Local martingale solutions and pathwise uniqueness for the three-dimensional stochastic inviscid primitive equations

Abstract

We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity, i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as $t>0$ and the vertical analytic radius increases as long as the solutions exist.

1 Introduction

In this paper, we consider the following three-dimensional stochastic primitive equations (PEs), which serves as one of the fundamental models in geophysical fluid dynamics,

$$\begin{aligned}&d V + (V\cdot \nabla V + w\partial _z V - \nu _h \Delta V - \nu _z \partial _{zz} V +f_0 V^\perp + \nabla p + f) dt = \sigma (V) dW , \end{aligned}$$

(1.1a)

$$\begin{aligned}&\partial _z p =0 , \end{aligned}$$

(1.1b)

$$\begin{aligned}&\nabla \cdot V + \partial _z w =0, \end{aligned}$$

(1.1c)

where the horizontal velocity field $V=(u, v)$, the vertical velocity w, and the pressure p are the unknown quantities. The non-negative constants $\nu _h$ and $\nu _z$ are horizontal and vertical viscosities, and we denote by $\nabla = (\partial _x, \partial _y)$ and $\Delta = \partial _{xx} + \partial _{yy}$. The parameter $f_0 \in {\mathbb {R}}$ stands for the speed and direction of rotation in the Coriolis force, and $V^\perp = (-v, u)$. The term $\sigma (V) dW$ represents the external forcing term driven by white noise, and f is a deterministic forcing term. The three-dimensional (3D) viscous PEs are derived by performing a formal asymptotic limit of the small aspect ratio (the ratio of the depth or the height to the horizontal length scale) from the Rayleigh-Bénard (Boussinesq) system. This limit is justified rigorously in [2, 37, 38]. Notice that we have omitted the coupling with the temperature and salinity in (1.1) to focus our attention towards difficulties arising from the ill-posedness in Sobolev spaces when $\nu _h=0$ [42].

To address the well-posedness of (1.1) in the inviscid case, i.e., $\nu _h=\nu _z=0$, one needs to overcome the following difficulties. We shall also discuss some key issues of our work below and explain how the restrictions can be relaxed when considering only vertical viscosity.

• (i) The first difficulty is that, even in the deterministic case, the PEs with $\nu _h=0$ are ill-posed in Sobolev spaces and Gevrey class of order $s>1$. For this reason, one has to work in the analytic function space. To the best of our knowledge, very little work has been done for stochastic partial differential equations in the analytic function space, as most of the results work in the classical Sobolev spaces.

• (ii) Unlike the stochastic PEs with $\nu _h>0$ where the global existence of solutions can be proven, we can only show the local existence of martingale solutions. The main reason is that, in the deterministic case, the solutions to the PEs with $\nu _h=0$ are unknown to exist globally when $\nu _z>0$, and have been shown to form singularity in finite time when $\nu _z=0$ [7, 13, 30, 48].

• (iii) Compared to the viscosity which can give “strong dissipation” by providing one gain in the derivative, the analytic framework only gives “weak dissipation” by providing one-half gain in the derivative. Therefore, we can not consider transport noise in any direction for the inviscid case, which requires one gain in the derivative. When we consider the vertical viscosity in (1.1), we can allow the existence of vertical transport noise. On the other hand, thanks to the “weak dissipation” effect from the analytic framework, one can consider the noises in a class that is larger than multiplicative noises; see Remark 2 for more details.

• (iv) In [6, 15, 43] where $\nu _h >0$, the existence of both martingale solutions (weak solution in the stochastic sense) and pathwise solutions (strong solution in the stochastic sense) has been shown. The authors first showed the existence of martingale solutions and the pathwise uniqueness to a modified system, then used a Yamada-Watanabe type argument (see [15, Proposition 2.2] or [27]) and applied a suitable stopping time to establish the existence of pathwise solutions to the original system. In fact, the “strong dissipation” from the horizontal viscosity can help control the nonlinear estimates. In contrast, the “weak dissipation” from the analytic framework is not strong enough to provide the same control on the nonlinear estimates in our case. Consequently, we are only able to show the pathwise uniqueness to the original system, but not to the modified system. This prevents us from showing the existence of pathwise solutions; see Propositions 3.6 and 4.6 and Remark 6 for more details.

• (v) For the inviscid PEs, the initial condition needs to be analytic in all directions, and the analytic radii of the solutions in all variables shrink. In terms of the case with vertical viscosity, the initial condition only needs to be analytic in the horizontal variables with Sobolev regularity in the z variable. Moreover, the solutions become analytic in z variable instantaneous when $t>0$, and the analytic radius in the z variable increases with time as long as the solutions exist (see (4.6)). This result is parallel to the deterministic case [39], and is due to the effect of vertical viscosity.

System (1.1) is usually studied under some proper boundary conditions. For example, one considers the domain to be ${\mathbb {T}}^2 \times [0,H]$ and

$$\begin{aligned} V \text { is periodic in } (x,y) \text { with period } 1, \qquad (\partial _z V,w)|_{z=0,H} = 0. \end{aligned}$$

(1.2)

Here the condition on $\partial _z V$ vanishes when $\nu _z = 0$.

In this paper, instead of ${\mathbb {T}}^2 \times [0,H]$, we consider the domain to be ${\mathbb {T}}^3$, and

$$\begin{aligned} V \text { is periodic in } (x,y,z) \text { with period } 1, \qquad V \text { is even in } z \text { and } w \text { is odd in } z.\nonumber \\ \end{aligned}$$

(1.3)

Observe that the space of periodic functions with such symmetry condition is invariant under the dynamics of system (1.1). If $H=\frac{1}{2}$, a solution to system (1.1) in ${\mathbb {T}}^3$ subject to (1.3) restricted on the horizontal channel ${\mathbb {T}}^2 \times [0,H]$ is a solution to system (1.1) subject to the boundary conditions (1.2). The same simplification has been done in [22]. Working in ${\mathbb {T}}^3$ allows us to use Fourier analysis, and makes the mathematical presentation simpler and more elegant.

Main results. The following two theorems are the main results of this paper, which concern the local existence of martingale solutions and pathwise uniqueness of the 3D stochastic hydrostatic Euler equations (2.11) and 3D stochastic hydrostatic Navier-Stokes equations (2.14). The notations in these theorems will become clear in Sect. 2.

Theorem 1.1

(Stochastic hydrostatic Euler) Suppose that the initial data of system (2.11) is given by a Borel measure $\mu _0$ satisfying (2.12) with some constant $M>0$. Let $\rho \ge M$, $\tau _0>0$, and $r>\frac{5}{2}$ be fixed. Assume and that the noise $\sigma$ and the external force f satisfy (2.2) and (2.3), respectively. Then there exist a time T, a function $\tau (t)$, and a stopping time $\eta$ (which may depend on $\rho$, $\tau _0$ and r), defined in (3.8), (3.7), and (3.30), respectively, such that there exists a local martingale solution $({\mathcal {S}}, W, V, \eta \wedge T)$ to system (2.11) in the sense of Definition 2.1. Moreover, system (2.11) also has pathwise uniqueness.

With the help of the vertical viscosity in the stochastic hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and show similar local existence of martingale solutions and pathwise uniqueness in the following theorem.

Theorem 1.2

(Stochastic hydrostatic Navier-Stokes) Suppose that the initial data of system (2.14) is given by a Borel measure $\mu _0$ satisfying (2.15) with constant $M>0$. Let $\rho \ge M$, $\tau _0>0$, and $r>\frac{5}{2}$ be fixed, and let $\gamma _0 = 0$. Assume and that the noise $\sigma$ and the external force f satisfy (2.4) and (2.5), respectively. Then there exist a time T, functions $\tau (t)$ and $\gamma (t)$, and a stopping time $\eta$ (which may depend on $\rho$, $\tau _0$ and r), defined in (4.8), (4.6), and (4.16), respectively, such that there exists a local martingale solution $({\mathcal {S}}, W, V, \eta \wedge T)$ to system (2.14) in the sense of Definition 2.2. Moreover, system (2.14) also has pathwise uniqueness.

Related literature. For the deterministic case, the global well-posedness of strong solutions to the 3D PEs with full viscosity was first established in [12], and later in [29, 32, 35]. In [8,9,10], the authors showed the global well-posedness of strong solutions to the 3D PEs with only horizontal viscosity. When $\nu _h = 0$, the inviscid PEs (i.e., $\nu _z=0$) are called the hydrostatic Euler equations, and with only vertical viscosity (i.e., $\nu _z>0$) are called the hydrostatic Navier-Stokes equations. It has been shown that the PEs with $\nu _h=0$ are linearly ill-posed in any Sobolev spaces and Gevrey class of order $s>1$ [42] (see also [28, 30]). With only vertical viscosity, to overcome the ill-posedness, one can consider additional weak dissipation [11], assume Gevrey regularity with some convex condition [21], or take the initial data to be analytic in the horizontal direction and only Sobolev in the vertical direction without any special structure [39, 41]. For the inviscid case, the ill-posedness can be overcomed by assuming either some special structures (local Rayleigh condition) on the initial data in 2D, or real analyticity in all directions for general initial data in both 2D and 3D [4, 5, 22, 26, 33, 34, 40]. While the strong solutions to the PEs with the horizontal viscosity $\nu _h>0$ exist globally in time, it has been shown that smooth solutions to the inviscid PEs can develop singularity in finite time [7, 13, 30, 48]. Whether the smooth solutions to the PEs with only vertical viscosity exist globally or blow up in finite time still remains open.

On the other hand, stochastic factors in primitive equations are essential in studying geophysical fluid dynamics. For example, introducing white noise terms to the system could account for numerical and empirical uncertainties. It can also provide probabilistic predictions, which is a range of possible scenarios associated with their likelihoods. Along this direction, the stochastic PEs with full viscosity were studied by [23, 24] in the 2D case, and by [6, 15, 16] in the 3D case. In the recent work [6], the global well-posedness of strong solutions (strong in both PDE and stochastic sense) was established with multiplicative and transport noise, under smallness assumption on the transport noise. The global existence of solutions is based on the deterministic result. With only horizontal viscosity, global existence and uniqueness of strong solutions have been established in [43], where the noise can be transported but only in the horizontal direction, and the global existence of solutions is also based on the deterministic result. The two situations above ($\nu _h>0$) allow the classical Sobolev framework as the PEs with horizontal viscosity are well-posed in Sobolev spaces in the deterministic case. Note that, in our work, as well as in many others in the literature, the stochastic term $\sigma (V) d W$ in (1.1) is in the Itô sense. If the model were understood in the Stratonovich sense, the approach performed below may still apply, by first converting it back to the Itô formalism under suitable conditions [17, 47]. This will add an correction dt term in (1.1) which contains the Fréchet derivative of $\sigma$ on V, leading to more involved assumptions and analysis.

The rest of the paper is organized as follows. In Sect. 2, we introduce the functional settings, the notations, the assumptions, and some preliminary results that will be used throughout the paper. In Sect. 3, we analyze the stochastic hydrostatic Euler equations (i.e., the inviscid case with $\nu _h=\nu _z=0$ in (1.1)), and prove Theorem 1.1. Section 4 focuses on the stochastic hydrostatic Navier-Stokes equations (i.e., the vertically viscous case with $\nu _h=0$ and $\nu _z>0$ in (1.1)) where we prove Theorem 1.2. We make conclusive remarks in Sect. 5 and provide the detailed nonlinear estimates in the appendix.

2 Preliminaries

The universal constant C appears in the paper may change from line to line. We shall use subscripts to indicate the dependence of C on other parameters, e.g., $C_r$ means that the constant C depends only on r.

2.1 Functional settings

Let ${\varvec{x}}:= ({\varvec{x}}',z) = (x_1, x_2, z)\in {\mathbb {T}}^3$, where ${\varvec{x}}'$ and z represent the horizontal and vertical variables, respectively, and ${\mathbb {T}}^3$ denotes the three-dimensional torus with unit length. Denote by

$$\begin{aligned} \Vert f \Vert :=\Vert f\Vert _{L^2({\mathbb {T}}^3)} = \left( \int _{{\mathbb {T}}^3} |f({\varvec{x}})|^2 d{\varvec{x}}\right) ^{\frac{1}{2}}, \end{aligned}$$

associated with the inner product $\langle f,g\rangle = \int _{{\mathbb {T}}^3} f({\varvec{x}})g({\varvec{x}}) d{\varvec{x}}$ for $f,g \in L^2({\mathbb {T}}^3)$. For a function $f \in L^2({\mathbb {T}}^3)$, let ${\hat{f}}_{{\varvec{k}}}$ be its Fourier coefficient such that

$$\begin{aligned} f({\varvec{x}}) = \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} {\hat{f}}_{{\varvec{k}}} e^{ i{\varvec{k}}\cdot {\varvec{x}}}, \qquad {\hat{f}}_{{\varvec{k}}} = \int _{{\mathbb {T}}^3} e^{- i{\varvec{k}}\cdot {\varvec{x}}} f({\varvec{x}}) d{\varvec{x}}. \end{aligned}$$

For $r\ge 0$, we define the following Sobolev $H^r$ norm and ${\dot{H}}^r$ semi-norm

$$\begin{aligned}&\Vert f \Vert _{H^r}:= \Big (\sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} (1+|{\varvec{k}}|^{2r}) |{\hat{f}}_{{\varvec{k}}}|^2 \Big )^{\frac{1}{2}}, \qquad \Vert f \Vert _{{\dot{H}}^r}:= \Big (\sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |{\varvec{k}}|^{2r} |{\hat{f}}_{{\varvec{k}}}|^2 \Big )^{\frac{1}{2}}, \end{aligned}$$

and we refer to [1] for more details about Sobolev spaces.

The divergence-free condition (1.1c) and the oddness of w in z imply that

$$\begin{aligned} \int _0^1 \nabla \cdot V ({\varvec{x}}',z)dz = 0. \end{aligned}$$

Moreover, we consider V even in the z variable. That is, $V\in {\mathcal {D}}_0$ where

$$\begin{aligned} {\mathcal {D}}_0 := \left\{ f\in L^2({\mathbb {T}}^3) : \int _0^1 \nabla \cdot f ({\varvec{x}}',z)dz = 0 ,\, f \text { is even in } z\right\} . \end{aligned}$$

2.1.1 Inviscid case

Let $\tau (t) \ge 0$ represent the radius of analyticity, and let $A = \sqrt{-(\Delta + \partial _{zz})}$ and $e^{\tau (t)A}$ be defined by, in terms of the Fourier coefficients,

$$\begin{aligned} \widehat{(Af)}_{{\varvec{k}}} := |{\varvec{k}}| {\hat{f}}_{{\varvec{k}}}, \ \ (\widehat{e^{\tau (t)A}f})_{{\varvec{k}}} := e^{\tau (t)|{\varvec{k}}|} {\hat{f}}_{{\varvec{k}}}, \qquad {\varvec{k}} \in 2\pi {\mathbb {Z}}^3. \end{aligned}$$

In this paper, the argument of $\tau (t)$ is frequently omitted if no confusion arises. Then for $\tau , r\ge 0$, $A^rf$ and $e^{\tau A} f$ are given by

$$\begin{aligned} A^rf := \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |{\varvec{k}}|^r {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}, \ \ e^{\tau A} f := \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} e^{\tau |{\varvec{k}}|} {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}. \end{aligned}$$

For $r\ge 0$, we define a family of normed spaces, parameterized by $\tau \ge 0$,

$$\begin{aligned} {\mathcal {D}}_{\tau ,r} := \{f\in {\mathcal {D}}_0: \Vert f\Vert _{\tau ,r} <\infty \}, \end{aligned}$$

where the equipped norm is

$$\begin{aligned} \Vert f\Vert _{\tau ,r}:=\Vert e^{\tau A} f\Vert _{H^r} = \Big (\sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} (1+|{\varvec{k}}|^{2r}) e^{2\tau |{\varvec{k}}|} |{\hat{f}}_{{\varvec{k}}}|^2 \Big )^{\frac{1}{2}}, \end{aligned}$$

and the corresponding semi-norm is

$$\begin{aligned} \Vert A^r e^{\tau A} f\Vert = \Vert e^{\tau A} f\Vert _{{\dot{H}}^r} = \Big (\sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |{\varvec{k}}|^{2r} e^{2\tau |{\varvec{k}}|} |{\hat{f}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}}. \end{aligned}$$

It is easy to see that

$$\begin{aligned} \Vert A^r e^{\tau A} f\Vert ^2 + \Vert f\Vert ^2 \le \Vert f\Vert _{\tau ,r}^2 \le 2(\Vert A^r e^{\tau A} f\Vert ^2 + \Vert f\Vert ^2). \end{aligned}$$

(2.1)

Notice that when $\tau =0$, one has ${\mathcal {D}}_{0,r} = H^r \cap {\mathcal {D}}_0.$

Remark 1

The space of analytic functions is defined as (cf. [36])

$$\begin{aligned} G^1({\mathbb {T}}^3) = \bigcup \limits _{\tau >0} {\mathcal {D}}_{\tau ,r}, \end{aligned}$$

which is a special case of the Gevrey class [18, 20, 36]. Since inviscid PEs are ill-posed in Sobolev spaces and in the Gevrey class $G^s$ of order $s>1$ [28, 30, 42], we shall focus on the Gevrey class of order $s=1$, which is the space of analytic function.

