ON THE EXISTENCE OF A SOLUTION

TO STOCHASTIC NAVIER-STOKES EQUATIONS

Marek Capiński and Szymon Peszat

IMUJ PREPRINT 1997/32

Abstract. We consider stochastic Navier–Stokes equations on a possibly unbounded domain O ⊆ ℝ^{d}, where d is equal to 2 or 3. First we prove the existence of a martingale solution for the initial value being a probability measure on the space of square integrable ℝ^{d}-valued functions. Then we show the existence of a spatially homogeneous solution to the equation on the whole ℝ^{d}, driven by a spatially homogeneous Wiener random field.

0. Introduction

Let O be a possibly unbounded open subset of ℝ^{d}, where d is equal to 2 or 3. We assume that O is connected with the boundary ∂O of class C^{2}. Let us fix a bounded time interval [0, T]. The paper is concerned with the existence of a martingal solution to the following system of stochastic Navier–Stokes equations

(0.1)  \begin{aligned}∂_{t}u − νΔu + ⟨u, ∇u⟩ + ∇p & = F (t, u) + G(t, u)Ẇ , \\ div u & = 0 \end{aligned}

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary: 60H15, 60G60; Secondary: 35R60, 76D05, 60J25.

Key words and phrases. Stochastic Navier–Stokes equations, spatially homogeneous measures.

The research was supported by KBN Grant 2PO3A 064 08

considered with the Dirichlet boundary condition u|_{∂O} = 0 in case O ≠ ℝ^{d}. In (0.1), u = (u_{1}, …, u_{d}) and p are unknown random fields on [0, T] × O, ν is a strictly positive constant, and W is a cylindrical Wiener process on a Hilbert space H. We interpret u(t, x) and p(t, x) as the velocity and pressure of an incompressible fluid at point x at time t. Then F (t, u) + G(t, u)Ẇ is the density of the force per unit volume. The force has the deterministic and random parts which depend on u.

There is an extensive literature concerning stochastic fluid mechanics. Stochastic Navier–Stokes equations in a bounded domain were investigated in a number of papers. In [3] the existence of a solution was proved for a constant diffusion coefficient and a one-dimensional Wiener process. The case of a more general diffusion term was studied using different methods in [2], [8], [9], [11], and [21]. The case of an unbounded domain, the present paper is concerned with, has only been considered in the deterministic setup. Some other problems for the stochastic Navier–Stokes equations were discussed in [12], [13], [20], [26], [33], and [34]. The one-dimensional case, the stochastic Burgers equation, was dealt with in [14] and [15]. Stochastic Korteweg–de Vries equation has been recently investigated in [5]. Two dimensional stochastic Euler equations were studied in [4], [7] and [10].

In the paper we consider two cases. First we assume that the initial value is a probability measure on the space of square integrable function on O. Then we consider the Navier–Stokes equations on ℝ^{d} with homogeneous Wiener random field and homogeneous initial data. In the former case our Theorem 1.1 provides sufficient conditions for the existence of a martingale solution to (0.1). We formulate the theorem in a rather abstract and general form. Then we apply it to the problem (0.1) with F and G being Nemitskii operators, and the noise W being either an L^{2}(O; ℝ^{m})-valued Wiener process, or, for O = ℝ^{d}, a finite system of independent spatially homogeneous Wiener random fields. In the case of the initial data being a spatially homogeneous probability measure, we assume that F and G are Nemitskii operators. The constructed solution lives in a weighted Sobolev space.

The fact that the domain is unbounded creates a difficulty. This is concerned with the embedding of the Sobolev space of functions with square integrable gradient into the L^{2}-space, which, unlike in the bounded case, is not compact. Compactness is crucial in proving the existence of solution and so it is necessary to introduce spaces with weights. Notice that if the initial value is a nontrivial spatially homogeneous probability measure, then it is not supported on the space of square integrable functions. This is another reason for introducing weighted spaces. This, however, brings in another difficulty. The nonlinear term ⟨u, ∇⟩u, which can be tackled as being orthogonal to the solution, has no longer this property in spaces with weights. Unbounded domain also forces some modification of the construction of a solution. The standard method for bounded domains uses Galerkin approximations obtained by solving projected equations on the finite-dimensional spaces spanned by the eigenfunctions of the Laplace operator. In our approach we adapt the Galerkin method employing special bases of spaces of square integrable functions with weights. These bases have to be Schauder bases of some weighted Sobolev and L^{q} spaces.

In the proof of the existence of a spatially homogeneous solution to (0.1) we employ a beautiful idea of Vishik and Fursikov from the paper [37] dealing with a statistical solution to deterministic Navier–Stokes equations in ℝ^{d} with a spatially homogeneous initial value. Namely, they perturb (0.1) by adding the term ε|u|^{4}u, they replace the pressure by −ε^{−1}div u, and the resulting problem has a unique solution. The limit passage ε → 0 for a statistical solution is possible due to some uniform estimates, where the properties of spatially homogeneous measures play a crucial role. We combine their approach with that of [11], where a method of solving stochastic Navier–Stokes equations in a bounded domain was given. Thus we prove the existence of a spatially homogeneous solution to (0.1) by solving first the following auxiliary equation

(0.2) ∂_{t}u^{ε} − νΔu^{ε} + ⟨u^{ε}, ∇⟩u^{ε} + ε|u^{ε}|^{4}u^{ε} − \frac{1}{ε}∇div u^{ε} = F (t, u^{ε}) + G(t, u^{ε})Ẇ .

In Section 5 we prove the existence of a solution to (0.2). Then, in Section 6, we prove the existence of a spatially homogeneous solution to (0.1) by making use of the weak relative compactness of {u^{ε}, ε > 0}.

1. Notation and formulation of the main results

Let 𝔘 = (Ω, F, (F_{t})_{t∈[0,T]}, ℙ) be a filtered probability space, and let H be a real separable Hilbert space.

Definition 1.1. By an (F_{t})-adapted cylindrical Wiener process on H we understand a mapping W : [0, T] × H → L^{2}(Ω, F, ℙ) satisfying the following conditions:

(i) for each t ∈ [0, T], W (t, ·) is a bounded linear operator acting from H into L^{2}(Ω, F, ℙ),

(ii) for all t ∈ [0, T], and ψ, φ ∈ H, 𝔼 W (t, ψ)W (t, φ) = t⟨ψ, φ⟩_{H}, (iii) for each ψ ∈ H, {W (t, ψ)}_{t∈[0,T]} is a real valued (F_{t})-adapted Wiener pro-

cess such that W (t, ψ) has covariance t|ψ|_{H}^{2}.

Let V be a separable Hilbert space. We denote by P_{T} (V) the space of all B([0, T]) × F-measurable V -valued processes σ such that:

(i) σ(t) is F_{t}-measurable for t ∈ [0, T],

(ii) 𝔼 ∫_{0}^{T} |σ(t)|_{V}^{2} dt < ∞. Let us denote by L_{(HS)}(H, V) the space of all Hilbert–Schmidt operators from H into V. Let σ ∈ P_{T} (L_{(HS)}(H, V)), and let W be an (F_{t})-adapted cylindrical Wiener process on H. Then for t ∈ [0, T] we define the stochastic integral ∫_{0}^{t} σ(s) dW (s) as the L^{2}(Ω, F_{t}; V, ℙ)-limit of the series ∑ ∫_{0}^{t} σ(s)ψ_{k} dW (s, ψ_{k}), where {ψ_{k}} is an orthonormal basis of H, and each term is understood as Itô’s integral with respect to the real valued Wiener process W (·, ψ_{k}). Note that ∫_{0}^{t} σ(s) dW (s), t ∈ [0, T], is a continuous martingale in V, with 0 mean and second moment satisfying

𝔼 |∫_{0}^{t} σ(s) dW (s)|_{V}^{2} = 𝔼 ∫_{0}^{t} ||σ(s)||_{L_{(HS)}(H,V)}^{2} ds.

Clearly, the stochastic integral does not depend on the particular choice of the orthonormal basis of H.

Let ρ ∈ ℝ. We write

ϑ_{ρ}(x) = ∏_{i=1}^{d}(1 + |x_{i}|^{2})^{−ρ}, x = (x_{1}, …, x_{d}) ∈ ℝ^{d}.

For brevity we denote by L_{ρ}^{q} and 𝕃_{ρ}^{q} the weighted spaces L^{q}(O, ϑ_{ρ}(x) dx) and L^{q}(O, ϑ_{ρ}(x) dx; ℝ^{d}), respectively.

Let us denote by C_{0,div}^{∞}(O; ℝ^{d}) the space consisting of all v ∈ C_{0}^{∞}(O; ℝ^{d}) such that div v = 0, and let H_{ρ} be the closure of C_{0,div}^{∞}(O; ℝ^{d}) in 𝕃_{ρ}^{2}. The scalar product and norm on L_{ρ}^{2}, H_{ρ} and 𝕃_{ρ}^{2} are denoted by ⟨·, ·⟩_{ρ} and | · |_{ρ}. Let us define the Sobolev spaces W_{ρ}^{1} and H_{ρ}^{1} as the completions of C_{0}^{∞}(O; ℝ^{d}) and C_{0,div}^{∞}(O; ℝ^{d}) with respect to the norm

( ∫_{O}(∑_{i=1}^{d} |∂_{x_{i}}v(x)|^{2} + |v(x)|^{2})ϑ_{ρ}(x) dx)^{1/2}.

Then we define W_{ρ}^{−1} and H_{ρ}^{−1} as the dual spaces (W_{ρ}^{1})^{∗} and (H_{ρ}^{1})^{∗}, where the duality is given by the identification of (𝕃_{ρ}^{2})^{∗} with 𝕃_{ρ}^{2} and (H_{ρ})^{∗} with H_{ρ}. The norm on W_{ρ}^{r} and H_{ρ}^{r}, r = −1, 1, is denoted by || · ||_{r,ρ}.

Throughout the paper we adopt the convention that if ρ = 0 then we write L^{q}, 𝕃^{q}, W^{r}, H^{r}, and || · ||_{r} of instead of L_{0}^{q}, 𝕃_{0}^{q}, W_{0}^{r}, H_{0}^{r}, and || · ||_{r,0}. The products on L^{2}, 𝕃^{2}, H, and ℝ^{d} are denoted by the same symbol ⟨·, ·⟩. Occasionally, ⟨·, ·⟩_{ρ} stands for the canonical bilinear form on W_{ρ}^{−1} × W_{ρ}^{1}.

In the paper F and G map [0, T] × 𝕃_{ρ}^{2} into W_{ρ}^{−1} and L_{(HS)}(H, 𝕃_{ρ}^{2}), respectively. They are supposed to be measurable, weakly continuous with respect to the second variable, and having linear growth. Here by the weak continuity of F and G we understand that for all t ∈ [0, T], ψ ∈ C_{0}^{∞}(O; ℝ^{d}), and φ ∈ H, the functions u ↦ ⟨F (t, u), ψ⟩_{ρ} and u ↦ ⟨G(t, u)φ, ψ⟩_{ρ} are continuous.

Definition 1.2. Let ρ ∈ ℝ, and let U_{0} be a probability measure on H_{ρ}. By an H_{ρ}- valued martingale solution to (0.1) starting from U_{0} we understand a triple (𝔘, W, u) consisting of a filtered probability space 𝔘 = (Ω, F, (F_{t})_{t∈[0,T]}, ℙ), an (F_{t})-adapted cylindrical Wiener process W on H, and a measurable (F_{t})-adapted stochastic process u such that:

(i) u ∈ L^{p}(Ω, F, ℙ; L^{∞}(0, T; H_{ρ})) for every p ∈ [1, ∞), (ii) u ∈ L^{2}(Ω × [0, T], F × B([0, T]), ℙ × dt; H_{ρ}^{1}), (iii) u(0) has the law U_{0}, (iv) for all t ∈ [0, T] and ψ ∈ C_{0}^{∞}(O; ℝ^{d}) with div (ψϑ_{ρ}) = 0 we have

⟨u(t), ψ⟩_{ρ} = ⟨u(0), ψ⟩_{ρ} + ν ∫_{0}^{t}⟨u(s), Δ(ϑ_{ρ}ψ)⟩ ds

+ ∫_{0}^{t}{∑_{i=1}^{d}⟨u(s)u_{i}(s), ∂_{x_{i}}(ψϑ_{ρ})⟩ + ⟨F (s, u(s)), ψ⟩_{ρ}} ds

+ ⟨∫_{0}^{t} G(s, u(s)) dW (s), ψ⟩_{ρ}.

Remark 1.1. In Definition 1.2 we make a specific choice of the space and the class of testing functions. The incompressibility condition div u = 0 is contained in the requirement that u belong to the divergence-free space H_{ρ}. Since div u = 0, the term ⟨u, ∇u_{i}⟩ in (0.1) has been replaced by div (uu_{i}).

The form of the equation appearing in (iv) is explained by the following heuristic argument. If we take the inner product in 𝕃_{ρ}^{2} of both sides of (0.1) with ψ such that div (ψϑ_{ρ}) = 0, then the pressure vanishes and after integrating by parts we obtain (iv). Going back, suppose that u is sufficiently regular. We move all the terms of the equation in condition (iv) of Definition 1.2 on one side and integrate by parts. Then we arrive at the relation of the form ⟨Θ(t; u), ψ⟩_{ρ} = 0 with a certain Θ. Since we have the following orthogonal decomposition

𝕃_{ρ}^{2} = \overline{\{∇p : p ∈ C_{0}^{∞}(O; ℝ^{d})\}} ⊕ \overline{\{ψ ∈ C_{0}^{∞}(O; ℝ^{d}) : div (ψϑ_{ρ}) = 0\}},

we conclude that Θ is in the form of the gradient of a certain function. Hence formally \frac{d}{dt}Θ(t; u) = ∇p(t), however, similarly as in the bounded domain case, Θ need not to be absolutely continuous with respect to the time variable.

The main result concerning the case of U_{0} being concentrated on H = H_{0} is the following theorem.

Theorem 1.1. Let U_{0} be a probability measure on H satisfying

∫_{H} |v|^{p} dU_{0}(v) < ∞ for every p ∈ [1, ∞).

Assume that F : [0, T] × 𝕃^{2} → W^{−1} and G : [0, T] × 𝕃^{2} → L_{(HS)}(H, 𝕃^{2}) are measurable, weakly continuous with respect to the second variable, and satisfy

||F (t, v)||_{−1} + ||G(t, v)||_{L_{(HS)}(H,𝕃^{2})} ≤ C(1 + |v|)

with some constant C independent of u. Then there exists an H-valued martingale solution to (0.1) starting from U_{0}.

Remark 1.2. In the 2-dimensional case, under the condition of Lipschitz continuity of F : [0, T] × 𝕃^{2} → W^{−1} and G : [0, T] × 𝕃^{2} → L_{(HS)}(H, 𝕃^{2}) with respect to the second variable, we can prove that the solution is unique. Due to the estimate of Proposition 2.2(iii) we can proceed in exactly the same way as for bounded domains, see [33]. Namely, let us consider the difference z(t) = u_{1}(t) − u_{2}(t) of two solutions with the same initial value and let us apply Itô’s formula to |z(t)|^{2}ξ(t), where ξ(t) = exp{−C ∫_{0}^{t} ||u_{1}(s)||_{1}^{2} ds} with C large enough. Straightforward estimations lead to

𝔼 (|z(t)|^{2}ξ(t)) ≤ C ∫_{0}^{t} 𝔼 (|z(s)|^{2}ξ(s)) ds,

which by means of Gronwall’s lemma implies z(t) = 0 a.s. as ℙ(ξ < ∞) = 1.

