Global well-posedness and interior regularity of 2D Navier–Stokes equations with stochastic boundary conditions

The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.


Introduction
Partial differential equations with boundary noise have been introduced by Da Prato and Zabczyck in the seminal paper [18].They showed that, also in the one dimensional case, the solutions of the heat equation with white noise Dirichlet or Neumann Boundary conditions have low regularity compared to the case of noise diffused inside the domain.In particular, in the case of Dirichlet boundary conditions the solution is only a distribution.Some improvements in the analysis of the interior regularity of the solutions of these problems and some nonlinear variants have been obtained exploiting specific properties of the heat kernel and of suitable nonlinearities.For some results in this direction we refer to [5,15,23,25,31].All these issues make the problem of treating non-linear partial differential equations with boundary noise coming from fluid dynamical models an, almost untouched, field of open problems.
Throughout the manuscript we fix a finite time horizon T > 0. Let a > 0, O = T × (0, a) and let T be the one dimensional torus.Finally, we denote by (1.1) Γ b = T × {0} and Γ u = T × {a}, the bottom and the upper part of the boundary of O, respectively.
In this paper we are interested in the global well-posedness and the interior regularity of the 2D Navier-Stokes equations with boundary noise for the unknown velocity field u(t, ω, x, z) = (u 1 , u 2 ) : R + × Ω × O → R 2 , formally written as where ∇u = (∂ j u i ) 2 i,j=1 , W H (t) is a H-cylindrical Brownian motion and h b (t, x) is a sufficiently regular forcing term; we refer to Section 1.1 below for the the relevant assumptions and definitions.To the best of our knowledge this is the first instance of a global well-posedness result for a fluid dynamical system driven by stochastic white in time boundary conditions.We refer to [12,13] for some homogenization results in the case of Navier-Stokes equations with dynamic boundary conditions driven by a stochastic forcing and to [14] for the local analysis of the three dimensional primitive equations with boundary noise.Finally, we refer to [21,22] for some limit behaviors of the model (1.2) with h b ẆH replaced by a highly oscillating and regular stationary random field.
Following the books by Pedlosky [41,42] and Gill [30], the model (1.2) is a good idealization of the velocity of the fluid in the ocean.In this scenario, the domain O = T × (0, a) can be considered a vertical slice of the ocean with depth a > 0 and we should interpret u 1 (resp.u 2 ) as the horizontal (resp.vertical) component of the velocity field u.Indeed even if, in principle, one should consider a free surface, instead of Γ u = T × {a}, depending on the time, the approximation of such surface as independent of the time, although highly unrealistic, is justified by the fact that the behavior of the fluid around the surface is in general very turbulent.Hence, as emphasized in [24], only a modelization is tractable and meaningful.The stochastic boundary condition appearing in (1.2) is interpreted as the physical law describing the driving mechanism on the atmosphere-ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force, see [38] for details.
1.1.Main Results.We begin by introducing some notation.Consider a complete filtered probability space (Ω, F, (F t ) t≥0 , P), a separable Hilbert space H and a cylindrical F−Brownian motion (W H (t)) t≥0 on H.We say that a process Φ is Fprogressive measurable if Φ| (0,t)×Ω is F t × B((0, t))-measurable for all t > 0, where B denotes the Borel σ-algebra.For the relevant notation on function spaces, we refer to Section 1.1.1.
Following the idea of [17] we split the analysis of (1.2) in two parts.First we consider the stochastic linear problem with non-homogeneous boundary conditions The solution to the above linear equation (1.3) can be treated in mild form as in [18,19].Secondly, denoting by v = u − w we will consider the Navier-Stokes equations with random coefficients (1.4) As discussed in [19,Chapter 13], if h b , u 0 , W H (t) would be regular enough, then u = v +w will be a classical solution of the Navier-Stokes equations with inhomogeneous boundary conditions (1.2).
To state our first result, we introduce some more notation.Here and below, we denote by of divergence free vector fields adapted to our framework, introduced rigorously in Section 2.1.Definition 1.3.A process u with paths P − a.s. in C([0, T ]; H) ∩ L 4 (0, T ; L 4 ) and progressively measurable with respect to these topologies, is a pathwise weak solution of (1.2) if u = v + w, where w has paths in C(0, T ; H) ∩ L 4 (0, T ; L 4 (O)), it is progressively measurable with respect to these topologies and is a mild solution of (1.3) while v has paths in C(0, T ; H) ∩ L 2 (0, T ; V ), it is progressively measurable with respect to these topologies and is a weak solution of (1.4).
The first main result of this paper reads as follows.
Theorem 1.4 (Global well-posedness).Let Hypothesis 1.1 be satisfied.Then for all u 0 ∈ H there exists a unique weak solution u to (1.2) in the sense of Definition 1.3.
According to Remark 1.2, the introduction of the non-Hilbertian setting is necessary in order to prove Theorem 1.4 above, at least with the tools introduced in this article.