2.1.2 Vertically viscous case

For the vertically viscous case, we consider $V\in {\mathcal {D}}_{\tau ,r,\gamma ,s}$ where

$$\begin{aligned} {\mathcal {D}}_{\tau ,r,\gamma ,s} := \Big \{ f\in {\mathcal {D}}_0, \, \Vert f\Vert _{\tau ,r,\gamma ,s} < \infty \Big \}. \end{aligned}$$

The equipped norm is defined by

$$\begin{aligned} \Vert f\Vert _{\tau ,r,\gamma ,s} := \left( \sum \limits _{{\varvec{k}}'\in 2\pi {\mathbb {Z}}^2, k_3 \in 2\pi {\mathbb {Z}}} \left( 1 + |{\varvec{k}}'|^{2r} + |k_3|^{2s}\right) e^{2\tau |{\varvec{k}}'|} e^{2\gamma |k_3|} |{\hat{f}}_{{\varvec{k}}',k_3}|^2 \right) ^{\frac{1}{2}}, \end{aligned}$$

where we denote by

$$\begin{aligned} {\varvec{k}} = ({\varvec{k}}',k_3), \qquad {\hat{f}}_{{\varvec{k}}} \equiv {\hat{f}}_{{\varvec{k}}',k_3} :=\int _{{\mathbb {T}}^3} e^{-i{\varvec{k}}' \cdot {\varvec{x}}'- ik_3 z} f({\varvec{x}}',z)\, d{\varvec{x}}' dz. \end{aligned}$$

Here $\tau \ge 0$ and $\gamma \ge 0$ represent the horizontal and vertical radius of analyticity, respectively. Notice that when $\tau =\gamma =0$, ${\mathcal {D}}_{0,r,0,s} = H_{{\varvec{x}}'}^r \cap H_z^s \cap {\mathcal {D}}_0.$ For ${\varvec{k}}= ({\varvec{k}}' ,k_3) \in 2\pi ({\mathbb {Z}}^2 \times {\mathbb {Z}}), ~ \tau ,\gamma \ge 0$, let $A_h = \sqrt{-\Delta }$, $A_z=\sqrt{-\partial _{zz}}$, and $e^{\tau A_h}e^{\gamma A_z}$ be defined by, in terms of the Fourier coefficients,

$$\begin{aligned} (\widehat{A_h f})_{{\varvec{k}}} := |{\varvec{k}}'| {\hat{f}}_{{\varvec{k}}}, \quad (\widehat{A_z f})_{{\varvec{k}}} := |k_3| {\hat{f}}_{{\varvec{k}}}, \quad (\widehat{e^{\tau A_h}e^{\gamma A_z} f})_{{\varvec{k}}} = e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} {\hat{f}}_{{\varvec{k}}}. \end{aligned}$$

Then for $\tau , \gamma , r, s\ge 0$, one has

$$\begin{aligned} A_h^r f:= & {} \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |{\varvec{k}}'|^r{\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}, \quad A_z^s f := \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |k_3|^s {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}, \quad \\ e^{\tau A_h} e^{\gamma A_z} f:= & {} \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}. \end{aligned}$$

In particular, we are interested in the case when $s=r$. Notice that when $\tau =\gamma =0$ one has ${\mathcal {D}}_{0,r,0,r} = {\mathcal {D}}_{0,r}= H^r\cap {\mathcal {D}}_0.$ Moreover, we have

$$\begin{aligned} \begin{aligned} \Vert f\Vert ^2 + \Vert A^r e^{\tau A_h} e^{\gamma A_z} f\Vert ^2 \le \Vert f\Vert _{\tau ,r,\gamma ,r}^2&\le 2(\Vert f\Vert ^2 + \Vert A^r e^{\tau A_h} e^{\gamma A_z} f\Vert ^2) . \end{aligned} \end{aligned}$$

2.2 Assumptions on noise and force terms

2.2.1 Inviscid case

For the inviscid case, for any $\tau \ge 0$ and $r>\frac{3}{2}$, let $\sigma : {\mathcal {D}}_{\tau ,r+\frac{1}{2}} \rightarrow L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r})$, and assume that the noise $\sigma$ satisfies the growth conditions and the Lipschitz continuity:

$$\begin{aligned} \begin{aligned} \Vert \sigma (V)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r})} \le C \Big (1+ \Vert V\Vert ^2_{\tau ,r+\frac{1}{2}} \Big )&, \\ \Vert \sigma (V) - \sigma (V^\#)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r})} \le C \Vert V - V^\#\Vert ^2_{\tau ,r+\frac{1}{2}}&, \qquad V, V^\# \in {\mathcal {D}}_{\tau ,r+\frac{1}{2}}. \end{aligned} \end{aligned}$$

(2.2)

where $L_2(X, Y)$ denotes the space of Hilbert-Schmidt operators from X to Y and we defer the precise definition of ${\mathscr {U}}$ to Sect. 2.3.

For the force f, with $\tau _0>0$ and $r>\frac{5}{2}$, we assume that

$$\begin{aligned} f \in L^2(0,\infty ; {\mathcal {D}}_{\tau _0,r}). \end{aligned}$$

(2.3)

2.2.2 Vertically viscous case

For the vertically viscous case, for any $\tau \ge 0$, $\gamma \ge 0$, and $r>\frac{3}{2}$, let $\sigma : {\mathcal {D}}_{\tau ,r+\frac{1}{2},\gamma ,r+1} \rightarrow L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r,\gamma ,r})$, and assume that the noise $\sigma$ satisfies the growth condition and the Lipschitz continuity:

$$\begin{aligned} \begin{aligned} \Vert \sigma (V)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r,\gamma ,r})}&\le C \Big (1+ \Vert V\Vert ^2_{\tau ,r+\frac{1}{2},\gamma ,r} \Big ) + \delta ^2 \Vert V\Vert ^2_{\tau ,r+\frac{1}{2},\gamma ,r+1} , \\ \Vert \sigma (V) - \sigma (V^\#)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r,\gamma ,r})}&\le C \Vert V - V^\#\Vert ^2_{\tau ,r+\frac{1}{2},\gamma ,r} + \delta ^2 \Vert V - V^\#\Vert ^2_{\tau ,r+\frac{1}{2},\gamma ,r+1}, \\&\quad V, V^\# \in {\mathcal {D}}_{\tau ,r+\frac{1}{2},\gamma ,r+1} , \end{aligned} \end{aligned}$$

(2.4)

with $\delta$ small enough depending on $\nu _z$, as indicated in Lemma 4.1 and Proposition 4.6.

For the force f, with $\tau _0>0$, $\gamma ^* \ge \frac{\nu _z\tau _0}{8}$, and $r>\frac{5}{2}$, we assume that

$$\begin{aligned} f \in L^2(0,\infty ; {\mathcal {D}}_{\tau _0,r,\gamma ^*,r}). \end{aligned}$$

(2.5)

Notice that f has large enough analytic radius in z, which will allow the growth of analytic radius in z for the solutions.

2.2.3 Examples of noise terms

To be more precise, we give the following two examples of $\sigma$ that satisfy assumptions (2.2) and (2.4), respectively. Let ${\mathscr {U}}$ be a separable Hilbert space with an orthonormal basis $\lbrace e_k \rbrace _{k = 1}^\infty$.

Example 1

(Noise satisfies (2.2)) Suppose that $\tau \le \tau ^*$. Let $\phi _k, \psi _k, \chi _k \in {\mathcal {D}}_{\tau ^*,r}$ be such that

$$\begin{aligned} \sum _{k = 1}^\infty \Vert \phi _k\Vert _{\tau ^*,r}^2 = \theta _1^2, \quad \sum _{k=1}^\infty \Vert \psi _k\Vert _{\tau ^*,r}^2 = \theta _2^2, \quad \sum _{k=1}^\infty \Vert \chi _k\Vert _{\tau ^*,r}^2 = \kappa ^2, \end{aligned}$$

for some $\theta _1, \theta _2, \kappa \ge 0$. Let us define

$$\begin{aligned} \sigma (V) \zeta= & {} \sum _{k=1}^\infty \zeta _k \left[ \phi _k A^{\frac{1}{2}} V + \psi _k V + \chi _k \right] , \qquad \zeta = \sum \limits _{k=1}^\infty \zeta _k e_k \in {\mathscr {U}}. \end{aligned}$$

(2.6)

When $V\in {\mathcal {D}}_{\tau ,r+\frac{1}{2}}$, thanks to Lemma A.3 and since $r>\frac{3}{2}$, $\sigma$ satisfies

$$\begin{aligned} \begin{aligned} \Vert \sigma (V) \Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r})}&= \sum _{k=1}^\infty \Vert \phi _k A^{\frac{1}{2}} V + \alpha _k V + \chi _k \Vert _{\tau ,r}^2 \\&\le C_r (\theta _1^2 \Vert V \Vert _{\tau ,r+\frac{1}{2}}^2 + \theta _2^2 \Vert V \Vert _{\tau ,r}^2 + \kappa ^2) \\&\le C_{r,\theta _1,\theta _2,\kappa }(1+\Vert V \Vert _{\tau ,r+\frac{1}{2}}^2). \end{aligned} \end{aligned}$$

Replacing V by $V^\#$ in (2.6), the Lipschitz continuity can be verified:

$$\begin{aligned} \begin{aligned} \Vert \sigma (V) - \sigma (V^\#)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r})}&= \sum _{k=1}^\infty \Vert \phi _k A^{\frac{1}{2}} (V-V^\#) + \psi _k (V-V^\#) \Vert _{\tau ,r}^2 \\&\le C_r (\theta _1^2 \Vert V - V^\# \Vert _{\tau ,r+\frac{1}{2}}^2 + \theta _2^2 \Vert V - V^\# \Vert _{\tau ,r}^2 ) \\&\le C_{r,\theta _1,\theta _2}\Vert V - V^\# \Vert _{\tau ,r+\frac{1}{2}}^2. \end{aligned} \end{aligned}$$

Example 2

(Noise satisfies (2.4)) Suppose that $\tau \le \tau ^*$ and $\gamma \le \gamma ^*$. Let $\phi _k, \psi _k, \varphi _k, \xi _k, \chi _k \in {\mathcal {D}}_{\tau ^*,r,\gamma ^*,r}$ be such that

$$\begin{aligned} \begin{aligned}&\sum _{k = 1}^\infty \Vert \phi _k\Vert _{\tau ^*,r,\gamma ^*,r} ^2 = \theta _1^2, \quad \sum _{k = 1}^\infty \Vert \psi _k\Vert _{\tau ^*,r,\gamma ^*,r} ^2+ \Vert \varphi _k\Vert _{\tau ^*,r,\gamma ^*,r}^2 = \theta _2^2, \\&\sum _{k=1}^\infty \Vert \xi _k\Vert _{\tau ^*,r,\gamma ^*,r}^2 = \theta _3^2, \quad \sum _{k=1}^\infty \Vert \chi _k \Vert _{\tau ^*,r,\gamma ^*,r}^2 = \kappa ^2, \end{aligned} \end{aligned}$$

for some $\theta _1, \theta _2, \theta _3, \kappa \ge 0$. Let us define

$$\begin{aligned} \sigma (V) \zeta= & {} \sum _{k=1}^\infty \zeta _k \left[ \phi _k A_h^{\frac{1}{2}} V + \psi _k A_h^{\frac{1}{2}} A_z V + \varphi _k A_z V + \xi _k V + \chi _k \right] , \qquad \nonumber \\ \zeta= & {} \sum \limits _{k=1}^\infty \zeta _k e_k \in {\mathscr {U}}. \end{aligned}$$

(2.7)

When $V\in {\mathcal {D}}_{\tau ,r+\frac{1}{2},\gamma ,r+1}$, thanks to Lemma A.4 and since $r>\frac{3}{2}$, $\sigma$ satisfies

$$\begin{aligned} \begin{aligned} \Vert \sigma (V) \Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r,\gamma ,r})}&= \sum _{k=1}^\infty \Vert \phi _k A_h^{\frac{1}{2}} V + \psi _k A_h^{\frac{1}{2}} A_z V + \varphi _k A_z V + \xi _k V + \chi _k \Vert _{\tau ,r,\gamma ,r}^2 \\&\le C_r\left( \theta _1^2 \Vert V \Vert _{\tau ,r+\frac{1}{2},\gamma ,r}^2 + \theta _2^2 \Vert V \Vert _{\tau ,r+\frac{1}{2},\gamma ,r+1}^2 \right. \\&\left. \quad + \theta _3^2 \Vert V \Vert _{\tau ,r,\gamma ,r}^2 + \kappa ^2 \right) \\&\le C_{r,\theta _1,\theta _3,\kappa }(1+\Vert V \Vert _{\tau ,r+\frac{1}{2},\gamma ,r}^2) + C_r\theta _2^2 \Vert V \Vert _{\tau ,r+\frac{1}{2},\gamma ,r+1}^2. \end{aligned} \end{aligned}$$

Replacing V by $V^\#$ in (2.7), the Lipschitz continuity can be verified:

$$\begin{aligned} \begin{aligned}&\Vert \sigma (V) - \sigma (V^\#)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r,\gamma ,r})} \\&\quad = \sum _{k=1}^\infty \Vert \phi _k A^{\frac{1}{2}} (V-V^\#) + \psi _k A_h^{\frac{1}{2}} A_z (V-V^\#) \\&\qquad + \varphi _k A_z (V-V^\#) + \xi _k (V-V^\#) \Vert _{\tau ,r,\gamma ,r}^2 \\&\quad \le C_r\left( \theta _1^2 \Vert V - V^\# \Vert _{\tau ,r+\frac{1}{2},\gamma ,r}^2 + \theta ^2_2 \Vert V - V^\# \Vert _{\tau ,r+\frac{1}{2},\gamma ,r+1}^2 + \theta _3^2 \Vert V - V^\# \Vert _{\tau ,r,\gamma ,r}^2 \right) \\&\quad \le C_{r,\theta _1,\theta _3}\Vert V - V^\# \Vert _{\tau ,r+\frac{1}{2},\gamma ,r}^2 + C_r\theta _2^2 \Vert V - V^\# \Vert _{\tau ,r+\frac{1}{2},\gamma ,r+1}^2. \end{aligned} \end{aligned}$$

When $\theta _2 \le \frac{\delta }{\sqrt{C_r}},$ $\sigma$ satisfies (2.4).

Remark 2

The assumptions on the noise for the inviscid case allow a class of noise that is larger than the set of the multiplicative noises (e.g., setting $\theta _1 >0$ in Example 1), but does not contain the transport noise. On the other hand, the assumptions for the vertically viscous case allow the transport noise in the vertical direction (e.g., $\theta _2 >0$ in Example 2). In general, if the system has the viscosity in some directions, then the transport noise in that direction is allowed since the viscosity can control the loss of the derivative. See, for example, [6] where the transport noise in all spatial directions is allowed since they consider full viscosity, and [43] where the authors consider the transport noise only in the horizontal direction since they only have horizontal viscosity. In our inviscid case, it is therefore natural to exclude transport noise.

2.3 Stochastic preliminaries

Let ${\mathcal {S}}= \left( \Omega , {\mathcal {F}}, {\mathbb {F}}, {\mathbb {P}}\right)$ be a stochastic basis with filtration ${\mathbb {F}}= \left( {\mathcal {F}}_t \right) _{t \ge 0}$. Let ${\mathscr {U}}$ be a separable Hilbert space and let W be an ${\mathbb {F}}$-adapted cylindrical Wiener process with reproducing kernel Hilbert space ${\mathscr {U}}$ on ${\mathcal {S}}$. Let $\lbrace e_k \rbrace _{k = 1}^\infty$ be an orthonormal basis of ${\mathscr {U}}$, then W may be formally written as $W = \sum _{k=1}^\infty e_kW^k$, where $W^k$ are independent 1D Wiener processes on ${\mathcal {S}}$.

Let X be another separable Hilbert space and denote by $L_2({\mathscr {U}}, X)$ the collection of Hilber-Schmidt operators from ${\mathscr {U}}$ into X. Given a predictable process $\Phi \in L^2\bigl (\Omega ; L^2\left( 0, T; L_2\left( {\mathscr {U}}, X \right) \right) \bigr )$, one may define the stochastic integral with respect to the cylindrical Wiener process by

$$\begin{aligned} \int _0^T \Phi \, dW = \sum _{k = 1}^\infty \int _0^T \Phi e_k \, dW^k. \end{aligned}$$

Note that the integral can also be extended to $\Phi$ with $\int _0^T \Vert \Phi \Vert ^2_{L_2\left( {\mathscr {U}}, X \right) } \, dt < \infty$, ${\mathbb {P}}$-almost surely, and we refer readers to [14, Section 4] for more details.

Let us also recall the definitions of Sobolev spaces with fractional time derivative, see e.g. [45]. Let X be a separable Hilbert space and let $t > 0$, $p > 1$ and $\alpha \in (0, 1)$. We define

$$\begin{aligned} W^{\alpha , p}(0, t; X) = \left\{ u \in L^p(0, t; X) \mid \int _0^t \int _0^t \frac{\vert u(s) - u(r) \vert _X^p}{|s - r|^{1+\alpha p}} \, dr \, ds < \infty \right\} \end{aligned}$$

and equip it with the norm

$$\begin{aligned} \Vert u \Vert ^{p}_{W^{\alpha , p}(0, t; X)} = \int _0^t \vert u(s) \vert ^p_X \, ds + \int _0^t \int _0^t \frac{\vert u(s) - u(r) \vert _X^p}{|s - r|^{1+\alpha p}} \, dr \, ds. \end{aligned}$$

In the sequel, we shall repeated use two versions of the Burkholder-Davis-Gundy inequality. For $\Phi \in L^2\left( \Omega ; L^2\left( 0, T; L_2\left( {\mathscr {U}}, X\right) \right) \right)$, one has

$$\begin{aligned} {\mathbb {E}}\sup _{t \in \left[ 0, T\right] } \left| \int _0^t \Phi \, dW \right| ^r_X \le C_{r} \, {\mathbb {E}}\left( \int _0^T \Vert \Phi \Vert _{L_2({\mathscr {U}}, X)}^2 \, dt \right) ^{r/2}. \end{aligned}$$

(2.8)

Moreover, if $p \ge 2$ and $\Phi \in L^p\left( \Omega ; L^p\left( 0, T; L_2\left( {\mathscr {U}}, X\right) \right) \right)$, then

$$\begin{aligned} {\mathbb {E}}\left| \int _0^\cdot \Phi \, dW \right| ^p_{W^{\alpha , p}(0, T; X)} \le c_{p} \, {\mathbb {E}}\int _0^T \Vert \Phi \Vert _{L_2({\mathscr {U}}, X)}^p \, dt, \end{aligned}$$

(2.9)

for $\alpha \in [0, 1/2)$. For proofs, see, for instance, [31] and [19, Lemma 2.1].