Let W^{Q} = (W_{1}^{Q}, …, W_{m}^{Q}) be an L^{2}(O; ℝ^{m})-valued Wiener process with a nuclear covariance operator Q. Then, see e.g. [17], W can be viewed as a cylindrical Wiener process on the space H_{Q} = Range Q^{1/2} equipped with the scalar product

⟨ψ, φ⟩_{H_{Q}} = ⟨u, v⟩_{L^{2}(O;ℝ^{m})},

where u, v ∈ (Ker Q^{1/2})^{⊥} ⊆ L^{2}(O; ℝ^{m}) are such that ψ = Q^{1/2}u, φ = Q^{1/2}v. Since Q^{1/2} is Hilbert–Schmidt, it is given be an integral kernel Q, that is

Q^{1/2}ψ(x) = (∑_{j=1}^{m} ∫_{O} Q_{1,j}(x, y)ψ_{j}(y) dy, …, ∑_{j=1}^{m} ∫_{O} Q_{m,j}(x, y)ψ_{j}(y) dy).

Now let W^{Q} and W Q̃ be two independent L^{2}(O; ℝ^{m})-valued Wiener processes with the covariance operators Q and Q̃, respectively. We denote by Q and Q̃ the integral kernels corresponding to Q^{1/2} and Q̃^{1/2}. Let

f, f_{i}, g_{j}, g̃_{j} : [0, T] × O × ℝ^{d} → ℝ^{d}, i = 1, …, d, j = 1, …, m.

Consider the following system of stochastic Navier–Stokes equations

(1.3)  \begin{matrix}∂_{t}u − νΔu + & ⟨u, ∇⟩u + ∇p = f(t, x, u) + ∑_{i=1}^{d} \frac{d}{dx_{i}} f_{i}(t, x, u) \\ + ∑_{j=1}^{m}\{g_{j}(t, x, u)Ẇ_{j}^{Q}(t) + g̃_{j}(t, x, u)Ẇ Q̃ ; j(t)\}, \\ div u = 0, \end{matrix}

with the Dirichlet boundary condition u|_{∂O} = 0 in case O ≠ ℝ^{d}. In (1.3), d/ dx_{i}f_{i}(t, x, u) denotes the total derivative with respect to the x_{i}-variable.

Since W^{Q} and W Q̃ are independent, W = (W^{Q}, W Q̃) is a cylindrical Wiener space on H = H_{Q} × HQ̃. By a solution to (1.3) we understand a solution to the problem (0.1) with F and G defined for x ∈ O, v : O → ℝ^{d}, and ψ = (ψ^{Q}, ψQ̃) =

((ψ_{1}^{Q}, …, ψ_{m}^{Q}), (ψQ̃ ; 1, …, ψQ̃ ; m)) ∈ H by the formulas

(1.4)  \begin{aligned}F (t, v)(x) & = f(t, x, v(x)) + ∑_{i=1}^{d} \frac{d}{dx_{i}} f_{i}(t, x, v(x)), \\ (G(t, v)ψ)(x) & = ∑_{j=1}^{m}\{g_{j}(t, x, v(x))ψ_{j}^{Q}(x) + g̃_{j}(t, x, v(x))ψQ̃ ; j(x)\}. \end{aligned}

In fact, we will show in Section 4 that under the conditions of the theorem below, F and G satisfy the conditions of Theorem 1.1, and consequently we will obtain the existence of a martingale solution to (1.3).

Theorem 1.2. Assume that:

(i) U_{0} is a probability measure on H such that ∫_{H} |v|^{p} dU_{0}(v) < ∞ for every p ∈ [1, ∞),

(ii) the integral kernels Q and Q̃ satisfy the estimate

∑_{i,j=1}^{m}[sup_{x∈O} ∫_{O} Q_{i,j}^{2}(x, y) dy + ∫_{O} ∫_{O} Q̃_{i,j}^{2}(x, y) dx dy] < ∞,

(iii) f, f_{i}, g_{j}, g̃_{j}, are measurable functions of t, x and v, continuous with respect to v, and there exist a constant L and a function γ ∈ L^{2} such that for all t, x, v one has

|f(t, x, v)| + ∑_{i=1}^{d} |f_{i}(t, x, v)| + ∑_{j=1}^{m} |g_{j}(t, x, v)| ≤ L(γ(x) + |v|),

∑_{j=1}^{m} |g̃_{j}(t, x, v)| ≤ L.

Then there exists a martingale H-valued solution to (1.3) starting from U_{0}.

Remark 1.3. Let F and G be given by (1.4). Recall that O may be unbounded. Thus, in particular constant functions need not be square integrable. The reason why we have introduced the function γ is that the assumptions of Theorem 1.2 should guarantee that F and G take values in the spaces W^{−1} and L_{(HS)}(H, 𝕃^{2}). Now note that from (iii) we have that g_{j}(t, ·, v(·)) ∈ 𝕃^{2} and g̃_{j}(t, ·, v(·)) ∈ 𝕃^{∞} for any v ∈ 𝕃^{2}. Then, see the proof of Theorem 1.2, the condition (ii) guarantees that the multiplication operator

M(v)ψ = ∑_{j=1}^{m} vψ_{j}

acts continuously from 𝕃^{2} into L_{(HS)}(H_{Q}, 𝕃^{2}), and from 𝕃^{∞} into L_{(HS)}(HQ̃, 𝕃^{2}).

The remaining part of this section is devoted to the Navier–Stokes equations on O = ℝ^{d} driven by an ℝ^{m}-valued spatially homogeneous Wiener random field. This field can be viewed as an example of the abstract cylindrical Wiener process introduced earlier.

First we set τ_{h}v(x) = v(x + h) for v ∈ L_{ρ}^{2} or v ∈ 𝕃_{ρ}^{2}, and x ∈ ℝ^{d}. A measure U on L_{ρ}^{2}, or 𝕃_{ρ}^{2} is said to be spatially homogeneous if and only if U = U ∘ τ_{−h} for every h ∈ ℝ^{d}. For ρ > 1/2 there are nontrivial spatially homogeneous probability measures on L_{ρ}^{2} and 𝕃_{ρ}^{2} whereas for ρ ≤ 1/2 only the measure concentrated on the constant 0-function is spatially homogeneous. A random field v on [0, T] × ℝ^{d} is spatially homogeneous if for all t ∈ [0, T], x_{1}, …, x_{n} ∈ ℝ^{d} and h ∈ ℝ^{d} the random vectors (v(t, x_{1}), …, v(t, x_{n})) and (v(t, x_{1} + h), …, v(t, x_{n} + h)) have the same distribution. Note that if v(t, ·) is a measurable spatially homogeneous random field such that 𝔼 |v(t, x_{0})|^{2} < ∞ for a certain x_{0} ∈ ℝ^{d}, then

𝔼 |v(t, x)|^{2} = 𝔼 |v(t, x_{0})|^{2} < ∞ for every x ∈ ℝ^{d},

and consequently, with probability one v(t, ·) belongs to L_{ρ}^{2} for any ρ > 1/2. In this case v(t, ·) is spatially homogeneous if and only if its law is spatially homogeneous.

Definition 1.3. Let 𝔘 = (Ω, F, (F_{t})_{t∈[0,T]}, ℙ) be a filtered probability space. By an ℝ^{m}-valued spatially homogeneous Wiener random field on [0, T] × ℝ^{d} we understand a system W_{j}, j = 1, …, m, of independent measurable real valued spatially homogeneous random fields on [0, T] × ℝ^{d} such that each W_{j} is Gaussian and Wiener, that is:

(i) The random vector (W_{j}(t_{1}, x_{1}), …, W_{j}(t_{n}, x_{n})) is Gaussian for an arbitrary finite sequence (t_{1}, x_{1}), …, (t_{n}, x_{n}) ∈ [0, T] × ℝ^{d}.

(ii) For each x ∈ ℝ^{d}, {W_{j}(t, x)}_{t∈[0,T]} is a real valued Wiener process with respect to the filtration (F_{t}).

Let W be an ℝ^{m}-valued spatially homogeneous Wiener random field on ℝ^{d}. Then, see [32], there are symmetric positive finite measures μ_{j}, j = 1, …, m, on ℝ^{d} such that for all j = 1, …, m, t, s ∈ [0, T], and x, y ∈ ℝ^{d} one has

𝔼 W_{j}(t, x)W_{j}(s, y) = t ∧ s \frac{1}{(2π)^{\frac{d}{2}}} ∫_{ℝ^{d}} exp\{i⟨x − y, z⟩\} dμ_{j}(z).

We call μ = (μ_{1}, …, μ_{m}) the spectral measure of W. Clearly, the distribution of W is uniquely determined by its spectral measure.

Stochastic evolution equations driven by a homogeneous Wiener process were studied in [19] and [32]. In [19] the spectral measure of the noise is assumed to be absolutely continuous with respect to Lebesgue measure. In our paper we do not need this assumption. In the absolutely continuous case the typical densities are a_{j} exp(−b_{j}|x|^{r_{j}}), a_{j}, b_{j}, r_{j} > 0, and a_{j}(b_{j} + |x|^{2})^{−r_{j}}, a_{j}, b_{j} > 0 and r_{j} > d/2. In the case where we have no density, a simple example is obtained by taking μ_{j} equal to the Dirac measure δ_{0}. Then W_{j}(t, x) does not depend on x and is a one dimensional Brownian motion. Some interesting examples can be constructed by considering the combinations of the above or the measures concentrated on manifolds.

Let W be an ℝ^{m}-valued spatially homogeneous Wiener random field on ℝ^{d} with a spectral measure μ. Let us denote by L_{(s)}^{2}(ℝ^{d}, μ_{j}) the subspace of L^{2}(ℝ^{d}, μ_{j}; ℂ) consisting of all ξ such that ξ(x) = \overline{ξ(−x)}, x ∈ ℝ^{d}. Set

H_{μ} = \{ξ̂μ = (ξ̂_{1}μ_{1}, …, { \atop ξ_{m}μ_{m}}) : ξ_{j} ∈ L_{(s)}^{2}(ℝ^{d}, μ_{j})\},

where v̂ denotes the Fourier transform of v. Then as each μ_{j} is finite we have H_{μ} ⊂ C_{b}(ℝ^{d}; ℝ^{m}). Clearly, H_{μ} equipped with the norm

⟨ξ̂μ, η̂μ⟩_{H_{μ}} = ∑_{j=1}^{m} ∫_{ℝ^{d}} ξ_{j}(x)\overline{η_{j}(x)} dμ_{j}(x)

is a real separable Hilbert space. Then, see [32],

(1.5)  W (t, ψ) = ∑_{j=1}^{m} ∫_{ℝ^{d}} W_{j}(t, x)ψ_{j}(x) dx, ψ = (ψ_{1}, …, ψ_{m}) ∈ H_{μ}

is a cylindrical Wiener process on H_{μ}. Note that having a cylindrical Wiener process W on H_{μ} we can easily find an ℝ^{m}-valued spatially homogeneous Wiener random field W with spectral measure μ such that (1.5) is fulfilled.

Let W be an ℝ^{m}-valued spatially homogeneous Wiener random field. Let

f, f_{i}, g_{j} : [0, T] × ℝ^{d} × ℝ^{d} → ℝ^{d}, i = 1, …, d, j = 1, …, m,

and let us consider the system of stochastic Navier–Stokes equations on ℝ^{d} driven by W,

(1.6)  \begin{matrix}∂_{t}u − νΔu + & ⟨u, ∇⟩u + ∇p = f(t, x, u) + ∑_{i=1}^{d} \frac{d}{dx_{i}} f_{i}(t, x, u) \\ + ∑_{j=1}^{m} g_{j}(t, x, u)Ẇ_{j}, \\ div u = 0. \end{matrix}

As in the case of (1.3), by a solution to (1.6) we understand a solution to the problem (0.1) with H = H_{μ}, and F and G given by the formulas

(1.7)  \begin{aligned}F (t, v)(x) & = f(t, x, v(x)) + ∑_{i=1}^{d} \frac{d}{dx_{i}} f_{i}(t, x, v(x)), \\ (G(t, v)ξ̂μ)(x) & = ∑_{j=1}^{m} g_{j}(t, x, v(x))ξ̂_{j}μ_{j}(x). \end{aligned}

It is worth pointing out that the stochastic integral term in Definition 1.2 can be interpreted as

∑_{i=1}^{d} ∑_{j=1}^{m} ∫_{ℝ^{d}} ∫_{0}^{t} g_{j,i}(s, x, u(s, x))W_{j}( ds, x)ψ_{i}(x)ϑ_{ρ}(x) dx,

where W_{j}( ds, x) means that we integrate in Itô’s sense with respect to the real valued Wiener process W_{j}(·, x), see [32, Section 1.2] for more details.

Theorem 1.3. Assume that:

(i) U_{0} is a probability measure on H such that ∫_{H} |v|^{p} dU_{0}(v) < ∞ for every p ∈ [1, ∞),

(ii) f, f_{i}, g_{j}, are measurable functions of t, x and v, continuous with respect to v, and there exist a constant L and a function γ ∈ L^{2} such that for all t, x, v one has

|f(t, x, v)| + ∑_{i=1}^{d} |f_{i}(t, x, v)| + ∑_{j=1}^{m} |g_{j}(t, x, v)| ≤ L(γ(x) + |v|).

Then there exists a martingale H-valued solution to (1.6) starting from U_{0}.

Our last result is concerned with spatially homogeneous solutions. As in Theorem 1.3 we assume that O = ℝ^{d}, W is an ℝ^{m}-valued spatially homogeneous Wiener random field, and F, G are given by (1.7). The initial value need not be concentrated on H. Thus we have to take ρ > 1/2 and the result is not covered by Theorem 1.1. We assume that ρ < 1 for some technical reasons, see Theorem C.1.

Theorem 1.4. Let 1/2 < ρ < 1, and let U_{0} be a homogeneous probability measure on H_{ρ} satisfying

∫_{H_{ρ}} |v|_{ρ}^{p} dU_{0}(v) < ∞ for every p ∈ [1, ∞).

Assume that the functions f, f_{i} and g_{i} are measurable and independent of the x-variable. Furthermore, assume that there exists a constant L such that for all t ∈ [0, T], and v, z ∈ ℝ^{d} and i, j one has

|f(t, v) − f(t, z)| + |f_{i}(t, v) − f_{i}(t, z)| + |g_{j}(t, v) − g_{j}(t, z)| ≤ L|v − z|,
|f(t, v)| + |f_{i}(t, v)| + |g_{j}(t, v)| ≤ L(1 + |v|).

Then there exists an H_{ρ}-valued martingale solution to (1.6) starting from U_{0}. Moreover, the random field u is spatially homogeneous.

2. Auxiliary results

For any q ∈ [2, ∞) we denote by S the semigroup on 𝕃_{0}^{q} generated by A = νΔ with the Dirichlet boundary conditions. Then S is holomorphic on any 𝕃_{0}^{q}-space. Moreover, for arbitrary ρ ≥ 0 and q ∈ [2, ∞), S has the unique extension to a holomorphic semigroup on 𝕃_{ρ}^{q}. To see this we set i_{ρ}v = ϑ_{−ρ/2}v. Obviously, for every q ∈ [1, ∞), i_{ρ} is an isometry between 𝕃_{0}^{q} and 𝕃_{qρ/2}^{q}. Consider now the operator Ã given by

Ã = i−1_{\frac{2ρ}{q}}Ai_{\frac{2ρ}{q}} = A + ν ∑_{j=1}^{d}(ϑ_{\frac{ρ}{q}} ∂_{x_{j}}ϑ_{\frac{−ρ}{q}} )∂_{x_{j}} + νϑ_{\frac{ρ}{q}} Δϑ_{\frac{−ρ}{q}} .