Remark 1.5 (Additional bulk forces).Without additional difficulties we could also consider in equation (1.2) an additive noise diffused inside the domain of the form The case q = p = 2 and λ = 0 is also allowed, see [20,Chapter 5].Here A q and γ stands for the Stokes operator on L q and the class of γ-radonifying operators, see Section 2.1 and [32, Chapter 9], respectively.Finally, X −λ,Aq is the extrapolated space or order λ w.r.t.A q as defined below, see (2.9).To see this, note that, under these assumptions, arguing as in Proposition 3.1 the solution w to can be obtained as a stochastic convolution.In particular, the above assumptions on h d imply that w is a progressively measurable process with values in C([0, T ]; H) ∩ L 4 (0, T ; L 4 ).Therefore this term adds no difficulties in order to analyze the wellposedness of equation (1.4).For this reason we prefer to not consider this classical source of randomness.
Remark 1.6 (Comparison with the literature).
(1) Theorem 1.4 shares strong similarities with [11,Theorem 1.2], which addresses the well-posedness of certain 2D deterministic Navier-Stokes equations with non-homogeneous non-smooth Navier-type boundary conditions.However, it is important to note that our model focuses on a different phenomenon than the one studied in [11].For this reason, contrary to us, they stress the regularity of the boundary condition of the normal trace of the velocity.From a mathematical viewpoint, the white noise appearing in equation (1.2) is rougher both in time and in space compared to the boundary conditions discussed in [11].However, as discussed in [18], Neumann boundary conditions are more regular than Dirichlet boundary conditions and allow us to treat rougher inputs.Due to these differences, the two results have different ranges of applicability and do not cover each other.Moreover, the tools introduced here differ significantly from the techniques involved in [11].(2) As discussed in the introduction, the first result in the direction of the analysis of fluid dynamical models with stochastic boundary conditions have been proved in [14,Theorem 5.1], where the authors established local well-posedness of 3D primitive equations with boundary noise modeling wind forces.Both their strategy and ours are based on the splitting technique introduced in [17].After showing suitable regularity properties of the stochastic convolution via stochastic maximal L p -regularity techniques (cf.Proposition 3.1 and [14, Proposition 4.3]), a thorough analysis of certain nonlinear models is required.In contrast, we conduct this analysis within a suitable Hilbertian framework, enabling us to derive energy estimates essential for establishing the global well-posedness of (1.2) (cf.Theorem 3.3 and [14, Section 5.3]).The difference between the global well-posedness result which we are able to obtain and [14, Theorem 5.1] can be seen as consequence of the fact that the 2D Navier-Stokes equations are globally well-posed in the weak setting, while the same cannot be asserted for the primitive equations (cf.[33]).Therefore, in order to prove their local well-posedness result, the authors in [14] need to work with a notion of solution which mixes strong and weak regularity in the space variables.As a byproduct of this fact we are able to consider a noise rougher in space compared to them.Additionally, a minor distinction lies in the boundary conditions applied to the bottom part of the domain Γ b .We introduce no-slip boundary conditions to accurately model the bottom of the ocean, a choice with theoretical underpinning in works such [21,22,30,41,42].In contrast, [14] considered some form of homogeneous Neumann boundary conditions, a choice related to the functional analytic setup of the primitive equations (cf.[14,Remark 3.3]).
Beyond the distinct justifications from a modeling perspective, our choice leads to differences in the analysis of the corresponding linear elliptic systems (cf.Section 2.2 and [14, Section 3.5]).
Secondly, we are interested in studying the interior regularity of the solution u provided by Theorem 1.4.
Our second main result reads as follows: Theorem 1.7 (Interior regularity).Let Hypothesis 1.1 be satisfied.Let u be the unique weak solution of (1.2) provided by Theorem 1.4.Then for all t 0 ∈ (0, T ) and O 0 ⊂ O such that dist(O 0 , ∂O) > 0, According to [47] (see also [37,Section 13.1]), it seems not possible to gain high-order interior time-regularity for the Navier-Stokes problem.This fact is in contrast to the case of the heat equation with white noise boundary conditions, see [16].The reason behind this is the presence of the unknown pressure P which, due to its non-local nature, provides a connection between the interior and the boundary regularity.Finally, let us mention that other techniques to bootstrap further interior space regularity (e.g.analyticity), such as the 'parameter' trick (see [8,9] and [43,Subsection 9.4]), seem not to work due to the presence of the noise on Γ u .Similarly to the proof of Theorem 1.4, we analyze the interior regularity of u combining the interior regularity of w and the interior regularity of v.The interior regularity of w is obtained introducing a proper weak formulation, see Definition 4.1 below.Instead the regularity of v is analyzed via a Serrin's argument exploiting the aforementioned regularity of w.
