors proved the existence of global solutions for the forced stochastic Navier-Stokes equations. As a consequence of the existence result, the Markov transition kernels are defined for data q 0 ∈ L 4 and u 0 ∈ H 1 .
In the absence of the potential Φ (that is Φ = 0), we prove that the stochastic model (1)- (7) has an invariant measure. The requirement of a vanishing potential Φ is due to the fact that the velocity u does not maintain a zero spatial average for all positive times regardless of whether or not the average of the initial velocity vanishes. The term q∇Φ forcing the velocity equation does not have a zero mean over T 2 , and so the expectation of the L 2 norm of the velocity might grow exponentially in time. The Krylov-Bogoliubov procedure is applied to prove the existence of an invariant measure after we obtain bounds for some s > 1 2 when u 0 = q 0 = 0 (The Krylov-Bogoliubov procedure is then applicable because of the compactness of H s ⊕ H 2 in L 4 ⊕ H 1 .) The required H s (T 2 ) regularity is difficult to obtain, due to the nonlinear terms involved in (1)- (7) together with the insufficient critical regularity obtained from the dissipative term. By contrast, in the subcritical case where the term Λq in the surface charge density equation is replaced by Λ α q for some α > 1, the desired bounds (8) are directly obtained due to the higher regularity of the dissipation. The existence of ergodic invariant measures for stochastic partial differential equations has been extensively studied. The existence of an invariant measure for the stochastic Navier-Stokes equations was obtained in [6]. In [5], global existence and uniqueness of strong solutions for the 2D stochastic Navier-Stokes equations on the two-dimensional torus was established and existence of invariant measures was obtained on the base of the Krylov-Bogoliubov averaging procedure. In [7], the authors proved existence of invariant measures for the 3D stochastic primitive equations by establishing moment bounds for strong solutions. In [3], existence and uniqueness of an ergodic invariant measure was obtained for the 2D fractionally dissipated periodic stochastic Euler equation by deriving moment bounds in Sobolev spaces that grow linearly in time.
This paper is organized as follows. In section 2, we prove that the system (1)-(7) has a unique global solution provided that the initial charge density has a zero spatial average and is L 4 integrable, the initial velocity is divergence-free and is weakly differentiable, and the noise is sufficiently regular. Then we define the semigroup associated with (1)- (7) in section 3 and we prove that it is weak Feller. In the absence of potential (Φ = 0), we prove in section 4 the existence of an invariant measure for the Markov transition kernels associated with the electroconvection model (1)-(7) based on the Krylov-Bogoliubov averaging procedure. Finally, we treat the stochastic subcritical case in section 5 and we obtain the existence of an invariant measure.

EXISTENCE AND UNIQUENESS OF SOLUTIONS
Let (Ω, F , P ) be a probability space. Let W (t, w) = (W 1 , ..., W n ) be a collection of independent standard Brownian motions. Let T > 0. We consider the Itô stochastic model on T 2 × [0, T ] × Ω, with initial data q(x, 0) = q 0 and u(x, 0) = u 0 . The unknowns q(x, t, w), u(x, t, w) = (u 1 (x, t, w), u 2 (x, t, w)), and p(x, t, w) depend on three different variables: position x ∈ T 2 , time t ∈ [0, T ], and outcome w ∈ Ω. The body forces f and the potential Φ depend only on the position variable x. The forces f are smooth, divergence-free and have a zero space average. The potential Φ is assumed to be smooth. The functionsg l (x) and g l (x) are assumed to be