The stochastic primitive equations with transport noise and turbulent pressure

In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal $L^2$-regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well.

geophysical flows used to describe oceanic and atmospheric dynamics. They are derived from the Navier-Stokes equations in domains where the vertical scale is much smaller than the horizontal scale by the small aspect ratio limit. Detailed information on the geophysical background for the various versions of the deterministic primitive equations are given e.g. in [Ped87,Val06]. The introduction of additive and multiplicative noise into models for geophysical flows can be used on the one hand to account for numerical and empirical uncertainties and errors and on the other hand as subgrid-scale parameterizations for data assimilation, and ensemble prediction as described in the review articles [Del04, FOB + 14, Pal19].
In this paper we are mainly concerned with stochastic perturbations of transport type. In the study of turbulent flows, transport noise has been introduced by R.H. Kraichanan in [Kra68,Kra94] and has been widely studied in the context of stochastic Navier-Stokes equations, see [MR01,MR04] for a physical justification and also [BCF91,BCF92,MR05,Fla08,AV21] and the references therein for related mathematical results. The aim of this paper is to give a systematic and detailed treatment of transport noise in the context of the primitive equations.
In describe the deterministic effect of the turbulent pressure. Finally, F v , F θ , G v,n and G θ,n are given maps depending on v, θ, ∇v and ∇θ describing deterministic and stochastic forces also taking into account lower order effects like the Coriolis force. For a physical justification of the occurrence of the turbulent pressure P n , i.e., a pressure term within the stochastic integral and for the related deterministic contribution ∂ γ P , we refer to [MR01] and [ In the horizontal directions periodicity is assumed. The results of the current paper also hold in case (1.2) are replaced by periodic boundary conditions, see Remark 3.7(c). When modeling the ocean, the system (1.1) is expanded by an equation for the salinity. This leads to terms resembling the ones for the temperature, and this does not lead to additional mathematical difficulties or more restrictive assumptions. Therefore we omit this coupling here to concentrate on the main features.
The mathematical analysis of the deterministic primitive equations (i.e., β n t = 0 in (1.1)) has been pioneered by J.L. Lions, R. Teman and S. Wang in a series of articles [LTW92a,LTW92b,LTW93]. There the existence of a global, weak solution to the primitive equations is proven for initial data in v 0 ∈ L 2 (O) and θ 0 ∈ L 2 (O). The uniqueness of these weak solutions remains an open problem until today, and only under additional regularity assumptions in the vertical direction they are known to be unique (see e.g. [Ju17]).
A landmark result on the global strong well-posedness of the deterministic primitive equations subject to homogeneous Neumann conditions on top and bottom for initial data in H 1 (O) was shown first by C. Cao and E.S. Titi in [CT07], and independently by R.M. Kobelkov [Kob07], via L ∞ (0, T ; H 1 (O)) a priori energy estimates. For mixed Dirchlet-Neumann conditions see also [KZ07]. A different approach to the deterministic primitive equations based on evolution equations has been introduced in [HK16, GGH + 20]. This approach is based on the analysis of the hydrostatic Stokes operator and the corresponding hydrostatic Stokes semigroup. For a survey on results concerning the deterministic primitive equations using energy estimates, we refer to [LT16] and for a survey concerning the approach based on evolution equations to [HH20].
Stochastic versions of the primitive equations have been studied by several authors. A global well-posedness result for pathwise strong solutions is established for multiplicative white noise in time by A. Debussche, N. Glatt-Holtz and R. Temam in [DGHT11] and the same authors with M. Ziane in [DGHTZ12]. There, a Galerkin approach is used to first show the existence of martingale solutions, and then a pathwise uniqueness result is deduced which leads together with a Yamada-Watanabe type result to the existence of a local pathwise solutions. The global existences of solutions is then shown by energy estimates where the noise is seen as a perturbation of the linear system. There, one of the difficulties is the handling of the pressure when proving L p -estimates for p > 2. This is overcome by considering the corresponding Stokes problem with the noise term and then proving estimates for the difference of the solution of the full non-linear problem and the solution of that Stokes problem. This difference solves a random partial differential equation where analytic tools can be used to estimate the pressure term. A disadvantage of this approach is that it requires the solution of the Stokes problem to be rather smooth and transport noise cannot be included for this reason.
In the recent work [BS21] by Z. Brzeźniak and J. Slavík a similar approach is used for the local existence, but instead of considering the Stokes problem, they impose conditions on the noise such that it does not act directly on the pressure when turning to the question of global existence. Hence, by using a hydrostatic version of the Helmholtz projection, they can apply deterministic estimates to the pressure. Transport noise acting on the full velocity field is therefore not included, only the vertical average of v can be transported by the noise. By our approach we can overcome both drawbacks at once, we can handle full transport noise acting directly on the pressure.
Let us mention some further results on the stochastic primitive equations. For additive noise there is a transformation such that the probabilistic dependence turns into a parameter for a deterministic system. For this case also the existence of a random pull-back attractor is known (see e.g. [GH09]). Logarithmic moment bounds in H 2 (O) are obtained in [GHKVZ14] and used to prove the existence of ergodic invariant measures in H 1 (O). A construction of weak-martingale solutions, that means martingale solutions the regularity of which in space and time is the one of a weak solution, by an implicit Euler scheme is given in [GHTW17]. Large deviation principles are known for small multiplicative noise (see e.g. [DZZ17]) and small times (see e.g. [DZ18]), for an extension to transport noise and moderate deviation principles see [Sla21]. The existence of a Markov selection is proven in [DZ17] for additive noise. For results in two dimensions we refer to [GHT11] and the references therein.