The paper is organized as follows.In Section 2 we will introduce the functional setting in order to deal with problem (1.2).In particular, we will introduce the corresponding of the classical spaces and operator needed to deal with Navier-Stokes equations with no-slip boundary condition to this more involved set of boundary conditions.Indeed, the Stokes operator associated to our problem generates an analytic semigroup which admits an H ∞ -calculus of angle strictly less than π 2 also in the non-Hilbertian setting.This is crucial in order to apply the Stochastic maximal L p -regularity results of [52], recalled in Section 2.4.The proof of Theorem 1.4 is the object of Section 3. In particular, in Section 3.1 we will consider the linear problem (1.3), while in Section 3.2 we will consider the nonlinear problem (1.4).The proof of Theorem 1.7 is the object of Section 4. In particular, in Section 4.1 we will study the interior regularity of the solution of the linear problem (1.3), while in Section 4.2 we will consider the nonlinear problem (1.4).We postpone some technical proofs related to the properties of the Stokes operator in the Appendix A.
1.1.1.Notation.Here we collect some notation which will be used throughout the paper.Further notation will be introduced where needed.By C we will denote several constants, perhaps changing value line by line.If we want to keep track of the dependence of C from some parameter ξ we will use the symbol C(ξ).Sometimes we will use the notation a ≲ b (resp.a ≲ ξ b), if it exists a constant such that a ≤ Cb (resp.a ≤ C(ξ)b).
Fix q ∈ (1, ∞).For an integer k ≥ 1, W k,q denotes the usual Sobolev spaces.In the non-positive and non-integer case s ∈ (−∞, ∞) ∖ N, we let W s,q := B s q,q where B s q,q is the Besov space with smoothness s, and integrability q and microscopic integrability q (in particular W 0,q ̸ = L q ).Moreover, H s,q denotes the Bessel potential spaces.Both Besov and Bessel potential spaces can be defined by means of Littlewood-Paley decompositions and restrictions (see e.g.[45], [44,Section 6]) or using the interpolation methods starting with the standard Sobolev spaces W k,q (see e.g.[10,Chapter 6]).Finally, we set A s,q (D; R d ) := (A s,q (D)) d for an integer d ≥ 1, a domain D and A ∈ {W, H}.
Let K and Y be a Hilbert and a Banach space, respectively.We denote by γ(K, Y ) the set of γ-radonifying operators, see e.g.[32,Chapter 9] for basic definitions and properties.If Y is Hilbert, then γ(K, Y ) coincides with the class of Hilbert-Schmidt operator from K to Y .Below, we need the following Fubini-type result: A s,q (D; K) = γ(K, A s,q (D)) for all s ∈ R, q ∈ (1, ∞), A ∈ {W, H}.
The above follows from [32, Theorem 9.3.6]and interpolation.

Preliminaries
2.1.The Stokes operator and its spectral properties.In this section we introduce the functional analytic setup in order to define all the object necessarily in the following.In order to improve the readability of the results we will just state the main results on the Stokes operator postponing the proofs to Appendix A.
Throughout this subsection we let q ∈ (1, ∞).Recall that O = T × (0, a) where a > 0. We begin by introducing the Helmholtz projection on L q (O; R 2 ), see e.g.[43,Subsection 7.4].Let f ∈ L q (O; R 2 ) and let ψ f ∈ W 1,q (O) be the unique solution to the following elliptic problem Here n denotes the exterior normal vector field on ∂O.Of course, the above elliptic problem is interpret in its natural weak formulation: By [43,Corollary 7.4.4], we have (the proof of such estimate can also be obtained by the Lax-Milgram theorem in Banach spaces [35,Theorem 1.1], see also the proof of Theorem 2.2 below).Then the Helmholtz projection is given by P q is defined as Next we define the Stokes operator on L q (O; R 2 ).For convenience of notation, we actually define A q as minus the Stokes operator so that A q is a positive operator for q = 2 (i.e.⟨A 2 u, u⟩ ≥ 0 for all u ∈ D(A 2 )).Let L q := P(L q (O; R 2 )).Then, we define the operator A q : D(A q ) ⊆ L q → L q where and A q u = −P q ∆u for u ∈ D(A q ).
In the main arguments we need stochastic maximal L q -regularity estimates for stochastic convolutions.By [52] (see also [3,53]), it is enough to show the boundedness of the H ∞ -calculus for A q .For the main notation and basic results on the H ∞ -calculus we refer to [43,Chapters 3 and 4] and [32,Chapter 10].
In the following, we let Theorem 2.1 (Boundedness H ∞ -calculus).For all q ∈ (1, ∞), the operator A q is invertible and has a bounded H ∞ -calculus of angle < π 2 .Moreover the domain of the fractional powers of A q is characterized as follows: (1) The above implies that −A q generates an analytic semigroup on L q .For convenience of notation, we will simply write A in place of A 2 .Moreover we define . We denote by ⟨•, •⟩ and ∥•∥ the inner product and the norm in H respectively.In the sequel we will denote by V * the dual of V and we will identify H with H * .Every time X is a reflexive Banach space such that the embedding X → H is continuous and dense, denoting by X * the dual of X, the scalar product ⟨•, •⟩ in H extends to the dual pairing between X and X * .We will simplify the notation accordingly.