2.4 Notion of solution

In this paper, we only consider martingale solutions (i.e., weak solutions in the stochastic sense), and adapt the definition from [6, 15].

2.4.1 Inviscid case

First, we consider the following inviscid version of system (1.1),

$$\begin{aligned}&d V + (V\cdot \nabla V + w\partial _z V +f_0 V^\perp + \nabla p + f)dt = \sigma (V) dW , \end{aligned}$$

(2.10a)

$$\begin{aligned}&\partial _z p =0 , \end{aligned}$$

(2.10b)

$$\begin{aligned}&\nabla \cdot V + \partial _z w =0, \end{aligned}$$

(2.10c)

$$\begin{aligned}&V(0) = V_0. \end{aligned}$$

(2.10d)

From the system above, one can write $P({\varvec{x}}')=p({\varvec{x}}',z)$ and $w=-\int _0^z \nabla \cdot V({\varvec{x}}',{\tilde{z}})d{\tilde{z}}$. Denoting by

$$\begin{aligned} Q(g,h) = g\cdot \nabla h - \left( \int _0^z (\nabla \cdot g) ({\tilde{z}}) d{\tilde{z}} \right) \partial _z h, \ \ F(g) := f_0 g^{\perp } + \nabla P + f, \end{aligned}$$

we can rewrite (2.10) as

$$\begin{aligned}&d V + \big [Q( V, V) + F(V) \big ]dt = \sigma (V) dW , \end{aligned}$$

(2.11a)

$$\begin{aligned}&V(0) = V_0. \end{aligned}$$

(2.11b)

We consider the initial data to be given by a Borel measure $\mu _0$ on ${\mathcal {D}}_{\tau _0,r}$ such that for some $M> 0$,

$$\begin{aligned} \mu _0\left( \left\{ V\in {\mathcal {D}}_{\tau _0,r}, \Vert V \Vert _{\tau _0,r} \ge M \right\} \right) = 0. \end{aligned}$$

(2.12)

In particular, this implies that for any $p \ge 1$,

$$\begin{aligned} \int _{{\mathcal {D}}_{\tau _0,r}} \Vert V \Vert _{\tau _0,r}^p \, d\mu _0(V) < \infty . \end{aligned}$$

(2.13)

Remark 3

The assumption on the initial condition (2.12) is to guarantee the stopping time (3.30) appearing in Corollary 3.5 to be positive almost surely. One could also follow the strategy in [6, 15] to weaken such assumption by considering an additional linear differential equation in the modified system (3.1). In order to simplify our presentation and focus on the difficulties discussed in Sect. 1, we take the assumption (2.12) in our paper.

Definition 2.1

Let $r>\frac{5}{2}$, $T>0$, $\tau (t)\ge 0$ for $t\in [0,T]$, and let $\mu _0$ satisfy (2.12) with some constant $M>0$. Assume that $\sigma$ and f satisfy (2.2) and (2.3), respectively. We call a quadruple $({\mathcal {S}}, W, V, \eta \wedge T)$ a local martingale solution if ${\mathcal {S}}= \left( \Omega , {\mathcal {F}}, {\mathbb {F}}, {\mathbb {P}}\right)$ is a stochastic basis, W is an ${\mathbb {F}}$-adapted cylindrical Wiener process with reproducing kernel Hilbert space ${\mathscr {U}}$, $\eta$ is an ${\mathbb {F}}$-stopping time and $V\left( \cdot \wedge \eta \right) : \Omega \times [0, T] \rightarrow {\mathcal {D}}_{\tau (t),r}$ is a progressively measurable process such that $\eta > 0$ ${\mathbb {P}}$-a.s.,

$$\begin{aligned} V\left( \cdot \wedge \eta \right) \in L^2\left( \Omega ; C\left( [0, T], {\mathcal {D}}_{\tau (t),r}\right) \right) , \qquad \mathbbm {1}_{[0, \eta ]}(\cdot ) V \in L^2\left( \Omega ; L^2\left( 0, T; {\mathcal {D}}_{\tau (t),r+\frac{1}{2}} \right) \right) , \end{aligned}$$

the law of V(0) is $\mu _0$ and V satisfies the following equality in ${\mathcal {D}}_0$ for all $t \in [0,T]$

$$\begin{aligned} V\left( t \wedge \eta \right) + \int _0^{t \wedge \eta } \big [Q(V,V)+F(V)\big ] ds = V(0) + \int _0^{t \wedge \eta } \sigma (V) \, dW. \end{aligned}$$

2.4.2 Vertically viscous case

For the vertically viscous version of system (1.1), we consider

$$\begin{aligned}&d V + \big [Q( V, V) + F(V) - \nu _z \partial _{zz} V\big ]dt = \sigma (V) dW , \end{aligned}$$

(2.14a)

$$\begin{aligned}&V(0) = V_0, \end{aligned}$$

(2.14b)

We consider the initial data to be given by a Borel measure $\mu _0$ on ${\mathcal {D}}_{\tau _0,r,0,r}$ such that for some $M>0$,

$$\begin{aligned} \mu _0\left( \left\{ V\in {\mathcal {D}}_{\tau _0,r,0,r}, \Vert V \Vert _{\tau _0,r,0,r} \ge M \right\} \right) = 0. \end{aligned}$$

(2.15)

In particular, this implies that for any $p \ge 1$,

$$\begin{aligned} \int _{{\mathcal {D}}_{\tau _0,r,0,r}} \Vert V \Vert _{\tau _0,r,0,r}^p \, d\mu _0(V) < \infty . \end{aligned}$$

(2.16)

Definition 2.2

Let $r>\frac{5}{2}$, $T>0$, $\tau (t),\gamma (t)\ge 0$ for $t\in [0,T]$, and let $\mu _0$ satisfy (2.15) with some constant $M>0$. Assume that $\sigma$ and f satisfy (2.4) and (2.5), respectively. We call a quadruple $({\mathcal {S}}, W, V, \eta \wedge T)$ a local martingale solution if ${\mathcal {S}}= \left( \Omega , {\mathcal {F}}, {\mathbb {F}}, {\mathbb {P}}\right)$ is a stochastic basis, W is an ${\mathbb {F}}$-adapted cylindrical Wiener process with reproducing kernel Hilbert space ${\mathscr {U}}$, $\eta$ is an ${\mathbb {F}}$-stopping time and $V\left( \cdot \wedge \eta \right) : \Omega \times [0, T] \rightarrow {\mathcal {D}}_{\tau (t),r,\gamma (t),r}$ is a progressively measurable process such that $\eta > 0$ ${\mathbb {P}}$-a.s.,

$$\begin{aligned} \begin{aligned}&V\left( \cdot \wedge \eta \right) \in L^2\left( \Omega ; C\left( [0, T], {\mathcal {D}}_{\tau (t),r,\gamma (t),r}\right) \right) , \\&\quad \mathbbm {1}_{[0, \eta ]}(\cdot ) A^r V \in L^2\left( \Omega ; L^2\left( 0, T; {\mathcal {D}}_{\tau (t),\frac{1}{2},\gamma (t),1}\right) \right) , \end{aligned} \end{aligned}$$

the law of V(0) is $\mu _0$ and V satisfies the following equality in ${\mathcal {D}}_0$ for all $t \in [0,T]$

$$\begin{aligned} V\left( t \wedge \eta \right) + \int _0^{t \wedge \eta } \big [Q(V,V)+F(V) - \nu _z \partial _{zz} V\big ] ds = V(0) + \int _0^{t \wedge \eta } \sigma (V) \, dW. \end{aligned}$$

2.5 Compactness theorem

We recall the following compactness result which are needed for our results. For proofs see [44, Theorem 5] and [19, Theorem 2.1], respectively.

Lemma 2.3

a) (Aubin-Lions-Simon Lemma). Let $X_2 \subset X \subset X_1$ be Banach spaces such that the embedding $X_2 \hookrightarrow \hookrightarrow X$ is compact and the embedding $X \hookrightarrow X_1$ is continuous. Let $p \in (1, \infty )$ and $\alpha \in (0, 1)$. Then the following embedding is compact

$$\begin{aligned} L^p(0, t; X_2) \cap W^{\alpha , p}(0, t; X_1) \hookrightarrow \hookrightarrow L^p(0, t; X). \end{aligned}$$

b) Let $X_2 \subset X$ be Banach spaces such that $X_2$ is reflexive and the embedding $X_2 \hookrightarrow \hookrightarrow X$ is compact. Let $\alpha \in (0, 1]$ and $p \in (1, \infty )$ be such that $\alpha p > 1$. Then the following embedding is compact

$$\begin{aligned} W^{\alpha , p}(0, t; X_2) \hookrightarrow \hookrightarrow C([0, t], X). \end{aligned}$$

3 The Inviscid case

In this section, we focus on system (2.11). In order to deal with the nonlinear terms in the energy estimates, we use the strategy introduced in [6, 15, 43] and consider the following modified system

$$\begin{aligned}&d V + \big [\theta _\rho (\Vert V\Vert _{\tau ,r})Q( V, V) + F(V) \big ]dt = \sigma (V) dW , \end{aligned}$$

(3.1a)

$$\begin{aligned}&V(0) = V_0, \end{aligned}$$

(3.1b)

where $\rho \ge M$ is a fixed constant with M appearing in the condition (2.12), and the function $\theta _\rho (x)\in C^\infty ({\mathbb {R}})$ is a non-increasing cut-off function such that

$$\begin{aligned} \mathbbm {1}_{[-\frac{\rho }{2}, \frac{\rho }{2}]} \le \theta _\rho (x) \le \mathbbm {1}_{[-\rho , \rho ]}. \end{aligned}$$

(3.2)

The main goal of this section is to prove Theorem 1.1. For this purpose, we fix $\rho \ge M$, $\tau _0>0$ and $r>\frac{5}{2}$ through this section. In Sects. 3.1 and 3.2, we first work on a Galerkin approximation system of the modified system (3.1) and derive some necessary energy estimates. In Sect. 3.3, by Prokhorov’s Theorem and Skorokhod Theorem, the existence of local martingale solution to the modified system (3.1) is established. By defining a suitable stopping time, we show the existence of local martingale solutions to the original system (2.11). Finally, in Sect. 3.4, we establish the pathwise uniqueness to the original system (2.11), which completes the proof of Theorem 1.1.

3.1 Galerkin scheme

We employ the Galerkin approximation procedure. For ${\varvec{k}}\in 2\pi {\mathbb {Z}}^3$, let

$$\begin{aligned} \phi _{{\varvec{k}}} = \phi _{k_1,k_2,k_3} := {\left\{ \begin{array}{ll} \sqrt{2}e^{ i\left( k_1 x_1 + k_2 x_2 \right) }\cos ( k_3 z) &{} \text {if} \; k_3\ne 0,\\ e^{ i\left( k_1 x_1 + k_2 x_2 \right) } &{} \text {if} \; k_3=0, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned}&{\mathcal {E}}:= \left\{ \phi \in L^2({\mathbb {T}}^3) \; | \; \phi = \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} a_{{\varvec{k}}} \phi _{{\varvec{k}}}, \; a_{-k_1, -k_2,k_3}=a_{k_1,k_2,k_3}^{*}, \; \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |a_{{\varvec{k}}}|^2 < \infty \right\} , \end{aligned}$$

where $a^*$ denotes the complex conjugate of a. Notice that ${\mathcal {E}}$ is a closed subspace of $L^2({\mathbb {T}}^3)$, and consists of real-valued functions which are even in z variable. For any $n\in {{\mathbb {N}}}$, denote by

$$\begin{aligned}&{\mathcal {E}}_n:= \left\{ \phi \in L^2({\mathbb {T}}^3)\; | \; \phi = \sum \limits _{|{\varvec{k}}|\le n} a_{{\varvec{k}}} \phi _{{\varvec{k}}}, \; a_{-k_1, -k_2, k_3}=a_{k_1,k_2, k_3}^{*} \right\} , \end{aligned}$$

the finite-dimensional subspaces of ${\mathcal {E}}$. For any function $f\in L^2({\mathbb {T}}^3)$, denote by

$$\begin{aligned} f_{{\varvec{k}}} :=\int _{{\mathbb {T}}^3} f({\varvec{x}})\phi ^*_{{\varvec{k}}}({\varvec{x}}) d{\varvec{x}}, \end{aligned}$$

and write $P_n f := \sum _{|{\varvec{k}}|\le n} f_{{\varvec{k}}} \phi _{{\varvec{k}}}$ for $n\in {{\mathbb {N}}}$. Then $P_n$ is the orthogonal projections from $L^2({\mathbb {T}}^3)$ to ${\mathcal {E}}_n$. Moreover, for $\tau \ge 0$ and $r\ge 0$, one has the following Poincaré inequality:

$$\begin{aligned} \begin{aligned} \Vert (I-P_n)f\Vert _{\tau ,r}^2&= \sum \limits _{|{\varvec{k}}|>n} (1+|{\varvec{k}}|^{2r})e^{2\tau |{\varvec{k}}|}|f_{{\varvec{k}}}|^2 \le \frac{1}{n} \sum \limits _{|{\varvec{k}}|>n} |{\varvec{k}}|(1+|{\varvec{k}}|^{2r})e^{2\tau |{\varvec{k}}|} |f_{{\varvec{k}}}|^2 \\&\le \frac{2}{n} \sum \limits _{|{\varvec{k}}|>n} (1+|{\varvec{k}}|^{2r+1})e^{2\tau |{\varvec{k}}|} |f_{{\varvec{k}}}|^2 = \frac{2}{n} \Vert (I-P_n)f\Vert _{\tau ,r+\frac{1}{2}}^2. \end{aligned} \end{aligned}$$

(3.3)

Denote by

$$\begin{aligned} Q_n(g , h) := P_n Q(g,h), \ \ F_n(g) := P_n F(g), \ \ \sigma _n := P_n \sigma . \end{aligned}$$

For each $n\in {{\mathbb {N}}}$, we consider the following Galerkin approximation of system (3.1) at order n as:

$$\begin{aligned}&d V_n + \big [\theta _\rho (\Vert V_n\Vert _{\tau ,r} ) Q_n(V_n, V_n) + F_n(V_n) \big ]dt = \sigma _n (V_n) dW , \end{aligned}$$

(3.4a)

$$\begin{aligned}&V_n(0) = P_n V_0 =: (V_n)_0. \end{aligned}$$

(3.4b)

Since all the terms in system (3.4) are locally Lipschitz, the local existence and uniqueness of the solution $V_n$ is a standard result.

For some function $\tau (t)$ to be determined later, with initial condition denote by $\tau _0$, i.e., $\tau (0) = \tau _0$, we define $U_n = e^{\tau A}V_n$, and therefore, $V_n = e^{-\tau A}U_n$. Since $dU_n = \dot{\tau }A U_n dt + e^{\tau A}dV_n$, system (3.4) is equivalent to

$$\begin{aligned}&d U_n + \big [\theta _\rho (\Vert U_n\Vert _{0,r} ) e^{\tau A} Q_n(e^{-\tau A}U_n, e^{-\tau A}U_n) + e^{\tau A}F_n(e^{-\tau A}U_n) -\dot{\tau } A U_n\big ]dt \nonumber \\&\quad = e^{\tau A}\sigma _n (e^{-\tau A}U_n) dW , \end{aligned}$$

(3.5a)

$$\begin{aligned}&U_n(0) = e^{\tau _0 A }(V_n)_0 = P_n \left( e^{\tau _0 A } V_0\right) , \end{aligned}$$

(3.5b)

and $U_n = e^{\tau A}V_n$ is the solution to system (3.5).

3.2 Energy estimates

Now we establish the main estimates needed to pass to the limit in the Galerkin system (3.4) (and equivalently, system (3.5)).