Aiming for noise as rough as possible we will first consider a strong-weak setting when investigating the system (1.1), meaning that the equations for v hold in the strong PDE sense and the one for θ in a weak sense, for the precise definitions of strong-weak solutions see Definition 3.3. Probabilistically, we are concerned with strong solutions. The reason for not considering both equations in the weak sense is that already in the deterministic case the uniqueness issue for the weak velocity equation is unsolved. We also investigate the strong-strong setting, see Definition 6.3 for this notion of solution, since this setting is the one for which most deterministic results have been proven. The main result is the global existence of solutions, see Theorem 3.6 for the strong-weak setting and Theorem 6.7 for the strong-strong setting. For the readers convenience we state here a simplified version. We write φ j := (φ j n ) n≥1 , ψ j := (ψ j n ) n≥1 , γ ℓ,m := (γ ℓ,m n ) n≥1 and R + = (0, ∞). Main Result (Simplified version). Let κ be constant, G k v,n = G θ,n = 0, F θ = 0, and F v = k 0 (v 2 , −v 1 ) for k 0 ∈ R be the Coriolis force. For all n ≥ 1 let the maps φ n , ψ n : R + × Ω × O → R 3 and γ n : R + × Ω × T 2 → R 2×2 be P ⊗ B-measurable, and let for some δ > 0 and all j ∈ {1, 2, 3}, ℓ, m ∈ {1, 2} be φ j ∈ L ∞ (R + × Ω; H 1,3+δ (O; ℓ 2 )), ψ j ∈ L ∞ (R + × Ω × O; ℓ 2 ), and γ ℓ,m ∈ L ∞ (R + × Ω; L 3+δ (T 2 ; ℓ 2 )), where φ 1 n and φ 2 n are assumed to be independent of x 3 . Furthermore, assume that there exists ν ∈ (0, 2) such that, almost surely (a.s.) for all t ∈ R + , x ∈ O and ξ ∈ R 3 the parabolicity conditions (Ω; X), and the notation for the function spaces see Section 2. In the above, we have not specified the unknowns w, P and P n as they are uniquely determined by v and θ due to the divergence free condition and the hydrostatic Helmholtz projection. Moreover, replacing the regularity assumption on ψ j by ψ j ∈ L ∞ (R + × Ω; H 1,3+δ (O; ℓ 2 )) and considering θ 0 ∈ L 0 F0 (Ω; H 1 (O)) we obtain the analogous result in the strong-strong setting. Let us compare our result with the above mentioned Kraichanan's turbulence theory. There one usually assumes that, for some (typically small) γ > 0, (1.4) φ j n ∈ H 3/2+γ (O) for all j ∈ {1, 2, 3} and n ≥ 1, cf. e.g. [MR05,Equation (1.3)]. Since H 3/2+γ (O) ֒→ H 1,3+δ (O) for some δ > 0 by Sobolev embedding, our noise is consistent with the regularity of the reproducing kernel Hilbert space of the Kraichanan noise. However, taking into account the summability in n ≥ 1 required in our main results, we can cover only the case of regular Krainchan noise. We refer to [GY21, Section 5] and the references therein for the terminology. For the relevance of Kraichnan's noise in the context of geophysics we refer to [Del04].
To prove our main results we take another point of view on the problem than in [BS21] and [DGHTZ12]. Here, we interpret the transport part of the noise as a part of the linearized system, and we only need to impose conditions guaranteeing that this linearization is parabolic. Compared to [BS21,DGHTZ12], this makes it possible to consider noise that transports the full velocity field, and moreover this leads even to weaker assumptions than in [BS21] and [DGHTZ12] in the setting where the noise is such that their results apply. The only smallness condition in our result is the parabolicity condition, which is optimal in the sense that when dropping it the system is not parabolic any more and thus loosens its smoothing properties. This condition origins already in the local existence theory and is by far weaker than the smallness conditions in [BS21,DGHTZ12], where the noise is handled as a nonlinear perturbation of the deterministic system. Also, to deduce the global existence of solutions in our case, no additional smallness has to be assumed compared to the local existence.
The proof of the local existence in Theorem 3.4 is based on the theory of critical spaces for stochastic evolution equations developed by the M.C. Veraar and the first author in [AV22a,AV22b]. To apply these results, we need to study the stochastic maximal L 2 -regularity estimates for the linearized problem elaborated in Section 4. The global existence result Theorem 3.6 is then obtained from the blow-up criteria of Theorem 3.4(2) and suitable energy estimates obtained in the spirit of C. Cao and E.S. Titi [CT07]. Here we actually follow the approach taken by the T. Kashiwabara and the second author in [HK16] (see also [HH20]) where the L 6 -estimates proven in [CT07] are replaced by the (apparently) weaker L 4 -estimates. The deterministic estimates are proven by splitting the velocity field into its vertical average v and the remainder v = v − v. A crucial observation is that the deterministic part of the turbulent pressure ∂ γ P does not appear in the equations for v since it is x 3independent in case that γ ℓ,m n is also x 3 -independent. Otherwise, the L 4 -estimate for v could not be shown in this way since ∂ γ P is a non-local operator in v.
Our noise and the corresponding stochastic integrals are in Itô-form, but in fluid mechanics, and in particular for geophysical flows, also the Stratonovich formulation is relevant, and it is seen as a more realistic model see e.g. [BF20, HL84, FOB + 14, Wen14] and the references therein, and in [FP22,DP22] transport noise of Stratonovich type in fluid dynamical models has been rigorously justified from additive noise by multiscale arguments. Also the modelling in [MR01] and [MR04] is based on a Stratonovich type of noise to describe the turbulent part of the velocity field which is then translated into an Itô formulation. To include such types of noise directly, we will consider the primitive equations with Stratonovich noise, see system (8.1). In a preparatory step we first extend our result on the Itô system (1.1) to the case of non-homogeneous viscosity and conductivity in Theorems 7.3 and 7.5. Based on a Stratonovich to Itô transformation we then can use these results to infer the local and global existence of the primitive equations with Stratonovich type noise in Theorems 8.3 and 8.5, respectively.
Overview. This paper is organized as follows. In Section 2 the notation is fixed and a reformulated version of the stochastic primitive equations is given. In Section 3 we give the precise notion of solution in the strong-weak setting and present the main result for this case. In Section 4 we consider a linearized system for the turbulent hydrostatic Stokes system with temperature, and show that it admits stochastic maximal L 2 -regularity. The proofs of the theorems from Section 3 are carried out in Section 5. The strong-strong setting is investigated in Section 6. In Section 7 we generalize our results to the case of non-homogeneous viscosity and conductivity. Finally, in Section 8 we show how our results imply also the well-posedness of the primitive equations with Stratonovich noise.
We also use the standard notations Next we introduce the function spaces for the velocity field. As a first step we introduce the two-dimensional Helmholtz projection denoted by P H acting on the horizontal variables Then the Helmholtz projection is given by The hydrostatic Helmholtz projection P : and its complementary projection is given by Qf One can check that P is an orthonormal projection on L 2 (O; R 2 ), and by construction, div H 0 −h (Pf (·, z))dz = 0 holds in the distributional sense for all f ∈ L 2 (O; R 2 ). Let be endowed with the norm f L 2 (O) := f L 2 (O;R 2 ) and for all k ≥ 1 we set If no confusion seems likely, we write simply L 2 , H k , H k , L 2 (ℓ 2 ) and H k ( ) and H k (O; ℓ 2 (N; R m )), where, we use the short hand notation ℓ 2 for ℓ 2 (N; R m ) or l 2 (N). The dual space of H 1 (O) is denoted by (H 1 (O)) * .