Theorem 2.1 could be known to experts.For the reader's convenience, we provide in Appendix A a complete and relatively short proof based on the recent strategy used in [36] for the H ∞ -calculus for the Stokes operator on Lipschitz domains [36,Theorem 16].
2.2.The Neumann map.Now we are interested in L q -estimates for the Neumann map, i.e. we are interested in studying the weak solutions of the elliptic problem To state the main result of this subsection, we need to formulate (2.3) in the weak setting.To this end, we argue formally.Take In particular, the RHS of (2.4) makes sense even in case g is a distribution if we interpret T g(x)φ 1 (x, a) dx = ⟨φ 1 (•, a), g⟩.Theorem 2.2.Let q ∈ (1, ∞), for all g ∈ W −1/q,q (Γ u ) there exists a unique Proof.We divide the proof into three steps.
Step 1: Proof of (2.5).Let A q be as in Section 2. We prove (2.5) by applying the Lax-Milgram theorem of [35,Theorem 1.1] to the form a : Recall that, by Theorem 2.1, Hence the Lax-Milgram theorem of of [35,Theorem 1.1] implies the existence of u as in (2.5) provided, for all v ∈ D(A The case ≳ of (2.8) follows from the Hölder inequality.To prove the opposite inequality, we argue by duality.We start by discussing some known facts about the "Sobolev tower" of spaces associated the operator A p : Here ∼ denotes the completion (since 0 ∈ ρ(A q ) by Theorem 2.1, we have that f → ∥A α q f ∥ L q is a norm for all α < 0).Since (A q ) * = A q ′ , it follows that (see e.g.[6, Chapter 5, Theorem 1.4.9]) Now we can proceed in the proof of ≲ in (2.8).Firstly, as D(A q ) → D(A 1/2 q ) is dense for all q ∈ (1, ∞), we can prove such inequality assuming v ∈ D(A q ).In the latter case, the duality (2.9) and the Hahn-Banach theorem imply the existence of g ∈ X −α,A q ′ of unit norm such that where in (i) we used that A , in (ii) that A q = −P q ∆ q and therefore P q in (iii) we used that no boundary terms appear due to the boundary conditions and Hence the case ≲ of (2.8) follows from the above chain of equality, the fact that D(A Step 2: Proof of (2.6).By Step 1, it suffices to prove the existence of a solution (u, π) ∈ W 2,q (O) × W 1,q (O)/R for which (2.3) holds.In case of g ∈ C ∞ (Γ u ), the conclusion follows from standard L 2 -theory and we will present the argument in this case at the end of the proof.In the remaining case we argue by density.Note that, arguing as in the proof of Proposition A.4, a localization argument and [43, Theorem 7.2.1](applied with time as a dummy variable) yield the following a-priori estimates for solutions (u, π) where ε > 0 is arbitrary and in the last step we applied Step 1.
The above shows ∥u∥ (2.3).Combining this, the density of C ∞ (Γ u ) in W 1−1/q,q (Γ u ), and the above mentioned solvability for g ∈ C ∞ (Γ u ); one readily obtains the existence of solutions to (2.3) Step 3: Proof of the regularity of (u, π) in case of g ∈ C ∞ (T).The proof of this fact follows the lines of Proposition A.2. First, by Lax-Milgram Lemma and [51, Proposition 1.1, Proposition 1.2], there exists a unique couple, (u, π) (2.12) Now, let us fix h > 0, extend periodically either u and g in the x direction and consider ϕ = τ h τ −h u as a test function in (2.10), where

Then by change of variables, it follows that
Therefore Since u ∈ V and (2.12) holds the right hand side of inequality (2.13) is uniformly bounded in h → 0 and this implies . Since u is divergence free and (2.11) holds, then Therefore ∥∇π∥ L 2 ≲ ∥g∥ C 1 (Γu) .Lastly, again by relation (2.11) Combining all the information obtained we get Iterating the argument one gets that (u, π) Next we denote by N the solution map defined by Theorem 2.2 which associate to a boundary datum g the velocity u solution of (2.3), i.e.N g := u.From the above result we obtain Corollary 2.3.Let N and H be the Neumann map and a Hilbert space, respectively.Then, for all q ≥ 2 and ε > 0, Proof.To begin, recall that W s,q (Γ u ; H) = γ(H, W s,q (Γ u )) for all s ∈ R and q ∈ (1, ∞), see Section 1.1.1.Hence, due to the ideal property of γ-radonifying operators [32, Theorems 9.1.10and 9.1.20],it is enough to consider the scalar case H = R.
Since by Sobolev embedding theorem V ⊂ L 4 , b is also defined and continuous on V × V × V .Moreover, by standard interpolation inequalities, Integrating by parts, the standard oddity relation below holds Lastly we introduce the operator defined by the identity for all ϕ ∈ V .When v ∈ V , we may also write Moreover, when u We have to define our notion of weak solution for problem (2.15).
Definition 2.4.Given u 0 ∈ H and f ∈ L 2 (0, T ; V * ), we say that The following results are simple adaptations of classical results, see for instance [27,39,50,51]. Moreover, where C is a constant independent of ε and ε ′ .