time-independent and square-integrable over the torus T 2 . The functions g l are assumed to be divergence-free and the functionsg l are assumed to have mean zero for all l ∈ {1, ..., n}. Here Λ is the periodic fractional Laplacian of order one and R = (R 1 , R 2 ) is the periodic Riesz transform. We show the existence of solutions for the stochastic system (9). For each ǫ ∈ (0, 1], we let J ǫ be the standard mollifier operator, and we let (q ǫ , u ǫ ) be the solution of the stochastic system with smoothed out initial data q ǫ 0 = J ǫ q 0 , u ǫ 0 = J ǫ u 0 . Proposition 1. Let ǫ ∈ (0, 1] and let T > 0. Let q 0 ∈ L 2 have mean zero over T 2 . Let u 0 ∈ L 2 be divergence-free. The stochastic system (10) has a solution (q ǫ , u ǫ ) on [0, T ] such that q ǫ has mean zero, u ǫ is divergence-free, and (q ǫ , u ǫ ) satisfies the following properties: and satisfies E sup (ii) Let p ∈ [4, ∞). Ifg l ∈ L 2 (T 2 ) for all l ∈ {1, ..., n}, then q ǫ is uniformly bounded in and satisfies (iii) If Ifg l ∈ L 2 (T 2 ) and g l ∈ L 2 (T 2 ) for all l ∈ {1, ..., n}, then u ǫ is uniformly bounded in and satisfies (iv) If q 0 ∈ L 4 (T 2 ) andg l ∈ L 4 (T 2 ) for all l ∈ {1, ..., n}, then q ǫ is uniformly bounded in and satisfies (v) Let p ∈ [8, ∞) be an even integer. If q 0 ∈ L 4 (T 2 ) andg l ∈ L 4 (T 2 ) for all l ∈ {1, ..., n}, then q ǫ is uniformly bounded in and satisfies (vi) Let p ≥ 4. If q 0 ∈ L 4 (T 2 ),g l ∈ L 4 (T 2 ), and g l ∈ L 2 (T 2 ) for all l ∈ {1, ..., n}, then u ǫ is uniformly bounded in and satisfies (vii) If q 0 ∈ L 4 (T 2 ), u 0 ∈ H 1 (T 2 ),g l ∈ L 4 (T 2 ), and g l ∈ H 1 (T 2 ) for all l ∈ {1, ..., n}, then u ǫ is uniformly bounded in L 2 (Ω; L ∞ (0, T ; H 1 (T 2 ))) ∩ L 2 (Ω; L 2 (0, T ; H 2 (T 2 ))) and satisfies For simplicity, we ignore the viscous term −ǫ∆q ǫ in the proof of proposition 1 below because it does not have any major contribution in estimating the solutions of the mollified system (10) and vanishes as we take the limit ǫ → 0.
Proof of (i). We apply Itô's lemma pointwise in x to the stochastic process F (X t (w)) where F (ξ) = ξ 2 and X t (w) = q ǫ , and we obtain Next we integrate in the space variable over T 2 . In view of the divergence-free condition obeyed by u ǫ , the nonlinear term vanishes, that is which yields the energy equality We estimate using the Hölder and Young inequalities. We obtain the differential inequality Integrating in time from 0 to t, we get We take the supremum over all t ∈ [0, T ], Now we apply the expectation E. In view of the martingale estimate (see [4]), we have This gives (12). Proof of (ii). Applying Itô's lemma to the process F (X t (w)) where X t (w) = q ǫ (t, w) 2 L 2 obeys (27) and F (ξ) = ξ p 2 , we derive the energy equality which yields the differential inequality In view of the bound we have where we used Young's inequality to estimate and Integrating in time (37) from 0 to t and taking the supremum over [0, T ], we obtain We estimate and we obtain (14).
Proof of (iii). We apply Itô's lemma pointwise in x to the processes F (u ǫ 1 (w)) and F (u ǫ 2 (w)) where F (ξ) = ξ 2 , we add the resulting equations, and we integrate in the space variable over the torus T 2 . We obtain the energy equality which implies where we used the cancellation due to the divergence-free condition satisfied by u ǫ . By Ladyzhenskaya's interpolation inequality and the boundedness of the Riesz transforms in L 4 , we estimate We also estimate and using Hölder's inequality followed by Young's inequality. We obtain the differential inequality hence for all s ∈ [0, t]. Integrating in time from 0 to t, we obtain We take the supremum in time over [0, T ] and apply E. Using the continuous Sobolev embedding and (14) with p = 4, we have and we obtain (16).