2.2. Probabilistic notation and function spaces. Here we collect the main probabilistic notation. Throughout the paper we fix a filtered probability space Moreover, (β n ) n≥1 = (β n t : t ≥ 0) n≥1 denotes a sequence of standard independent Brownian motions on the above mentioned probability space. We will denote by B ℓ 2 the ℓ 2 -cylindrical Brownian motion uniquely induced by (β n ) n≥1 via By P and B we denote the progressive and the Borel σ-algebra, respectively. Moreover, we say that a map Φ : (Ω) we denote the space of F 0 -measureable functions and by L 2 P the L 2 -space with respect to the progressive σ-algebra.
2.3. Reformulation of the primitive equations. As it is well-known the primitive equations can be formulated equivalently in terms of the unknown v = (v k ) 2 k=1 : [0, ∞) × Ω × O → R 2 which contains only the first two components of the unknown velocity field u. Indeed, the divergence-free condition and (1.3) are equivalent to a.s. for all t ∈ R + and x H ∈ T 2 . Moreover, we get by integrating the third equation Thus the pressure depends linearly on the temperature θ. In the physical literature p is usually called the surface pressure. Hence, (1.1)-(1.3) turns into where w(v) is given by (2.3) and complemented with the boundary conditions (1.2).

Local and global existence in the strong-weak setting
In this section we study the stochastic primitive equations in the strong-weak setting, i.e. in case the equation for v is understood in the strong setting and the one for θ in the weak one. The latter means that equation for θ will be formulated in its natural weak (analytic) form. In Section 6, we also consider the case where both equations are understood in the strong setting (referred here as the strongstrong setting). Compared to the strong-strong setting, the choice made in this section has two basic advantage. Firstly, the energy estimates needed in our main global existence result are simpler, and secondly, we can allow a rougher noise in the equation for the temperature θ.
We begin by reformulating the problem. Applying the hydrostatic Helmholtz projection P to the first equation in (2.4) it is, at least formally, equivalent to Here α ∈ R is given, w(v) is as in (2.3) and a.s. for all t ∈ R + , Finally, let us note that in the stochastic part of the equation for the velocity field v, (in general) the operator P cannot be removed since it may happen that div H 3.1. Main assumptions and definitions. We begin by listing the main assumptions which are in force in this section.
Assumption 3.1. There exist M, δ > 0 for which the following hold.
A similar reformulation holds for the condition on ψ. In particular, (5) is optimal in the parabolic setting. (d) (7) contains the optimal growth assumptions on the nonlinearities which ensure existence and uniqueness of (local) solutions for data cf. the proof of Theorem 3.4 in Subsection 5.1.
To formulate (3.1)-(3.2) in the strong-weak setting, we regard the equation for θ in its natural weak analytic formulation on the dual space (H 1 (O)) * . To this end, the basic observation is given by the following formal integration by parts . Note that the volume and boundary integrals disappear since div H v + ∂ 3 w = 0 and w(·, 0) = w(·, −h) = 0 on T 2 , respectively. The right hand side in (3.4) naturally defines an element in (H 1 (O)) * by setting Below, we will use the more suggestive notation div H (vθ) + ∂ 3 (w(v)θ) = T θ . To complete the reformulation of the equation for θ it remains to replace the Laplace operator ∆ by its weak formulation ∆ w R in case of Robin boundary conditions, i.e. (3.6) where ·, · denotes the duality pairing for H 1 (O) and (H 1 (O)) * . Note that the above definition is consistent with a formal integration by parts using the Robin boundary conditions for θ in (3.2). Since the trace operator f → f | T 2 ×{0} is bounded on H 1 (O) with values in L 2 (T 2 ), the previous definition in (3.6) makes sense.
Next we state our main result of this section concerning global existence of solutions to (3.1)-(3.2). To this end, we also need the following assumptions.
Then the L 2 -maximal strong-weak solution ((v, θ), τ ) to (3.1)-(3.2) provided by Theorem 3.4 is global in time, i.e. τ = ∞ a.s. In particular The proof of Theorems 3.4 and 3.6 will be given in Subsections 5.1 and 5.2, respectively. As a key tool in the proofs, we need suitable estimates for the linearized problem of (3.1)-(3.2) which will be proven in Section 4. Before giving the proofs, below we collect some comments on Assumption 3.5.
(a) Assumption 3.5(1) contains additional assumptions on φ 1 n , φ 2 n but not on φ 3 n as compared to Assumption 3.1. Roughly speaking, this means that the transport noise can be very rough in the vertical direction while the horizontal part is two-dimensional. (b) Taking ξ = (ξ 1 , ξ 2 , 0) in Assumption 3.1(5) we also have that there exists ν ∈ (0, 2) such that, a.s. for all x ∈ O, t ∈ R + and ξ ∈ R 2 , This implies that we also have parabolicity for the subsystem (5.21) below obtained from the first equation in (3.1) after averaging in the x 3 -variable. (c) (Periodic boundary conditions). The results of Theorems 3.4 and 3.6 also hold in case the boundary conditions (3.2) are replaced by the periodic ones. To see this it is enough to ignore the boundary terms appearing in the integration by part arguments in the proofs below. The same applies to the results of Sections 6-8. For brevity, we do not repeat this observation in the following.
The proof of Proposition 4.1 will be given in Section 4.2 below.

4.2.
Proof of Proposition 4.1. Here we prove Proposition 4.1. To focus on the main difficulties, we only discuss the case γ ℓ,m n ≡ 0 since the operator P γ,φ v can be shown to be of lower order type provided Assumption 3.1(2) holds. For details, we refer to Remark 4.3 below.
Proof of Proposition 4.1 -Case γ ℓ,m n ≡ 0. Let us set To prove the claim, we employ the method of continuity as in [AV22b, Proposition 3.13]. Let us denote the strong Neumann Laplacian by Here we use that P∆ N = ∆ N P = ∆ N . It is well-known that ∆ N is self-adjoint. Similarly, by form methods, one can check that the weak Robin Laplacian ∆ w R defined in (3.6) is self-adjoint as well. Thus, it is well-established that ((−∆ N , −∆ w R ), 0) ∈ SMR • 2 (T ), see e.g. [DPZ92, Theorem 6.14] which applies up to a shift, and compare also with [AV22a, Section 3.2].
For all λ ∈ [0, 1] and U = (v, θ) ∈ X 1 , we set Let L 2 (ℓ 2 , X 1/2 ) be the space of all Hilbert-Schmidt operators endowed with its natural norm. By the previous considerations and the method of continuity in [AV22b, Proposition 3.13 and Remark 3.14], it remains to prove the existence of C > 0 such that, for each stopping time τ : and each L 2 -strong-weak solution We split the proof of (4.9) into two steps. The key observation is that v does not appear in the equation for θ. Thus, first we prove an estimate for θ and then we use the latter to obtain the estimate for v.