Since Ã is a uniformly elliptic operator with coefficients from C_{b}^{∞}(ℝ^{d}), it generates a holomorphic semigroup S̃ on 𝕃_{0}^{p}. Let S_{ρ} = i_{2ρ/q}S̃i_{2ρ/q}^{−1}. Then S_{ρ} is holomorphic on 𝕃_{ρ}^{q} as it is unitarily equivalent to S̃. Note that A is the generator of S_{ρ}, and that S_{ρ} is an extension of S. In what follows we will use the same symbol to denote the semigroup on 𝕃_{0}^{q} and its extension to 𝕃_{ρ}^{q}.

It is easy to check that each H_{ρ} is invariant with respect to S. Consequently, the restriction of S to H_{ρ} is also a holomorphic semigroup. It will cause no confusion if we use the same letter for the semigroup and its restriction to H_{ρ}. Similarly, we denote by A the generator of S on 𝕃_{ρ}^{2} and H_{ρ}.

Let λ belong to the resolvent set of A. Using the same methods as in [32, Section 3] one may easily show that the Sobolev spaces W_{ρ}^{1} and H_{ρ}^{1} introduced in Section 1 are identical with the domain of (−A + λI)^{\frac{1}{2}} in 𝕃_{ρ}^{2} and H_{ρ}, respectively.

For any positive real r we define W_{ρ}^{r} and H_{ρ}^{r} as the domain of (−A+λI)^{\frac{r}{2}} , where A is considered as the generator of a holomorphic semigroup on 𝕃_{ρ}^{2} and H_{ρ}. The spaces W_{ρ}^{r} and H_{ρ}^{r} are equipped with the graph norm || · ||_{r,ρ}. Identifying (𝕃_{ρ}^{2})^{∗} with 𝕃_{ρ}^{2}, and (H_{ρ})^{∗} with H_{ρ} we set W_{ρ}^{−r} = (W_{ρ}^{r})^{∗} and H_{ρ}^{−r} = (H_{ρ}^{r})^{∗}. In the paper we write W^{r}, H^{r}, and || · ||_{r} instead of W_{0}^{r}, H_{0}^{r}, and || · ||_{r,0}.

Let ρ ∈ ℝ and α ∈ [0, 1). Set

(2.8) J_{α}^{ρ}(v)(t) = ∫_{0}^{t}(t − s)^{−α}S(t − s)v(s) ds, v ∈ L^{1}(0, T; 𝕃_{ρ}^{1}), t ∈ [0, T].

The following propositions play an important role in the proof of the existence of a martingale solution. Their proofs are postponed into Appendix A.

Proposition 2.1. Let 0 ≤ ρ < κ, and let J_{α}^{ρ} be given by (2.8).

(i) For an arbitrary p > 1, J_{0}^{ρ} can be extended to a compact operator acting from L^{p}(0, T; W_{ρ}^{−1}) into L^{2}(0, T; 𝕃_{κ}^{2}). (ii) For arbitrary α ∈ [0, 1) and p ∈ (1, ∞) such that α + p^{−1} < 1, J_{α}^{ρ} is a compact operator from L^{p}(0, T; 𝕃_{ρ}^{2}) into L^{2}(0, T; 𝕃_{κ}^{2}).

(iii) If 5ρ/3 < κ, then J_{0}^{ρ} is a compact operator acting from L^{6/5}(0, T; 𝕃_{ρ}^{6/5}) into L^{2}(0, T; 𝕃_{κ}^{2}).

For u ∈ C_{0}^{∞}(O; ℝ^{d}) and x ∈ ℝ^{d} write

(2.9)  B_{0}(v)(x) = (∑_{i=1}^{d} ∂_{x_{i}}(v_{i}v_{1})(x), …, ∑_{i=1}^{d} ∂_{x_{i}}(v_{i}v_{d})(x)).

Proposition 2.2. Let B_{0} be given by (2.9). Then:

(i) for any ρ ∈ ℝ, B_{0} has a unique extension to a continuous mapping B acting from 𝕃_{ρ}^{4} into W_{ρ}^{−1}. Moreover, there exists a constant C such that

||B(v)||_{−1,ρ} ≤ C|v|_{𝕃_{ρ}^{4}}^{2} for v ∈ 𝕃_{ρ}^{4}.

(ii) B_{0} has a unique extension to a continuous mapping B acting from W^{1} into W^{−1}. Moreover, there exists a constant C such that

||B(v)||_{−1} ≤ C|v|^{\frac{1}{2}} ||v||1^{\frac{3}{2}} for v ∈ W^{1}.

(iii) If d = 2 then there is a constant C such that

||B(v)||_{−1} ≤ C|v|||v||_{1} for v ∈ W^{1}.

3. Proof of Theorem 1.1

Let B be the mapping constructed in Proposition 2.2. A martingale solution to (0.1) is found using a modification of the classical Galerkin method. The idea is this: consider a sequence of equations in which F, B, and G are replaced by Π_{n}F, Π_{n}B(Π_{n}·) and Π_{n}G for a properly chosen system of projections {Π_{n}}. The goal is to show that the laws of the sequence of solutions are tight in a suitable space and to extract a convergent subsequence. Skorokhod’s theorem allows us to replace the convergence of laws by pointwise convergence of the stochastic processes after changing the underlying probability space. Hence we can find that the limit is a solution to our equation.

By Theorem B.1 from Appendix B there exists an orthonormal basis {e_{k}} of H such that {e_{k}} ⊂ C_{b}^{∞}(O; ℝ^{d}) ∩ H^{2}, {e_{k}} is a Schauder basis of H^{1}, and

(3.10)  sup_{n} ||Π_{n}||_{L(H^{1},H^{1})} < ∞.

Let n ∈ ℕ. As {e_{k}} ⊂ H^{1} = (H^{−1})^{∗} we can extend each Π_{n} to H^{−1} putting

Π_{n}v = ∑_{k=1}^{n}⟨u, e_{k}⟩e_{k},

where ⟨·, ·⟩ is now considered as the dual form on H^{−1} × H. Note that

||Π_{n}||_{L(H^{−1},H^{−1})} = sup_{||v||_{−1}≤1} ||Π_{n}v||_{−1} = sup_{||u||_{1}≤1} sup_{||v||_{−1}≤1} |⟨Π_{n}v, u⟩|

= sup_{||u||_{1}≤1} sup_{||v||_{−1}≤1} |⟨v, Π_{n}u⟩| ≤ ||Π_{n}||_{L(H^{1},H^{1})}.

Hence, from (3.10), we have

(3.11)  sup_{n} ||Π_{n}||_{L(H^{−1},H^{−1})} < ∞.

For t ∈ [0, T] and v ∈ H we set F^{(n)}(t, v) = Π_{n}F (t, v), G^{(n)}(t, v) = Π_{n}G(t, v), and B^{(n)}(v) = Π_{n}B(Π_{n}v).

Let 𝔘 = (Ω, F, (F_{t})_{t∈[0,T]}, ℙ) be a filtered probability space, let W be an (F_{t})- adapted cylindrical Wiener process on H, and let u(0) be an F_{0}-measurable Hvalued random element with the law U_{0}. Consider the following stochastic evolution equation

(3.12)  \begin{matrix}du^{(n)} = & (Au^{(n)} + B^{(n)}(u^{(n)}) + F^{(n)}(t, u^{(n)})) dt \\ + G^{(n)}(t, u^{(n)}) dW, \\ u^{(n)}(0) = & Π_{n}u(0). \end{matrix}

By a solution to (3.12) we mean the so-called mild solution, that is a measurable (F_{t})-adapted process u^{(n)}, having continuous trajectories in H and such that for t ∈ [0, T],

u^{(n)}(t) = S(t)u^{(n)}(0) + ∫_{0}^{t} S(t − s){B^{(n)}(u^{(n)}(s)) + F^{(n)}(s, u^{(n)}(s))} ds
+ ∫_{0}^{t} S(t − s)G^{(n)}(s, u^{(n)}(s)) dW (s).

It is known that, see [17], that every mild solution is also a weak solution, that is

⟨u^{(n)}(t), ψ⟩ = ⟨u^{(n)}(0), ψ⟩ + ∫_{0}^{t}⟨u^{(n)}(s), Aψ⟩ ds
+ ∫_{0}^{t}⟨B^{(n)}(u^{(n)}(s)) + F^{(n)}(s, u^{(n)}(s)), ψ⟩ ds
+ ⟨∫_{0}^{t} G^{(n)}(s, u^{(n)}(s)) dW (s), ψ⟩

for all t ∈ [0, T] and ψ ∈ Dom A^{∗} = Dom A. Note that C_{0}^{∞}(O; ℝ^{d}) ⊂ Dom A.

Lemma 3.1. Assume that there are constants L and K such that for all t ∈ [0, T], and u, v ∈ 𝕃^{2} we have

||F (t, v) − F (t, u)||_{−1} + ||G(t, v) − G(t, u)||_{L_{(HS)}(H,𝕃^{2})} ≤ L|v − u|,
||F (t, v)||_{−1} + ||G(t, v)||_{L_{(HS)}(H,𝕃^{2})} ≤ K(1 + |v|).

Then for every n ∈ ℕ there exists a unique solution u^{(n)} to (3.12). Moreover, for any p ≥ 2,

(3.13)  𝔼 sup_{t∈[0,T]} |u^{(n)}(t)|^{p} ≤ C(p)

and

(3.14)  𝔼 (|u^{(n)}(t)|^{2} + ∫_{0}^{t} ||u^{(n)}(s)||_{1}^{2} ds) ≤ C,

the constants C(p) and C being independent of n, L, and t ∈ [0, T].

Proof. We consider (3.12) as an equation on H with locally Lipschitz nonlinearities. Then it is easy to show that for an arbitrary probability space there exists a unique, possibly exploding, mild solution u^{(n)} of (3.12). Since u^{(n)}(0) ∈ Dom A, and B^{(n)}, F^{(n)} are continuous mapping from H into Dom A, and since G^{(n)} is a continuous mapping from H into L_{(HS)}(H, Dom A), u^{(n)} is a local strong solution to (3.12). In particular, u^{(n)} has continuous trajectories in Dom A. Set

Ψ_{n}(t) = Au^{(n)}(t) + B^{(n)}(u^{(n)}(t)) + F^{(n)}(t, u^{(n)}(t)).

We locally have

du^{(n)} = Ψ_{n} dt + G^{(n)}(t, u^{(n)}) dW.

We now show (3.14) which guarantees that u^{(n)} is a global solution to (3.12). Let p ≥ 2. Applying Itô’s formula to the function H_{p}(v) = |v|^{p} we obtain

d|u^{(n)}|^{p} = Φ_{p}(t, u^{(n)}) dt + H′_{p}(u^{(n)})G^{(n)}(t, u^{(n)}) dW,

where

Φ_{p}(t, v) := ⟨H′_{p}(v), Ψ_{n}(t)⟩ + \frac{1}{2}Trace H′′_{p}(v)G^{(n)}(t, v)G^{(n)}(t, v)^{∗}.

Then we have

Φ_{p}(t, v) ≤ p|v|^{p−2}(⟨Av, v⟩ + |⟨F^{(n)}(t, v), v⟩| + |⟨B^{(n)}(v), v)⟩|

+ \frac{p − 1}{2}||G^{(n)}(t, v)||_{L_{(HS)}(H,H)}^{2}).

As A is self-adjoint, for v ∈ Dom A we have ⟨Av, v⟩ = −ν||v||_{1}^{2}. Integrating by parts and using the fact that div v = 0 we obtain ⟨B^{(n)}(v), v⟩ = 0. From (3.11) we have

|⟨F^{(n)}(t, v), v⟩| + \frac{p − 1}{2}||(G^{(n)}(t, v))||_{L_{(HS)}(H,H)}^{2}

≤ c_{1}(1 + |v|)||v||_{1} + c_{2}(1 + |v|^{2})

≤ \frac{ν}{2} ||v||_{1}^{2} + c_{3}(1 + |v|^{2}).

Hence

Φ_{p}(t, v) ≤ p|v|^{p−2}(−\frac{ν}{2} ||v||_{1}^{2} + c_{3}(1 + |v|^{2})).

Now let p = 2. Then taking the mathematical expectation we get

𝔼 (|u^{(n)}(t)|^{2} + ν ∫_{0}^{t} ||u^{(n)}(s)||_{1}^{2} ds) ≤ 𝔼 |u(0)|^{2} + 2c_{3}t + 2c_{3} 𝔼 ∫_{0}^{t} |u^{(n)}(s)|^{2} ds.

Applying Gronwall’s lemma we obtain (3.14).

Now let p ≥ 2. Then using the maximal inequality for martingales we obtain

𝔼 sup_{s∈[0,t]} |u^{(n)}(s)|^{2p} ≤ 3 𝔼 |u^{(n)}(0)|^{2p}

+ 3 (c_{3}p 𝔼 ∫_{0}^{t} |u^{(n)}(s)|^{p−2}(1 + |u^{(n)}(s)|^{2}) ds)^{2}
+ c_{4} 𝔼 ∫_{0}^{t} ||H′_{p}(u^{(n)}(s))G^{(n)}(s, u^{(n)}(s))||_{L_{(HS)}(H,H)}^{2} ds
≤ c_{5} + 3𝔼 |u^{(n)}(0)|^{2p} + c_{5} 𝔼 ∫_{0}^{t} |u^{(n)}(s)|^{2p} ds,

c_{5} being independent of n and t ∈ [0, T]. Hence Gronwall’s lemma gives (3.13). □

Proof of Theorem 1.1. We first prove the theorem under the additional assumption that F, G satisfy the conditions of Lemma 3.1.

For each n let u^{(n)} be the solution to (3.12). Let us fix α ∈ (1/2, 1) and p > 2 such that α + p^{−1} < 1. Recall that the operators J_{α}^{ρ}, ρ ∈ ℝ were introduced in Section 2. In this section we write J_{0} and J_{α} instead of J_{0}^{0} and J_{α}^{0}. Let

Y_{n}^{α}(t) = \frac{sin(α − 1)π}{π} ∫_{0}^{t}(t − s)^{α−1}S(t − s)G^{(n)}(s, u^{(n)}(s)) dW (s).

Then by the factorization result from [16, Proposition 1] we have

∫_{0}^{t} S(t − s)G^{(n)}(s, u^{(n)}(s)) dW (s) = J_{α}(Y_{n}^{α})(t),

and consequently

u^{(n)}(t) = S(t)Π_{n}u(0) + J_{0}(B^{(n)}(u^{(n)}) + F^{(n)}(·, u^{(n)}))(t) + J_{α}(Y_{n}^{α})(t).

Our first goal is to show that the laws of u^{(n)}, n ∈ ℕ, are tight in L^{2}(0, T; H_{κ}), where κ > 0. Let us fix such a κ. Clearly, the sequence {S(·)Π_{n}u(0)} converges, and it is enough to show that the laws of the remaining sequence

y_{n} = J_{0}(B^{(n)}(u^{(n)}) + F^{(n)}(·, u^{(n)})) + J_{α}(Y_{n}^{α})

are tight. To this end, for each δ > 0 we will find a relatively compact set Λ_{δ} in L^{2}(0, T; H_{κ}) such that

(3.15)  ℙ(y_{n} ∈ Λ_{δ}) ≥ 1 − δ  for every n.