Theorem 2.6.For every u 0 ∈ H and f ∈ L 2 (0, T ; V * ) there exists a unique weak solution of equation (2.15).It satisfies then the corresponding unique solutions (u n ) n∈N converge to the corresponding solution u in C ([0, T ] ; H) and in L 2 (0, T ; V ).

2.4.
Stochastic Maximal L p -regularity.Let H and (W H (t)) t≥0 be a Hilbert space and a cylindrical F−Brownian motion on H, respectively.The following result was proven in [52], see also [53,Section 7] and [4, Section 3] for additional references.Below, for a Banach space Y , H s,q (R + ; Y ) denotes the Y -valued Bessel potential space on R + with smoothness s ∈ R and integrability q; such space can be defined either by complex interpolation (see e. Theorem 2.7.Let X be a Banach space isomorphic to a closed subspace of L q (D, µ) where q ∈ [2, +∞) and (D, A , µ) is a σ-finite measure space.Let A be an invertible operator and assume that it admits a bounded H ∞ calculus of angle < π/2 on X and let (S(t)) t≥0 the bounded analytic semigroup generated by −A.For all F−adapted G ∈ L p (R + × Ω; γ(H; X)) the stochastic convolution process is well defined in X, takes values in the fractional domain D(A 1/2 ) almost surely and for all 2 < p < +∞ the following space-time regularity estimate holds: with a constant C θ independent of G.
For extensions of the above result we refer to [3,40] for the weighted case, and to [2, Subsection 6.2] for the case of homogeneous spaces.However, the latter situations will not be considered here.

Well-Posedness
3.1.Stokes Equations.As discussed in Section 1.1, we start by considering the linear problem (1.3).According to [18,19], the mild solution w of the former problem is formally given by Here A q is (minus) the Stokes operator with homogeneous boundary conditions, and (S q (t)) t≥0 its corresponding semigroup (cf.Theorem 2.1).
Next step is to prove that w(t) is well defined in some functional spaces and has some regularities useful to treat the nonlinearity of the Navier-Stokes equations.Proposition 3.1.Let α ∈ [0, 1  q ] and assume that h b : (0, T ) × Ω → W −α,q (Γ u ; H) is F-progressive measurable with P − a.s.paths in L p (0, T ; W −α,q (Γ u )).Then the process w defined in (3.1) is a well defined process with P − a.s.paths in )) for all θ ∈ [0, 1  2 ), ε > 0. In particular, if h b satisfies Hypothesis 1.1, then w has P − a.s.trajectories in C([0, T ]; H) ∩ L 4 (0, T ; L 4 ).

Auxiliary Navier-Stokes Type Equations.
Having solved the Stokes problem we introduce the auxiliary variable This first equation in the above system has the form with the affine function For each ω ∈ Ω fixed, the Navier-Stokes structure is preserved and the auxiliary equation for v with homogeneous boundary conditions is solvable similarly to the classical Navier-Stokes equations.Therefore, let us introduce the notion of weak solution of the deterministic problem (1.4) with random coefficients.Recall that A and b are (minus) the Stokes operator on L 2 and defined in (2.16), respectively.Definition 3.2.Given u 0 ∈ H and w ∈ L 4 0, T ; L 4 , we say that If (u n 0 ) n∈N is a sequence in H converging to u 0 ∈ H and (w n ) n∈N is a sequence in L 4 0, T ; L 4 converging to w ∈ L 4 0, T ; L 4 , then the corresponding unique solutions (v n ) n∈N converge to the corresponding solution v in C ([0, T ] ; H) and in L 2 (0, T ; V ).
Proof.We split the proof into several steps.
Step 1: Uniqueness.Let v (i) be two solutions.The function + w .
Step 2: Existence.Define the sequence (v n ) by setting v 0 = 0 and for every n ≥ 0, given v n ∈ C ([0, T ] ; H) ∩ L 2 (0, T ; V ), let v n+1 be the solution of equation (2.15) with initial condition u 0 and with in place of f .In particular for every ϕ ∈ D (A).In order to claim that this definition is well done, we notice that B (v n , w) , B (w, v n ) , B (w, w) ∈ L 2 (0, T ; V * ) by Lemma 2.5.
Then let us investigate the convergence of (v n ).First, let us prove a bound.From the previous identity and Theorem 2.6 we get It gives us (using Lemma 2.5) Choosing a small constant ε, one can find R > ∥u 0 ∥ 2 and T small enough, depending only from ∥u 0 ∥ and ∥w∥ L 4 (0,T ;L 4 ) , such that if then the same inequalities hold for v n+1 .Set w n = v n − v n−1 , for n ≥ 1.From the identity above, Again as above, since we may rewrite it as One can check as above the applicability of Theorem 2.6 and get As above we deduce Now we work under the bounds (3.3) and deduce, using the Gronwall lemma, for T , depending only from ∥u 0 ∥ and ∥w∥ L 4 (0,T ;L 4 ) , possibly smaller than the previous one, It implies that the sequence (v n ) is Cauchy in C 0, T ; H ∩L 2 0, T ; V .The limit v has the right regularity to be a weak solution and satisfies the weak formulation; in the identity above for v n+1 and v n we may prove that All these convergences can be proved easily by recalling the definition of b.Similarly, we can pass to the limit in the energy identity.After proving existence and uniqueness in [0, T ] we can reiterate the existence procedure and in a finite number of steps cover the interval [0, T ].