Since 1 + ∆ w R is a sectorial operator (see e.g. [HNVW17, Definition 10.1.1] for the definition of this notion), where I is the identity operator. Since D(∆ w R ) = H 1 and D((∆ w R ) 1/2 ) = L 2 , (4.11) also holds with (H 1 ) * replaced by either H 1 or L 2 . For each k ≥ 1, let Note that E k and ∆ w R commute on (H 1 ) * . Thus, applying E k to (4.10) we have, a.s. for all t ∈ [0, τ ], Since θ k ∈ L 2 ((0, τ ) × Ω; H 2 ) by the regularity of the strong Robin Laplacian, we have ∆ w R θ k = ∆θ k in the strong sense and we may apply Itô's formula to compute where we also integrated by parts and used that ). Thus, we may take k → ∞ in the previous identity, and obtain a.s. for all t ∈ [0, T ], (4.14) where, as above, ·, · denotes the duality pairing for H 1 and (H 1 ) * . Let 0 ≤ s ≤ t ≤ T . Note that for all r ∈ (0, 1 2 ) by the continuity of the trace map, we have Let ν be as in Assumption 3.1 and fix ν ′ ∈ (ν, 2). Thus, for some c ν > 0, Since ν ′ < 2 and E[III T ] = 0, taking the expectation in (4.14) and using standard interpolation inequalities, one has where C 0 > 0 is independent of τ, f θ , g θ and λ.
s., there is no need for an approximation argument), an integration by parts and using the argument performed in Step 1 and (4.20), one can check that, L 2 (0,τ ;H 1 (ℓ 2 )) , and the implicit constant in (4.21) is independent of λ, θ, f v , f θ , g v and g θ .
To complete the proof of this step, it remains to show where as above, the implicit constant in (4.21) is independent of λ, θ, f v , f θ , g v and g θ . To this end, we apply Itô's formula to v → ∇v 2 L 2 . Set v τ (t) := v(t ∧ τ ). Using an approximation argument similar to Step 1 and an integration by parts, one has a.s. for all t ∈ [0, T ], Thus, taking t = T and the expected value in the previous formula we have By (4.20) and the Cauchy-Schwartz inequality we have, for all ε > 0, We claim that, for some c < 2 and C > 0 (both independent of λ, It is easy to see that, if (4.25) holds, then (4.22) follows by combining (4.21), (4.23) and (4.24) with ε ∈ (0, 2 − c).

PRIMITIVE EQUATIONS WITH TRANSPORT NOISE AND TURBULENT PRESSURE 21
Let us note that (4.27) has to be used twice. First to show that for ℓ, m ∈ {1, 2}, which provides (4.21) and the second one to show that which, in combination with the Kadlec formula of Lemma A.1, yields (4.25). In both cases, one chooses ε > 0 small enough to absorb the leading terms in the LHS of the corresponding estimate.

Proof of the main results in the strong-weak setting
In this section we have collected the proofs of our main results in the strong-weak setting. Namely, the proofs of Theorems 3.4 and 3.6 are given in Subsection 5.1 and 5.2, respectively. 5.1. Proof of Theorem 3.4. To prove Theorem 3.4 we employ the results in [AV22a,AV22b], and therefore we reformulate (3.1) as a semilinear stochastic evolution equation. To this end, let ) be as in Remark 4.2(a) and for all U := (v, θ) ∈ X 1 we set for the non-linearities where w(v) is as in (2.3) and 3). With the above notation, (3.1)-(3.2) can be reformulated as a stochastic evolution equation on X 0 of the form Here [·, ·] β denotes the complex interpolation functor.
Step 1: Let us begin by noticing that F 1 is a bilinear map, and therefore to prove (5.2) it is enough to consider the case (v ′ , θ ′ ) = 0. To this end, we note that where in the last step we used Sobolev embeddings. The remaining terms in F 1 can be estimated as in [HK16, Lemma 5.1] for p = 2. For the reader's convenience we include some details. Note that, for all v ∈ H 2 , Analogously, one can check that θ L 2 (−h,0;L 4 (T 2 )) θ H 1/2 for all θ ∈ H 1 . Thus, using the previous estimates, we get By (5.3), the previous estimates yield (5.2).
can be estimated as in Step 2, we only consider the G v,n -term. By the chain rule Arguing as in Step 2, one can check that to estimate ( By Assumption 3.1(7) and the Hölder inequality, for all v, v ′ ∈ H 2 , I(v, v ′ ) is less or equal than ; where in the last estimate we have used Sobolev embeddings. Since X 2/3 ֒→ H 4/3 × H 1/3 , the claim of this step follows.
Step 4: Conclusion. Due to Steps 1-3 and Proposition 4.1, the existence of an L 2 -maximal strong-weak solution to (3.1) and Theorem 3.4(1) follow from [AV22a, Theorem 4.8], where we set m F = 3 and m G = 2 which correspond the to number of different terms on the right hand side when estimating F and G, respectively, as done in Steps 1-3. Moreover, each of these five terms involves numbers ρ j describing the power in the estimates of the non-linearities, and β j , ϕ j indicating the order in the estimates of the non-linearities in terms of interpolation spaces X βj and X ϕj , respectively. Here, by Steps 1-3 we can chose in [AV22a, Theorem 4.8] ρ 1 = 1, ρ 2 = ρ 4 = 4, ρ 3 = 2/3, ρ 5 = 2, and where one also uses that Theorem 3.4(2) follows from [AV22b, Theorem 4.11] and Proposition 4.1.
5.2. Proof of Theorem 3.6. The key ingredient in the proof is the following energy estimate. Recall that N k has been defined in (3.9).
The proof of Proposition 5.1 will be divided into two parts. Firstly, in Subsection 5.2.1 we prove a standard L 2 -energy estimate for L 2 -maximal strong-weak solutions to (3.1)-(3.2) and in Subsection 5.2.2 we prove Proposition 5.1. 5.2.1. An L 2 -energy estimate. The aim of this subsection is to prove the following result. Recall that N k is as in (3.9).
Lemma 5.2 (L 2 -energy estimate). Let the assumptions of Proposition 5.1 be satisfied. Let ((v, θ), τ ) be the L 2 -maximal strong-weak solution to (3.1)-(3.2) provided by Theorem 3.4. Then for each T ∈ (0, ∞) there exists c T > 0 independent of v 0 , v, θ 0 , θ such that Proof. For the reader's convenience we split the proof into several steps. Below T ∈ (0, ∞) is fixed and ·, · denotes the duality pairing for H 1 and (H 1 ) * . Recall that L 2 ֒→ (H 1 ) * and the embedding is given by ϕ, f : where inf ∅ := τ . By progressive measurability of (v, θ) (see Definition 3.3) and Theorem 3.4(1), for each k ≥ 1, τ k is a stopping time and lim k→∞ P(τ k = τ ) = 1. Therefore, by Fatou's lemma, it is enough to prove (5.6) for τ replaced by τ k provided c T is independent of k ≥ 1. Note that uniformly in Ω. In particular, all the integrals appearing below are finite. By Gronwall's and Fatou's lemma, it is enough to prove the existence of c T independent of k ≥ 1 such that, for all t ∈ [0, T ], To shorten the notation, in the following steps, we set σ := τ k .