We put Λ_{δ} = Λ(R) for R sufficiently large, where

Λ(R) = J_{0}(K_{1}(R)) + J_{α}(K_{2}(R)),

K_{i}(R) being balls with radius R in L^{5/4}(0, T; H^{−1}) and L^{p}(0, T; H), respectively. Note that, by Proposition 2.1(i) and (ii), Λ(R) is compact. Thus to prove that for each δ > 0 there exists an R such that (3.15) holds it is enough to show that

(3.16)  sup_{n} 𝔼 ∫_{0}^{T} ||B^{(n)}(u^{(n)}(t))||−1^{\frac{5}{4}} dt < ∞,

(3.17)  sup_{n} 𝔼 ∫_{0}^{T} ||F^{(n)}(t, u^{(n)}(t))||−1^{\frac{5}{4}} dt < ∞,

(3.18)  sup_{n} 𝔼 ∫_{0}^{T} |Y_{n}^{α}(t)|^{p} dt < ∞.

To show (3.16) note that, by (3.11), Proposition 2.2(ii), and (3.10) we have

||B^{(n)}(u^{(n)})||_{−1} ≤ c_{1} ||B(Π_{n}u^{(n)})||_{−1} ≤ c_{2} |Π_{n}u^{(n)}(t)|^{\frac{1}{2}} ||Π_{n}u^{(n)}(t)||1^{\frac{3}{2}}

≤ c_{3} |u^{(n)}(t)|^{\frac{1}{2}} ||u^{(n)}(t)||1^{\frac{3}{2}} .

Hence

𝔼 ∫_{0}^{T} ||B^{(n)}(u^{(n)}(t))||−1^{\frac{5}{4}} dt ≤ c_{3} 𝔼 ∫_{0}^{T} |u^{(n)}(t)|^{\frac{5}{8}} ||u^{(n)}(t)||1^{\frac{15}{8}} dt

≤ c_{3} 𝔼 |u^{(n)}|L^{∞}(0,T;H)^{\frac{5}{8}}|u^{(n)}|L^{\frac{15}{8}} (0,T;H_{1})^{\frac{15}{8}}

≤ c_{3} (𝔼 |u^{(n)}|_{L^{∞}(0,T;H)}^{10})^{\frac{1}{16}} (𝔼 |u^{(n)}|_{L^{\frac{15}{8}} (0,T;H_{1})}^{2})^{\frac{15}{16}} , and applying (3.13) and (3.14) we obtain (3.16).

Clearly (3.17) follows from (3.11), and (3.14). To show (3.18) note that

𝔼 ∫_{0}^{T} |Y_{n}^{α}(t)|^{p} dt

= ∫_{0}^{T} 𝔼 |\frac{sin(α − 1)π}{π} ∫_{0}^{t}(t − s)^{α−1}S(t − s)G^{(n)}(s, u^{(n)}(s)) dW (s)|^{p} dt

≤ c_{4} ∫_{0}^{T} 𝔼 (∫_{0}^{t}(t − s)^{2α−2}||S(t − s)G^{(n)}(s, u^{(n)}(s))||_{L_{(HS)}(H,H)}^{2} ds)^{\frac{p}{2}} dt.

Thus (3.18) follows from (3.13).

Let L(u^{(n)}) denote the law of u^{(n)} on L^{2}(0, T; 𝕃_{κ}^{2}). According to (3.15) the sequence L(u^{(n)}), n ∈ ℕ, is tight, and hence, by the Prokhorov theorem, it is relatively weakly compact. Let H̃ be a Hilbert space such that H ↪ H̃ with a Hilbert–Schmidt imbedding. Then since W is a process with continuous trajectories in H̃, the sequence L(u^{(n)}, W), n ∈ ℕ of the laws of (u^{(n)}, W) on L^{2}(0, T; 𝕃_{κ}^{2})× C([0, T]; H̃) is also relatively weakly compact. So there is a probability P^{0} on L^{2}(0, T; 𝕃_{κ}^{2}) × C([0, T]; H̃) such that L(u^{(n_{l})}, W) converges weakly to P^{0} for a subsequence {n_{l}}, which we will denote by {n}. By the Skorokhod imbedding, see [24], there exists a probability space (Ω^{1}, F^{1}, (F_{t}^{1}), ℙ^{1}) and a sequence of random elements {v^{(n)}, W^{(n)}} on Ω^{1} such that the laws of (u^{(n)}, W) and (v^{(n)}, W^{(n)}) are the same, and (v^{(n)}, W^{(n)}) converges ℙ^{1}-almost surely in L^{2}(0, T; 𝕃_{κ}^{2})×C([0, T]; H̃) to a certain random element (ũ, W̃). Note that for every n, W^{(n)} is a cylindrical Wiener processes on H, and v^{(n)} is the solution of (3.12) driven by W^{(n)}. Taking a subsequence we may assume that v^{(n)}(t, x) converges to ũ(t, x), ℙ^{1} × dt × dx - almost surely. Thus, by (3.13) and Fatou’s lemma we obtain that for all p, q ∈ [1, ∞),

𝔼^{1} (∫_{0}^{T} |ũ(t)|^{q} dt)^{\frac{p}{q}} ≤ lim inf_{n→∞} 𝔼^{1} (∫_{0}^{T} |v^{(n)}(t)|^{q} dt)^{\frac{p}{q}} ≤ CT^{\frac{p}{q}} < ∞.

Hence, we have

(3.19)  ũ ∈ ⋂_{p∈[1,∞)} L^{p}(Ω^{1}, F^{1}, ℙ^{1}; L^{∞}(0, T; H)).

Now since for every q the sequence |v^{(n)} − ũ|_{L^{q}(0,T;H)}^{q} is uniformly integrable, we have

(3.20)  𝔼^{1} ∫_{0}^{T} |v^{(n)}(t) − ũ(t)|^{q} dt → 0 for q ∈ [1, ∞),

and, in particular, we may assume that

∫_{0}^{T} |v^{(n)}(t) − ũ(t)|^{2} dt → 0, ℙ^{1} - a.s..

Now, as {v^{(n)}} is bounded in L^{2}(Ω^{1} × [0, T], F^{1} × B([0, T]), ℙ^{1} × dt; H^{1}), we have

(3.21)  ũ ∈ L^{2}(Ω^{1} × [0, T], F^{1} × B([0, T]), ℙ^{1} × dt; H^{1}).

To this end we note that the embedding H^{1} ↪ H_{κ} is compact, hence, see [31, Lemma 2.2], there exist an orthonormal basis {e_{k}^{κ}} of H_{κ} and a sequence {λ_{k}} such that for any v ∈ H_{κ} we have

v ∈ H^{1} ⟺ ∑_{k=1}^{∞}⟨v, e_{k}^{κ}⟩_{κ}^{2}λ_{k}^{2} < ∞,

and, if v ∈ H^{1} then

||v||_{1}^{2} = ∑_{k=1}^{∞}⟨v, e_{k}^{κ}⟩_{κ}^{2}λ_{k}^{2}.

Applying Fatou’s lemma we obtain

𝔼^{1} ∫_{0}^{T} ∑_{k=1}^{∞}⟨ũ(t), e_{k}^{κ}⟩_{κ}^{2}λ_{k}^{2} dt ≤ lim inf_{n} 𝔼^{1} ∫_{0}^{T} ∑_{k=1}^{∞}⟨v^{(n)}(t), e_{k}^{κ}⟩_{κ}^{2}λ_{k}^{2} dt < ∞.

Taking into account (3.19) and (3.21) we see that ũ satisfies the regularity conditions (i) and (ii) of Definition 1.2.

For n ∈ ℕ we write

M_{n}(t) = ∫_{0}^{t} G^{(n)}(s, v^{(n)}(s)) dW^{(n)}(s), t ∈ [0, T].

Then M_{n} is a square integrable martingale in H, and its quadratic variation is given by

≪ M_{n} ≫ (t) = ∫_{0}^{t} G^{(n)}(s, v^{(n)}(s))G^{(n)}(s, v^{(n)}(s))^{∗} ds.

From (3.20) we have

lim_{n,m→∞} 𝔼^{1} ∫_{0}^{T} ||G^{(n)}(t, v^{(n)}(t)) − G^{(m)}(t, v^{(m)}(t))||_{L_{(HS)}(H,H)}^{2} dt = 0.

Since there is a constant C such that for all n, m,

𝔼^{1} sup_{t∈[0,T]} |M_{n}(t) − M_{m}(t)|^{2}

≤ C 𝔼^{1} ∫_{0}^{T} ||G^{(n)}(t, v^{(n)}(t)) − G^{(m)}(t, v^{(m)}(t))||_{L_{(HS)}(H,H)}^{2} dt, the sequence M_{n}, n ∈ ℕ, converges in L^{2}(Ω^{1}, F^{1}, ℙ^{1}; C([0, T]; H)) to a certain martingale M. Taking if necessary a subsequence we may assume that the convergence is also ℙ^{1} - almost surely. It is easy to see that

≪ M ≫ (t) = ∫_{0}^{t} ΠG(s, ũ(s))G^{∗}(s, ũ(s))Π ds, t ∈ [0, T],

where Π is the projection of 𝕃^{2} onto H. Using now the representation theorem [17, Theorem 8.2] we can find a filtered probability space (Ω̃, F̃, (F̃_{t}), ℙ̃) and an (F_{t}^{1} × F̃_{t})-adapted cylindrical Wiener process W on H, such that

M(t, ω, ω̃) = ∫_{0}^{t} ΠG(s, u(s, ω, ω̃)) dW(s, ω, ω̃),

where M(t, ω, ω̃) = M(t, ω), and u(s, ω, ω̃) = ũ(s, ω). Clearly, the process u is adapted to the filtration (F_{t}^{1}×F̃_{t}), u(0) has the law U_{0}, and u satisfies the conditions (i) and (ii) of Definition 1.2. What is left is to show that u satisfies the integral equation in (iv). Fix t ∈ [0, T] and ψ ∈ C_{0}^{∞}(O; ℝ^{d}) satisfying div ψ = 0. Set v^{(n)}(t, ω, ω̃) = v^{(n)}(t, ω). Since F and G are continuous we have that ℙ^{1} × ℙ̃ - almost surely:

⟨v^{(n)}(t), ψ⟩ → ⟨u(t), ψ⟩, ⟨Π_{n}v^{(n)}(0), ψ⟩ → ⟨u(0), ψ⟩,

∫_{0}^{t}{⟨v^{(n)}(s), νΔψ⟩ + ∑_{i=1}^{d}⟨(Π_{n}v^{(n)}(s)(Π_{n}v^{(n)})_{i}(s), ∂_{x_{i}}Π_{n}ψ⟩} ds

→ ∫_{0}^{t}{⟨u(s), νΔψ⟩ + ∑_{i=1}^{d}⟨u(s)u_{i}(s), ∂_{x_{i}}ψ⟩} ds,

∫_{0}^{t}⟨F (s, v^{(n)}(s)), Π_{n}ψ⟩ ds → ∫_{0}^{t}⟨F (s, u(s)), ψ⟩ ds,

⟨∫_{0}^{t} G^{(n)}(s, v^{(n)}(s)) dW^{(n)}(s), ψ⟩

→ ⟨Π ∫_{0}^{t} G(t, u(s)) dW(s), ψ⟩ = ⟨∫_{0}^{t} G(t, u(s)) dW(s), ψ⟩.

Thus the desired conclusion follows from the fact that each v^{(n)} is a solution to (3.12).

Let us note that the solution constructed satisfies the estimates

𝔼 sup_{t∈[0,T]} |u(t)|^{p} ≤ C(p),

𝔼 (|u(t)|^{2} + ∫_{0}^{t} ||u(s)||_{1}^{2} ds) ≤ C,

with C(p), C independent of the Lipschitz constant L for F and G.

Finally, to get rid of the Lipschitz condition, we approximate continuous F and G by Lipschitz mappings F_{(l)} and G_{(l)} with the same growth constant K. Thus we obtain a sequence of solutions u_{(l)} and using the same argument as in the first step of the proof we construct the required solution. □

4. Proofs of Theorems 1.2 and 1.3

Proof of Theorem 1.2. It is enough to show that the mappings F and G given by (1.4) satisfy the assumptions of Theorem 1.1. Clearly, F maps [0, T]×𝕃^{2} into W^{−1}. It has linear growth, and it is weakly measurable, and hence measurable. Since f and f_{i} are continuous with respect to the third variable, F is weakly continuous with respect to v. Similarly, as g_{j} and g̃_{j} are measurable and continuous with respect to v, we have measurability and weak continuity of G. In order to show that G is Hilbert–Schmidt and its norm satisfies the linear growth estimate from Theorem 1.1 we take two orthonormal bases {e_{k}} and {ẽ_{k}} of (Ker Q^{1/2})^{⊥} and (Ker Q̃^{1/2})^{⊥}, respectively. Then for T ∈ [0, T] and v ∈ 𝕃^{2} we have

∑_{k} ∫_{O} |G(t, v)(Q^{1/2}e_{k}, Q̃^{1/2}ẽ_{k})(x)|^{2} dx

= ∑_{k} ∫_{O} | ∑_{i,j=1}^{m} ∫_{O}{g_{j}(t, x, v(x))Q_{i,j}(x, y)(e_{k}(y))_{i}

+ g̃_{j}(t, x, v(x))Q̃_{i,j}(x, y)(ẽ_{k}(y))_{i}} dy|^{2} dx

≤ C_{1} ∑_{i,j=1}^{m} ∫_{O} ∫_{O}{|g_{j}(t, x, v(x))|^{2}Q_{i,j}^{2}(x, y)

+ |g̃_{j}(t, x, v(x))|^{2}Q̃_{i,j}^{2}(x, y)} dx dy

≤ C_{2}(|v|^{2} + 1) ∑_{i,j=1}^{m}[sup_{x∈O} ∫_{O} Q_{i,j}^{2}(x, y) dy + ∫_{O} ∫_{O} Q̃_{i,j}^{2}(x, y) dx dy].

Hence there is a constant C < ∞ such that for all t ∈ [0, T] and v ∈ 𝕃^{2} one has

||G(t, v)||_{L_{(HS)}(H,𝕃^{2})} ≤ C(1 + |v|),

which is the desired conclusion. □

Let ρ ∈ ℝ. For ψ ∈ H_{μ} and a function v : ℝ^{d} → ℝ^{m} we write

M(v)ψ = ∑_{j=1}^{m} v_{j}ψ_{j}.

For a proof of the lemma below see [32, Lemma 4.1(ii)].

Lemma 4.1. For every ρ ∈ ℝ, M is a bounded linear operator acting from 𝕃_{ρ}^{2} into the space of Hilbert–Schmidt operators L_{(HS)}(H_{μ}, L_{ρ}^{2}).

Proof of Theorem 1.3. Using the arguments from the proof of Theorem 1.2 one can easily show that the mapping F given by (1.7) satisfies the condition from Theorem 1.1. Let

G_{i}(t, v)(x) = (g_{1}^{i}(t, x, v(x)), …, g_{m}^{i}(t, x, v(x))),

where g_{j}^{i} denotes the i^{th}-coordinate of g_{j}. Note that G = (G_{1}, …, G_{d}) is measurable and weakly continuous with respect to v mapping from [0, T] × 𝕃^{2} into the m-fold product 𝕃^{2} × … × 𝕃^{2}. Thus the desired properties of G follows from Lemma 4.1 and the representation

G(t, v)ψ = (M(G_{1}(t, v))ψ, …, M(G_{d}(t, v))ψ).