Step 3: Continuous dependence on the data.Let v n (resp.v) the unique solution of (1.4) with data u n 0 , w n (resp.u 0 , w).Since u n 0 → u 0 in H (resp. w n → w in L 4 (0, T ; L 4 )) the family (u n 0 ) n∈N is bounded in H (resp. the family (w n ) n∈N is bounded in L 4 (0, T ; L 4 )), by (3.2) one can show easily that the family (v We can easily bound the right hand side of relation (3.4) by Young's inequality and Hölder's inequality obtaining Applying Gronwall's inequality to relation (3.5) the claim follows immediately.□ Remark 3.4.Freezing the variable ω ∈ Ω and solving (1.4) for each ω does not allow to obtain information about the measurability properties of v.However, measurability of v with respect of the progressive σ-algebra follows from the continuity of the solution map with respect to u 0 and w.Therefore we have the required measurability properties for v with w being the mild solution of (1.3).In particular v has P-a.s.paths in C(0, T ; H) ∩ L 2 (0, T ; V ), it is progressively measurable with respect to these topologies and ⟨v (t) , ϕ⟩ − for each t ∈ [0, T ], P − a.s.
Note that the last term in (4.1) is well-defined as α < 1/2 and q ′ < 2.
Lemma 4.3.Let Hypothesis 1.1 be satisfied.There exists a unique weak solution of (1.3) in the sense of Definition 4.1 and it is given by the formula (3.1).
Proof.We split the proof into two steps.
Step 1: There exists a unique weak solution of (1.3) and it is necessarily given by the mild formula (3.1).Let ψ ∈ C 1 ([0, T ]; D(A)).Arguing as in the first step of the proof of [27,Theorem 1.7], see also [26,Proposition 17], one can readily check that w satisfies for each t ∈ [0, T ], P − a.s.The stochastic integral in the relation above is welldefined arguing as in Remark 4.2.Now consider ϕ ∈ D(A 2 ) and use Recalling the definition of the Neumann map N , (4.3) can be rewritten as Then, exploiting the self-adjointness property of S q and A q we have that weak solutions of (1.3) satisfy the mild formulation.Therefore they are unique.
Step 2: The mild formula (3.1) is a weak solution of (1.3) in the sense of Definition 4.1.We begin by noticing that w has the required regularity due to Proposition 3.1.Let us test our mild formulation (3.1) against functions ϕ ∈ D(A 2 ) ⊆ D(A 2 q ′ ).It holds, since S q ′ (t)| H = S(t), A q ′ | D(A) = A and exploiting self-adjointness property of S q and A q ⟨w(t), ϕ⟩ where in the last step we used the definition of Neumann map.In order to complete the proof of this step it is enough to show that The double integrals in (4.6) can be exchanged via stochastic Fubini's Theorem, see [20], since < +∞ Therefore the double integral in the right hand side of (4.6) can be rewritten as Inserting this expression in (4.6), (4.5) holds and the proof is complete.□ Thanks to the weak formulation guaranteed by Lemma 4.3 we can easily obtain the interior regularity of the linear stochastic problem (1.3).Let N 0 be the P null measure set where at least one between w / ∈ C([0, T ]; H) ∩ L 4 (0, T ; L 4 ), v / ∈ C([0, T ]; H) ∩ L 2 (0, T ; V ), (4.1) and (3.6) is not satisfied.In the following we will work pathwise in Ω ∖ N 0 even if not specified.Corollary 4.4.Let Hypothesis 1.1 be satisfied.Let w be the unique weak solution of (1.3) in the sense of Definition 4.1.Then, for all 0 < t 1 ≤ t 2 < T, x 0 ∈ O, r > 0 such that dist(B(x 0 , r), ∂O) > 0, and use ∇ ⊥ ψ as test function in (4.1).This implies that ω w is a distributional solution of the heat equation Since ω w solves the heat equation in distributions, a standard localization argument and regularity results for the heat equation (see e.g.[48, Chapter 6, Section 1]) imply that Let us now consider a test function ϕ ∈ C ∞ c (B(x 0 , r + ε)) identically equal to one on B(x 0 , r + ε/2).Since div w = 0, we have that ŵ = ϕw solves the elliptic problem Therefore, by standard elliptic regularity theory From the fact that ϕ ≡ 1 on B(x 0 , r + ε/2) it follows that Therefore, the required regularity of w is established by inductively reiterating this argument and by considering test functions ϕ ∈ C ∞ c (B(x 0 , r + ε 2 2n )) identically equal to one on B(x 0 , r + ε 2 2n+1 ).