Step 2: L ∞ (L 2 )-and L 2 (H 1 )-estimate for v, see (5.12). As in Step 1, the idea is to apply Itô's formula to v → v 2 L 2 and use an argument similar to the one used in Step 1 of Proposition 4.1. As in (5.9) we have the following cancellation where J κ is as in (4.3), and where we use that the hydrostatic Helmholtz projection P is orthogonal on L 2 (O; R 2 ), in particular P L (L 2 ) = 1.

PRIMITIVE EQUATIONS WITH TRANSPORT NOISE AND TURBULENT PRESSURE 27
(ii) , where in (i) we applied the Hölder inequality, in (ii) the boundeedness of Q and in (iii) the Young's and standard interpolation inequalities.
The remaining terms can be estimated as in Step 1. Thus, choosing ε small enough, one can check that Assumption 3.5(2) yields, for all t ∈ [0, T ], where C T,v > 0 is independent of θ, θ 0 , v, v 0 and k ≥ 1.
More precisely, the above means that (5.8) follows by multiplying (5.10) by (2C T,θ ) −1 and then adding the estimate with (5.12). On the RHS of the resulting estimate the term 1 2 E σ∧t 0 ∇v(s) 2 L 2 ds appears and can be adsorbed into the LHS since σ = τ k and therefore E σ∧t 0 ∇v(s) 2 L 2 ds ≤ k a.s. by (5.7). 5.2.2. Higher order energy estimates and proof of Proposition 5.1. Through this subsection, we assume that the assumptions of Proposition 5.1 holds, and in particular T ∈ (0, ∞) is fixed.
where in (i) we used the Hölder inequality, in (ii) the embedding H 1 (ℓ 2 ) ֒→ L 6 (ℓ 2 ) and (5.14). Thus the previous estimates, Assumption 3.5(2) and (5.13) ensure that, for some C k independent of v 0 , v, 2 L 2 (0,ℓ k ;H 1 (ℓ 2 )) ≤ C k a.s. Following [CT07], we derive from (3.1) a coupled system of SPDEs for the unknowns Recall that P H denotes the Helmholtz projection on L 2 (T 2 ; R 2 ) which acts on the horizontal variable x H ∈ T 2 where x = (x H , x 3 ) ∈ O, see Subsection 2.1. Since Pv = P H v, applying the vertical avarage in (3.1) and using Assumption 3.5(1), for all k ≥ 1, (v, ℓ k ) is an L 2 -local strong solution to (5.21) where φ n,H := (φ 1 n , φ 2 n ). Here we also used that which follows from v = 0, (2.3) and an integration by parts, and by Assumption 3.5(1), Let us also note that the first equation in (5.21) and v 0 ∈ H 1 imply div H v = 0.
Here, by L 2 -local strong solution to (5.21) we understand that (v, ℓ k ) solves (5.21) in its natural integral form, cf. Definition 3.3.
Analogously, noticing that Pz − P H z = z − z for all z ∈ L 2 , one can readily check that ( v, ℓ k ) is an L 2 -local strong solution to (5.23) Here we used that ∂ 3 v = ∂ 3 v. Note also that w(v) = w( v) since div H v = 0. This fact will be used frequently in the following.
With this preparation we can prove an intermediate estimate which is the key ingredient in the proof of Proposition 5.1.

Lemma 5.3 (An intermediate estimate).
Let the assumptions of Proposition 5.1 be satisfied. Let ℓ k be as in (5.13) and let ((v, θ), τ ) be the L 2 -maximal strong-weak solution to (3.1)-(3.2). Then there exists a sequence of constants (C k ) k≥1 such that where, for each t ∈ [0, τ ), (5.25) In (5.25), with a slight abuse of notation, we wrote H k (T 2 ) instead of H k (T 2 ; R 2 ). We will use the same notation also below if no confusion seems likely.
Proof of Lemma 5.3. We begin by collecting some useful facts. By Definition 3.3, for each j ≥ 1, the following is a stopping time Note that, by Definition 3.3 and the definition of the τ j 's, To prove (5.24), it is enough to show that for each k ≥ 1 there exists C 0,k > 0 independent of j and v, v 0 such that, for each j ≥ 1 and any stopping times 0 ≤ η ≤ ξ ≤ τ j ∧ ℓ k , where N v,θ is as in (5.18). Recall the Sobolev embedding H 1 ֒→ L 6 . Thus all the integrals in (5.28) are finite due to ξ ≤ τ j and (5.27). Let us first prove the sufficiency of (5.28). Due to (5.14), for each j, k ≥ 1, Therefore the stochastic Gronwall's lemma in [GHZ09, Lemma 5.3] with τ = ℓ k ∧ τ j applies to (5.28), and it ensures the existence of C(k, T, C 0,k ) > 0 independent of j, v, v 0 such that where we also used that E[X 0 ] 1 + E v 0 4 H 1 by Sobolev embeddings. Recall that ℓ k ≤ τ a.s. for all k ≥ 1. Thus the claimed estimate follows by taking j → ∞ in (5.29) using that C in (5.29) is independent of j ≥ 1 and the second in (5.27).
The proof of (5.28) will be divided into several steps. The argument is an extension of the one in [HH20, Subsection 1.4.3] for the deterministic case. Recall that η, ξ are stopping times such that 0 ≤ η ≤ ξ ≤ τ j ∧ ℓ k a.s. for some j, k ≥ 1.
Step 3: An L ∞ t (L 4 x )-estimate for v, see (5.54) below. As in the previous step we set v η,ξ := v((· ∨ η) ∧ ξ). By the Sobolev embedding H 1 ֒→ L 4 and ξ ≤ τ j , we have v η,ξ ∈ C([0, T ]; L 4 ) a.s. and v η,ξ C([0,T ];L 4 ) ≤ j a.s. The Itô's formula applied to v → v 4 L 4 , gives a.s. for all t ∈ [0, T ], Let us remark that, to justify the above identity, one needs a standard approximation argument. More precisely to prove (5.45), one applies the Itô's formula to for all y ∈ R, m ≥ 1, and R m := m(m + 1 + ∆ N ) −1 (here ∆ N denotes the Neumann Laplacian on L 2 ) and then taking the limit as m → ∞ in the obtained equality. Since M m ∈ C 2 b (R 2 ), M m has quadratic growth at infinity, R m → I strongly in H k for k ∈ {0, 1} and D(∆ N ) ֒→ H 2 ֒→ L ∞ , the Itô's formula can be applied. By (5.26) and the fact that ξ ≤ τ j a.s., the limit as m → ∞ can be justified by recalling that H 1 ֒→ L 6 and noticing that, for all y = (y 1 , y 2 ) ∈ R 2 , i, j ∈ {1, 2}, where C > 0 is independent of m ≥ 1 and δ i,j is the Kronecker's delta.
where in the last inequality we used (5.18), (5.19) and ξ ≤ ℓ k a.s. Using the previous estimates in (5.47), we have a.s.