Lemma 3.1

Let $\tau _0>0$, $p\ge 2$, and $r>\frac{5}{2}$ be fixed. Suppose that $V_0 \in L^p(\Omega ; {\mathcal {D}}_{\tau _0,r})$ is an ${\mathcal {F}}_0$-measurable random variable. Suppose that $\sigma$ and f satisfy (2.2) and (2.3), respectively. Let $V_n$ be the solution to system (3.4). Then there exist a decreasing function $\tau (t)$ defined in (3.7) and a constant T defined in (3.8) such that the following hold:

• (i) $\sup \limits _{n\in {{\mathbb {N}}}} {\mathbb {E}} \Big [\sup \limits _{s\in [0,T]} \Vert V_n\Vert _{\tau ,r}^p + \int _0^T \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ds \Big ] \le C_p (1+{\mathbb {E}} \Vert V_0\Vert _{\tau _0,r}^p) e^{C_p T};$

• (ii) For $\alpha \in [0,1/2)$, $\int _0^{\cdot } e^{\tau A}\sigma _n(V_n) dW$ is bounded in $L^p\left( \Omega ; W^{\alpha , p}(0, T; {\mathcal {D}}_{0,r-1}) \right) ;$

• (iii) If moreover $p\ge 4$, then $e^{\tau A}V_n - \int _0^\cdot e^{\tau A}\sigma _n(V_n) dW$ is bounded in $L^2\left( \Omega ; W^{1, 2}(0,\right. \left. T; {\mathcal {D}}_{0,r-1}))\right) .$

Proof

Applying the finite dimensional Itô formula to (3.4a) produces, for $p\ge 2$,

$$\begin{aligned} \begin{aligned} d\Vert V_n\Vert ^p =&- p \theta _\rho (\Vert V_n\Vert _{\tau ,r} ) \Vert V_n\Vert ^{p-2} \big \langle Q_n(V_n, V_n) , V_n\big \rangle dt - p \Vert V_n\Vert ^{p-2} \big \langle F_n (V_n), V_n\big \rangle dt \\&+\frac{p}{2} \Vert \sigma _n(V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} \Vert V_n\Vert ^{p-2} dt + \frac{p(p-2)}{2} \big \langle \sigma _n(V_n), V_n \big \rangle ^2 \Vert V_n\Vert ^{p-4} dt \\&+ p \big \langle \sigma _n(V_n), V_n \big \rangle \Vert V_n\Vert ^{p-2} dW \\ :=&A_1 dt + A_2 dt + A_3 dt + A_4 dt + A_5 dW. \end{aligned} \end{aligned}$$

Thanks to the periodic boundary condition, integration by parts gives $A_1=0$. By integration by parts, thanks to the Cauchy-Schwartz inequality, Young’s inequality, and (2.3), one obtains that

$$\begin{aligned} \begin{aligned} \int _0^t A_2 ds&\le \int _0^t p \Vert V_n\Vert ^{p-2} \Big |\big \langle F_n (V_n), V_n\big \rangle \Big | ds = \int _0^t p \Vert V_n\Vert ^{p-2} \Big |\big \langle f, V_n\big \rangle \Big | ds \\&\le \int _0^t p \Vert V_n\Vert ^{p-2} (\Vert f\Vert ^2 + \Vert V_n\Vert ^2) ds \le \frac{1}{4} \sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p \\&\quad + C_p\left( 1 + \int _0^t \Vert V_n\Vert ^p ds\right) . \end{aligned} \end{aligned}$$

From the assumptions (2.2), since $r>\frac{5}{2}$, one has $\Vert \sigma (V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} \le \Vert \sigma (V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r-\frac{1}{2}})} \le C (1+ \Vert V_n\Vert _{\tau ,r}^2)$. Then using the Cauchy-Schwartz inequality and Young’s inequality yields that

$$\begin{aligned} \int _0^t | A_3|+| A_4| ds \le C_p \int _0^t (1+ \Vert V_n\Vert _{\tau ,r}^2 ) \Vert V_n\Vert ^{p-2} ds\le C_p \int _0^t(1+ \Vert V_n\Vert _{\tau ,r}^p ) ds. \end{aligned}$$

Finally, the Burkholder-Davis-Gundy inequality, together with the Cauchy-Schwartz inequality, Young’s inequality, and the property of $\sigma$ in (2.2) lead to

$$\begin{aligned} \begin{aligned} {\mathbb {E}} \sup \limits _{s\in [0,t]} \Big |\int _0^s A_5 dW \Big |&\le C_p {\mathbb {E}}\Big (\int _0^t \Vert V_n\Vert ^{2(p-1)} \Vert \Vert \sigma _n(V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} \Big )^{\frac{1}{2}} \\&\le C_p {\mathbb {E}}\Big (\int _0^t \Vert V_n\Vert ^{2(p-1)} (1+ \Vert V_n\Vert ^2 + \Vert A^{\frac{1}{2}} V_n\Vert ^2 ) \Big )^{\frac{1}{2}} \\&\le \frac{1}{4} {\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p + C_p{\mathbb {E}} \int _0^t (1+ \Vert V_n\Vert _{\tau ,r}^p) ds . \end{aligned} \end{aligned}$$

Next, for the seminorm $\Vert A^r e^{\tau A} V_n\Vert$, applying the finite dimensional Itô formula yields, for $p\ge 2$,

$$\begin{aligned} \begin{aligned}&d\Vert A^r e^{\tau A} V_n\Vert ^p - p\dot{\tau } \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \Vert A^{r+\frac{1}{2}} e^{\tau A} V_n\Vert ^2 dt \\&\quad = - p \theta _\rho (\Vert V_n\Vert _{\tau ,r} ) \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \big \langle A^r e^{\tau A} Q_n(V_n, V_n) , A^r e^{\tau A} V_n\big \rangle dt \\&\qquad - p \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \big \langle A^r e^{\tau A} F_n (V_n), A^r e^{\tau A}V_n\big \rangle dt \\&\qquad +\frac{p}{2} \Vert A^r e^{\tau A} \sigma _n(V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} \Vert A^r e^{\tau A}V_n\Vert ^{p-2} dt \\&\qquad + \frac{p(p-2)}{2} \big \langle A^r e^{\tau A} \sigma _n(V_n), A^r e^{\tau A} V_n \big \rangle ^2 \Vert A^r e^{\tau A}V_n\Vert ^{p-4} dt \\&\qquad + p \big \langle A^r e^{\tau A} \sigma _n(V_n), A^r e^{\tau A} V_n \big \rangle \Vert A^r e^{\tau A}V_n\Vert ^{p-2} dW \\&\quad := B_1 dt + B_2 dt + B_3 dt + B_4 dt + B_5 dW. \end{aligned} \end{aligned}$$

For the nonlinear terms $B_1$, Lemma A.1 and properties of the cut-off function $\theta _\rho$ imply that

$$\begin{aligned} \begin{aligned} \int _0^t | B_1| ds&= p \theta _\rho (\Vert V_n\Vert _{\tau ,r} ) \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \Big |\big \langle A^r e^{\tau A} Q_n(V_n, V_n) , A^r e^{\tau A}V_n\big \rangle \Big |ds \\&\le C_{r,p} \int _0^t\theta _\rho (\Vert V_n\Vert _{\tau ,r} ) \Vert V_n\Vert _{\tau ,r} \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \Vert A^{r+\frac{1}{2}} e^{\tau A} V_n\Vert ^2 ds \\&\le C_{r,p}\int _0^t \rho \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \Vert A^{r+\frac{1}{2}} e^{\tau A} V_n\Vert ^2 ds. \end{aligned} \end{aligned}$$

Thanks to the periodic boundary condition, using integration by parts, the Cauchy-Schwartz inequality, Young’s inequality, and (2.3), one obtains

$$\begin{aligned} \begin{aligned} \int _0^t B_2 ds&\le \int _0^t p \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \Big |\big \langle A^r e^{\tau A} F_n (V_n), A^r e^{\tau A}V_n\big \rangle \Big | ds \\&= \int _0^t p \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \Big |\big \langle A^r e^{\tau A} f, A^r e^{\tau A}V_n\big \rangle \Big | ds \\&\le C_p \left( 1 + \int _0^t \Vert V_n\Vert _{\tau ,r}^p ds \right) + \frac{1}{4} \sup \limits _{s\in [0,t]} \Vert A^r e^{\tau A} V_n\Vert ^p. \end{aligned} \end{aligned}$$

From (2.2), we know that $\Vert A^r e^{\tau A} \sigma (V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} \le \Vert \sigma (V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r})}\le C \Big (1+ \Vert V_n\Vert ^2 + \Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 \Big )$, and thus

$$\begin{aligned} \begin{aligned} \int _0^t | B_3|+| B_4| ds&\le C_p \int _0^t (1+ \Vert V_n\Vert ^2 + \Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ) \Vert A^r e^{\tau A}V_n\Vert ^{p-2} ds \\&\le \int _0^t C_p (1+\Vert V_n\Vert _{\tau ,r}^p + \Vert A^r e^{\tau A}V_n\Vert ^{p-2} \Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2)ds. \end{aligned} \end{aligned}$$

Finally, using the Burkholder-Davis-Gundy inequality, the Cauchy-Schwartz inequality, Young’s inequality, and the property of $\sigma$ in (2.2), we deduce that

$$\begin{aligned} \begin{aligned} {\mathbb {E}} \sup \limits _{s\in [0,t]} \Big |\int _0^s B_5 dW \Big | \le&C_p {\mathbb {E}}\Big (\int _0^t \Vert A^r e^{\tau A} V_n\Vert ^{2(p-1)} \Vert \sigma _n(V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_{\tau ,r})} \Big )^{\frac{1}{2}} \\ \le&C_p {\mathbb {E}}\Big (\int _0^t \Vert A^r e^{\tau A} V_n\Vert ^{2(p-1)} (1+ \Vert V_n\Vert ^2 + \Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ) \Big )^{\frac{1}{2}} \\ \le&\frac{1}{4} {\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert A^r e^{\tau A} V_n\Vert ^p + C_p {\mathbb {E}} \int _0^t (1+ \Vert V_n\Vert _{\tau ,r}^p \\&+ \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ) ds. \end{aligned} \end{aligned}$$

Combining the estimates of $A_1$ to $A_5$ and $B_1$ to $B_5$, thanks to (2.1), one obtains that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert V_n\Vert _{\tau ,r}^p + {\mathbb {E}} \int _0^t \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ds \\&\quad \le {\mathbb {E}} \sup \limits _{s\in [0,t]} (2\Vert V_n\Vert ^2 + 2\Vert A^r e^{\tau A} V_n\Vert ^2)^{\frac{p}{2}} + {\mathbb {E}} \int _0^t \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ds \\&\quad \le C_p \Big ( {\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p + {\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert A^r e^{\tau A} V_n\Vert ^p\Big ) \\&\qquad + {\mathbb {E}} \int _0^t \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ds \\&\quad \le C_p {\mathbb {E}} \int _0^t \big (\dot{\tau } + C_{r,p} (\rho +1)\big ) \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ds \\&\qquad + C_{p}{\mathbb {E}} \Big [ 1 + \Vert (V_n)_0\Vert _{\tau _0,r}^p + \int _0^t \big (1 + \Vert V_n\Vert _{\tau ,r}^p \big ) ds \Big ]. \end{aligned} \end{aligned}$$

(3.6)

Here we have added the term ${\mathbb {E}} \int _0^t \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ds$ on both hand sides in order to get the corresponding regularity. Let the deterministic function $\tau (t)$ satisfy

$$\begin{aligned} \tau (t) = \tau _0 - C_{r,p}(\rho + 1) t, \end{aligned}$$

(3.7)

and define the time

$$\begin{aligned} T = \frac{\tau _0}{2C_{r,p}(\rho + 1)}. \end{aligned}$$

(3.8)

Observe that $\tau (t)>0$ and $\dot{\tau } + C_{r,p} (\rho +1) = 0$ for $t\in [0,T]$, and therefore (3.6) gives

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \Big [\sup \limits _{s\in [0,t]} \Vert V_n\Vert _{\tau ,r}^p + \int _0^t \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ds \Big ] \\&\quad \le C_{p}{\mathbb {E}} \Big [ 1+ \Vert V_0\Vert _{\tau _0,r}^p + \int _0^t \big (1 + \Vert V_n\Vert _{\tau ,r}^p \big ) ds \Big ]. \end{aligned} \end{aligned}$$

Thanks to the Gronwall inequality [25, Lemma 5.3], for $p\ge 2$ and $t=T$, one has

$$\begin{aligned}&{\mathbb {E}} \Big [\sup \limits _{s\in [0,T]} \Vert V_n\Vert _{\tau ,r}^p + \int _0^T \Vert A^r e^{\tau A} V_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} e^{\tau A}V_n\Vert ^2 ds \Big ] \nonumber \\&\quad \le C_p(1+ {\mathbb {E}} \Vert V_0\Vert _{\tau _0,r}^p) e^{C_p T}. \end{aligned}$$

(3.9)

For part (ii), the fractional Burkholder-Davis-Gundy inequality (2.9), (2.2), and the estimate (3.9) produce

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left| \int _0^\cdot e^{\tau A}\sigma _n\left( V_n\right) \, dW \right| ^p_{W^{\alpha , p}\left( 0, T; {\mathcal {D}}_{0,r-1}\right) }&\le C_p {\mathbb {E}}\int _0^T \Vert e^{\tau A}\sigma _n\left( V_n\right) \Vert ^p_{L_2\left( {\mathscr {U}}, {\mathcal {D}}_{0,r-1}\right) } \, ds \\&\le C_p {\mathbb {E}}\left[ \int _0^T 1 + \Vert V_n\Vert _{\tau ,r}^p \, ds \right] < \infty . \end{aligned} \end{aligned}$$

Finally for (iii), from (3.4a) (or (3.5a)), for $t\in [0,T]$, one has

$$\begin{aligned} \begin{aligned}&e^{\tau A}V_n(t) - \int _0^t e^{\tau A}\sigma _n\left( V_n\right) \, dW \\&\quad = e^{\tau _0 A}V_n(0) - \int _0^t \theta _\rho (\Vert V_n\Vert _{\tau ,r} ) e^{\tau A} Q_n(V_n, V_n) + e^{\tau A} F_n\left( V_n\right) - \dot{\tau }A e^{\tau A} V_n \, ds. \end{aligned} \end{aligned}$$

Then by Lemma A.3 and $r>\frac{5}{2}$, the Minkowski inequality, and Young’s inequality, from the properties of $\sigma$, F, and the cut-off function $\theta _\rho$, we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\left\| e^{\tau A}V_n - \int _0^\cdot e^{\tau A} \sigma _n\left( V_n\right) \, dW \right\| ^2_{W^{1, 2}\left( 0, T; {\mathcal {D}}_{0,r-1}\right) } \\&\quad \le C_{T,f_0,p,\rho ,r} {\mathbb {E}}\left[ 1+ \Vert V_0\Vert _{\tau _0,r}^2 + \sup \limits _{t\in [0,T]}\Vert V_n(t)\Vert _{\tau ,r}^4+ \int _0^T \Vert f\Vert _{\tau ,r}^2 \, ds \right] < \infty , \end{aligned} \end{aligned}$$

(3.10)

where the pressure gradient disappears since $\nabla P$ is orthogonal to the space ${\mathcal {D}}_{0,r-1}.$ Note that $p\ge 4$ is required due to applying the estimate (3.9) to the term $\sup \limits _{t\in [0,T]}\Vert V_n(t)\Vert _{\tau ,r}^4$ in (3.10). $\square$

Remark 4

The energy estimates in Lemma 3.1 hold up to a finite time T defined in (3.8). Due to this fact, the martingale solutions to the modified system (3.1), constructed below in Proposition 3.4, can only exist on [0, T] (see Proposition 3.3 and 3.4, below). Alternatively, one can replace the cut-off function $\theta _\rho$ by $\theta _{\rho \tau (t)}$ in the modified system (3.1), where $\tau (t)$ is the radius of analyticity, then energy estimates similar to Lemma 3.1 will hold for the new modified system with $\theta _{\rho \tau (t)}$ for any time $t\ge 0$. Moreover, based on this, the martingale solutions to this new modified system will exist globally in time. Nevertheless, the solution to the original one will still be local after applying a stopping time (see Corollary 3.5).

Corollary 3.2

With the same assumptions as in Lemma 3.1. Let $\tau (t)$ and T be defined in (3.7) and (3.8), respectively. Suppose that $U_n = e^{\tau A}V_n$ is the solution to system (3.5), and we denote by $U_0 = e^{\tau _0A} V_0$. Then

• (i) $\sup \limits _{n\in {{\mathbb {N}}}} {\mathbb {E}} \Big [\sup \limits _{s\in [0,T]} \Vert U_n\Vert _{0,r}^p + \int _0^T \Vert A^r U_n\Vert ^{p-2}\Vert A^{r+\frac{1}{2}} U_n\Vert ^2 ds \Big ] \le (1+ {\mathbb {E}} \Vert U_0\Vert _{0,r}^p) e^{C_p T};$

• (ii) For $\alpha \in [0,1/2)$, $\int _0^{\cdot } e^{\tau A}\sigma _n(e^{-\tau A}U_n) dW$ is bounded in $L^p\bigl ( \Omega ; W^{\alpha , p}(0, T; {\mathcal {D}}_{0,r-1}) \bigr );$

• (iii) If moreover $p\ge 4$, then $U_n - \int _0^\cdot e^{\tau A}\sigma _n(e^{-\tau A}U_n) dW$ is bounded in $L^2\bigl ( \Omega ; W^{1, 2}(0, T; {\mathcal {D}}_{0,r-1})\bigr ) .$

3.3 Local existence of martingale solutions

In this subsection, we establish the existence of the martingale solutions to system (2.11). First, we need to construct the local martingale solutions to the modified system (3.1). For this purpose, we follow similar procedures as in [6, 15]. However, since our spaces ${\mathcal {D}}_{\tau (t),r}$ are changing with time, in order to use Aubin-Lions Lemma 2.3, one needs to work with $U_n$ instead of $V_n$ since $U_n$ lives in classical Sobolev spaces.

Given an initial distribution $\mu _0$ satisfying (2.13) for some $p\ge 4$ (which is implied by (2.12)), for some stochastic basis ${\mathcal {S}}= \left( \Omega , {\mathcal {F}}, {\mathbb {F}}, {\mathbb {P}}\right)$, let $U_0= e^{\tau _0A}V_0$ be an ${\mathcal {F}}_0$-measurable ${\mathcal {D}}_{0,r}$-valued random variable with law $\mu _0$. For fixed $\rho \ge M$, let $U_n$ be the solutions to the approximating system (3.5) with $U_n(0) = P_nU_0$ on the stochastic basis ${\mathcal {S}}$. Let ${\mathscr {U}}_0$ be an auxiliary Hilbert space such that the embedding ${\mathscr {U}}\subseteq {\mathscr {U}}_0$ is Hilbert-Schmidt. Define

$$\begin{aligned}&{\mathcal {X}}_{U} = L^2\left( 0, T; {\mathcal {D}}_{0,r} \right) \cap C\left( \left[ 0, T\right] ; {\mathcal {D}}_{0,r-\frac{3}{2}}\right) , \quad {\mathcal {X}}_W = C\left( \left[ 0, T\right] ; {\mathscr {U}}_0\right) ,\\&{\mathcal {X}}= {\mathcal {X}}_{U} \times {\mathcal {X}}_W, \end{aligned}$$

where T is defined in (3.8). Notice that here ${\mathcal {D}}_{0,r}$ and ${\mathcal {D}}_{0,r-\frac{3}{2}}$ are independent of time. Let $\mu _{U}^n$, $\mu ^n_W$ and $\mu ^n$ be laws of $U_n$, W and $(U_n, W)$ on ${\mathcal {X}}_{U}$, ${\mathcal {X}}_W$ and ${\mathcal {X}}$, respectively, in other words

$$\begin{aligned} \mu ^n_{U}(\cdot ) = {\mathbb {P}}\left( \left\{ U_n \in \cdot \right\} \right) , \qquad \mu _W^n(\cdot ) = {\mathbb {P}}\left( \left\{ W \in \cdot \right\} \right) , \qquad \mu ^n = \mu _{U}^n \otimes \mu _W^n. \end{aligned}$$

(3.11)

The proof of the existence of the local martingale solutions to the modified system (3.1) will be shown once we prove the following two propositions, which are following [15, Proposition 4.1 and Proposition 7.1] or [6, Proposition 3.2 and Proposition 3.3].