□ 4.2.Auxiliary Navier-Stokes Type Equations.In order to deal with the interior regularity of (1.4) we perform a Serrin type argument, see [37,47].The regularity of w guaranteed by Corollary 4.4 will play a crucial role to treat the linear terms appearing in (1.4).We start with the following lemma.This means that ω is a distributional solution in (0, T )×O of the partial differential equation Since dist(B(x 0 , r), ∂O) > 0, 0 < t 1 ≤ t 2 < T we can find ε small enough such that 0 Therefore g ∈ L 2 (0, T ; H −1 (R 2 )) + L 1 (0, T ; L 2 (R 2 )) P − a.s.Then, arguing as in the first step of the proof of [37,Theorem 13.2], the fact that ω is a distributional solution of (4.7) implies that ω ∈ C([0, T ]; From the regularity of ω, by standard elliptic regularity theory (see for example [7]), it follows that ϕv ∈ By Corollary 4.4 and relation (4.9) it follows that ĝ ∈ L 2 (0, T ; Proof.Since dist(B(x 0 , r), ∂O) > 0, 0 < t 1 ≤ t 2 < T we can find ε small enough such that 0 ) such that it is equal to one in [t 1 + ε/2, t 2 + ε/2] × B(x 0 , r + ε/2).From Lemma 4.5 and Sobolev embedding theorem we know that v ∈ C([t 1 − ε, t 2 + ε]; L ∞ (B(x 0 , r + ε); R 2 )) P − a.s.Denoting by arguing as in the proof of Lemma 4.5, it follows that ω is a distributional solution in (0, T ) × B(x 0 , r + ε) of From the regularity of ω, v, ω, ω w , w, then g ∈ L 2 (t 1 − ε, t 2 + ε; H −1 (B(x 0 , r + ε))) P − a.s.By standard regularity theory for the heat equation, see for example Step 2 in [37, Theorem 13.1], a solution of (4.11) with g ∈ by standard elliptic regularity theory (see for example [7]), ))) P − a.s.

Reiterating the argument as in
This implies that ∥∇π∥ L 2 ≲ ∥f ∥.Lastly u 1 satisfies which completes the proof.□ We are ready to prove Proposition A.1.[46].The second identity in (A.1) follows analogously, where one uses the argument in [28] (see also [49,Proposition 5.5,Chapter 17]) to deduce D(A γ ) = H 2γ (O) from the first identity in (A.1).□ A.2. Bounded H ∞ -calculus for Laplace operators.In this subsection we prove the boundedness of the H ∞ -calculus for B q .The basic idea is to use the product structure of the domain O and to write B q u = (L q,R u 1 , L q,D u 2 ) where

Proof of Proposition
Proposition A.3 (Bounded H ∞ -calculus for Laplace operators).Let q ∈ (1, ∞) and let O be as above.Then −L q,D and −L q,R have a bounded H ∞ -calculus of angle 0. In particular B q has a bounded H ∞ -calculus of angle 0.
The above statement also holds for the Neumann Laplacian, but it will not be needed below.
Proof.We divide the proof into three steps.In the first step, we exploit the product structure of our domain to reduce the problem to a one dimensional situation.
On L 2 (O) considers the operator (ℓ q,P f )(x, z) = (ℓ q,D f (x, •))(z), with the corresponding natural domains.It is clear that both −ℓ q,P has a bounded H ∞ -calculus of angle 0 with domain D(A q ) = D(ℓ (x) q,D ) ∩ D(ℓ (z) q,P ) = D(L q,D ) where the last equality follows from elliptic regularity.
Step 2: −L q,D has a bounded H ∞ -calculus of angle 0. By rescaling and translation we may replace (0, a) by (−π, π).Let ℓ q,P be the Laplacian on the periodic torus T = (−π, π) (as measure space) with domain W 2,q (T).Let It is clear that Y ⊆ L 2 (T) is closed, and (λ − ℓ q,P ) −1 : Y → Y for all λ ∈ ρ(L q,D ).Now note that L q,D is the part of ℓ q,P on Y , i.e.D(L q,D ) = {f ∈ D(ℓ q,P ) ∩ Y : ℓ q,P f ∈ Y }, L q,D f = ℓ q,P f for all f ∈ D(L q,D ).Now the claim of Step 1 follows from [32,Proposition 10.2.18] and the periodic version of [32,Theorem 10.2.25].

Now the claim follows from
Step 1 and the definition of H ∞ -calculus.□ A.3.R-sectoriality for the Stokes operator.For the notion of R-boundedness of a family of linear operators we refer to [32,Chapter 8].For a family of linear operators J , the R-bound is denoted by R(J ).As in [32, Chapter 10], we said that operator T on a Banach space X is said to be R-sectorial if there exists ϕ ∈ (0, π) such that ρ The R-sectoriality angle is the infimum over all ϕ ∈ (0, π) for which the above holds.The main result of this subsection reads as follows.