Note that, on the RHS of the resulting estimate, the following terms appear: (5.55) 1
The first term follows from the RHS of (5.38) and the RHS of (5.54) and the second one from the RHS of (5.38) and the RHS of (5.44). However, in the LHS of the resulting estimate we get 1
We are now in the position to prove Proposition 5.1.
Proof of Proposition 5.1. Let ℓ k and X t , Y t be as in (5.13) and (5.25), respectively. Recall that lim k→∞ P(ℓ k = τ ) = 1. By Lemma 5.3, for each k ≥ 1 there exists R k > 0 for which the stopping times As explained at the beginning of this subsection it is sufficient to prove (5.15). The idea is to use the stochastic Gronwall's lemma as in the proof of Lemma 5.3. To this end, we need a localization argument. For any j ≥ 1, let τ j be the stopping time defined as Reasoning as in the proof of Lemma 5.3, by the stochastic Gronwall's lemma [GHZ09,Lemma 5.3] and (5.56), it is enough to show the existence of C 0,k > 0 such that for each j ≥ 1 and each stopping times 0 ≤ η ≤ ξ ≤ τ j ∧ µ k , Let η, ξ be stopping time such that 0 ≤ η ≤ ξ ≤ τ j ∧ µ k . Recall that (v, ξ) is an L 2 -local solution to (5.16) (see also the text below it). Thus, by Proposition 4.1 (cf. Remark 4.2(b)), there is a constant C 0 > 0 independent of η, ξ, j, k such that where f and g n are as in (5.17). Note that the last two terms on the right hand side of (5.58) are finite due to (5.19). It remains to estimate the second and the third term on the RHS of (5.58). Writing Reasoning as in Step 1 of Lemma 5.3 (see (5.37)), one has E (v·∇ H )v 2 L 2 (η,ξ;L 2 (T 2 )) ≤ R 2 k . Moreover, by Sobolev embeddings, where in (i) we used the interpolation inequality f L 3 f 1/2 H 1 and Young's inequality.

Local and global existence in the strong-strong setting
In this section we study (2.4) in the strong-strong, i.e. in case the equations for v and θ both are understood in the strong setting. Compared to the strong-weak setting analyzed in Section 3, we need additional assumptions on ψ. However, we can allow F v and G v,n to depend on ∇θ and θ, respectively. As in Section 3, we begin by reformulating the problem (2.4). To this end, we apply the hydrostatic Helmholtz projection P to (2.4), and at least formally, (2.4) is equivalent to (6.1) complemented with the following boundary conditions Here α ∈ R is given, the subscript H stands for the horizontal part (see Subsection 2.1) and a.s. for all t ∈ R + and x = ( As in Section 3, in the stochastic part of the equation for the velocity field v, the operator P cannot be (in general) removed and the term P γ defined in (6.3) coincides with ∂ γ P n in (2.4) since Q[(φ n · ∇)v + G v,n (·, v, θ)] = ∇ P n . 6.1. Main assumptions and definitions. We begin by listing the main assumption of this section. Below we employ the notation introduced in Subsection 2.1.
Assumption 6.1. There exist M, δ > 0 for which the following hold.
(2) As in Remark 3.2(c), (4) is equivalent to the usual stochastic parabolicity and therefore (4) is optimal in the parabolic setting.
(3) (6) contains the optimal growth assumptions on the nonlinearities which allows to prove local existence for data in H 1 (O) × H 1 (O), cf. the proof of Theorem 6.4 below.
We are in position to define L 2 -strong-strong solutions to (6.1)-(6.2). Recall that B ℓ 2 is as in Subsection 2.1. For notational convenience, we set (6.5) Definition 6.3 (L 2 -strong-strong solutions). Let Assumption 6.1 be satisfied.
(1) (Pathwise regularity) There exists a sequence of stopping times (τ k ) k≥1 such that, a.s. for all k ≥ 1, one has 0 ≤ τ k ≤ τ , lim k→∞ τ k = τ and Finally, we turn our attention to the existence of global strong-strong solutions to (6.1)-(6.2). To formulate our global existence result, we need the following Assumption 6.5. Let Assumption 6.1 be satisfied.
Remark 6.6. Assumption 6.5 should be compared with Assumption 3.5. Note that (1) has already been discussed in Remark 3.7 below Assumption 3.5 and that (2) is symmetric w.r.t. v and θ. As before Assumption 6.1(4) implies the parabolicity condition from Remark 3.7(b).
Next we state our main result on global existence to (6.1)-(6.2) in the strongstrong setting. Recall that H 2 N and H 2 R have been defined in (6.5). Theorem 6.7 (Global existence). Let Assumption 6.5 be satisfied, and let v 0 ∈ L 0 F0 (Ω; H 1 (O)), and θ 0 ∈ L 0 F0 (Ω; H 1 (O)). Then the L 2 -maximal strong-strong solution ((v, θ), τ ) to (6.1)-(6.2) provided by Theorem 6.4 is global in time, i.e. τ = ∞ a.s. In particular H 1 (O)) a.s. The proofs of Theorems 6.4 and 6.7 follow the strategy used in the proof of Theorems 3.4 and 3.6. As in the proof of Theorem 3.4, to show Theorem 6.4 we employ the results in [AV22a,AV22b]. In particular, we need to prove stochastic maximal L 2 -regularity estimates for the linearization of (6.1)-(6.2). Such estimates will be proven in Subsection 6.3. The proof of Theorem 6.7 is similar to the one of Theorem 3.6 where we have followed the arguments in [CT07,HK16]. In the present case, we need to prove also L 2 t (H 2 x )-and L ∞ t (H 1 x )-estimates for the temperature θ to apply the blow-up criteria of Theorem 6.4(2). 6.3. Stochastic maximal L 2 -regularity. In this subsection we study maximal L 2 -regularity estimates for the linearized problem of (6.1)-(6.2), see [AV22a, Section 3] and the references therein for the general theory of stochastic maximal L p -regularity.
Here, we study maximal L 2 -regularity estimates for where P γ,φ and J κ are as in (4.2) and (4.3), respectively. Moreover ). Let τ be a stopping time. Recall that H 2 N and H 2 R are as in (6.5). We say that a.s. for all t ∈ [0, τ ]. Here B ℓ 2 is as in equation (2.2).