Proposition 3.3

Let $\mu _0$ be a probability measure on ${\mathcal {D}}_{0,r}$ satisfying

$$\begin{aligned} \int _{{\mathcal {D}}_{0,r}} \Vert U\Vert _{0,r}^p d\mu _0(U)<\infty \end{aligned}$$

(3.12)

with $p\ge 4$ and let $(\mu ^n)_{n \ge 1}$ be the measures defined in (3.11). Then there exists a probability space $(\tilde{\Omega }, \tilde{{\mathcal {F}}}, \tilde{{\mathbb {P}}})$, a subsequence $n_k \rightarrow \infty$ as $k \rightarrow \infty$ and a sequence of ${\mathcal {X}}$-valued random variables $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ such that

• (i) $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ converges in ${\mathcal {X}}$ to $(\tilde{U}, \tilde{W}) \in {\mathcal {X}}$ almost surely,

• (ii) $\tilde{W}_{n_k}$ is a cylindrical Wiener process with reproducing kernel Hilbert space ${\mathscr {U}}$ adapted to the filtration $\left( {\mathcal {F}}_t^{n_k} \right) _{t \ge 0}$, where $\left( {\mathcal {F}}_t^{n_k} \right) _{t \ge 0}$ is the completion of $\sigma (\tilde{W}_{n_k}, \tilde{U}_{n_k}; 0 \le s \le t)$,

• (iii) for $t\in [0,T]$, each pair $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ satisfies the equation

  $$\begin{aligned} \begin{aligned}&d {\tilde{U}}_{n_k} + \big [\theta _\rho (\Vert {\tilde{U}}_{n_k}\Vert _{0,r} ) e^{\tau A} Q_{n_k}(e^{-\tau A}{\tilde{U}}_{n_k}, e^{-\tau A}{\tilde{U}}_{n_k}) \\&\qquad + e^{\tau A}F_{n_k}(e^{-\tau A}{\tilde{U}}_{n_k}) -\dot{\tau } A {\tilde{U}}_{n_k}\big ]dt \\&\quad = e^{\tau A}\sigma _{n_k} (e^{-\tau A}{\tilde{U}}_{n_k}) d{\tilde{W}}_{n_k} \end{aligned} \end{aligned}$$

  (3.13)

  where $\tau$ and T are defined in (3.7) and (3.8), respectively.

Remark 5

In [6, Proposition 3.2], the authors required a condition similar to (3.12) with $p\ge 8$. The reason is that the estimate (3.10) in [6] requires such a higher integrability condition. The estimation therein corresponds to (3.10) in our derivation, which only requires $p\ge 4$.

Proof

For part (i), with Lemmas 2.3 and 3.1, one has the tightness of $\lbrace \mu ^n \rbrace _{n\ge 1}$ in ${\mathcal {X}}$ following the argument in [15, Lemma 4.1] with the spaces D(A), V, H, and $V'$ therein being replaced by ${\mathcal {D}}_{0,r+\frac{1}{2}}$, ${\mathcal {D}}_{0,r}$, ${\mathcal {D}}_{0,r-1}$, and ${\mathcal {D}}_{0,r-\frac{3}{2}}$. Thus $\lbrace \mu ^n \rbrace _{n\ge 1}$ is weakly compact by Prokhorov’s theorem. Then, the first assertion follows immediately by the Skorokhod Theorem, see, e.g., [14, Theorem 2.4].

Parts (ii) and (iii) follow from the proof in [6, Proposition 3.2]; see also [3, Section 4.3.4].

$\square$

Proposition 3.4

For $\tau (t)$ and T defined in (3.7) and (3.8), respectively, let $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ be a sequence of ${\mathcal {X}}$-valued random variables on a probability space $(\tilde{\Omega }, \tilde{{\mathcal {F}}}, \tilde{{\mathbb {P}}})$ such that

• (i) $(\tilde{U}_{n_k}, \tilde{W}_{n_k}) \rightarrow (\tilde{U}, \tilde{W})$ in the topology of ${\mathcal {X}}$ $\tilde{{\mathbb {P}}}$-almost surely, that is,

  $$\begin{aligned} \tilde{U}_{n_k} \rightarrow \tilde{U} \ \text {in} \ L^2\left( 0, T; {\mathcal {D}}_{0,r}\right) \cap C\left( \left[ 0, T \right] , {\mathcal {D}}_{0,r-\frac{3}{2}}\right) , \ \tilde{W}_{n_k} \rightarrow \tilde{W} \ \text {in} \ C\left( \left[ 0, T\right] ; {\mathscr {U}}_0 \right) , \end{aligned}$$

• (ii) $\tilde{W}_{n_k}$ is a cylindrical Wiener process with reproducing kernel Hilbert space ${\mathscr {U}}$ adapted to the filtration $\left( {\mathcal {F}}_t^{n_k} \right) _{t \ge 0}$ that contains $\sigma (\tilde{W}_{n_k}, \tilde{U}_{n_k}; 0 \le s \le t)$,

• (iii) each pair $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ satisfies (3.13).

Let

$$\begin{aligned} {\tilde{V}}_{n_k} = e^{-\tau A} {\tilde{U}}_{n_k}, \ \ {\tilde{V}} = e^{-\tau A} {\tilde{U}}, \end{aligned}$$

(3.14)

and let $\tilde{{\mathcal {F}}}_t$ be the completion of $\sigma (\tilde{W}(s), \tilde{V}(s), 0 \le s \le t)$ and $\tilde{{\mathcal {S}}} = (\tilde{\Omega }, \tilde{{\mathcal {F}}}, ( \tilde{{\mathcal {F}}}_t )_{t \ge 0}, \tilde{{\mathbb {P}}})$. Then $(\tilde{{\mathcal {S}}}, \tilde{W}, \tilde{V},T)$ is a local martingale solution to the modified system (3.1) on the time interval [0, T]. Moreover, $\tilde{V}$ satisfies

$$\begin{aligned}&\tilde{V} \in L^2\left( {\tilde{\Omega }}; C\left( [0, T]; {\mathcal {D}}_{\tau (t),r} \right) \right) , \qquad \nonumber \\&\quad \Vert A^{r+\frac{1}{2}} e^{\tau (t)A} \tilde{V} \Vert ^2 \Vert A^{r} e^{\tau (t)A} \tilde{V}\Vert ^{p-2} \in L^1\left( {\tilde{\Omega }}; L^1\left( 0, T \right) \right) . \end{aligned}$$

(3.15)

Proof

First, due to the relation (3.14) and the assumpstion (iii), one has

$$\begin{aligned}&d {\tilde{V}}_{n_k} + \big [\theta _\rho (\Vert {\tilde{V}}_n\Vert _{\tau ,r} ) Q_{n_k}({\tilde{V}}_{n_k}, {\tilde{V}}_{n_k}) + F_{n_k}({\tilde{V}}_{n_k}) \big ]dt = \sigma _{n_k} ({\tilde{V}}_{n_k}) d{\tilde{W}}_{n_k}. \end{aligned}$$

Therefore, for any $\phi \in {\mathcal {D}}_0$ and $t\in [0,T]$, one has

$$\begin{aligned} \begin{aligned}&\Big \langle {\tilde{V}}_{n_k}(t), \phi \Big \rangle + \int _0^t \Big \langle \theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) Q_{n_k}({\tilde{V}}_{n_k}, {\tilde{V}}_{n_k}) + F_{n_k}({\tilde{V}}_{n_k}) ,\phi \Big \rangle ds \\&\quad = \Big \langle {\tilde{V}}_{n_k}(0), \phi \Big \rangle + \int _0^t \Big \langle \sigma _{n_k}({\tilde{V}}_{n_k}) , \phi \Big \rangle d\tilde{W}_{n_k}. \end{aligned} \end{aligned}$$

(3.16)

From assumption (i), we deduce that

$$\begin{aligned} {\tilde{V}}_{n_k} \rightarrow {\tilde{V}} \text { in } L^2\left( 0, T; {\mathcal {D}}_{\tau (t),r}\right) \cap C\left( \left[ 0, T \right] , {\mathcal {D}}_{\tau (t),r-\frac{3}{2}}\right) \; \tilde{{\mathbb {P}}}-a.s.. \end{aligned}$$

(3.17)

Similarly as in [15, Section 7.1], by Corollary 3.2 and (3.14), one can establish that

$$\begin{aligned} \begin{aligned}&\tilde{V} \in L^2 \left( {\tilde{\Omega }}; L^2\big ( 0,T; {\mathcal {D}}_{\tau (t),r+\frac{1}{2}} \big ) \cap L^\infty \big ( 0,T; {\mathcal {D}}_{\tau (t),r} \big )\right) , \\&\tilde{V}_{n_k} \rightharpoonup \tilde{V} \text { in } L^2 \left( {\tilde{\Omega }}; L^2\big ( 0,T; {\mathcal {D}}_{\tau (t),r+\frac{1}{2}} \big ) \right) , \\&\tilde{V}_{n_k} \overset{*}{\rightharpoonup } \tilde{V} \text { in } L^2 \left( {\tilde{\Omega }}; L^\infty \big ( 0,T; {\mathcal {D}}_{\tau (t),r} \big ) \right) . \end{aligned} \end{aligned}$$

(3.18)

Moreover, from Lemma 3.1 with $p >2$, one has the following uniform integrability for $\tilde{V}_{n_k}$:

$$\begin{aligned} \begin{aligned} \sup \limits _{k\in {\mathbb {N}}} {\tilde{{\mathbb {E}}}} \left[ \left( \int _0^T \Vert {\tilde{V}}_{n_k}\Vert ^2_{\tau ,r} ds \right) ^{\frac{p}{2}} \right] \le C_T \sup \limits _{k\in {\mathbb {N}}} {\tilde{{\mathbb {E}}}} \sup \limits _{t\in [0,T]} \Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r}^p < \infty . \end{aligned} \end{aligned}$$

(3.19)

The Vitali Convergence Theorem, together with (3.17) and (3.19), implies that

$$\begin{aligned} \tilde{V}_{n_k} \rightarrow \tilde{V} \text { in } L^2 \left( {\tilde{\Omega }}; L^2\big ( 0,T; {\mathcal {D}}_{\tau (t),r} \big ) \right) . \end{aligned}$$

(3.20)

Consequently, a further subsequence, still denoted by ${\tilde{V}}_{n_k}$ with a slightly abuse of notation, converges a.a. in $(0,T)\times {\tilde{\Omega }}$:

$$\begin{aligned} \tilde{V}_{n_k} \rightarrow \tilde{V} \text { in } {\mathcal {D}}_{\tau (t),r} \text { for a.a. } (t,\omega ) \in (0,T)\times {\tilde{\Omega }}. \end{aligned}$$

(3.21)

Convergence of the linear terms: The convergence of the initial condition is straightforward by (3.17):

$$\begin{aligned} \Big \langle {\tilde{V}}_{n_k}(0), \phi \Big \rangle \rightarrow \Big \langle {\tilde{V}}(0), \phi \Big \rangle = \Big \langle {\tilde{V}}_0, \phi \Big \rangle . \end{aligned}$$

(3.22)

For any $t\in [0,T]$, thanks to the Cauchy-Schwarz inequality and (2.3), we know that

$$\begin{aligned} \begin{aligned}&\Big | \int _0^t\Big \langle F_{n_k}(\tilde{V}_{n_k}) - F(\tilde{V}) , \phi \Big \rangle ds \Big | \\&\quad = \Big | \int _0^t\Big \langle f_0 (\tilde{V}_{n_k}^{\perp } - \tilde{V}^{\perp }) + (P_{n_k}\nabla P - \nabla P) + (P_{n_k}f - f) , \phi \Big \rangle ds \Big | \\&\quad \le f_0 \Big | \int _0^t\Big \langle \tilde{V}_{n_k}^{\perp } - \tilde{V}^{\perp }, \phi \Big \rangle ds \Big | + \Big | \int _0^t\Big \langle f , P_{n_k}\phi - \phi \Big \rangle ds \Big | \\&\quad \le C_{f_0} \Vert \phi \Vert \int _0^T \Vert \tilde{V}_{n_k}^{\perp } - \tilde{V}^{\perp }\Vert \; ds + C \Vert P_{n_k}\phi - \phi \Vert \int _0^T \Vert f\Vert ds \\&\quad \le C_{f_0,T} \Vert \phi \Vert \left( \int _0^T \Vert \tilde{V}_{n_k}^{\perp } - \tilde{V}^{\perp }\Vert ^2 \; ds \right) ^{\frac{1}{2}} + C_T \Vert P_{n_k}\phi - \phi \Vert . \end{aligned} \end{aligned}$$

Therefore, by (3.21) and the property of $P_{n_k}$, for a.a. $(t,\omega ) \in (0,T)\times {\tilde{\Omega }}$

$$\begin{aligned} \int _0^t\Big \langle F_{n_k}(\tilde{V}_{n_k}) , \phi \Big \rangle ds \rightarrow \int _0^t\Big \langle F(\tilde{V}) , \phi \Big \rangle ds . \end{aligned}$$

(3.23)

Convergence of the nonlinear terms: For $t\in [0,T]$, one has

$$\begin{aligned} \begin{aligned}&\Big |\int _0^t \Big \langle \theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) Q_{n_k}(\tilde{V}_{n_k}, \tilde{V}_{n_k}) - \theta _\rho (\Vert \tilde{V}\Vert _{\tau ,r} ) Q(\tilde{V}, \tilde{V}) ,\phi \Big \rangle ds \Big | \\&\quad \le \Big |\int _0^t \Big \langle \theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) \big ( Q_{n_k}(\tilde{V}_{n_k}, \tilde{V}_{n_k}) - Q_{n_k}(\tilde{V}, \tilde{V}) \big ) ,\phi \Big \rangle ds \Big | \\&\qquad + \Big |\int _0^t \Big \langle \theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) Q(\tilde{V}, \tilde{V}) \big ) ,P_{n_k}\phi - \phi \Big \rangle ds \Big | \\&\qquad + \Big |\int _0^t \Big \langle \big (\theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) - \theta _\rho (\Vert \tilde{V}\Vert _{\tau ,r} \big ) Q(\tilde{V}, \tilde{V}) \big ) , \phi \Big \rangle ds \Big | := I_1 + I_2 + I_3. \end{aligned} \end{aligned}$$

For the term $I_1$, by the Hölder inequality and the Sobolev inequality, together with (3.21), one obtains that for a.a. $(t,\omega ) \in (0,T)\times {\tilde{\Omega }}$,

$$\begin{aligned} \begin{aligned} I_1&\le C \Vert P_{n_k}\phi \Vert \int _0^T \Vert Q(\tilde{V}_{n_k}, \tilde{V}_{n_k}) - Q(\tilde{V}, \tilde{V}) \big )\Vert ds \\&\le C \int _0^T \Vert Q(\tilde{V}_{n_k}- \tilde{V}, \tilde{V}_{n_k}) \Vert + \Vert Q( \tilde{V}, \tilde{V}_{n_k} - \tilde{V}) \Vert ds \\&\le C \int _0^T \Vert \tilde{V}_{n_k}- \tilde{V}\Vert _{0,2} (\Vert \tilde{V}_{n_k}\Vert _{0,2} + \Vert \tilde{V}\Vert _{0,2} ) \\&\le C \left( \int _0^T \Vert \tilde{V}_{n_k}- \tilde{V}\Vert _{0,2}^2 \right) ^{\frac{1}{2}} \left( \int _0^T \Vert \tilde{V}_{n_k}\Vert ^2_{0,2} + \Vert \tilde{V}\Vert ^2_{0,2} \right) ^{\frac{1}{2}} \rightarrow 0. \end{aligned} \end{aligned}$$

For the term $I_2$, by the Hölder inequality and the Sobolev inequality, we have

$$\begin{aligned} \begin{aligned} I_2&\le C \Vert \tilde{V}\Vert _{0,2}^2 \int _0^T \Vert P_{n_k}\phi - \phi \Vert ds \rightarrow 0, \end{aligned} \end{aligned}$$

for a.a. $(t,\omega ) \in (0,T)\times {\tilde{\Omega }}$. For the term $I_3$, thanks to (3.21) and since $\theta _\rho$ is smooth, one has

$$\begin{aligned} \theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) \rightarrow \theta _\rho (\Vert \tilde{V}\Vert _{\tau ,r} \big ), \end{aligned}$$

(3.24)

for a.a. $(t,\omega ) \in (0,T)\times {\tilde{\Omega }}$. Using (3.18) yields

$$\begin{aligned} {\mathbb {E}}\int _0^T \Big | \Big \langle \big (\theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) - \theta _\rho (\Vert \tilde{V}\Vert _{\tau ,r} \big ) Q(\tilde{V}, \tilde{V}) \big ) , \phi \Big \rangle \Big |dt \le C{\mathbb {E}}\int _0^T\Vert {\tilde{V}}\Vert _{0,2}^2 \Vert \phi \Vert dt <\infty . \end{aligned}$$

The dominated convergence theorem together with (3.24) yield that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\int _0^T I_3 dt \le C_T {\mathbb {E}}\int _0^T \Big | \Big \langle \big (\theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) - \theta _\rho (\Vert \tilde{V}\Vert _{\tau ,r} \big ) Q(\tilde{V}, \tilde{V}) \big ) , \phi \Big \rangle \Big | dt \rightarrow 0. \end{aligned} \end{aligned}$$

Thinning the sequence if necessary, we conclude that $I_3\rightarrow 0$ for a.a. $(t,\omega ) \in (0,T)\times {\tilde{\Omega }}$. Combining the estimates of $I_1$ to $I_3$, for a.a. $(t,\omega ) \in (0,T)\times {\tilde{\Omega }}$,

$$\begin{aligned}&\int _0^t \Big \langle \theta _\rho (\Vert {\tilde{V}}_{n_k}\Vert _{\tau ,r} ) Q_{n_k}(\tilde{V}_{n_k}, \tilde{V}_{n_k}) + F_{n_k}(\tilde{V}_{n_k}) ,\phi \Big \rangle ds \nonumber \\&\quad \rightarrow \int _0^t \Big \langle \theta _\rho (\Vert \tilde{V}\Vert _{\tau ,r} ) Q(\tilde{V}, \tilde{V}) + F(\tilde{V}) ,\phi \Big \rangle ds. \end{aligned}$$