Proof.Fix q ∈ (1, ∞).For simplicity we first prove the statement for a shifted Stokes operator and in a second step we conclude by a simple translation argument.

t 0 b4. Interior Regularity 4 . 1 .
(v (s) + w (s) , ϕ, v (s) + w (s)) ds = ⟨u 0 , ϕ⟩ − t 0 ⟨v (s) , Aϕ⟩ ds P − a.s.(3.6) for every ϕ ∈ D (A) and t ∈ [0, T ].Proof of Theorem 1.4.It follows immediately combining Proposition 3.1, Theorem 3.3 and Remark 3.4.□Stokes Equations.As in the proof of Theorem 1.4, by a stopping time argument we may assume that h b is also L p (Ω)-integrable, cf. the beginning of the proof of Proposition 3.1.This fact will be used below without further mentioning it.We start showing a lemma, concerning the relation between the mild and the weak formulation of (1.3) defined below.Definition 4.1.Let Hypothesis 1.1 be satisfied.A stochastic process w is a weak solution of (1.3) if it is F-progressively measurable with P − a.s.paths in w ∈ C([0, T ]; H) ∩ L 4 (0, T ; L 4 ) and for each ϕ ∈ D(A)

.
Step 3 in [37, Theorem 13.1] the thesis follows.□ Proof of Theorem 1.7.The claim follows by Corollary 4.4, Corollary 4.6 and a localization argument.Moreover, to obtain the claimed smoothness up to time t = T , let us consider the extension by 0 of h b on [0, T + 1], i.e.h b (t) = h b (t), if t ∈ (0, T ), 0, if t ∈ (T, T + 1).Let u be the unique weak solution (1.2) provided by Theorem 1.4 with T replaced by T + 1.Then, by Corollary 4.4, Corollary 4.6 and a standard covering argument, for all t 0 ∈ (0, T ),O 0 ⊂ O such that dist(O 0 , ∂O) > 0, u ∈ C([t 0 , T ]; C ∞ (O 0 ; R 2 )) P − a.s.(4.13)Now, let u be the unique weak solution of (1.2) provided by Theorem 1.4.By uniqueness, we have u = u| [0,T ] and the conclusion follows from(4.13).□ in the x direction and consider ϕ = τ h τ −h u as a test function in (A.3), whereτ h v = v(x+h,z)−v(x,z)h Then by change of variables it follows that∥τ −h ∇u∥ ≤ C∥f ∥.The relation above implies that∥∂ x ∇u∥ ≤ C∥f ∥.Let us now consider ϕ = ∂ x ψ, ψ ∈ D(O) as test function in (A.3).Therefore arguing as above it follows that ∂ x π ∈ L 2 (O) and ∥∂ x π∥ L 2 ≲ ∥f ∥.Since equation (A.4) is satisfied in the sense of distribution and u is divergence free it follows that

A. 1 .
Step 1: A and B are a positive self-adjoint operators.As above we only discuss the operator A. The positivity of A is clear.Next, note that, integrating by parts⟨Au, v⟩ = ⟨u, Av⟩ ∀u, v ∈ D(A).This means that A is symmetric.It remains to show that D(A * ) = D(A) and ∀u ∈ D(A), A * u = Au.By definition D(A * ) = {u ∈ H : F : D(A) ⊆ H → R, F (v) = ⟨u, Av⟩ has a linear bounded extension on H}.For each u ∈ D(A * ), F (v) = ⟨u, Av⟩ = ⟨f u , v⟩ therefore A * u = f u .In particular, ∀u ∈ D(A * ) ⟨u, Av⟩ = ⟨A * u, v⟩.Thanks to the fact that A is symmetric we haveD(A) ⊆ D(A * ).Given now v ∈ D(A * ), f v = A * v ∈ H,let us consider the boundary value problem (A.2) with forcing term equal to f v .By Proposition A.2 it has a unique solution (w, π) ∈ D(A) × H 1 (O), this implies that Aw = f v = A * v.For each z ∈ H, let us consider the boundary value problem (A.2) with forcing term equal to z.By Proposition A.2 it exists a unique S z ∈ D(A) such that As z = z.Therefore ⟨z, w − v⟩ = ⟨As z , w − v⟩ = ⟨s z , Aw − A * v⟩ = 0 thanks to the fact that A is symmetric.Since z is arbitrary, then v = w and the claim follows.Step 2: Proof of (A.1).We begin by proving the first identity in (A.1).Note that D(B γ ) = D((B * ) γ ) for γ < 1/2 follows from [34, Theorem 1.1] and Step 1.By Step 1 and [32, Proposition 10.2.23], B has bounded H ∞ -calculus and in particular B has the bounded imaginary powers property, [43, Subsection 3.4].By [43, Theorem 3.3.7],D( have bounded H ∞ -calculus of angle 0. Now by sum of commuting operators [43, Corollary 4.5.8], the sum operator −A q := −ℓ (A) ⊆ {λ ∈ C | | arg λ| ≥ π − ϕ} and R(λ(λ − T ) −1 | | arg λ| > π − ϕ) < ∞.