Proof of (iv). We apply Itô's lemma pointwise in x to the stochastic process F (X t (w)) where X t (w) = q ǫ and F (ξ) = ξ 4 . We have Integrating in the space over T 2 , we obtain the energy equality We note that due to the divergence-free condition for u ǫ . By the nonlinear Poincaré inequality for the fractional Laplacian in L 4 applied to the mean zero function q ǫ (see [1,3]), we have Using Hölder's inequality with exponents 4, 4 3 and Young's inequality with exponents 4, 4 3, we get We also bound using Hölder and Young inequalities. Putting (57)-(61) together, we obtain the differential inequality Consequently, We take the supremum over [0, T ] and then we apply E. We estimate and we obtain (18). Proof of (v). We apply Itô's lemma to the process By Hölder's inequality with exponents 4 3, 4 and Young's inequality with exponents p (p−2), p 2, we have We obtain Integrating (67) in time from 0 to t, taking the supremum over [0, T ], applying E, and estimating we obtain (20). Proof of (vi). Using Itô's lemma, we derive the energy equality By Young's inequality with exponents p (p − 2) and p 2, and Similarly, using Young's inequality with exponents p (p − 1) and p, and By Ladyzhenskaya's interpolation inequality and the boundedness of the Riesz transforms in L 4 (T 2 ), we have This yields the differential inequality and thus We integrate in time from 0 to t, take the supremum over [0, T ], and apply E. We obtain We estimate Putting (77) and (78) together, and using (14) and (20), we obtain (22). Proof of (vii). We write the equation satisfied by ∇u ǫ , apply Itô's lemma, and integrate in the space variable. We obtain the energy equality The nonlinear term for the velocity vanishes, that is and using Hölder's inequality, we obtain An application of Young's inequality yields the differential inequality We integrate (82) in time from 0 to t, take the supremum in time, and then apply E. We obtain We estimate the martingale term Putting (83) and (84) together, and using (12) and (18), we get (24). Now we prove the existence of solutions for the stochastic electroconvection model (9). The proof uses ideas from [9], where the authors investigated and determined the limiting drift for the stochastic 2D Navier-Stokes equations. Theorem 1. Let T > 0. Let q 0 ∈ L 4 have mean zero over T 2 , and let u 0 ∈ H 1 be divergence-free. Supposeg l ∈ L 4 and g l ∈ H 1 for all l ∈ {1, ..., n} . Then there exists a pair (q, u) such that u ∈ L 2 (Ω; L ∞ (0, T ; H 1 (T 2 ))) ∩ L 2 (Ω, L 2 (0, T ; H 2 (T 2 ))), for any ξ ∈ H 1 (T 2 ) and a.e. w ∈ Ω, and for any v ∈ H 1 (T 2 ) and a.e. w ∈ Ω.
Proof: Let F 1 (q ǫ , u ǫ ) = u ǫ ⋅ ∇q ǫ (89) and 13 We note that using Ladyzhenskaya's interpolation inequality, and using the boundedness of the Riesz transforms in L 2 (T 2 ). As a consequence of Proposition 1, F 1 and F 2 are uniformly bounded in L 2 (Ω, L 2 (0, T ; H −1 (T 2 ))). Therefore, up to subsequences, u ǫ converges weakly to some function u in q ǫ converges weakly to some function q in and F 1 (q ǫ , u ǫ ) and F 2 (q ǫ , u ǫ ) converge weakly to some functions F 1 and F 2 , respectively, in Now we write the equations satisfied by (q ǫ , u ǫ ) and (q, u) as We show that F (q, u) = F 0 (100) for almost every w ∈ Ω.

Remark 1. Uniqueness of solutions is obtained as for the deterministic system [1, Theorem 2].
Indeed, if we suppose the existence of two different solutions, and we write the equations obeyed by their difference, then we obtain deterministic equations which are independent of the noise.

ELECTROCONVECTION SEMIGROUP
that is, F t is the smallest σ-algebra for which W s is measurable for all s ≤ t. Let τ be the stopping time random variable with respect to F t . For u 0 = u τ (x, w) and q 0 = q τ (x, w), we consider the electroconvection model (9) in its variational form Theorem 2. Let τ be a stopping time with respect to F t , and let (q τ , u τ ) be F τ measurable random variables such that 0 ≤ τ ≤ T , u τ ∈ L 2 (Ω; H 1 (T 2 )) and q τ ∈ L 4 (Ω; L 4 (T 2 )). Supposeg l ∈ L 4 and g l ∈ H 1 for all l ∈ {1, ..., n}. Then there exists a solution (q, u) of (127) satisfying Proof: The proof of (128) is similar to the proof of (18). Indeed, we integrate (62) from τ to t, we take the supremum over the time interval [τ, T ], and then we apply E. We estimate the noise term as in (64) and we obtain (128). As for the bound (129), the proof is similar to the proof of (24). Indeed, we integrate the differential inequality (82) from τ to t, we take the supremum over [τ, T ], and we take the expectation in w. We use (128) to estimate the charge density terms, and we bound the noise term as in (84). This gives (129).
The proof is based on the same ideas used to prove (112). We omit further details.