(3.25)

Convergence of the stochastic terms: Using (2.2) and the Poincaré inequality (3.3), we have

$$\begin{aligned} \begin{aligned}&\Vert \sigma _{n_k}({\tilde{V}}_{n_k}) - \sigma ({\tilde{V}}) \Vert _{L^2\left( 0, T; L_2\left( {\mathscr {U}}, {\mathcal {D}}_0\right) \right) }^2\\&\quad \le C \left( \Vert \sigma ({\tilde{V}}_{n_k}) - \sigma ({\tilde{V}}) \Vert _{L^2\left( 0, T; L_2\left( {\mathscr {U}}, {\mathcal {D}}_{0,r-\frac{1}{2}}\right) \right) }^2 \right. \\&\qquad \left. + \Vert (I-P_{n_k}) \sigma ({\tilde{V}}_{n_k}) \Vert _{L^2\left( 0, T; L_2\left( {\mathscr {U}}, {\mathcal {D}}_{0,r-1}\right) \right) }^2 \right) \\&\quad \le C \left( \Vert {\tilde{V}}_{n_k} - {\tilde{V}} \Vert _{L^2(0, T; {\mathcal {D}}_{0,r})}^2 + \frac{2}{n_k} \Vert \sigma ({\tilde{V}}_{n_k}) \Vert _{L^2(0, T; L_2({\mathscr {U}}, {\mathcal {D}}_{0,r-\frac{1}{2}}))}^2 \right) \\&\quad \le C \left( \Vert {\tilde{V}}_{n_k} - {\tilde{V}} \Vert _{L^2(0, T; {\mathcal {D}}_{0,r})}^2 + \frac{2}{n_k} \int _0^T 1 + \Vert {\tilde{V}}_{n_k} \Vert _{0,r}^2 \, dt \right) . \end{aligned} \end{aligned}$$

Therefore, with Lemma 3.1 and the convergence result (3.17) we have

$$\begin{aligned} \sigma _{n_k}({\tilde{V}}_{n_k}) \rightarrow \sigma ({\tilde{V}}) \text { in } L^2\left( 0,T; L_2\left( {\mathscr {U}}, {\mathcal {D}}_0\right) \right) , \; \tilde{{\mathbb {P}}}-a.s.. \end{aligned}$$

In particular, this implies the convergence in probability in $L^2\left( 0, T; L_2\left( {\mathscr {U}}, {\mathcal {D}}_0\right) \right) .$ Thanks to [15, Lemma 2.1], from item (i), we obtain that

$$\begin{aligned} \int _0^\cdot \sigma _{n_k}({\tilde{V}}_{n_k}) \, d\tilde{W}_{n_k} \rightarrow \int _0^\cdot \sigma ({\tilde{V}}) \, d\tilde{W}, \end{aligned}$$

(3.26)

in probability in $L^2\left( 0,T; {\mathcal {D}}_0\right)$. Thanks to Lemma 3.1 and (2.2), for $p>2$, we get

$$\begin{aligned} \sup _{k \in {\mathbb {N}}} {\mathbb {E}}\left[ \int _0^T \Vert \sigma _{n_k}({\tilde{V}}_{n_k}) \Vert _{L_2\left( {\mathscr {U}}, {\mathcal {D}}_0\right) }^{2} \, ds \right] ^{\frac{p}{2}} \le C_T \sup _{k \in {\mathbb {N}}} {\mathbb {E}}\left[ 1 + \sup _{t \in [0,T]} \Vert {\tilde{V}}_{n_k} \Vert _{0,r}^{p} \right] <\infty . \nonumber \\ \end{aligned}$$

(3.27)

(3.27) together with (2.8) and the Vitali convergence theorem yield that the convergence (3.26) occurs in the space $L^2(\tilde{\Omega }; L^2\left( 0,T; {\mathcal {D}}_0\right) )$. Then for any ${\mathcal {R}} \subset {\tilde{\Omega }} \times [0,T]$ measurable, one has

$$\begin{aligned} \lim _{k \rightarrow \infty }{\mathbb {E}}\int _0^T \chi _{{\mathcal {R}}} \left( \int _0^t \Big \langle \sigma _{n_k}(\tilde{V}_{n_k}) , \phi \Big \rangle d{\tilde{W}}_{n_k} \right) dt= {\mathbb {E}}\int _0^T \chi _{{\mathcal {R}}} \left( \int _0^t \Big \langle \sigma (\tilde{V}) , \phi \Big \rangle d {\tilde{W}} \right) dt. \end{aligned}$$

This implies that for a.a. $(t, \omega ) \in [0,T] \times \tilde{\Omega }$,

$$\begin{aligned} \int _0^t \Big \langle \sigma _{n_k}({\tilde{V}}_{n_k}), \phi \Big \rangle d\tilde{W}_{n_k} \rightarrow \int _0^t \Big \langle \sigma ({\tilde{V}}), \phi \Big \rangle d\tilde{W}. \end{aligned}$$

(3.28)

Applying the convergences (3.21), (3.22), (3.23), (3.25) and (3.28) to (3.16), we infer that for all $\phi \in {\mathcal {D}}_0$ and for a.a. $(t, \omega ) \in [0,T] \times \tilde{\Omega }$,

$$\begin{aligned} \Big \langle {\tilde{V}}(t), \phi \Big \rangle + \int _0^t \Big \langle \theta _\rho (\Vert \tilde{V}\Vert _{\tau ,r} ) Q(\tilde{V}, \tilde{V}) + F(\tilde{V}) ,\phi \Big \rangle ds = \Big \langle {\tilde{V}}_0, \phi \Big \rangle + \int _0^t \Big \langle \sigma ({\tilde{V}}) , \phi \Big \rangle d\tilde{W}. \end{aligned}$$

Therefore $(\tilde{{\mathcal {S}}}, \tilde{W}, \tilde{V}, T)$ is a local martingale solution to the modified system (3.1).

Proof of the regularity (3.15): The proof of continuity of ${\tilde{V}}$ in time in the space ${\mathcal {D}}_{\tau (t),r}$ follows similarly as in [15, Section 7.3]. For completeness, we highlight the main steps. By the property of $\sigma$ in (2.2) and the regularity of ${\tilde{V}}$ (3.18), we have

$$\begin{aligned} \sigma ({\tilde{V}}) \in L^2\left( \tilde{\Omega }; L^2\left( 0,T; L_2\left( {\mathscr {U}}, {\mathcal {D}}_{\tau (t),r}\right) \right) \right) . \end{aligned}$$

Therefore, the solution to

$$\begin{aligned} dZ = \sigma ({\tilde{V}}) \, d\tilde{W}, \qquad Z(0) = {\tilde{V}}_0, \end{aligned}$$

satisfies

$$\begin{aligned} Z \in L^2\left( \tilde{\Omega }; C\left( [0,T]; {\mathcal {D}}_{\tau (t),r}\right) \right) \cap L^2\left( \tilde{\Omega }; L^2\left( 0,T; {\mathcal {D}}_{\tau (t),r+\frac{1}{2}}\right) \right) . \end{aligned}$$

(3.29)

Defining ${\bar{V}} = {\tilde{V}} - Z$, by (3.1a) we have ${\mathbb {P}}$-almost surely

$$\begin{aligned} \tfrac{d}{dt} {\bar{V}} + \theta (\Vert {\tilde{V}} \Vert _{\tau ,r}) Q({\tilde{V}}, {\tilde{V}}) + F({\tilde{V}}) = 0, \qquad {\bar{V}}(0) = 0. \end{aligned}$$

Thanks (3.18) and Lemma A.3,

$$\begin{aligned} \ \theta (\Vert {\tilde{V}} \Vert _{\tau ,r}) Q({\tilde{V}}, {\tilde{V}}) \text { and } F({\tilde{V}}) \in L^2\left( \tilde{\Omega }; L^2\left( 0,T; {\mathcal {D}}_{\tau (t),r-\frac{1}{2}}\right) \right) \end{aligned}$$

and thus

$$\begin{aligned} \frac{d}{dt} A^{r}e^{\tau (t)A} {\bar{V}}= & {} A^{r}e^{\tau (t)A} \frac{d}{dt} {\bar{V}} + \dot{\tau }A^{r+1}e^{\tau (t)A} {\bar{V}} \in L^2\left( \tilde{\Omega }; L^2\left( 0,T; {\mathcal {D}}_{0,\frac{1}{2}}'\right) \right) , \\&A^{r}e^{\tau (t)A} {\bar{V}} \in L^2\left( \tilde{\Omega }; L^2\left( 0,T; {\mathcal {D}}_{0,\frac{1}{2}}\right) \right) , \end{aligned}$$

where ${\mathcal {D}}_{0,\frac{1}{2}}'$ is the dual space of ${\mathcal {D}}_{0,\frac{1}{2}}$. Since ${\mathcal {D}}_{0,\frac{1}{2}} \subset {\mathcal {D}}_0 \equiv {\mathcal {D}}_0' \subset {\mathcal {D}}_{0,\frac{1}{2}}'$, by the Lions-Magenes Lemma (e.g., [46, Lemma 1.2, Chapter 3]), we infer that $A^{r}e^{\tau (t)A} {\bar{V}} \in L^2\left( \tilde{\Omega }; C\left( \left[ 0,T \right] ; {\mathcal {D}}_0\right) \right)$. Similarly, one can show that ${\bar{V}} \in L^2\left( \tilde{\Omega }; C\left( \left[ 0,T \right] ; {\mathcal {D}}_0\right) \right)$, and thus ${\bar{V}} \in L^2\left( \Omega ; C\left( \left[ 0,T \right] ; {\mathcal {D}}_{\tau (t),r}\right) \right)$. This together with (3.29) imply that $\tilde{V} \in L^2\left( \Omega ; C\left( \left[ 0,T \right] ; {\mathcal {D}}_{\tau (t),r}\right) \right)$. The second part of (3.15) follows directly from (3.18). $\square$

Corollary 3.5

Suppose that $\mu _0$ satisfy (2.12) with constant $M>0$. Let $\rho \ge M$, and let $(\tilde{{\mathcal {S}}}, \tilde{W}, \tilde{V},T)$ be the local martingale solution to the modified system (3.1) given in Proposition 3.4. Let

$$\begin{aligned} \eta = \inf \left\{ t \ge 0 \mid \Vert \tilde{V}\Vert _{\tau ,r} \ge \frac{\rho }{2} \right\} . \end{aligned}$$

(3.30)

Then $(\tilde{{\mathcal {S}}}, \tilde{W}, \tilde{V}, \eta \wedge T)$ is a local martingale solution to the problem (2.11). Moreover,

$$\begin{aligned}&\tilde{V}\left( \cdot \wedge \eta \right) \in L^2 \left( {\tilde{\Omega }}; C\left( [0, T], {\mathcal {D}}_{\tau (t),r} \right) \right) , \quad \\&\quad \mathbbm {1}_{[0, \eta ]} \Vert A^{r+\frac{1}{2}} e^{\tau (t)A} \tilde{V} \Vert ^2 \Vert A^{r} e^{\tau (t)A} \tilde{V}\Vert ^{p-2} \in L^1\left( {\tilde{\Omega }}; L^1\left( 0, T \right) \right) . \end{aligned}$$

3.4 Pathwise uniqueness

In this section, we establish the pathwise uniqueness for the original system (2.11). The following proposition is similar to [15, Proposition 5.1] and [6, Proposition 3.5]. However, in their results the uniqueness of martingales solutions to the corresponding modified equations is proven, while in our case, due to the different nonlinear estimates, only the uniqueness to the original system is obtained.

Proposition 3.6

Let $\tau (t)$ and T defined as in (3.7) and (3.8), respectively. Suppose that $\sigma$ and f satisfy (2.2) and (2.3), respectively. Let ${\mathcal {S}}= \left( \Omega , {\mathcal {F}}, \left( {\mathcal {F}}_t \right) _{t \ge 0}, {\mathbb {P}}\right)$ and W be fixed. Suppose that there exist two local martingale solutions $\left( {\mathcal {S}}, W, V^1,T\right)$ and $\left( {\mathcal {S}}, W, V^2,T\right)$ to the modified system (3.1). Correspondingly, $\left( {\mathcal {S}}, W, V^1, \eta _1\wedge T\right)$ and $\left( {\mathcal {S}}, W, V^2, \eta _2\wedge T\right)$ are the two local martingale solutions to the original system (2.11). Denote $\Omega _0 = \left\{ V^1(0) \equiv V^2(0) \right\} \subseteq \Omega$, and $\eta = \eta _1 \wedge \eta _2$. Then

$$\begin{aligned} {\mathbb {P}}\left( \left\{ \mathbbm {1}_{\Omega _0}\left( V^1(t\wedge \eta ) - V^2(t \wedge \eta )\right) = 0 \ \text {for all} \ t \in [0,T] \right\} \right) = 1. \end{aligned}$$

Proof

Let $R = V^1 - V^2$ and ${\bar{R}} = \mathbbm {1}_{\Omega _0} R$. Let $\eta ^n$ be the stopping time defined by

$$\begin{aligned} \eta ^n = \inf \left\{ t \ge 0 : \int _0^t \sum \limits _{i=1}^2\Vert V^i\Vert _{\tau ,r} ^2 + \Vert A^{r+\frac{1}{2}}e^{\tau A} V^i\Vert ^2\, ds \ge n \right\} , \end{aligned}$$

where the function $\tau (t)$ is defined in (3.7). Notice that both $\tau (t)$ and the existence time T defined in (3.8) are independent of the initial data, thus both $V^1$ and $V^2$ are defined on the same interval [0, T]. Since $V_0 \in L^p\left( \Omega ; {\mathcal {D}}_{\tau _0,r}\right)$ with $p \ge 4$, from the estimates in Lemma 3.1 we deduce that $\lim \limits _{n\rightarrow \infty }\eta ^n \ge T$ ${\mathbb {P}}$-a.s. and therefore it suffices to show that

$$\begin{aligned} {\mathbb {E}}\sup _{s \in \left[ 0, \eta \wedge \eta ^n \wedge t \right] } \Vert {\bar{R}}(s) \Vert _{\tau ,r}^2 = 0, \end{aligned}$$

for all $t \in (0,T]$ and $n \in {\mathbb {N}}$ such that $\eta ^n \le T$. Subtracting the equation of $V^2$ from the one of $V^1$ one has

$$\begin{aligned} \begin{aligned}&dR + \left[ Q(V^1,V^1) - Q(V^2,V^2) + f_0 R^\perp + \nabla (P^1-P^2)\right] \, dt \\&\quad = \left[ \sigma (V^1) - \sigma (V^2) \right] \, dW, \\&R(0) = V^1(0) - V^2(0). \end{aligned} \end{aligned}$$

Fix $n \in {\mathbb {N}}$ and let $\eta _a$, $\eta _b$ be stopping times such that $0 \le \eta _a \le \eta _b \le \eta \wedge \eta ^n \wedge T$. We now calculate

$$\begin{aligned} {\mathbb {E}}\sup _{s \in \left[ \eta _a, \eta _b\right] } \Vert {\bar{R}} \Vert ^2&\le {\mathbb {E}}\Vert {\bar{R}}(\eta _a) \Vert ^2 \nonumber \\&\quad + 2 {\mathbb {E}}\int _{\eta _a}^{\eta _b} \mathbbm {1}_{\Omega _0}\left| \Big \langle Q(V^1,V^1) - Q(V^2,V^2), {\bar{R}}\Big \rangle \, ds \right| \nonumber \\&\quad + 2 {\mathbb {E}}\int _{\eta _a}^{\eta _b} \left| \Big \langle f_0 {{\bar{R}}}^\perp + \mathbbm {1}_{\Omega _0}\nabla (P^1-P^2), {\bar{R}}\Big \rangle \right| \, ds \nonumber \\&\quad + 2 {\mathbb {E}}\sup _{s \in \left[ \eta _a, \eta _b\right] } \left| \int _{\eta _a}^{s} \mathbbm {1}_{\Omega _0}\Big \langle \sigma (V^1) - \sigma (V^2) , {\bar{R}} \Big \rangle dW \right| \nonumber \\&\quad + {\mathbb {E}}\int _{\eta _a}^{\eta _b} \mathbbm {1}_{\Omega _0}\Vert \sigma (V^1) - \sigma (V^2) \Vert _{L_2\left( {\mathscr {U}}, {\mathcal {D}}_0\right) }^2 \, ds \nonumber \\&= {\mathbb {E}}\Vert {\bar{R}}(\eta _a) \Vert ^2 + A_1 + A_2 + A_3 + A_4 . \end{aligned}$$

(3.31)

where the Itô’s Lemma, integrating in time and taking supremums, multiplying by $\mathbbm {1}_{\Omega _0}$ and taking the expected value have been successively applied.