□

5. Solution of the auxiliary equation

In this section we deal with equation (0.2) with fixed ε > 0. Recall that here O = ℝ^{d}, W is an ℝ^{m}-valued spatially homogeneous Wiener random field with a spectral measure μ, and we identify W with a cylindrical Wiener process W on H_{μ}. Let us fix 1/2 < ρ < 1. We assume that the nonlinear mappings F and G are given by (1.7). Since we will pass to the limit ε → 0, it is sufficient to consider ε ∈ (0, 1], which will be convenient for a technical reason. Let us first explain what we mean by a solution to (0.2).

Definition 5.1. Let U_{0} be a probability measure on 𝕃_{ρ}^{2}. By an 𝕃_{ρ}^{2}-valued martingale solution to (0.2) starting from U_{0} we understand a triple (𝔘, W, u) consisting of a filtered probability space 𝔘 = (Ω, F, (F_{t})_{t∈[0,T]}, ℙ), (F_{t})-adapted cylindrical Wiener process W on H_{μ}, and an adapted stochastic process u such that:

(i) u ∈ L^{p}(Ω, F, ℙ; L^{∞}(0, T; 𝕃_{ρ}^{2})) for every p ∈ [1, ∞), (ii) u ∈ L^{2}(Ω × [0, T], F × B([0, T]), ℙ × dt; W_{ρ}^{1}) ∩ L^{6}(Ω, F, ℙ; L^{6}(0, T; 𝕃_{ρ}^{6})), (iii) u(0) has the law U_{0}, (iv) for all t ∈ [0, T] and ψ ∈ C_{0}^{∞}(ℝ^{d}; ℝ^{d}),

⟨u(t), ψ⟩_{ρ} = ⟨u(0), ψ⟩_{ρ} + ν ∫_{0}^{t}⟨u(s), Δ(ϑ_{ρ}ψ)⟩ ds

+ ∫_{0}^{t} ∑_{i=1}^{d}⟨u(s)u_{i}, ∂_{x_{i}}(ψϑ_{ρ})⟩ ds

+ ∫_{0}^{t}\{−⟨ε|u(s)|^{4}u(s), ψ⟩_{ρ} + \frac{1}{ε}⟨u(s), ∇div (ψϑ_{ρ})⟩\} ds

+ ∫_{0}^{t}⟨F (s, u(s)), ψ⟩_{ρ} ds + ⟨∫_{0}^{t} G(s, u(s)) dW (s), ψ⟩_{ρ}.

Note that if (𝔘, W, u) is a martingale solution to (0.2) then for each ψ from C_{0}^{∞}(ℝ^{d}; ℝ^{d}) the real valued process ⟨u(t), ψ⟩_{ρ}, t ∈ [0, T] has continuous trajectories.

Definition 5.2. Let U_{0} be a probability measure on 𝕃_{ρ}^{2}. We say that equation (0.2) has a unique martingale solution starting from U_{0} if for any two 𝕃_{ρ}^{2}-valued martingale solutions (𝔘, W, u) and (𝔘̃, W̃ , ũ) to (0.2) starting from U_{0}, and for all t ∈ [0, T] and ψ ∈ C_{0}^{∞}(ℝ^{d}; ℝ^{d}) the random variables ⟨u(t), ψ⟩_{ρ} and ⟨ũ(t), ψ⟩_{ρ} have the same distribution.

The main existence result of this section is

Theorem 5.1. Let U_{0} be a probability measure on 𝕃_{ρ}^{2} such that

∫_{𝕃_{ρ}^{2}} |v|_{ρ}^{p} dU_{0}(v) < ∞ for every p ∈ [1, ∞).

Assume that the functions f, f_{i} and g_{j} appearing in (1.7) satisfy the condition from Theorem 1.4. Then there exists a unique 𝕃_{ρ}^{2}-valued martingale solution (𝔘, W, u) to (0.2) starting from U_{0}.

As in the proof of Theorem 1.1 we shall use a modification of the Galerkin method. Let us denote by S^{l} the space of all v ∈ C^{l}(ℝ^{d}; ℝ^{d}) such that

sup_{|α|≤l} sup_{x∈ℝ^{d}} |(1 + |x|^{2})^{n}∂^{α}v(x)| < ∞ for all n ∈ ℕ.

By Theorem C.1 from Appendix C, there exists an orthonormal basis {e_{k}} of 𝕃_{ρ}^{2}, such that {e_{k}} ⊂ S^{3}, and {e_{k}} is a Schauder basis of all spaces 𝕃_{ρ}^{q}, 2 ≤ q ≤ 6, and W_{ρ}^{1}. Let Π_{n} be the orthogonal projection onto the spaces span {e_{1}, …, e_{n}}. Then by Lemma C.1 we have

(5.22)  sup_{n}(||Π_{n}||_{L(W_{ρ}^{1},W_{ρ}^{1})} + ||Π_{n}||_{L(𝕃_{ρ}^{q},𝕃_{ρ}^{q})})< ∞,

for any 2 ≤ q ≤ 6. Thus, see the proof of (3.11) we also have

(5.23)  sup_{n} ||Π_{n}||_{L(W_{ρ}^{−1},W_{ρ}^{−1})} < ∞.

Using the arguments from the proof of Theorem 1.3 one can easily prove the following lemma.

Lemma 5.1. Assume that f, f_{i} and g_{j} satisfy the condition in Theorem 1.4. Then there is a constant K such that for all t ∈ [0, T] and v, u ∈ 𝕃_{ρ}^{2} one has

||F (t, v) − F (t, u)||_{−1,ρ} + ||G(t, v) − G(t, u)||_{L_{(HS)}(H_{μ},𝕃_{ρ}^{2})} ≤ K|v − u|_{ρ},
||F (t, v)||_{−1,ρ} + ||G(t, v)||_{L_{(HS)}(H_{μ},𝕃_{ρ}^{2})} ≤ K(1 + |v|_{ρ}).

Let (Ω, F, (F_{t})_{t∈[0,T]}, ℙ) be a filtered probability space. Assume that W is an (F_{t})-adapted cylindrical Wiener process on H_{μ}, and that u(0) is an F_{0}-measurable random variable in 𝕃_{ρ}^{2} with the law U_{0}. For v ∈ C_{b}^{3}(ℝ^{d}; ℝ^{d}) we write

(5.24)  D(v) = \frac{1}{ε}∇div v and K(v) = −ε|v|^{4}v.

Write B^{(n)}(·) = Π_{n}B(Π_{n}·), K^{(n)}(·) = Π_{n}K(Π_{n}·), D^{(n)}(·) = Π_{n}D(Π_{n}·), F^{(n)} = Π_{n}F, and G^{(n)} = Π_{n}G.

Consider the following stochastic evolution equation

(5.25)  \begin{matrix}du^{(n)} = & (Au^{(n)} + B^{(n)}(u^{(n)}) + K^{(n)}(u^{(n)}) + D^{(n)}(u^{(n)}) \\ + F^{(n)}(t, u^{(n)})) dt + G^{(n)}(t, u^{(n)}) dW, \\ u^{(n)}(0) = & Π_{n}u(0). \end{matrix}

Lemma 5.2. For every n ∈ ℕ there exists a strong solution u^{(n)} to (5.25). Moreover, for any p ≥ 2,

(5.26)  sup_{n} 𝔼 sup_{t∈[0,T]} |u^{(n)}(t)|_{ρ}^{p} < ∞

and

(5.27)

𝔼(|u^{(n)}(t)|_{ρ}^{2} + ∫_{0}^{t} ||u^{(n)}(s)||_{1,ρ}^{2} ds + \frac{1}{ε} ∫_{0}^{t} |div Π_{n}u^{(n)}(s)|_{ρ}^{2} ds

+ ε ∫_{0}^{t} |Π_{n}u^{(n)}(s)|_{𝕃_{ρ}^{6}}^{6} ds) ≤ \frac{C}{ε^{2}} exp(\frac{C}{ε} ),

the constant C being independent of n, ε and t ∈ [0, T].

Proof. Using the arguments from the proof of Lemma 3.1 we can obtain the existence of a local strong solution u^{(n)}. Thus it is enough to show (5.27). In the proof of the theorem we adopt the convention that all constants c_{1}, c_{2}, … are independent of n and ε. Applying Itô’s formula to the function H(v) = |v|_{ρ}^{2} we obtain

(5.28)  \begin{aligned}|u^{(n)}(t)|_{ρ}^{2} & = |u^{(n)}(0)|_{ρ}^{2} + ∫_{0}^{t} Φ(s, u^{(n)}(s)) dt \\ & + ∫_{0}^{t} 2⟨u^{(n)}(s), G^{(n)}(s, u^{(n)}(s)) dW (s)⟩_{ρ}. \end{aligned}

where

Φ(s, u^{(n)}) = ||G^{(n)}(t, u^{(n)})||_{L_{(HS)}(H_{μ},𝕃_{ρ}^{2})}^{2}

+ ⟨(Au^{(n)} + B^{(n)}(u^{(n)}) + D^{(n)}(u^{(n)}) + K^{(n)}(u^{(n)}) + F^{(n)}(s, u^{(n)})), u^{(n)}⟩_{ρ}.

Now, let v ∈ Dom A. Then

⟨Av + D^{(n)}(v), v⟩_{ρ} = ⟨νΔv + \frac{1}{ε}Π_{n}∇div Π_{n}v, v⟩_{ρ}

≤ −ν||v||_{1,ρ}^{2} + c_{1}|v|_{ρ}||v||_{1,ρ} − \frac{1}{ε}|div Π_{n}v|_{ρ}^{2} + \frac{c_{2}}{ε} |div Π_{n}v|_{ρ}|Π_{n}v|_{ρ}.

Clearly,

−ν||v||_{1,ρ}^{2} + c_{1}|v|_{ρ}||v||_{1,ρ} ≤ −\frac{ν}{2} ||v||_{1,ρ}^{2} + c_{3}|v|_{ρ}^{2},

and

−\frac{1}{ε}|div Π_{n}v|_{ρ}^{2} + c_{2} \frac{1}{ε}|div Π_{n}v|_{ρ}|Π_{n}v|_{ρ} ≤ −\frac{1}{2ε}|div Π_{n}v|_{ρ}^{2} + \frac{c_{4}}{ε} |v|_{ρ}^{2}.

Hence

⟨Av + D^{(n)}(v), v⟩_{ρ} ≤ −\frac{ν}{2} ||v||_{1,ρ}^{2} − \frac{1}{2ε}|div Π_{n}v|_{ρ}^{2} + c_{5}|v|_{ρ}^{2}

By Proposition 2.2(i) we have

⟨B^{(n)}(v) + K^{(n)}(v), v⟩_{ρ} ≤ c_{6}|Π_{n}v|_{𝕃_{ρ}^{4}}^{2}||Π_{n}v||_{1,ρ} − ε|Π_{n}v|_{𝕃_{ρ}^{6}}^{6}

≤ c_{7}|Π_{n}v|_{𝕃_{ρ}^{6}}^{4} + \frac{ν}{8} ||v||_{1,ρ}^{2} − ε|Π_{n}v|_{𝕃_{ρ}^{6}}^{6}.

Using now the elementary inequality

c_{7}r^{4} − \frac{ε}{2}r^{6} ≤ \frac{c_{8}}{ε^{2}} , r ∈ ℝ,

we get

⟨B^{(n)}(v) + K^{(n)}(v), v⟩_{ρ} ≤ \frac{c_{8}}{ε^{2}} + \frac{ν}{8} ||v||_{1,ρ}^{2} − \frac{ε}{2}|Π_{n}v|_{𝕃_{ρ}^{6}}^{6}.

From Lemma 5.1 and (5.22) we have

|⟨F^{(n)}(t, v), v⟩_{ρ}| + ||(G^{(n)}(t, v))||_{L_{(HS)}(H_{μ},𝕃_{ρ}^{2})}^{2} ≤ c_{8}(|v|_{ρ}^{2} + ||v||_{1,ρ}|v|_{ρ} + 1)

≤ c_{9}|v|_{ρ}^{2} + \frac{ν}{8} ||v||_{1,ρ}^{2} + c_{9}.

Adding up, we obtain

Φ(t, v) ≤ −\frac{ν}{4} ||v||_{1,ρ}^{2} − \frac{1}{ε}|div Π_{n}v|_{ρ}^{2} − \frac{ε}{2}|Π_{n}v|_{𝕃_{ρ}^{6}}^{6} + \frac{c_{10}}{ε} |v|_{ρ}^{2} + \frac{c_{10}}{ε^{2}} .

Thus taking the mathematical expectation in (5.28) we get

𝔼 (|u^{(n)}(t)|_{ρ}^{2} + \frac{ν}{4} ∫_{0}^{t} ||u^{(n)}(s)||_{1,ρ}^{2} ds + \frac{1}{ε} ∫_{0}^{t} |div Π_{n}u^{(n)}(s)|_{ρ}^{2} ds

+ \frac{ε}{2} ∫_{0}^{t} |Π_{n}u^{(n)}(s)|_{𝕃_{ρ}^{6}}^{6} ds) ≤ 𝔼 |u(0)|_{ρ}^{2} + \frac{c_{10}}{ε} ∫_{0}^{t} 𝔼 |u^{(n)}(s)|_{ρ}^{2} ds + \frac{c_{10}}{ε^{2}} t.

Applying Gronwall’s lemma we obtain (5.27).

Proceeding as above, but using Itô’s formula to the function |v|_{ρ}^{p} for p ≥ 2 one can obtain (5.26), see the proof of (3.13). □

Proof of Theorem 5.1 - existence. The proof is based on the same idea as the proof of Theorem 1.1. For each n let u^{(n)} be the solution to (5.24). The first step is to show that the sequence L(u^{(n)}), n ∈ ℕ, of the laws of u^{(n)} is tight in L^{2}(0, T; H_{κ}) for a fixed κ > 5ρ/3. Write

y_{n}(t) = S(t)u^{(n)}(0) + ∫_{0}^{t} S(t − s){B^{(n)}(u^{(n)}(s)) + F^{(n)}(s, u^{(n)}(s))} ds
+ ∫_{0}^{t} S(t − s)G^{(n)}(s, u^{(n)}(s)) dW (s),
z_{n}(t) = ∫_{0}^{t} S(t − s){D^{(n)}(u^{(n)}(s)) + K^{(n)}(s, u^{(n)}(s))} ds
= J_{0}^{ρ}(D^{(n)}(u^{(n)}))(t) + J_{0}^{ρ}(K^{(n)}(u^{(n)}))(t).

Then u^{(n)} = y_{n} + z_{n}. Using the arguments from the proof of Theorem 1.1 we can show the tightness of the laws L(y_{n}), n ∈ ℕ, and it remains to prove the tightness of L(z_{n}), n ∈ ℕ. Recall, see Proposition 2.1 (i) and (iii), that J_{0}^{ρ} is a compact operator acting from L^{2}(0, T; W_{ρ}^{−1}) into L^{2}(0, T; 𝕃_{κ}^{2}), and from L^{6/5}(0, T; 𝕃_{ρ}^{6/5}) into L^{2}(0, T; 𝕃_{κ}^{2}). Thus it is enough to show that

(5.29)  sup_{n} 𝔼 ∫_{0}^{T} ||D^{(n)}(u^{(n)}(t))||_{−1,ρ}^{2} dt < ∞,

(5.30)  sup_{n} 𝔼 ∫_{0}^{T} |K^{(n)}(u^{(n)}(t))|𝕃 \frac{6}{5} ; ρ^{\frac{6}{5}} dt < ∞.