Proposition 2. Let (q τ , u τ ) be an initial data satisfying the conditions of Theorem 2. Supposẽ g l ∈ L 4 and g l ∈ H 1 for all l ∈ {1, ..., n}. Then the unique solution (q, u) of (127) obeys where c 7 is a positive constant depending only on Φ, f , g, andg, and c 8 is a positive constant depending only on Φ.
Proof: By Itô's lemma, we have and We add the equations (133) and (134). Integrating by parts, we have and using the cancellation (u ⋅ ∇u, u) L 2 = 0, (136) we obtain the differential equation From (137), we arrive at the differential inequality Letting Integrating in time from τ to t, taking the supremum over [τ, T ], applying the expectation E in w, and using martingale estimates, we obtain (132). We consider the space consisting of vectors (ξ, v) where ξ ∈ H − 1 2 has mean zero and v ∈ L 2 is divergence-free, and we consider the space V = L 4 (T 2 ) ⊕ H 1 (T 2 ) (142) consisting of vectors (ξ, v) where ξ ∈ L 4 has mean zero and v ∈ H 1 is divergence-free. We define the norms ⋅ H and ⋅ V by and (145) We point out that continuity of h on the space (V, ⋅ H ) means that if (ξ n , v n ) ∈ V converges to (ξ, v) in the norm ⋅ H , then h(ξ n , v n ) converges to h(ξ, v).
We note that the uniqueness of solutions in V (see [1]) imply that (Φ(t, s), t ≥ s ≥ 0) is indeed a semigroup. Moreover, (Φ(t, s), t ≥ s ≥ 0) is a H-Markov Feller semigroup: and if t n → s, then In view of the continuity property given in Theorem 3, we have q(t, s; ξ n ) − q(t, s; ξ) and u(t, s; v n ) − u(t, s; v) L 2 → 0.
(152) Since h is continuous on (V, ⋅ H ), we conclude that h(q(t, s; ξ n ), u(t, s; v n )) → h(q(t, s; ξ), u(t, s; v)) and hence E {h(q(t, s; ξ n ), u(t, s; v n )} → E {h(q(t, s; ξ), u(t, s; v))} by the Lebesgue Dominated Convergence Theorem (which can be applied due to the growth condition (145), the bound (132), and the convergence (150)) yielding the boundedness of the sequence of initial datum (ξ n , v n ) in the H-norm. Now, suppose that {t n } ∞ n=1 is a sequence of positive times converging to s, and (ξ, v) ∈ V. Noting that the solution (q(t, s; ξ), u(t, s; v)) of (127) belongs to the space we obtain E {h(q(t n , s; ξ), u(t n , s; v))} → E {h(ξ, v)} (156) due to the continuity of h in (V, ⋅ H ) and the Lebesgue Dominated Convergence Theorem. This ends the proof of Theorem 4.

EXISTENCE OF AN INVARIANT MEASURE IN THE ABSENCE OF POTENTIAL
In this section, we consider the electroconvection system We note that if the initial charge density and velocity are assumed to have a zero spatial average, then the solution (q, u) will have mean zero over T 2 for all positive times t ≥ 0. LetL p (T 2 ) andḢ s (T 2 ) be the spaces of L p (T 2 ) and H s (T 2 ) functions with zero spatial averages respectively. Let H and V be the spaces of L 2 (T 2 ) and H 1 (T 2 ) functions that are divergencefree and mean zero respectively. LetḢ respectively. See [1] for details on the notation and functional setting. We note thatV is compactly embedded inḢ. We define the Markov transition kernels {P t } t≥0 associated to the electroconvection model (157) as P t (q 0 , u 0 , A) = P((q(t, q 0 ), u(t, u 0 )) ∈ A).
(162) These kernels are defined onV and areḢ-Feller as shown in Theorem 4.
We will show that the solution (q, u) of (157) lies in and the bounds are linear in T , hence the Krylov-Bogoliubov procedure can be applied in order to prove the existence of an invariant measure. The rigorous estimates in this section can be done by taking a viscous system approximating (157), deriving the bounds for the mollified solution, and then inheriting them to the solution of (157) using the lower semi-continuity of the norms. We present formal proofs, omitting the approximation. We need the following propositions: Proposition 3. Let p be an even integer such that p ∈ {4} ∪ [8, ∞). Let q 0 ∈L 4 . Supposeg l ∈L 4 for all l ∈ {1, ..., n}. Then there exist a positive constant Γ 1 depending only on q 0 L 4 , p and some universal constants, and a positive constant Γ 2 depending only ong, p and some universal constants such that Proof: The p-th power of the L 4 norm of q obeys the energy inequality Integrating in time from 0 to t and applying E, we obtain the desired bound (164). and hold for all t ≥ 0. Here Γ 3 = Γ 5 = 0 if u 0 = q 0 = 0.