For $A_1$, thanks to the the Hölder inequality and the Sobolev inequality:

$$\begin{aligned} A_1&\le 2 {\mathbb {E}}\int _{\eta _a}^{\eta _b} \left| \Big \langle Q({{\bar{R}}}, V^1) + Q(V^2, {{\bar{R}}}) , {\bar{R}} \Big \rangle \right| \, ds \\&\le C {\mathbb {E}}\int _{\eta _a}^{\eta _b} \left( \Vert V^1\Vert _{0,2} + \Vert V^2\Vert _{0,2} \right) \Vert {{\bar{R}}}\Vert _{0,2}^2 ds. \end{aligned}$$

Since $\langle {{\bar{R}}}^\perp , {{\bar{R}}}\rangle = 0$ and $\langle \mathbbm {1}_{\Omega _0} \nabla (P^1-P^2), {{\bar{R}}}\rangle = 0$, one has $A_2= 0$. Regarding $A_3$, the Burkholder-Davis-Gundy inequality (2.8), the Lipschitz continuity of in $\sigma$ (2.2), and the Young inequality give

$$\begin{aligned} \begin{aligned} A_3 \le&C {\mathbb {E}}\left( \int _{\eta _a}^{\eta _b} \mathbbm {1}_{\Omega _0}\Vert \sigma (V^1) - \sigma (V^2) \Vert _{L_2\left( {\mathscr {U}}, {\mathcal {D}}_0\right) }^2 \Vert {\bar{R}} \Vert ^2 \, ds \right) ^{1/2}\le \frac{1}{2} {\mathbb {E}}\sup _{s \in \left[ \eta _a, \eta _b\right] } \Vert {\bar{R}} \Vert ^2 \\&+ C {\mathbb {E}}\int _{\eta _a}^{\eta _b} \Vert {\bar{R}} \Vert _{0,r}^2 ds. \end{aligned} \end{aligned}$$

Finally, the integral $A_4$ is estimated using the Lipschitz continuity of $\sigma$ in (2.2). We get

$$\begin{aligned} A_4 \le C {\mathbb {E}}\int _{\eta _a}^{\eta _b} \Vert {\bar{R}} \Vert _{0,r}^2 \, ds. \end{aligned}$$

For the higher order part, thanks to (3.18), one is allowed to take inner product of $dA^{r}e^{\tau A} R$ with $A^{r}e^{\tau A} R$ and follow a similar derivation of (3.31) to get

$$\begin{aligned}&{\mathbb {E}}\sup _{s \in \left[ \eta _a, \eta _b\right] } \Vert A^r e^{\tau A} {\bar{R}} \Vert ^2 \\&\quad \le {\mathbb {E}}\Vert A^r e^{\tau (\eta _a) A} {\bar{R}}(\eta _a) \Vert ^2 + 2 {\mathbb {E}}\int _{\eta _a}^{\eta _b} {\dot{\tau }} \Vert A^{r+\frac{1}{2}} e^{\tau A} {\bar{R}} \Vert ^2 ds \\&\qquad + 2 {\mathbb {E}}\int _{\eta _a}^{\eta _b} \mathbbm {1}_{\Omega _0}\left| \Big \langle A^r \Big (e^{\tau A} Q(V^1,V^1) - Q(V^2,V^2)\Big ), A^r e^{\tau A}{\bar{R}}\Big \rangle \, ds \right| \\&\qquad + 2 {\mathbb {E}}\int _{\eta _a}^{\eta _b} \left| \Big \langle f_0 A^r e^{\tau A} {{\bar{R}}}^\perp + \mathbbm {1}_{\Omega _0} A^r e^{\tau A} \nabla (P^1-P^2), A^r e^{\tau A} {\bar{R}}\Big \rangle \right| \, ds \\&\qquad + 2 {\mathbb {E}}\sup _{s \in \left[ \eta _a, \eta _b\right] } \left| \int _{\eta _a}^{s} \mathbbm {1}_{\Omega _0}\Big \langle A^r e^{\tau A} \Big (\sigma (V^1) - \sigma (V^2) \Big ) , A^r e^{\tau A}{\bar{R}} \Big \rangle dW \right| \\&\qquad + {\mathbb {E}}\int _{\eta _a}^{\eta _b} \mathbbm {1}_{\Omega _0}\Vert \sigma (V^1) - \sigma (V^2) \Vert _{L_2\left( {\mathscr {U}}, {\mathcal {D}}_{\tau , r}\right) }^2 \, ds \\&\quad = {\mathbb {E}}\Vert {\bar{R}}(\eta _a) \Vert ^2 + 2 {\mathbb {E}}\int _{\eta _a}^{\eta _b} {\dot{\tau }} \Vert A^{r+\frac{1}{2}} e^{\tau A} {\bar{R}} \Vert ^2 ds + B_1 + B_2 + B_3 + B_4. \end{aligned}$$

Similar to $A_1-A_4$, by applying Lemma A.1 and since $\eta _b \le \eta$, one obtains that

$$\begin{aligned} \begin{aligned} B_1&\le 2 {\mathbb {E}}\int _{\eta _a}^{\eta _b} \left| \Big \langle A^r e^{\tau A} \Big (\ Q({{\bar{R}}}, V^1) + Q(V^2, {{\bar{R}}}) \Big ) ,A^r e^{\tau A} {\bar{R}} \Big \rangle \right| \, ds \\&\le C_r {\mathbb {E}}\int _{\eta _a}^{\eta _b} \rho \Vert A^{r+\frac{1}{2}} e^{\tau A} {\bar{R}} \Vert ^2 ds \\&\quad + C_r {\mathbb {E}}\int _{\eta _a}^{\eta _b} \sum \limits _{i=1}^2 \Vert A^{r+\frac{1}{2}} e^{\tau A} V^i \Vert \Vert {\bar{R}} \Vert _{\tau ,r} \Vert A^{r+\frac{1}{2}} e^{\tau A} {\bar{R}} \Vert ds \\&\le C_r{\mathbb {E}}\int _{\eta _a}^{\eta _b} (\rho + 1) \Vert A^{r+\frac{1}{2}} e^{\tau A} {\bar{R}} \Vert ^2 ds + C_r{\mathbb {E}}\int _{\eta _a}^{\eta _b} \sum \limits _{i=1}^2\Vert A^{r+\frac{1}{2}} e^{\tau A} V^i \Vert ^2 \Vert {\bar{R}} \Vert _{\tau ,r}^2 ds. \end{aligned} \end{aligned}$$

(3.32)

Since $\langle f_0 A^r e^{\tau A} {{\bar{R}}}^\perp , A^r e^{\tau A} {\bar{R}}\rangle = 0$ and $\langle \mathbbm {1}_{\Omega _0} A^r e^{\tau A} \nabla (P^1-P^2), A^r e^{\tau A} {\bar{R}}\rangle = 0$, one has $B_2= 0$. Regarding $B_3$, the Burkholder-Davis-Gundy inequality (2.8), the Lipschitz continuity of $\sigma$ in (2.2), and the Young inequality give

$$\begin{aligned} \begin{aligned} B_3 \le&C {\mathbb {E}}\left( \int _{\eta _a}^{\eta _b} \mathbbm {1}_{\Omega _0}\Vert \sigma (V^1) - \sigma (V^2) \Vert _{L_2\left( {\mathscr {U}}, {\mathcal {D}}_{\tau ,r}\right) }^2\Vert A^{r} e^{\tau A} {\bar{R}} \Vert ^2 \, ds \right) ^{1/2} \\ \le&\frac{1}{2} {\mathbb {E}}\sup _{s \in \left[ \eta _a, \eta _b\right] } \Vert A^{r} e^{\tau A} {\bar{R}} \Vert ^2 + C {\mathbb {E}}\int _{\eta _a}^{\eta _b} \Vert {\bar{R}} \Vert _{\tau ,r}^2 + \Vert A^{r+\frac{1}{2}} e^{\tau A} {\bar{R}} \Vert ^2 ds. \end{aligned} \end{aligned}$$

Finally, the integral $B_4$ is estimated using the Lipschitz continuity of $\sigma$ in (2.2). Therefore,

$$\begin{aligned} B_4 \le C {\mathbb {E}}\int _{\eta _a}^{\eta _b} \Vert {\bar{R}} \Vert _{\tau ,r}^2 +\Vert A^{r+\frac{1}{2}} e^{\tau A} {\bar{R}} \Vert ^2 ds. \end{aligned}$$

Collecting the estimates of $A_1$ to $A_4$ and $B_1$ to $B_4$ gives

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\sup _{s \in \left[ \eta _a, \eta _b\right] } \Vert {\bar{R}}(s) \Vert _{\tau (s),r}^2 \\&\quad \le C{\mathbb {E}}\Vert {\bar{R}} (\eta _a) \Vert _{\tau (\eta _a),r}^2 + C {\mathbb {E}}\int _{\eta _a}^{\eta _b} \left( {\dot{\tau }} + C_r\left( \rho +1 \right) \right) \Vert A^{r+\frac{1}{2}} e^{\tau A} {\bar{R}} \Vert ^2 \, ds \\&\qquad + C_r{\mathbb {E}}\int _{\eta _a}^{\eta _b} \left( 1 + \sum \limits _{i=1}^2 \left( \Vert V^i \Vert _{\tau ,r}^2 + \Vert A^{r+\frac{1}{2}} e^{\tau A} V^i \Vert ^2 \right) \right) \Vert {\bar{R}} \Vert _{\tau ,r}^2 \, ds. \end{aligned} \end{aligned}$$

(3.33)

By taking the constant $C_{r,\rho }$ appearing in the definition of $\tau$ in (3.7) large enough so that $C_{r,\rho } \ge C_r$ where $C_r$ appearing in (3.33), one obtains that ${\dot{\tau }} + C_r\left( \rho +1 \right) \le 0$. With this and (3.33) one can apply the stochastic Gronwall Lemma (see [25, Lemma 5.3]) to conclude the proof. $\square$

Remark 6

Unlike [6, 15, 43] where the authors were able to show the strong uniqueness of the martingale solutions to the modified system, we are only able to show it for the original system. The reason is that we need the help of stopping times to control the terms $\Vert A^{r} e^{\tau A} V^i \Vert$ in (3.32). With stopping times, we are considering the solutions to the original system instead of the modified system. The essential reason behind this is that the horizontal viscosity can help control the nonlinear estimates in [6, 15, 43], while the dissipation from analytic framework is not strong enough to provide the same control in our case.

4 The Vertically Viscous Case

In this section, we study system (2.14), i.e., the stochastic PEs with vertical viscosity. For some fixed $\rho \ge M$ with M appearing in the condition (2.15), and the cut-off function $\theta _\rho$ defined in (3.2), we consider the following modified system

$$\begin{aligned}&d V + \left[ \theta _\rho \left( \Vert V\Vert _{\tau ,r,\gamma ,r} \right) Q( V, V) + F(V) - \nu _z \partial _{zz} V\right] dt = \sigma (V) dW , \end{aligned}$$

(4.1a)

$$\begin{aligned}&V(0) = V_0, \end{aligned}$$

(4.1b)

where the functions $\tau (t)$ and $\gamma (t)$ in $\Vert V\Vert _{\tau ,r,\gamma ,r}$ will be determined later. Following the same setup as in Sect. 3.1, for each $n\in {{\mathbb {N}}}$, we consider the following Galerkin approximation of system (4.1) at order n as:

$$\begin{aligned}&d V_n + \left[ \theta _\rho \left( \Vert V_n\Vert _{\tau ,r,\gamma ,r} \right) Q_n(V_n, V_n) + F_n(V_n) - \nu _z \partial _{zz} V_n \right] dt = \sigma _n (V_n) dW , \end{aligned}$$

(4.2a)

$$\begin{aligned}&V_n(0) = P_n V_0 := (V_n)_0. \end{aligned}$$

(4.2b)

Define $U_n = e^{\tau A_h}e^{\gamma A_z}V_n$, and therefore, $V_n = e^{-\tau A_h}e^{-\gamma A_z}U_n$. Since $dU_n = (\dot{\tau }A_h U_n + {\dot{\gamma }} A_z U_n)dt + e^{\tau A_h}e^{\gamma A_z}dV_n$, system (4.2) is equivalent to

$$\begin{aligned}&d U_n + \Big [\theta _\rho \left( \Vert U_n\Vert _{0,r} \right) e^{\tau A_h} e^{\gamma A_z} Q_n(e^{-\tau A_h} e^{-\gamma A_z}U_n, e^{-\tau A_h} e^{-\gamma A_z}U_n) \nonumber \\&\qquad + e^{\tau A_h} e^{\gamma A_z}F_n(e^{-\tau A_h} e^{-\gamma A_z}U_n) -\dot{\tau }A_h U_n - {\dot{\gamma }} A_z U_n - \nu _z \partial _{zz} U_n\Big ]dt \end{aligned}$$

(4.3a)

$$\begin{aligned}&\quad = e^{\tau A_h} e^{\gamma A_z}\sigma _n (e^{-\tau A_h} e^{-\gamma A_z}U_n) dW , \end{aligned}$$

(4.3b)

$$\begin{aligned}&U_n(0) = e^{\tau _0 A_h }V_n(0), \end{aligned}$$

(4.3c)

and $U_n = e^{\tau A_h}e^{\gamma A_z}V_n$ is the solution to system (4.3). Notice that initially the analytic radius in z is $\gamma _0 =0$, thus $e^{\gamma _0 A_z} = 1$.

This section aims to provide results that are parallel to the inviscid case summarized in Theorem 1.2. For this purpose, we fix $\rho >M$, $\tau _0>0$, $\gamma _0=0$, and $r>\frac{5}{2}$ through this section. In Sect. 4.1, we establish the energy estimates of the Galerkin systems (4.2) and (4.3). Section 4.2 will focus on the local existence martingale solutions and pathwise uniqueness to the original system (2.14), where we only highlight some necessary details as most of the proofs are similar to the inviscid case given in Sects. 3.3 and 3.4. This will complete the proof of Theorem 1.2.

4.1 Energy estimates

We establish the main estimates needed to pass to the limit in the Galerkin system (4.2) (and equivalently, system (4.3)). Recall that when $\tau = \gamma =0$ and $r=s$, ${\mathcal {D}}_{0,r,0,s} = {\mathcal {D}}_{0,r}$.

Lemma 4.1

Let $\tau _0>0, \gamma _0 = 0$, $p\ge 2$, and $r>\frac{5}{2}$ be fixed. Suppose that $V_0 \in L^p(\Omega ; {\mathcal {D}}_{\tau _0,r,0,r})$ be an ${\mathcal {F}}_0$-measurable random variable. Assume that $\sigma$ satisfies (2.4) with $\delta$ satisfying (4.5), and let f satisfies (2.5). Let $V_n$ be the solutions of system (4.2). Then there exist functions $\tau (t)$ and $\gamma (t)$ defined in (4.6), and a constant T defined in (4.8) such that the following hold:

$$\begin{aligned}&\text {(i)} \quad {\mathbb {E}} \Big [\sup \limits _{t\in [0,T]} \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p + \int _0^T \nu _z \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 ds \\&\quad + \int _0^T \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ds \Big ] \le C_p(1\\&\quad + {\mathbb {E}} \Vert V_0\Vert _{\tau _0,r,0,r}^p) e^{C_p T}. \end{aligned}$$

(ii) Let $\alpha \in [0,1/2)$. Then $\int _0^{\cdot } e^{\tau A_h} e^{\gamma A_z}\sigma _n(V_n) dW$ is bounded in $L^p\left( \Omega ; W^{\alpha , p}(0, \right. \left. T; {\mathcal {D}}_{0,r-1}) \right) ;$

(iii) If moreover $p\ge 4$, $e^{\tau A_h} e^{\gamma A_z}V_n - \int _0^\cdot e^{\tau A_h} e^{\gamma A_z}\sigma _n(V_n) dW$ is bounded in $L^2\left( \Omega ; W^{1, 2}(0, T; {\mathcal {D}}_{0,r-1}))\right) .$

Remark 7

Different from the inviscid case where the radius of analyticity is shrinking on all variables, from (4.6) below, one will see that the radius of analyticity in the z variable starts from 0 and is increasing thanks to the vertically viscousity.

Proof

For $\Vert V_n\Vert$, by applying the finite dimensional Itô formula to (4.2a), for $p\ge 2$, one has

$$\begin{aligned} \begin{aligned}&d\Vert V_n\Vert ^p + p\nu _z\Vert A_z V_n\Vert ^2 \Vert V_n\Vert ^{p-2} \\&\quad = - p \theta _\rho \left( \Vert V_n\Vert _{\tau ,r,\gamma ,r} \right) \Vert V_n\Vert ^{p-2} \big \langle Q_n(V_n, V_n) , V_n\big \rangle dt - p \Vert V_n\Vert ^{p-2} \big \langle F_n (V_n), V_n\big \rangle dt \\&\qquad +\frac{p}{2} \Vert \sigma _n(V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} \Vert V_n\Vert ^{p-2} dt + \frac{p(p-2)}{2} \big \langle \sigma _n(V_n), V_n \big \rangle ^2 \Vert V_n\Vert ^{p-4} dt \\&\qquad + p \big \langle \sigma _n(V_n), V_n \big \rangle \Vert V_n\Vert ^{p-2} dW \\&\quad := A_1 dt + A_2 dt + A_3 dt + A_4 dt + A_5 dW. \end{aligned} \end{aligned}$$

For $\Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert$, applying the finite dimensional Itô formula gives, for $p\ge 2$,

$$\begin{aligned}&d\Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^p + p\nu _z \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A^r A_z e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^2 \\&\qquad - p\big ( \dot{\tau } \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^2 \\&\qquad + \dot{\gamma } \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_z^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 \big ) dt \\&\quad = - p \theta _\rho \left( \Vert V_n\Vert _{\tau ,r,\gamma ,r} \right) \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \big \langle A^r e^{\tau A_h} e^{\gamma A_z} Q_n(V_n, V_n) , A^r e^{\tau A_h} e^{\gamma A_z} V_n\big \rangle dt \\&\qquad - p \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \big \langle A^r e^{\tau A_h} e^{\gamma A_z} F_n (V_n), A^r e^{\tau A_h} e^{\gamma A_z}V_n\big \rangle dt \\&\qquad +\frac{p}{2} \Vert A^r e^{\tau A_h} e^{\gamma A_z}\sigma _n(V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} dt \\&\qquad + \frac{p(p-2)}{2} \big \langle A^r e^{\tau A_h} e^{\gamma A_z} \sigma _n(V_n), A^r e^{\tau A_h} e^{\gamma A_z} V_n \big \rangle ^2 \Vert A^r e^{\tau A_h} e^{\gamma _z}V_n\Vert ^{p-4} dt \\&\qquad + p \big \langle A^r e^{\tau A_h} e^{\gamma A_z} \sigma _n(V_n), A^r e^{\tau A_h} e^{\gamma A_z} V_n \big \rangle \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} dW \\&\quad := B_1 dt + B_2 dt + B_3 dt + B_4 dt + B_5 dW. \end{aligned}$$