From (5.22) and (5.23) there is a constant c such that for every n,

||D^{(n)}(u^{(n)}(t))||_{−1,ρ} = \frac{1}{ε}||Π_{m}∇div Π_{n}u^{(n)}(t)||_{−1,ρ} ≤ \frac{c}{ε}||u^{(n)}||_{1,ρ},

and (5.29) follows from (5.27). Similarly,

|K^{(n)}(u^{(n)}(t))|𝕃 \frac{6}{5} ; ρ^{\frac{6}{5}} = ε^{\frac{6}{5}} |Π_{n}\{|Π_{n}u^{(n)}(t)|^{4}Π_{n}u^{(n)}(t)\}|𝕃 \frac{6}{5} ; ρ^{\frac{6}{5}} ≤ cε^{\frac{6}{5}} |Π_{n}u^{(n)}(t)|_{𝕃_{ρ}^{6}}^{6},

and (5.30) follows from (5.27). Having the tightness of {L(u^{(n)})} in L^{2}(0, T; 𝕃_{κ}^{2}), and using the Skorokhod imbedding and the arguments from the proof of Theorem 1.1 we may construct a filtered probability space (Ω^{1}, F^{1}, (F_{t}^{1}), ℙ^{1}), and a sequence (v^{(n)}, W^{(n)}) such that for any n, W^{(n)} is cylindrical Wiener process on H_{μ} defined on Ω^{1}, v^{(n)} is a solution to (5.24) driven by W^{(n)}, and v^{(n)}, n ∈ ℕ, converges ℙ^{1} × dt × dx - almost surely to a certain u. Then u is adapted, and u(0) has the law U_{0}. Using now Fatou’s lemma we obtain

u ∈ L^{p}(Ω^{1}; F^{1}, ℙ^{1}; L^{∞}(0, T; 𝕃_{ρ}^{2})) for p ∈ [1, ∞).

The uniform integrability of |v^{(n)} − u|_{L^{q}(0,T;𝕃_{ρ}^{2})}^{q}, n ∈ ℕ, gives that

(5.31)  lim_{n→∞} 𝔼^{1} ∫_{0}^{T} |v^{(n)}(t) − u(t)|_{𝕃_{ρ}^{2}}^{q} dt = 0 for q ∈ [2, ∞).

Note that

lim_{n→∞} 𝔼^{1} ∫_{0}^{T} |Π_{n}v^{(n)}(t) − u(t)|_{ρ}^{2} dt = 0.

Thus taking if necessary a subsequence we may assume that

Π_{n}v^{(n)} → u, ℙ^{1} × dt × dx − almost surely.

Consequently, (5.27) and Fatou’s lemma yield

u ∈ L^{6}(Ω^{1}, ℙ^{1}, F^{1}; L^{6}(0, T; 𝕃_{ρ}^{6})).

Moreover, as for any 2 ≤ q < 6 the sequence |Π_{n}v^{(n)} − u|_{𝕃_{ρ}^{q}}^{q}, n ∈ ℕ, is uniformly integrable we have

(5.32)  lim_{n→∞} 𝔼^{1} ∫_{0}^{T} |Π_{n}v^{(n)}(t) − u(t)|_{𝕃_{ρ}^{q}}^{q} dt = 0 for 2 ≤ q < 6.

Let us observe now that the martingales

M_{n}(t) = ∫_{0}^{t} G^{(n)}(s, v^{(n)}(s)) dW^{(n)}(s), t ∈ [0, T], n ∈ ℕ,

converge in L^{2}(Ω^{1}, F^{1}, ℙ^{1}; C([0, T]; 𝕃_{ρ}^{2})) to a martingale M satisfying

≪ M ≫ (t) = ∫_{0}^{t} G(s, u(s))G(s, u(s))^{∗} ds, t ∈ [0, T].

Taking into account the representation theorem we may assume that M is of the form

M(t) = ∫_{0}^{t} G(s, u(s)) dW(s), t ∈ [0, T],

where W is a cylindrical Wiener process on H_{μ}. Let us now fix ψ from C_{0}^{∞}(ℝ^{d}; ℝ^{d}). It remains to note that (5.31) and (5.32) guarantee the convergence of each term appearing in the weak form of (5.24) to the corresponding term in (iv) from Definition 5.1. □

Proof of Theorem 5.1 - uniqueness. We prove pathwise uniqueness, which by [24, Section 4.3] implies uniqueness in law. Take two solutions u^{ε} and v^{ε} with the same initial value and write w^{ε} = u^{ε} − v^{ε}. As in the proof of the existence we fix an orthonormal basis {e_{k}} of 𝕃_{ρ}^{2}, such that {e_{k}} ⊂ S^{3} and {e_{k}} is a Schauder basis of all spaces 𝕃_{ρ}^{q}, 2 ≤ q ≤ 6, and W_{ρ}^{1}. Let Π_{n} be the orthogonal projection onto the space spanned on {e_{1}, …, e_{n}}. Using Itô’s formula and integrating by parts we get

\frac{1}{2}|Π_{n}w^{ε}(t)|_{ρ}^{2} = ν ∫_{0}^{t}⟨Δ(ϑ_{ρ}Π_{n}w^{ε}(s)), w^{ε}(s)⟩ ds

+ \frac{1}{ε} ∫_{0}^{t}⟨∇div (ϑ_{ρ}Π_{n}w^{ε}(s)), w^{ε}(s)⟩ ds

+ ∫_{0}^{t} ∑_{i=1}^{d}⟨∂_{x_{i}}(ϑ_{ρ}Π_{n}w^{ε}(s)), u_{i}^{ε}(s)u^{ε}(s) − v_{i}^{ε}(s)v^{ε}(s)⟩ ds

− ε ∫_{0}^{t}⟨Π_{n}w^{ε}(s), |u^{ε}(s)|^{4}u^{ε}(s) − |v^{ε}(s)|^{4}v^{ε}(s)⟩_{ρ} ds

+ ∫_{0}^{t}⟨Π_{n}w^{ε}(s), F (s, u^{ε}(s)) − F (s, v^{ε}(s)⟩_{ρ} ds

+ \frac{1}{2} ∫_{0}^{t} ||Π_{n}(G(s, u^{ε}(s)) − G(s, v^{ε}(s)))||_{L_{(HS)}(H_{μ},𝕃_{ρ}^{2})}^{2} ds

+ ∫_{0}^{t}⟨Π_{n}w^{ε}(s), (G(s, u^{ε}(s)) − G(s, v^{ε}(s))) dW (s)⟩_{ρ}.

We can again integrate by parts, and due to the regularity properties of solutions and since we deal with a basis which is a Schauder basis in all relevant spaces we may pass to the limit with n → ∞. Then the first term on the right hand side of the resulting equation can be estimated by

−\frac{ν}{2} ∫_{0}^{t} ||w^{ε}(s)||_{1,ρ}^{2} ds + C_{1} ∫_{0}^{t} |w^{ε}(s)|_{ρ}^{2} ds.

The second term is equal to

−\frac{1}{ε} ∫_{0}^{t} ∫_{ℝ^{d}} div (ϑ_{ρ}(x)w^{ε}(s, x))div w^{ε}(s, x) dx ds

and hence can be estimated by C_{2}ε^{−1} ∫_{0}^{t} |w^{ε}(s)|_{ρ}^{2} ds. We estimate the third term by

−\frac{ν}{4} ∫_{0}^{t} ||w^{ε}(s)||_{1,ρ}^{2} ds + C_{3} ∫_{0}^{t} ∫_{ℝ^{d}} |w^{ε}(s, x)|^{2}(|u^{ε}(s, x)|^{2} + |v^{ε}(s, x)|^{2})ϑ_{ρ}(x) dx ds

≤ −\frac{ν}{4} ∫_{0}^{t} ||w^{ε}(s)||_{1,ρ}^{2} ds + \frac{C_{4}}{ε} ∫_{0}^{t} |w^{ε}(s)|_{ρ}^{2} ds

+ \frac{ε}{2} ∫_{0}^{t} ∫_{ℝ^{d}} |w^{ε}(s, x)|^{2}(|u^{ε}(s, x)|^{4} + |v^{ε}(s, x)|^{4})ϑ_{ρ}(x) dx ds.

The fourth term is equal to

− ε ∫_{0}^{t} ∫_{ℝ^{d}}{|u^{ε}(s, x)|^{6} + |v^{ε}(s, x)|^{6}

− (|u^{ε}(s, x)|^{4} + |v^{ε}(s, x)|^{4})⟨u^{ε}(s, x), v^{ε}(s, x)⟩ϑ_{ρ}(x)} dx ds

and due to the inequality a^{6} + b^{6} ≥ 2^{−1}(a^{4} + b^{4})(a^{2} + b^{2}) this term is less or equal to

−\frac{ε}{2} ∫_{0}^{t} ∫_{ℝ^{d}}(|u^{ε}(s, x)|^{4} + |v^{ε}(s, x)|^{4})|w^{ε}(s, x)|^{2}ϑ_{ρ}(x) dx ds.

So the sum of the first four terms is less or equal to

C(ε) ∫_{0}^{t} |w^{ε}(s)|_{ρ}^{2} ds − \frac{ν}{4} ∫_{0}^{t} ||w^{ε}(s)||_{1,ρ}^{2} ds,

C(ε) being independent of t. For the next two terms we make use of the Lipschitz condition on F and G, and we easily estimate them by

C_{5} ∫_{0}^{t} |w^{ε}(s)|_{ρ}^{2} ds + \frac{ν}{4} ∫_{0}^{t} ||w^{ε}(s)||_{1,ρ}^{2} ds.

Therefore, putting the above together and taking the mathematical expectation we get

𝔼 |w^{ε}(t)|_{ρ}^{2} ≤ (C(ε) + C_{5}) ∫_{0}^{t} 𝔼 |w^{ε}(s)|_{ρ}^{2} ds,

and consequently by Gronwall’s lemma, w^{ε} = 0 a.s., which is the desired conclusion. □

As a consequence of the uniqueness of the solution to (0.2) and the fact that the coefficients of (0.2) do not depend on the space variable we have

Proposition 5.1. Assume that the conditions of Theorem 5.1 are fulfilled. Additionally, we assume that the initial measure U_{0} is spatially homogeneous. Then the solution is a spatially homogeneous process.

Proof. Let U_{0} be a spatially homogeneous measure on 𝕃_{ρ}^{2}, and let (𝔘, W, u^{ε}) be a solution to (0.2) starting from U_{0}. Since we have the uniqueness in law, it is enough to observe that for any h ∈ ℝ^{d}, (𝔘, τ_{h}W, τ_{h}u^{ε}) is a solution to (0.2) starting from U_{0} ∘ τ_{h} = U_{0}. Clearly, τ_{h}W(t), t ∈ [0, T], is an ℝ^{m}-valued spatially homogeneous Wiener random field with the spectral measure μ. Thus τ_{h}W = τ_{h}W is a cylindrical Wiener process on H_{μ}. Since F and G do not depend on x-variable, one has τ_{h}F (s, u^{ε}(s)) = F (s, τ_{h}u^{ε}(s)), and

τ_{h} ∫_{0}^{t} G(s, u^{ε}(s)) dW (s) = ∫_{0}^{t} G(s, τ_{h}u^{ε}(s)) d(τ_{h}W)(s).

From this, and from the fact that ⟨τ_{h}u^{ε}(t), ψ⟩_{ρ} = ⟨u^{ε}(t), τ_{−h}(ϑ_{ρ}ψ)⟩ one can easily deduce the desired conclusion. □

6. Limit passage

The limit passage requires some estimates, uniform with respect to ε, for the solution u^{ε} to (0.2), see Lemma 6.2. In their proof we will make use the fact that for any t the law of u^{ε}(t) is a spatially homogeneous probability measure. For further references we present the following lemma, whose proof is given in [38], see Théoréme 2.1, p. 537.

Lemma 6.1. Let z ∈ C(ℝ^{(d+1)d}; ℝ), ρ > 1/2, and let U be a spatially homogeneous probability measure on W_{ρ}^{1}. Let

Z(u)(x) = z(u(x), ∇u) for u ∈ W_{ρ}^{1}, x ∈ ℝ^{d}.

Suppose that Z(u) ∈ L_{loc}^{1}(ℝ^{d}) for every u ∈ W_{ρ}^{1}, and assume that the mappings

u ↦ ∫_{ℝ^{d}} Z(u(x))ψ(x) dx, u ↦ ∫_{ℝ^{d}} |Z(u(x))|ψ(x) dx

are U-integrable for any ψ ∈ C_{0}^{∞}(ℝ^{d}). Then there is a constant C such that for any ψ ∈ L^{1}(ℝ^{d}) one has

∫_{W_{ρ}^{1}} ∫_{ℝ^{d}} Z(u(x))ψ(x) dx dU(u) = C ∫_{ℝ^{d}} ψ(x) dx.

Lemma 6.2. Assume that the conditions of Theorem 5.1 are fulfilled. Additionally, we assume that the initial measure U_{0} is spatially homogeneous. Let ε ∈ (0, 1], and let u^{ε} be a solution to (0.2) starting from U_{0}. Then there is a constant C independent of ε such that for any p ≥ 2,

(6.33)  𝔼 sup_{t∈[0,T]} |u^{ε}(t)|_{ρ}^{p} ≤ C

and

(6.34) 𝔼(|u^{ε}(t)|_{ρ}^{2} + ∫_{0}^{t} ||u^{ε}(s)||_{1,ρ}^{2} ds + \frac{1}{ε} ∫_{0}^{t} |div u^{ε}|_{ρ}^{2} ds + ε ∫_{0}^{t} |u^{ε}|_{𝕃_{ρ}^{6}}^{6} ds) ≤ C.

Proof. Using Itô’s lemma and limit arguments, see the proof of the uniqueness, we get

\frac{1}{2}|u^{ε}(t)|_{ρ}^{2} = \frac{1}{2}|u^{ε}(0)|_{ρ}^{2} − ν ∑_{i=1}^{d} ∫_{0}^{t}⟨∂_{x_{i}}(ϑ_{ρ}u^{ε}(s)), ∂_{x_{i}}u^{ε}(s)⟩ ds

− \frac{1}{ε} ∫_{0}^{t}⟨div (ϑ_{ρ}u^{ε}(s)), div u^{ε}(s)⟩ ds

− ∫_{0}^{t}{∑_{i=1}^{d}⟨u^{ε}(s), ∂_{x_{i}}(u_{i}^{ε}(s)u^{ε}(s))⟩_{ρ} − ε|u^{ε}(s)|_{𝕃_{ρ}^{6}}^{6}} ds

+ ∫_{0}^{t}\{⟨u^{ε}(s), F (s, u^{ε}(s))⟩_{ρ} + \frac{1}{2}||G(s, u^{ε}(s))||_{L_{(HS)}(H_{μ},𝕃_{ρ}^{2})}^{2}\} ds

+ ∫_{0}^{t}⟨u^{ε}(s), G(s, u^{ε}(s)) dW (s)⟩_{ρ}.

The first integral on the right hand side is estimated by

−\frac{ν}{2} ∫_{0}^{t} ||u^{ε}(s)||_{1,ρ}^{2} ds + C_{1} ∫_{0}^{t} |u^{ε}(s)|_{ρ}^{2} ds.

The second is equal to

−\frac{1}{ε} ∫_{0}^{t} ∫_{ℝ^{d}}\{|div u^{ε}(s, x)|^{2}ϑ_{ρ}(x) + u^{ε}(s, x)div u^{ε}(s, x) ∑_{i=1}^{d} ∂_{x_{i}}ϑ_{ρ}(x)\} dx ds.