Proof: The L 2 norm of ∇u obeys In view of Hölder's inequality, Young's inequality, and the boundedness of the Riesz transforms on L 4 (T 2 ), we get the energy inequality Integrating in time from 0 to t and applying E, we obtain In view of the bound (164) applied with p = 4, we obtain (166). By Itô's lemma, we have From (172), we obtain the differential inequality and by Young's inequality, we get We integrate in time from 0 to t and we apply E. In view of the bound (164) applied with p = 8, we obtain (167).
Proposition 5. Let u 0 ∈ V and q 0 ∈L 4 . Suppose g l ∈ V andg l ∈L 4 for all l ∈ {1, ..., n}. Then there exist a positive constant Γ 7 depending only on ∇u 0 L 2 , q 0 L 4 ,g, g, f and some universal constants, and a positive constant Γ 8 depending only on f, g,g and some universal constants such that holds for all t ≥ 0. Here Γ 7 = 0 if u 0 = q 0 = 0.
Proof: The stochastic process ∇u 4 The 4-th power of the L 2 norm of ∇u evolves according to (171) whereas the 4-th power of the L 4 norm of q evolves according to Consequently, the product ∇u 4 L 2 q 4 L 4 satisfies the energy equality which yields the energy inequality in view of the Poincaré inequality for the fractional Laplacian in L 4 . By the Cauchy-Schwartz inequality, Young's inequality and the Poincaré inequality applied to the mean zero function ∇u, we estimate The boundedness of the Riesz transforms on L 4 (T 2 ) yields We bound 2 q 4 L 4 ∇u 2 and 4 q 4 using Young's inequality. Finally, we estimate Putting (179)-(184) together, we end up with the differential inequality d q 4 L 4 ∇u 4 L 2 + c ∇u 4 L 2 q 4 L 4 dt + q 4 L 4 ∇u 2 L 2 ∆u 2 L 2 dt ≤ K 1 (g) ∇u 2 L 2 ∆u 2 L 2 dt + K 2 (f, g) q 4 L 4 dt + K 3 (g,g) + C q 12 L 4 dt (185) where K 1 > 0 is a constant depending only ong, K 2 > 0 is a constant depending only on f and g, K 3 > 0 is a constant depending only on g andg, and C is a positive universal constant. We integrate in time from 0 to t and we apply E. The bound (164) applied with p = 4 and p = 12 together with the bound (167) gives the desired estimate (175).
We integrate in time from 0 to t and we apply E. In view of (167) and (175), we obtain (186). The above propositions give for all T ≥ 0. Now, let R > 0, and let B R be the ball of radius R inḢ 3 2 (T 2 ) ⊕ (Ḣ 2 (T 2 ) ∩ H) which is compact inV. By Chebyshev's inequality, as R → ∞ in view of the bounds (193) and (194) that are linear in T . Therefore, the family {ν T } is tight, ending the proof of Proposition 7. As a consequence of the Krylov-Bogoliubov averaging procedure, we obtain Theorem 5. Suppose that g l ∈ H 1 (T 2 ) andg l ∈ H 1 (T 2 ) for all l ∈ {1, ..., n}. There exists an invariant measure for the Markov transition kernels associated with (157).
The existence and uniqueness of solutions is obtained as for the critical case (when the fractional Laplacian is of order 1).
The solution (q, u) of (196) obeys and T 0 E ∆u 2 L 2 dt ≤ Γ 12 ( ∇u 0 L 2 , q 0 L 4 ) + Γ 13 (f, g,g)T for all T ≥ 0, where Γ 11 is a positive constant depending only ong and some universal constants, Γ 12 is a positive constant depending only on the initial data, and Γ 13 is a positive constant depending only on f, g,g and some universal constants. In view of the compactness of H α 2 (T 2 ) in L 4 (T 2 ) for α > 1 (which does not hold in the critical case), the Krylov-Bogoliubov averaging procedure implies automatically the existence of an invariant measure. Theorem 6. Suppose g l ∈ H 1 (T 2 ) andg l ∈ L 4 (T 2 ) for all l ∈ {1, ..., n}. Then there exists an invariant measure for the Markov transition kernels associated with (196).