By [AV22b, Proposition 3.9 and 3.12], Proposition 6.8 also proves maximal L 2estimates with non-trivial initial data and also the starting time 0 can be replaced by any stopping time τ with values in [0, T ] where T ∈ (0, ∞).
Proof of Proposition 6.8. The proof is similar to the one of Proposition 4.1. As in the latter case we only consider the case γ ℓ,m n ≡ 0. Thus, it is enough to prove a priori estimates (uniform in λ ∈ [0, 1]) for strong solutions of the following problem As in the proof of Proposition 4.1, we first estimate the temperature θ and then we use this estimate for estimating v. Using also (4.19) one can see that the argument in Step 2 of Proposition 4.1 can be reproduced almost verbatim to get an estimate for v. Thus it remains to prove the estimate for the temperature. Let us remark that, if α = 0 in (6.2) (i.e. θ also satisfied Neumann boundary conditions), then the estimate for the temperature can be performed again as in Step 2 Proposition 4.1.
By Itô's formula applied to F α , we have a.s. for all t ∈ [0, T ] and k ≥ 1, Next, we want to take k → ∞ in the previous identity. To this end, let us recall that the trace operator H 1/2+s ∋ f → f (·, 0) ∈ L 2 (T 2 ) is bounded for all s > 0.
6.5. Proof of Theorem 6.7. The strategy of the proof is similar to the one used for Theorem 3.6. As in the proof of Theorem 3.6, the global existence result of Theorem 6.7 is a consequence of the blow-up criterion in Theorem 6.4(2) and the following energy estimate. Recall that N k has been defined in (6.7).
(1) P(µ k = τ ∧ T ) → 1 as k → ∞; (2) For each k ≥ 1 there exists C k,T > 0 (possibly depending on v 0 , v, θ 0 , θ) such that Theorem 6.7 follows from Proposition 6.9 in the same way as Theorem 3.6 follows from Proposition 5.1. To avoid repetitions, we only give a sketch of the proof of Proposition 6.9, since it is an easy extension of the one given for Proposition 5.1.
Due to (6.20), one can check that the proof of Proposition 5.1 can be repeated also in the strong-strong setting. In particular, there exists a sequence of stopping times (µ ′ k ) k≥1 with values in [0, T ] such that, a.s. for all k ≥ 1, one has µ ′ k ≤ τ , lim k→∞ P(µ ′ k = τ ) = 1 and where c k,T is a constant (possibly) depending on v 0 , v and k ≥ 1.

Inhomogeneous viscosity and conductivity
In this section we show how the results of the Sections 3 and 6 can be extended to the case where the Laplacians ∆ appearing in the first two equations in (2.4) are replaced by elliptic second order differential operators with (t, ω, x)-dependent coefficients which will be denoted by L v and L θ , respectively. In the strong-weak setting, to accommodate the weak setting for the θ-equation, L θ is chosen to be a differential operator in divergence form.
Differential operators with (t, ω, x)-dependent coefficients can be useful to model inhomogeneous viscosity of the fluid and/or thermal conductivity. Moreover, if one considers the stochastic primitive equations with transport noise in Stratonovich form (see Section 8 below), then differential operators with (t, ω, x)-dependent coefficients appears naturally, and the principal part of such operators have coefficients (7.1) a i,j φ := δ i,j + 1 2 n≥1 φ j n φ i n , and a i,j ψ := δ i,j + 1 2 n≥1 ψ j n ψ i n , respectively. Here δ i,j is Kronecker's delta and i, j ∈ {1, 2, 3}. Let us stress that the Stratonovich formulation is often used in the physical literature (see e.g. [HL84,FOB + 14,Wen14]) and can be studied using Itô's calculus by translating the Strotonivch integration into an Itô's ones plus additional correction terms. Such correction terms lead to consider the system (2.4) modified to include variable viscosity and conductivity given by (7.1). This section is organized as follows. In Subsection 7.1 we state our main results on the stochastic primitive equations in the strong-weak setting and in Subsection 7.2 we provide the corresponding proofs. For brevity, we do not state any result in the strong-strong setting. However, the latter situation can be handled by extending the estimates for the strong-weak setting as we shown in Section 6 for the case of homogeneous viscosity and/or conductivity, see Remark 7.6 below for more comments.
7.1. The strong-weak setting. Here we extend the results of Section 3 to the case of variable viscosity and/or conductivity. More precisely, here we consider the primitive equations with transport noise in the strong-weak setting: complemented with the following boundary conditions Here P γ (·, v) and w(v) are as in (3.3) and (2.3), respectively, and Here, 0-th order terms in (7.4) can be added as well under suitable integrability conditions on the coefficients. Since it turns out not to be useful when dealing with the primitive equations with Stratonovich noise (see Section 8), we will not consider such terms.
(a) By (2) and Sobolev embeddings, for all i, j ∈ {1, 2, 3} we have s. for all t ∈ R + ; (b) The regularity assumption on the trace a i,j v (t, ·, 0) in (3) is motivated by Lemma A.2 which will be needed in the proofs of the results below; (c) (4) is the usual stochastic parabolicity condition, cf. Remark 3.2(c).
The notion of L 2 -maximal strong-weak solution to (7.2)-(7.3) can be defined as in Definition 3.3 where the weak Robin Laplacian ∆ w R has to be substitute by the weak formulation of L θ , which will be denoted by L w θ and it is defined as for all θ, ϕ ∈ H 1 (O) and t ∈ R + . Here ·, · denotes the pairing in the duality (H 1 (O)) * × H 1 (O). Note that the above formula is consistent with a formal integration by part using (7.3) and the second statement in Assumption 7.1(3).
As in Subsection 3.2, for global existence we need additional assumptions.
Assumption 7.4. Let Assumption 3.5 be satisfied. Suppose that (1) a.s. for all n ≥ 1, (2) a.s. for all n ≥ 1, x H ∈ T 2 and t ∈ R + , 2}. Remarks on Assumption 3.5 can be found in Remark 3.7. Note that Assumption 7.4(1) is the analogue of Assumption 3.5(1). Note that Assumption 7.4(2) implies the first condition in Assumption 7.1(3), and it will be needed to extend the L 2estimate of Lemma 5.2 to the case of inhomogeneous viscosity and/or conductivity.
The following is an extension of Theorem 3.6.
The proofs of Theorems 7.3 and 7.5 are given in Subsection 7.2 below and consist of a variation of the one in Section 5 given for Theorems 3.4 and 3.6. The major difference is that we will use the Kaldec's formula in Lemma A.2 instead of the one in Lemma A.1.
Remark 7.6 (Inhomogeneous viscosity and/or conductivity -Strong-strong setting). As in Section 6, one sees that the results of Theorems 7.3 and 7.5 extend to the strong-strong setting under the following minor modifications: (a) The local existence result of Theorem 7.3 holds provided we take into account the following modifications.