Integrating by parts we can show that the first term of the third integral is equal to

\frac{1}{2} ∫_{0}^{t} ∫_{ℝ^{d}}\{div u^{ε}(s, x)|u^{ε}(s, x)|^{2}ϑ_{ρ}(x) − |u^{ε}(s, x)|^{2} ∑_{i=1}^{d} u_{i}^{ε}(s, x)∂_{x_{i}}ϑ_{ρ}(x)\} dx ds,

and next we use the Schwartz and Young inequalities to get

|∫_{ℝ^{d}} |u^{ε}(s, x)|^{2}div u^{ε}(s, x)ϑ_{ρ}(x) dx| ≤ \frac{1}{2ε}|div u^{ε}(s)|_{ρ}^{2} + \frac{ε}{2}|u^{ε}(s)|_{L_{ρ}^{6}}^{6} + C_{2}|u^{ε}(s)|_{ρ}^{2}.

By the linear growth of F and G we estimate the forth integral by

C_{3} + C_{4} ∫_{0}^{t} |u^{ε}(s)|_{ρ}^{2} ds + \frac{ν}{4} ∫_{0}^{t} ||u^{ε}(s)||_{1,ρ}^{2} ds.

Thus taking the mathematical expectation we have

(6.35)  𝔼(|u^{ε}(t)|_{ρ}^{2} + ∫_{0}^{t}\{\frac{ν}{4} ||u^{ε}(s)||_{1,ρ}^{2} + \frac{1}{2ε}|div u^{ε}(s)|_{ρ}^{2} + \frac{ε}{2}|u^{ε}(s)|_{ρ}^{6}\} ds)

≤ 𝔼 |u^{ε}(0)|_{ρ}^{2} + C_{5} + C_{6} 𝔼 ∫_{0}^{t} |u^{ε}(s)|_{ρ}^{2} ds

− 𝔼 ∫_{0}^{t} ∫_{ℝ^{d}} ∑_{i=1}^{d}\{\frac{1}{2}|u^{ε}(s, x)|^{2}u_{i}^{ε}(s, x) + \frac{1}{ε}div u^{ε}(s, x)u_{i}^{ε}(s, x)\}∂_{x_{i}}ϑ_{ρ}(x) dx ds.

By Proposition 5.1, the laws L(u^{ε}(s)) of u^{ε}(s), s ≥ 0, are spatially homogeneous. Taking into account the regularity of u^{ε} we see that each L(u^{ε}(s)) is supported on W_{ρ}^{1} ∩ 𝕃_{ρ}^{6}. Hence, by Lemma 6.1 we have

𝔼 ∫_{ℝ^{d}} ∑_{i=1}^{d}\{\frac{1}{2}|u^{ε}(s, x)|^{2}u_{i}^{ε}(s, x) + \frac{1}{ε}div u^{ε}(s, x)u_{i}^{ε}(s, x)\}∂_{x_{i}}ϑ_{ρ}(x) dx

= C ∫_{ℝ^{d}} ∑_{i=1}^{d} ∂_{x_{i}}ϑ_{ρ}(x) dx = 0,

and (6.35) and Gronwall’s lemma give (6.34). Applying Itô’s formula to the function u ↦ |Π_{n}u|_{ρ}^{p}, and then letting n → ∞ one can prove (6.33). Almost all calculations in the proofs of (6.33) and (6.34) are the same. However in the case of (6.33) one should apply a maximal inequality for the stochastic term, as in the proofs of (3.13) and (5.26). □

Proof of Theorem 1.4. Let κ > 5ρ/3, α ∈ (1/2, 1), and let u^{ε} be a solution to (0.2). The estimates in Lemma 6.1 enables us to show the tightness of L(u^{ε}), ε ∈ (0, 1] in the space L^{2}(0, T; 𝕃_{κ}^{2}). For, we have the equivalence between mild and weak solutions to stochastic evolution equations, see [17]. Thus u^{ε} is also a mild solution to (0.2), and

u^{ε}(t) = S(t)u(0) + J_{0}^{ρ}(B(u^{ε}) + F (·, u^{ε}))(t) + J_{0}^{ρ}(D^{ε}(u^{ε}) + K^{ε}(u^{ε}))(t)
+ J_{α}^{ρ}(Y_{ε}^{α})(t),

where: D^{ε}(u^{ε}) = ε^{−1}∇div u^{ε}, K^{ε}(u^{ε}) = −ε|u^{ε}|^{4}u^{ε}, and J_{0}^{ρ} and J_{α}^{ρ} are defined in Section 2, and

Y_{ε}^{α}(t) = \frac{sin(α − 1)π}{π} ∫_{0}^{t}(t − s)^{α−1}S(t − s)G(s, u^{ε}(s)) dW (s).

Using estimates (6.33) and (6.34) and arguments from the proof of Theorem 1.1 one can show the tightness of J_{0}^{ρ}(B(u^{ε}) + F (·, u^{ε})), and J_{α}^{ρ}(Y_{ε}^{α}), ε ∈ (0, 1]. Using the same estimates and arguments from the proof of Theorem 5.1, see (5.29) and (5.30) one can show the tightness of the remaining sequence J_{0}^{ρ}(D^{ε}(u^{ε}) + K^{ε}(u^{ε})), ε ∈ (0, 1]. Thus by Prokhorov’s and Skorokhod’s theorems there exist a filtered probability space (Ω^{1}, F^{1}, (F_{t}^{1}), ℙ^{1}), and sequences ε_{n} ↓ 0, (v^{(n)}, W^{(n)}), such that every v^{(n)} is a solution to (0.2) with ε = ε_{n}, driven by a cylindrical process W^{(n)}. Moreover, there is a process u such that

v^{(n)} → u, ℙ^{1} × dt × dx - almost surely

and

(6.36)  lim_{n→∞} ∫_{0}^{T} |v^{(n)}(t) − u(t)|_{ρ}^{p} dt = 0 for p ∈ [2, ∞).

We have to show now that u is a divergence free process. This is a consequence of that fact that the sequence

\frac{1}{ε} 𝔼^{1} ∫_{0}^{T} |div v^{(n)}(t)|_{ρ}^{2} dt, n ∈ ℕ,

is bounded, see (6.34). For, we first note that the divergence operator acts from 𝕃_{ρ}^{2} into W_{ρ}^{−1}. It is easy to see that H_{ρ} is its kernel, and that H_{ρ}^{1} = W_{ρ}^{1} ∩ H. Hence it is enough to observe that for every ψ ∈ C_{0}^{∞}(ℝ^{d}; ℝ^{d}),

𝔼^{1} ∫_{0}^{T} ⟨div u(t), ψ⟩_{ρ}^{2} dt = 𝔼^{1} ∫_{0}^{T} ⟨u(t), ∇(ψϑ_{ρ})⟩^{2} dt

≤ lim sup_{n→∞} 𝔼^{1} ∫_{0}^{T} ⟨v^{(n)}(t), ∇(ψϑ_{ρ})⟩^{2} dt

≤ lim sup_{n→∞} ε_{n} sup_{m} \frac{1}{ε_{m}} 𝔼^{1} ∫_{0}^{T} ⟨v^{(m)}(t), ∇(ψϑ_{ρ})⟩^{2} dt = 0.

Thus u satisfies the desired regularity properties from Definition 1.2. Clearly u(0) has the law U_{0}. Using now the representation theorem we may assume that

∫_{0}^{t} G^{(n)}(s, v^{(n)}(s)) dW^{(n)}(s) → ∫_{0}^{t} G(s, u(s)) dW(s),

where W is a cylindrical Wiener process on H_{μ}, and the convergence is in the space L^{2}(Ω^{1}, F^{1}, ℙ^{1}; C([0, T]; H_{ρ})).

It remains to show that the integral equation from Definition 1.2(v) is fulfilled for any ψ ∈ C_{0}^{∞}(ℝ^{d}; ℝ^{d}) such that div (ψϑ_{ρ}) = 0. We have to show the convergence of seven terms. Five of them are very similar to those listed at the and of the proof of Theorem 1.1. The differences are that now we have no projections, and that the scalar product on H is replaced by the scalar product on 𝕃_{ρ}^{2}. Obviously, (6.36) guarantees the convergence of these terms in L^{2}(Ω^{1}, F^{1}, ℙ^{1}). What is left is to show that for the two remaining terms we have:

(6.37)  lim_{n→∞} \frac{1}{ε_{n}} 𝔼^{1} ∫_{0}^{T} ⟨∇divv^{(n)}(t), ψ⟩_{ρ} dt = 0

and

(6.38)  lim_{n→∞} ε_{n} 𝔼^{1} ∫_{0}^{T} ⟨|v^{(n)}(t)|^{4}v^{(n)}(t), ψ⟩_{ρ} dt = 0.

Since div (ψϑ_{ρ}) = 0 we have ⟨∇divv^{(n)}(t), ψ⟩_{ρ} ≡ 0, and consequently (6.37). Now we prove (6.38). Since ψ is bounded it is enough to show that

lim_{n→∞} ε_{n} 𝔼^{1} ∫_{0}^{T} |v^{(n)}(t)|_{𝕃_{ρ}^{5}}^{5} dt = 0.

To this end for p ∈ [2, ∞) set

L_{p} = L^{p}(Ω^{1} × [0, T] × ℝ^{d}, F^{1} × B([0, T]) × B(ℝ^{d}), ℙ^{1} × dt × ϑ_{ρ}(x) dx; ℝ^{d}).

We have to show that εn^{\frac{1}{5}} |v^{(n)}|_{L_{5}} → 0 as n → ∞. By (6.34), the sequence εn^{\frac{1}{6}} |v^{(n)}|_{L_{5}}, n ∈ ℕ is bounded. Since L_{p} is an L^{p}-space with respect to a finite measure, there is a constant c such that

lim sup_{n→∞} εn^{\frac{1}{5}} |v^{(n)}|_{L_{5}} ≤ c lim sup_{n→∞} εn^{\frac{1}{5}} |v^{(n)}|_{L_{6}} ≤ c lim sup_{n→∞} ε_{n}^{\frac{1}{5} − \frac{1}{6}} εn^{\frac{1}{6}} |v^{(n)}|_{L_{6}}

≤ c sup_{n} εn^{\frac{1}{6}} |v^{(n)}|_{L_{6}} lim sup_{n→∞} ε_{n}^{\frac{1}{5} − \frac{1}{6}} = 0,

which completes the proof. □

A. Appendix

We precede the proof of Proposition 2.1 by a sequence of lemmas. Recall, see Section 2, that the operator i_{ρ} given by i_{ρ}v = ϑ_{−ρ/2}v is an isometry between 𝕃^{q} and 𝕃_{qρ/2}^{q}. In fact we have something more.

Lemma A.1. Let ρ ≥ 0. Then for any r ∈ ℝ the mapping i_{ρ} is an isomorphism between W^{r} = W_{0}^{r} and W_{ρ}^{r}.

Proof. First let r = 2 and introduce the semigroup S̃(t) = i_{ρ}^{−1}S(t)i_{ρ} on 𝕃^{2}, where S is considered on L_{ρ}^{2}. Clearly, the generator of S̃ is equal to i_{ρ}^{−1}Ai_{ρ}. On the other hand, it is a uniformly elliptic operator with coefficients from C_{b}^{∞}(O). Therefore, i_{ρ}^{−1}(W_{ρ}^{2}) = W^{2}. Considering the powers of A we can show in the same way that i_{ρ} is an isomorphism between W^{r} and W_{ρ}^{r} for even r. Finally, using standard interpolation argument we may extend the result to an arbitrary r. □

Further we will need the following result on Sobolev imbeddings.

Lemma A.2. (i) For all 0 ≤ κ < ρ and r < s the imbedding W_{κ}^{s} ↪ W_{ρ}^{r} is compact. (ii) Let ρ ≥ 0, q ∈ [2, ∞), and let r ∈ [0, \frac{d}{2}) be such that \frac{d}{2} − r ≤ \frac{d}{q}. Then W_{ρ}^{r} is continuously imbedded into 𝕃_{qρ/2}^{q}.

Proof of (i). Drawing on Lemma A.1 we may assume that κ = 0. Take a sequence {v_{n}} bounded in W^{s}. Employing Sobolev’s compact imbedding theorem for bounded domains we can extract from {(−νΔ + I)^{s/2}v_{n}} a subsequence convergent almost everywhere. Since each sequence bounded in 𝕃^{2} is uniformly integrable in 𝕃_{ρ}^{2} the proof is complete. □

Proof of (ii). It is well known, see [1, Lemma 5.14], that W^{r} is continuously imbedded into 𝕃^{q}. The required result follows now from Lemma A.1. □

The lemma below is a version of the result formulated and proved in [16] as Lemma 2, see also [18, Proposition A.1.1].

Lemma A.3. Let ρ ∈ ℝ, and let α ∈ [0, 1) and p ≥ 1 be such that α + p^{−1} < 1. Then for each γ ∈ [0, 1−α−p^{−1}), J_{α}^{ρ} is a bounded linear operator from L^{p}(0, T; 𝕃_{ρ}^{2}) into C^{γ}([0, T]; 𝕃_{ρ}^{2}).

For the convenience of the reader we recall Dubinsky’s criterion for compactness. Its proof can be found in [38, Theorem 4.1, p. 132].

Lemma A.4. Let E_{0}, E_{1} and E be reflexive Banach spaces such that the imbeddings E_{0} ↪ E ↪ E_{1} are continuous and the imbedding E_{0} ↪ E is compact. Let p ∈ (1, ∞) and let M be a bounded set in L^{p}(0, T; E_{0}) consisting of equicontinous functions in C([0, T]; E_{1}). Then M is relatively compact in L^{p}(0, T; E) and C([0, T]; E_{1}).

Proof of Proposition 2.1.

Proof of (i). Clearly we may assume that 1 < p < 2. Let r > 0 be such that \frac{1}{p}(r + 1) < 1. Then since the semigroup S is analytic on 𝕃_{ρ}^{2}, there exists a constant c_{1} such that, see [29],

∫_{0}^{T} ||J_{0}^{ρ}(v)(t)||_{r,ρ}^{2} dt = ∫_{0}^{T} ∥∫_{0}^{t} S(t − s)v(s) ds∥_{r,ρ}^{2} dt
≤ c_{1} ∫_{0}^{T} (∫_{0}^{t}(t − s)^{− \frac{r+1}{2}} ||v(s)||_{−1,ρ} ds)^{2} dt.

Now using twice Jensen’s inequality we get

∫_{0}^{T} ||J_{0}^{ρ}(v)(t)||_{r,ρ}^{2} dt ≤ c_{2} ∫_{0}^{T} (∫_{0}^{t}(t − s)^{− \frac{r+1}{2}} ||v(s)||_{−1,ρ}^{p} ds)^{\frac{2}{p}} dt
≤ c_{3}(∫_{0}^{T} ||v(t)||_{−1,ρ}^{p} dt)^{\frac{2}{p}} .