7.2. Proof of Theorems 7.3 and 7.5. In this subsection we collect the proofs of Theorems 7.3 and 7.5. As the arguments are similar to the one used for Theorems 3.4 and 3.6, respectively, we only give a sketch.
Proof of Theorem 7.3 -Sketch. To prove Theorem 7.3 one can follow verbatim the one of Theorem 3.4 using the results in [AV22a,AV22b] once the stochastic maximal L 2 -regularity result of Proposition 4.1 holds in the present case, namely where ∆ w R and ∆ are replaced by L w θ and L v respectively. Let us note that, in the proof of Proposition 4.1, the structure of the operators plays a role in the integration by parts arguments used. Below, we provide some comments which allows one to extend the argument given in Proposition 4.1 for the case of homogeneous viscosity and/or conductivity. In particular, as in Proposition 4.1, we will assume that (v, θ) ∈ L 2 ((0, τ ) × Ω; H 2 N × H 1 ) ∩ L 2 (Ω; C([0, τ ]; H 1 × L 2 )), where τ : Ω → [0, T ] is a stopping time and T ∈ (0, ∞) is fixed. To begin we comment how to extend Step 1 in Proposition 4.1. Let E k be as in (4.12). Note that E k is symmetric since ∆ w R is self-adjoint. Thus, a.e. on [0, τ ] × Ω, . The above estimate and the Lebesgue dominated convergence theorem yield an identity similar to the one in (4.14). Note that, in all the remaining terms in (4.14), the Lebesgue dominated convergence theorem can be applied since θ ∈ L 2 ((0, τ ) × Ω; H 1 ) ∩ L 2 (Ω; C([0, τ ]; L 2 )) by assumption.
To extend the estimate in (4.26) similarly as for the θ-equation, note that, by Assumption 3.1(2) (see Remark 3.2(a)), one sees that (7.10) holds for ψ, a θ replaced by φ, a v with R v > 1 depending only on K, δ. Thus, one obtains the estimate, By the modified Kadlec's formula of Lemma A.2, one can estimate the first term on the RHS of (7.12) by 2 for any 1 < R ′ v < R v and it can be adsorbed in the LHS of the modified Itô's identity for ∇v τ due to (7.11) and that R ′ v > 1.
Proof of Theorem 7.5 -Sketch. To prove Theorem 7.5 one can follow the argument used in Subsection 5.2. Indeed, since the blow-up criteria in Theorem 7.3(2) holds, it remains to show that the energy estimate of Proposition 5.1 also holds in the case of inhomogeneous viscosity and/or conductivity. Note that the L 2 -estimate of Lemma 5.2 follows similarly where one also need to use Assumption 7.4(2) to integrate by parts in the v-equation, cf. (7.5).
Also the content of Lemma 5.3 holds in this case. For simplicity, we consider the case b j v ≡ 0, the general case is analogous. To see that Lemma 5.3 also holds in this case, note that, Assumption 7.4(1) is needed to obtain an equation for v similar to (5.21) and also in Step 3 where we estimate v 3 = ∂ 3 v to get where in (i) we used that P[ is independent of x 3 and ∂ 3 v(·, −h) = ∂ 3 v(·, 0) = 0 on T 2 . While (ii) follows by the symmetry of the matrix a = (a i,j ) 3 i,j=1 and the fact that, for all f ∈ C 3 (O) such that ∂ 3 f (·, −h) = ∂ 3 f (·, 0) = 0 on T 2 , where f 3 := ∂ 3 f and we have used Assumption 7.4(1) in the first integration by part. As a concluding remark, let us stress that the last terms in (7.13) are of lower-order type. Indeed, ∂ j a i,j ∈ H 1,3+η (O) by Assumption 7.1(2), and one can reason as in the proof of (4.26) to get, for each ε > 0, where K is as in Assumption 7.4(2). Having extended Lemmas 5.2 and 5.3, the extension of the Proposition 5.1 in the present case follows verbatim from the one given in Subsection 5.2 since Proposition 4.1 holds also in the case of inhomogeneous viscosity and/or conductivity (cf. the proof of Theorem 7.3).
In order to make this section as clear as possible, in contrast to Sections 3, 6 and 7, in (8.1) we do not consider lower order terms in the stochastic part keeping only the transport terms (φ n ·∇)v • dβ n t and (ψ n ·∇)θ • dβ n t which are the most relevant from a physical point of view (see [BF20,MR01,MR04] and the references therein) and mathematically the lower order terms are easier to handle. However, our methods can be also extended to the case of lower-order stochastic perturbations.
As common in SPDEs, the Stratonovich integration in P[(φ n · ∇)v] • dβ n t and (ψ n · ∇)θ • dβ n t will be understood as an Itô's ones plus some correction terms. As remarked at the beginning of Section 7 latter terms yield (in general) non-constant viscosity and/or conductivity and therefore (8.1) fits into the scheme of such section.
This section is organized as follows. In Subsection 8.1 we study the equations (8.1)-(8.2) in the strong-weak setting and in Subsection 8.2 we provide the corresponding proofs. For brevity, we do not give the explicit statements in the strongstrong setting. The modifications needed in the latter situation is commented on in Remark 8.7 below.
8.1. The strong-weak setting in the Stratonovich case. Here we analyze (8.1)-(8.2) in the strong-weak setting under the following Assumption 8.1. There exists M, δ > 0 for which the following hold.
As usual, under additional assumptions we obtain a global existence result.
The proofs of Theorems 8.3 and 8.5 as well as (8.4)-(8.5) will be given in Subsection 8.2 below. We conclude this subsection with a few remarks.
By the product rule the first term on the right hand side of (8.15) is equivalent to P[L φ v] dt. It remains to show that the second term on the right hand side of (8.14) is equivalent to P[P φ v] dt. For j ∈ {1, 2}, note that (φ n · ∇)∂ j P n (i) = (φ n,H · ∇ H )∂ j P n = ∂ j (φ n,H · ∇ H ) P n − where φ n,H = (φ 1 n , φ 2 n ) and in (i) we used that P n is independent of x 3 . By Assumption 8.1(4), one has P ∇ H [(φ n,H · ∇ H ) P n ] = P H ∇ H [(φ n,H · ∇ H ) P n ] = 0 and therefore Since ∂ i P n = (Q[(φ n · ∇)v]) i , the previous identity shows that the second term on the RHS of (8.14) is equivalent to P[P φ v] dt as desired.
It remains to prove Theorems 8.3 and 8.5.
Proof of Theorem 8.5. The claim follows from Theorem 7.5 noticing that Assumption 7.4 are satisfied with the choice (8.9) due to Assumption 8.4. In particular, note that Assumption 8.1(4) and 8.4(1) ensure that Assumption 7.4(1) and (2) hold, respectively.