Hence J_{0}^{ρ} has an extension to a bounded operator acting from L^{p}(0, T; W_{ρ}^{−1}) into L^{2}(0, T; W_{ρ}^{r}). By Lemma A.3 for any γ ∈ [0, 1 − p^{−1}), J_{0}^{ρ} is a bounded linear operator from L^{p}(0, T; W_{ρ}^{−1}) into C^{γ}([0, T]; W_{ρ}^{−1}). Now note that both spaces 𝕃_{κ}^{2} and W_{ρ}^{−1} are continuously and densely imbedded into W_{2ρ−κ}^{−1}. Thus Lemma A.2(i) and Dubinsky’s criterion applied to the triple E_{0} = W_{ρ}^{r}, E = 𝕃_{κ}^{2}, and E_{1} = W_{2ρ−κ}^{−1}, give the desired conclusion. □

Proof of (ii). By Lemma A.3, J_{α}^{ρ} is a bounded operator from L^{p}(0, T; 𝕃_{ρ}^{2}) into C^{γ}([0, T]; 𝕃_{ρ}^{2}), where γ < 1−α−p^{−1}. Proceeding as in the proof of (i) one may easily show that for r sufficiently small but strictly positive, J_{α}^{ρ} is a bounded operator from L^{p}(0, T; 𝕃_{ρ}^{2}) into L^{2}(0, T; W_{ρ}^{r}), and consequently the desired conclusion follows again from Dubinsky’s criterion applied to E_{0} = W_{ρ}^{r}, E = 𝕃_{κ}^{2}, and E_{1} = 𝕃_{κ}^{2}. □

Proof of (iii). Assume that r̃ < −1. Recall that in the present paper the dimension d is equal to 2 or 3. Thus by Lemma A.2(ii) we have W_{ρ/3}^{−r̃} ↪ 𝕃_{ρ}^{6}, and consequently

by the duality we have 𝕃_{ρ}^{6/5} ↪ W r̃ ; ρ̃ where ρ̃ = 5ρ/3. Hence the proof of (iii) will be completed as soon as we find an r̃ < −1 such that J_{0}^{ρ} is has an extension to a compact operator acting from L^{6/5}(0, T; W r̃ ; ρ̃) into L^{2}(0, T; 𝕃_{κ}^{2}). By Lemmas A.3, A.2(i) and A.4 it is enough to show that J_{0}^{ρ} is a bounded operator from L^{6/5}(0, T; W r̃ ; ρ̃) into L^{2}(0, T; W r ; ρ̃) for a certain r > 0. Take r > 0 and r̃ < −1 such that \frac{5}{6}(r − r̃) < 1. Then, as S is analytic, there exists a constant c_{1} such that

∫_{0}^{T} ||J_{0}^{ρ}(v)(t)||_{r,ρ̃}^{2} dt = ∫_{0}^{T} ∥∫_{0}^{t} S(t − s)v(s) ds∥_{r,ρ̃}^{2} dt
≤ c_{1} ∫_{0}^{T} (∫_{0}^{t}(t − s)^{− \frac{r−r̃}{2}} ||v(s)||_{r̃,ρ̃} ds)^{2} dt.

Applying now Jensen’s inequality we get

≤ c_{1} ∫_{0}^{T} ((∫_{0}^{t}(t − s)^{− \frac{r−r̃}{2}} ds)^{\frac{1}{3}} (∫_{0}^{t}(t − s)^{− \frac{r−r̃}{2}} ||v(s)||r̃,ρ̃^{\frac{6}{5}} ds)^{\frac{5}{3}} ) dt

≤ c_{2} ∫_{0}^{T} (∫_{0}^{t}(t − s)^{− \frac{r−r̃}{2}} ||v(s)||r̃,ρ̃^{\frac{6}{5}} ds)^{\frac{5}{3}} dt,

and again by Jensen’s inequality

≤ c_{2} ∫_{0}^{T} (∫_{0}^{t}(t − s)^{− \frac{5(r−r̃)}{6}} ||v(s)||r̃,ρ̃^{\frac{6}{5}} ds)(∫_{0}^{t} ||v(s)||r̃,ρ̃^{\frac{6}{5}} ds)^{\frac{2}{3}} dt

≤ c_{3}(∫_{0}^{T} ||v(t)||r̃,ρ̃^{\frac{6}{5}} dt)^{\frac{5}{3}} ,

which completes the proof. □

Proof of Proposition 2.2. Proof of (i). First note that for each ρ ∈ ℝ there exist a constant c_{ρ} > 0 such that

|∂_{x_{i}}ϑ_{ρ}(x)| ≤ c_{ρ}|ϑ_{ρ}(x)| for x ∈ ℝ^{d}, i = 1, …, d.

Now let v, u ∈ C_{0}^{∞}(ℝ^{d}; ℝ^{d}). Then integrating by parts and using Hölder’s inequality and the estimate above we find a constant C independent of v and u such that

|⟨B_{0}(v), u⟩_{ρ}| ≤ C|v|_{𝕃_{ρ}^{4}}^{2}||u||_{1,ρ},

which easily gives the desired conclusion. □ Proof of (ii). Since d = 2, 3, Lemma A.2(ii) yields that W^{\frac{3}{4}} ↪ 𝕃^{4}. Hence the desired conclusion follows from (i) and the following interpolation inequality

||v||_{\frac{3}{4}} ≤ c|v|^{\frac{1}{4}} ||v||1^{\frac{3}{4}} for v ∈ W^{1}.

□ Proof of (iii). By Lemma A.2(ii) we have W^{\frac{1}{2}} ↪ 𝕃_{ρ}^{4}. Thus the desired conclusion follows from (i) and the interpolation inequality

||v||_{\frac{1}{2}} ≤ c|v|^{\frac{1}{2}} ||v||1^{\frac{1}{2}} for v ∈ W^{1}.

□

B. Appendix

This appendix deals with the existence of an orthonormal basis of H which is a Schauder basis of H^{1}. This result is needed in the proof of Theorem 1.1.

Theorem B.1. There exists an orthonormal basis {e_{k}} of H such that: (i) {e_{k}} ⊂ C_{b}^{∞}(O; ℝ^{d}) ∩ H^{2}, (ii) for every v ∈ H^{1}, the sequence {Π_{n}v} converges in H^{1} to v, (iii) there exists a constant C such that for every n, ||Π_{n}||_{L(H^{1},H^{1})} ≤ C.

Proof. Let A be the Stokes operator, that is the generator of S on H, and let E( dλ) be the spectral projection corresponding to I − A. Set

X_{m} = Range E([m, m + 1)), m ∈ ℕ.

Then X_{m}, m ∈ ℕ are closed subspaces of H. For very m let {e_{m,l}} be an orthonormal basis of X_{m}. Put the set {e_{m,l} : m, l ∈ ℕ} into a sequence {e_{k}}. We will show that {e_{k}} has the desired properties. First it is an orthonormal basis of H as X_{m} is orthogonal to X_{m′} if m ≠ m′, and as

X =^{def} ⋃_{m∈ℕ} X_{m}

is dense in H. Then {e_{k}} ⊂ C_{b}^{∞}(O; ℝ^{d}) ∩ H^{2} because

X ⊂ ⋂_{r∈[0,∞)} H^{r} ⊂ C_{b}^{∞}(O; ℝ^{d}).

Let Π_{n} be the projection onto the space span {e_{1}, …, e_{n}}. Then, since on every X_{m} the norms | · | and || · ||_{1} are equivalent we have

||Π_{n}v − v||_{1} → 0 for every v ∈ X.

Hence, since X is dense, the proof will be complete as soon as we can show that

sup_{n∈ℕ} ||Π_{n}||_{L(H^{1},H^{1})} < ∞.

Let v ∈ X. Then

||Π_{n}||_{1}^{2} = ⟨(I − A)Π_{n}u, Π_{n}u⟩ = ∑_{m=1}^{∞} ∫_{m}^{m+1} λ⟨E( dλ)Π_{n}u, Π_{n}u⟩

\begin{matrix}≤ ∑_{m=1}^{∞}(m + 1)|E([m, m + 1))Π_{n}u|^{2} = ∑_{m=1}^{∞}(m + 1)|Π_{n}E([m, m + 1))u|^{2} \\ ≤ ∑_{m=1}^{∞} \frac{m + 1}{m} ∫_{m}^{m+1} λ⟨E( dλ)u, u⟩ ≤ 2||u||_{1}^{2}, \end{matrix}

which gives the desired conclusion. □

C. Appendix

In the proof of the existence of a solution to (0.2) we fix 1/2 < ρ < 1, and we project the equation onto the spaces span {e_{1}, …, e_{n}}, n ∈ ℕ, generated by the elements of a Schauder basis of W_{ρ}^{1}, 𝕃_{ρ}^{2} and 𝕃_{ρ}^{q}, 2 ≤ q ≤ 6. Its existence follows from the following theorem.

Recall that S^{l} denotes the space of all v ∈ C^{l}(ℝ^{d}; ℝ^{d}) such that

sup_{|α|≤l} sup_{x∈ℝ^{d}} |(1 + |x|^{2})^{n}∂^{α}v(x)| < ∞ for every n ∈ ℕ.

Theorem C.1. For all l ∈ ℕ and ρ ∈ (1/2, 1) there is an orthonormal basis Ψ_{ρ} of L_{ρ}^{2}(ℝ^{d}) such that Ψ_{ρ} ⊂ S^{l} and Ψ_{ρ} is an unconditional Schauder basis of the spaces W_{ρ}^{1}(ℝ^{d}) and L_{ρ}^{q}(ℝ^{d}), q ∈ [2, ∞).

Proof. We prove the theorem for d = 1. Then taking tensor products one can generalize the result to the case of d ≥ 1. As in the whole paper we write L^{q}, L_{ρ}^{q}, and W_{ρ}^{1} instead of L^{q}(ℝ^{d}), L_{ρ}^{q}(ℝ^{d}), and W_{ρ}^{1}(ℝ^{d}). We divide the proof into two steps. First we show the existence of an orthonormal basis Ψ of L^{2} such that Ψ ⊂ S^{l} and Ψ is an unconditional Schauder basis of the spaces W^{1} and L_{κ}^{q}, where

(C.1)  1 < q < ∞ and \frac{1}{2} − \frac{q}{2} < κ < \frac{1}{2}.

We construct this basis using the theory of wavelets. Recall, see [28, p. 21 and p. 71] that an l-regular multiresolution approximation of L^{2} is an increasing sequence V_{j}, j ∈ ℤ of closed linear subspaces of L^{2} such that:

(i) ⋂_{j=−∞}^{∞} V_{j} = {0} and ⋃_{j=−∞}^{∞} V_{j} is dense in L^{2},

(ii) for all v ∈ L^{2} and j ∈ ℤ, x ↦ v(x) ∈ V_{j} ⟺ x ↦ v(2x) ∈ V_{j+1}, (iii) for all v ∈ L^{2} and k ∈ ℤ, x ↦ v(x) ∈ V_{0} ⟺ x ↦ v(x − k) ∈ V_{0}, (iv) there is a function φ ∈ V_{0} ∩ S^{l} such that the sequence φ(· − k), k ∈ ℤ is an

orthonormal basis of V_{0}. Let {V_{j}} be an l-regular multiresolution approximation of L^{2}. We denote by W_{j} the orthogonal complement of V_{j} in V_{j+1}.

According to [28, Theorem 1, p. 72] for any l-regular multiresolution approximation {V_{j}} of L^{2} there is a function ψ ∈ W_{0} ∩ S^{l} such that ψ(· − k), k ∈ ℤ, is an orthonormal basis of W_{0}. For our purposes we take V_{0} being the space of l-splines on ℝ, that is the space of all v ∈ L^{2} ∩ C^{l−1} such that the restriction of v to each interval (k, k + 1), k ∈ ℤ, is a polynomial of degree ≤ l.

Let ℕ_{0} = ℕ ∪ {0}. For j ∈ ℕ_{0}, k ∈ ℤ, and x ∈ ℝ we write ψ_{(j,k)}(x) = 2^{j/2}ψ(2^{j}x − k). Note that the family {ψ_{(j,k)}, k ∈ ℤ} is an orthonormal basis of W^{j}. The following orthonormal basis of L^{2},

Ψ = {φ(· − k), k ∈ ℤ} ∪ {ψ_{(j,k)}, j ∈ ℕ_{0}, k ∈ ℤ}

in called a wavelet basis arising from {V_{j}}.

Our goal is to show that Ψ has the required properties. The fact that it is an unconditional Schauder basis of W^{1} follows from [28, Theorem 2, p. 172]. Let q and κ satisfy (C.1). Then, as ϑ_{κ} is an A_{q}-basis, see [35, Chapter 5], [22, Theorem 4] yields that Ψ is an unconditional Schauder basis of L_{κ}^{q}.

Let ρ ∈ (1/2, 1), let i_{ρ/2}v = ϑ_{−ρ/2}v, and let Ψ_{ρ} = {j_{ρ/2}v, v ∈ Ψ}. Since i_{ρ/2} is an isometry between L^{2} and L_{ρ}^{2}, and an isomorphism between W^{1} and W_{ρ}^{1}, see Lemma A.1, we have that Ψ_{ρ} is an orthonormal basis of L_{ρ}^{2}, and Ψ_{ρ} is an unconditional basis of W_{ρ}^{1}. Note that i_{ρ/2} is an isometry between L_{κ}^{q} and L_{κ+qρ/2}^{q}. Thus to show that Ψ_{ρ} is an unconditional basis of the spaces L_{ρ}^{q}, q ∈ [2, ∞), it is enough to observe that for an arbitrary q ∈ [2, ∞) there is a κ such that (C.1) holds true and ρ = κ + qρ/2. □

Lemma C.1. Let E be equal to either W_{ρ}^{1}, or 𝕃_{ρ}^{p} for a q ∈ [2, ∞), and let {e_{k}} be an orthonormal basis of 𝕃_{ρ}^{2} and a Schauder basis of E. Let

Π_{n}v = ∑_{k=1}^{n}⟨v, e_{k}⟩_{ρ}e_{k} and Π_{n,n+m}v = ∑_{k=n}^{n+m}⟨v, e_{k}⟩_{ρ}e_{k}.

Then:

(i) For every v ∈ E, |Π_{n}v − v|_{E} → 0 as n → ∞.

(ii) sup_{n,m} ||Π_{n,n+m}||_{L(E,E)} < ∞.

Proof. By the Banach–Steinhaus theorem (ii) follows from (i). Let v ∈ E, and let {v^{k}} be the coefficients of v in the basis {e_{k}}. Write,

Π̃_{n}v = ∑_{k=1}^{n} v^{k}e_{k} and Π̃_{n,n+m}v = ∑_{k=n}^{n+m} v^{k}e_{k}.

Then |Π̃_{n}v − v|_{E} → 0 as n → ∞. Moreover, the uniqueness of the coefficients in an orthonormal basis implies that Π_{n}v = Π̃_{n}v and Π_{n,n+m}v = Π̃_{n,n+m}v for v ∈ E. Thus the proof of (i) will be complete as soon as we show that Π̃_{n} and Π̃_{n,n+m} are continuous linear operators on E. This follows from the facts that under the identification (𝕃_{ρ}^{2})^{∗} = 𝕃_{ρ}^{2} we have {e_{k}} ⊂ E^{∗}, and e_{k}[e_{l}] = ⟨e_{k}, e_{l}⟩_{ρ} is equal to 0 for k ≠ l and 1 for k = l. □

Acknowledgements

We would like to thank Professors Z. Brzeźniak and S. Naboko for their help in the proof of Theorem B.1, and Professors Z. Ciesielski, A. Kamont and P. Wojtaszczyk for their help in the proof of Theorem C.1.

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Marek Capiński: Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

E-mail address: capinski@im.uj.edu.pl

Szymon Peszat: Institute of Mathematics, Polish Academy of Sciences, Św. Tomasza 30/7, Kraków 31-027, Poland

E-mail address: napeszat@cyf-kr.edu.pl