By integration by parts and due to the boundary condition and incompressible condition, $A_1 = 0$. For the nonlinear term $B_1$, Lemma A.2, Young’s inequality and the property of cut-off function $\theta _\rho$ together give

$$\begin{aligned} |B_1| \le&C_{p,r} \theta _\rho \left( \Vert V_n\Vert _{\tau ,r,\gamma ,r} \right) \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \\&\times \left( \Vert V_n\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V\Vert ^2 \right. \\&\left. + \Vert V_n\Vert _{\tau ,r,\gamma ,r} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V\Vert \right) \\ \le&C_{p,r,\nu _z} (1+\rho ^2) \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V\Vert ^2 \\&+ \frac{p\nu _z}{8} \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 . \end{aligned}$$

For $A_2$ and $B_2$, by integration by parts, thanks to the periodic boundary condition, the Cauchy-Schwartz inequality, and Young’s inequality, and noticing that $\gamma (T) = \frac{\nu _z}{8} T \le \frac{\nu _z}{8} \tau _0 \le \gamma ^*$ from (2.5), (4.6), and (4.8), one obtains that

$$\begin{aligned}&\int _0^t |A_2| + |B_2| ds \\&\quad \le C_p\int _0^t \Vert V_n\Vert _{\tau ,r,\gamma ,r}^{p-2} \Big ( \Big |\big \langle A^r e^{\tau A_h} e^{\gamma A_z} f, A^r e^{\tau A_h} e^{\gamma A_z}V_n\big \rangle \Big | + \Big |\big \langle f, V_n\big \rangle \Big | \Big )ds \\&\quad \le \int _0^t C_p \Vert V_n\Vert _{\tau ,r,\gamma ,r}^{p-2} \left( \Vert f\Vert _{\tau _0,r,\gamma (T),r}^2 + \Vert V_n\Vert _{\tau ,r,\gamma ,r}^2 \right) ds \\&\quad \le \frac{1}{4} \left( \sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p + \sup \limits _{s\in [0,t]}\Vert A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^p \right) \\&\qquad + C_p\left( 1 + \int _0^t \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p ds\right) . \end{aligned}$$

From the assumptions on $\sigma$ in (2.4), we know that

$$\begin{aligned}&\Vert \sigma (V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} + \Vert A^r e^{\tau A_h} e^{\gamma A_z} \sigma (V_n)\Vert ^2_{L_2({\mathscr {U}}, {\mathcal {D}}_0)} \\&\quad \le C (1+ \Vert V_n\Vert _{\tau ,r,\gamma ,r}^2 + \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ) + \delta ^2 \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2. \end{aligned}$$

Therefore, by the Cauchy-Schwarz inequality,

$$\begin{aligned}&|A_3|+| A_4| + |B_3|+|B_4| \\&\quad \le C_p (1+ \Vert V_n\Vert _{\tau ,r,\gamma ,r}^2 + \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 \\&\qquad + \delta ^2 \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 )\Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \\&\quad \le C_p(1+ \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p) + C_p \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \left( \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 \right. \\&\qquad \left. + \delta ^2 \Vert \partial _zV_n\Vert _{\tau ,r,\gamma ,r}^2\right) . \end{aligned}$$

Finally, thanks to the Burkholder-Davis-Gundy inequality, from the Cauchy-Schwartz inequality, Young’s inequality, and the property of in $\sigma$ (2.4), we deduce that

$$\begin{aligned}&{\mathbb {E}} \sup \limits _{s\in [0,t]} \Big |\int _0^s A_5 + B_5 dW \Big | \\&\quad \le C_p {\mathbb {E}}\left( \int _0^t \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{2(p-1)} \left( 1+ \Vert V_n\Vert _{\tau ,r,\gamma ,r}^2 + \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 \right. \right. \\&\qquad \left. \left. + \delta ^2 \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 \right) \right) ^{\frac{1}{2}} \\&\quad \le \frac{1}{4} \left( {\mathbb {E}}\sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p + {\mathbb {E}} \sup \limits _{s\in [0,t]}\Vert A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^p \right) \\&\qquad + C_p{\mathbb {E}} \int _0^t \left( 1+ \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p + \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \left( \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 \right. \right. \\&\qquad \left. \left. + \delta ^2 \Vert \partial _zV_n\Vert _{\tau ,r,\gamma ,r}^2\right) \right) ds . \end{aligned}$$

Notice that since $|k_3|^a\le |k_3|^b$ for $0\le a \le b$ and $k_3\in {{\mathbb {Z}}}$, one has

$$\begin{aligned} \begin{aligned} \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2&\le 2 \Vert A_z A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 \Vert A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^{p-2}. \end{aligned} \end{aligned}$$

Combining all estimates above produces

$$\begin{aligned}&{\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p + {\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^p \nonumber \\&\qquad + {\mathbb {E}} \int _0^t \frac{p}{2}\nu _z \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 ds \nonumber \\&\qquad + {\mathbb {E}} \int _0^t \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ds \nonumber \\&\quad \le {\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p + {\mathbb {E}} \sup \limits _{s\in [0,t]} \Vert A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^p\nonumber \\&\qquad + {\mathbb {E}} \int _0^t p\nu _z \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ds\nonumber \\&\qquad + {\mathbb {E}} \int _0^t \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ds \nonumber \\&\quad \le p {\mathbb {E}} \int _0^t \dot{\gamma } \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_z^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ds \nonumber \\&\qquad + p \big (\dot{\tau } + C_{p,r,\nu _z} (\rho ^2+1)\big ) {\mathbb {E}} \int _0^t \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V\Vert ^2 ds \nonumber \\&\qquad + {\mathbb {E}} \int _0^t \left( \frac{p}{8}\nu _z + C_p \delta ^2 \right) \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 ds \nonumber \\&\qquad + \frac{1}{2} \left( {\mathbb {E}}\sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p + {\mathbb {E}} \sup \limits _{s\in [0,t]}\Vert A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^p \right) \nonumber \\&\qquad + C_p {\mathbb {E}} \int _0^t \left( 1+ \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p\right) ds. \end{aligned}$$

(4.4)

Under the assumption that

$$\begin{aligned} \delta ^2 \le \frac{ p\nu _z}{8 C_p}, \end{aligned}$$

(4.5)

by taking

$$\begin{aligned} \tau (t) = - C_{p,r,\nu _z} (\rho ^2+1) t, \text { and } \gamma (t)=\frac{1}{8}\nu _z t, \end{aligned}$$

(4.6)

one can rewrite (4.4) as

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \left( {\mathbb {E}}\sup \limits _{s\in [0,t]} \Vert V_n\Vert ^p + {\mathbb {E}} \sup \limits _{s\in [0,t]}\Vert A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^p \right) \\&\qquad + {\mathbb {E}} \int _0^t \frac{p}{8} \nu _z\Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 ds \\&\qquad + {\mathbb {E}} \int _0^t \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ds \\&\quad \le C_p \left( 1+{\mathbb {E}} \int _0^t \left( 1+ \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p\right) ds \right) . \end{aligned} \end{aligned}$$

(4.7)

Now define the time T by

$$\begin{aligned} T = \frac{\tau _0}{2C_{p,r,\nu _z}(\rho ^2 + 1)}, \end{aligned}$$

(4.8)

then one observes that $\tau (T) = \frac{\tau _0}{2}$ and $\tau (t)>0$ for $t\in [0,T]$. Based on (4.7), we have for $t\in [0,T]$,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\sup \limits _{s\in [0,t]} \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p + {\mathbb {E}} \int _0^t \nu _z \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 ds \\&\qquad + {\mathbb {E}} \int _0^t \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ds \\&\quad \le C_p \left( 1+{\mathbb {E}} \int _0^t \left( 1+ \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p\right) ds \right) . \end{aligned} \end{aligned}$$

(4.9)

Thanks to the Gronwall inequality [25, Lemma 5.3], for $p\ge 2$ and $t=T$, one has

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \Big [\sup \limits _{t\in [0,T]} \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p + \int _0^T \nu _z \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 ds \\&\qquad + \int _0^T \Vert A^r e^{\tau A_h} e^{\gamma A_z}V_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} V_n\Vert ^2 ds \Big ] C_p\\&\quad \le (1+ {\mathbb {E}} \Vert V_0\Vert _{\tau _0,r,0,r}^p) e^{C_p T}. \end{aligned} \end{aligned}$$

(4.10)

Regarding part (ii), we apply the fractional Burkholder-Davis-Gundy inequality (2.9), (2.4), and the estimate (4.10) to obtain

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\left| \int _0^\cdot e^{\tau A_h }e^{\gamma A_z}\sigma _n\left( V_n\right) \, dW \right| ^p_{W^{\alpha , p}\left( 0, T; {\mathcal {D}}_{0,r-1}\right) } \\&\quad \le C_p {\mathbb {E}}\int _0^T \Vert e^{\tau A_h }e^{\gamma A_z}\sigma _n\left( V_n\right) \Vert ^p_{L_2\left( {\mathscr {U}}, {\mathcal {D}}_{0,r-1}\right) } \, ds \\&\quad \le C_{p,\delta } {\mathbb {E}}\left[ \int _0^T 1 + \Vert V_n\Vert _{\tau ,r,\gamma ,r}^p \, ds \right] < \infty . \end{aligned} \end{aligned}$$

For part (iii), from (4.2a) (or (4.3b)), for $t\in [0,T]$, one has

$$\begin{aligned} \begin{aligned}&e^{\tau A_h }e^{\gamma A_z}V_n(t) - \int _0^t e^{\tau A_h }e^{\gamma A_z}\sigma _n\left( V_n\right) \, dW \\&\quad = e^{\tau _0 A_h } V_n(0) - \int _0^t \Big [\theta _\rho (\Vert V_n\Vert _{\tau ,r,\gamma ,r} ) e^{\tau A_h }e^{\gamma A_z} Q_n(V_n, V_n) \\&\qquad + e^{\tau A_h }e^{\gamma A_z} F_n\left( V_n\right) - \nu _z e^{\tau A_h }e^{\gamma A_z} \partial _{zz} V_n \\&\qquad - \dot{\tau }A_h e^{\tau A_h} e^{\gamma A_z} V_n - \dot{\gamma }A_z e^{\tau A_h} e^{\gamma A_z} V_n \Big ]\, ds. \end{aligned} \end{aligned}$$

Then, with Lemma A.4 and $r>\frac{5}{2}$, the Minkowski inequality, and Young’s inequality, and the properties of $\sigma$, F and the cut-off function $\theta _\rho$, we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\left\| e^{\tau A_h }e^{\gamma A_z}V_n - \int _0^\cdot e^{\tau A_h }e^{\gamma A_z} \sigma _n\left( V_n\right) \, dW \right\| ^2_{W^{1, 2}\left( 0, T; {\mathcal {D}}_{0,r-1}\right) } \\&\quad \le C_{T,f_0,p,\rho ,r,\nu _z} {\mathbb {E}}\left[ 1+ \Vert V_0\Vert _{\tau _0,r,0,r}^2 + \sup \limits _{t\in [0,T]}\Vert V_n\Vert _{\tau ,r,\gamma ,r}^4\right. \\&\qquad \left. + \int _0^T \Vert f\Vert _{\tau _0,r,\gamma (T),r}^2 + \Vert \partial _z V_n\Vert _{\tau ,r,\gamma ,r}^2 \, ds \right] < \infty , \end{aligned} \end{aligned}$$

(4.11)

where the pressure gradient disappears since $\nabla P$ is orthogonal to the space ${\mathcal {D}}_{0,r-1}$. Again, we require a higher moment $p \ge 4$ here due to applying item (i) to $\Vert V_n\Vert _{\tau ,r,\gamma ,r}^4$ in (4.11). $\square$

Corollary 4.2

With the same assumptions as in Lemma 3.1. Let $\tau (t)$ and $\gamma (t)$ be defined in (4.6), and let T be defined in (4.8). Suppose that $U_n = e^{\tau A_h }e^{\gamma A_z}V_n$ be the solutions of system (4.3), and we denote by $U_0 = e^{\tau _0 A_h } V_0$. Then

$$\begin{aligned}&\text {(i)} \quad {\mathbb {E}} \Big [\sup \limits _{t\in [0,T]} \Vert U_n\Vert _{0,r}^p + \int _0^T \nu _z \Vert A^r U_n\Vert ^{p-2} \Vert \partial _z U_n\Vert _{0,r}^2 ds \\&\quad + \int _0^T \Vert A^r U_n\Vert ^{p-2} \Vert A_h^{\frac{1}{2}} A^r U_n\Vert ^2 ds \Big ] \le C_p (1+ {\mathbb {E}} \Vert U_0\Vert _{0,r}^p) e^{C_p T}. \end{aligned}$$

(ii) For $\alpha \in [0,1/2)$, $\int _0^{\cdot } e^{\tau A_h}e^{\gamma A_z}\sigma _n(e^{-\tau A_h }e^{-\gamma A_z}U_n) dW$ is bounded in $L^p\left( \Omega ; W^{\alpha , p}\right. \left. (0, T; {\mathcal {D}}_{0,r-1}) \right) ;$

(iii) If $p\ge 4$, $U_n - \int _0^\cdot e^{\tau A_h }e^{\gamma A_z}\sigma _n(e^{-\tau A_h }e^{-\gamma A_z}U_n) dW$ is bounded in $L^2\left( \Omega ; W^{1, 2}(0,\right. \left. T; {\mathcal {D}}_{0,r-1}))\right) .$

4.2 Local existence of martingale solution and pathwise uniqueness

This section follows similar to Sects. 3.3 and 3.4, once energy estimates are obtained above. Thus we give detail only when necessary.

Recall that when $\tau = \gamma =0$ and $r=s$, ${\mathcal {D}}_{0,r,0,s} = {\mathcal {D}}_{0,r}$. Given an initial distribution $\mu _0$ satisfying (2.16) for some $p\ge 4$ (which is implied by (2.15)), for some stochastic basis ${\mathcal {S}}= \left( \Omega , {\mathcal {F}}, {\mathbb {F}}, {\mathbb {P}}\right)$, let $U_0= e^{\tau _0 A_h} V_0$ be an ${\mathcal {F}}_0$-measurable ${\mathcal {D}}_{0,r}$-valued random variable with law $\mu _0$. With the same settings as in Sect. 3.3, we have the following two propositions similar to Proposition 3.3 and Proposition 3.4

Proposition 4.3

Let $\mu _0$ be a probability measure on ${\mathcal {D}}_{0,r}$ satisfying

$$\begin{aligned} \int _{{\mathcal {D}}_{0,r}} \Vert U\Vert _{0,r}^p d\mu _0(U)<\infty \end{aligned}$$

with $p\ge 4$ and let $(\mu ^n)_{n \ge 1}$ be the measures defined in (3.11). Then there exists a probability space $(\tilde{\Omega }, \tilde{{\mathcal {F}}}, \tilde{{\mathbb {P}}})$, a subsequence $n_k \rightarrow \infty$ as $k \rightarrow \infty$ and a sequence of ${\mathcal {X}}$-valued random variables $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ such that

• (i) $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ converges in ${\mathcal {X}}$ to $(\tilde{U}, \tilde{W}) \in {\mathcal {X}}$ almost surely,

• (ii) $\tilde{W}_{n_k}$ is a cylindrical Wiener process with reproducing kernel Hilbert space ${\mathscr {U}}$ adapted to the filtration $\left( {\mathcal {F}}_t^{n_k} \right) _{t \ge 0}$, where $\left( {\mathcal {F}}_t^{n_k} \right) _{t \ge 0}$ is the completion of $\sigma (\tilde{W}_{n_k}, \tilde{U}_{n_k}; s \le t)$,

• (iii) for $t\in [0,T]$, each pair $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ satisfies the equation

  $$\begin{aligned}&d {\tilde{U}}_{n_k} + \big [\theta _\rho (\Vert {\tilde{U}}_{n_k}\Vert _{0,r} ) e^{\tau A_h} e^{\gamma A_z} Q_{n_k}(e^{-\tau A_h} e^{-\gamma A_z}{\tilde{U}}_{n_k}, e^{-\tau A_h} e^{-\gamma A_z}{\tilde{U}}_{n_k}) \nonumber \\&\qquad + e^{\tau A_h} e^{\gamma A_z}F_{n_k}(e^{-\tau A_h} e^{-\gamma A_z}{\tilde{U}}_{n_k}) -\dot{\tau }A_h {\tilde{U}}_{n_k} - {\dot{\gamma }} A_z {\tilde{U}}_{n_k} - \nu _z \partial _{zz} {\tilde{U}}_{n_k}\big ]dt \nonumber \\&\quad = e^{\tau A_h} e^{\gamma A_z}\sigma _{n_k} (e^{-\tau A_h} e^{-\gamma A_z}{\tilde{U}}_{n_k}) d{\tilde{W}}_{n_k}, \end{aligned}$$

  (4.12)

  where $\tau$ and $\gamma$ are defined in (4.6) and T is defined in (4.8).

The proof of Proposition 4.3 is almost identical to that of Proposition 3.3, so we omit the details.

Proposition 4.4

For $\tau (t), \gamma (t)$ defined in (4.6) and T defined in (4.8), let $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ be a sequence of ${\mathcal {X}}$-valued random variables on a probability space $(\tilde{\Omega }, \tilde{{\mathcal {F}}}, \tilde{{\mathbb {P}}})$ such that

• (i) $(\tilde{U}_{n_k}, \tilde{W}_{n_k}) \rightarrow (\tilde{U}, \tilde{W})$ in the topology of ${\mathcal {X}}$, $\tilde{{\mathbb {P}}}$-almost surely, that is,

  $$\begin{aligned} \tilde{U}_{n_k} \rightarrow \tilde{U} \ \text {in} \ L^2\left( 0, T; {\mathcal {D}}_{0,r}\right) \cap C\left( \left[ 0, T \right] , {\mathcal {D}}_{0,r-\frac{3}{2}}\right) , \ \tilde{W}_{n_k} \rightarrow \tilde{W} \ \text {in} \ C\left( \left[ 0, T\right] ; {\mathscr {U}}_0 \right) , \end{aligned}$$

• (ii) $\tilde{W}_{n_k}$ is a cylindrical Wiener process with reproducing kernel Hilbert space ${\mathscr {U}}$ adapted to the filtration $\left( {\mathcal {F}}_t^{n_k} \right) _{t \ge 0}$ that contains $\sigma (\tilde{W}_{n_k}, \tilde{U}_{n_k}; s \le t)$,

• (iii) each pair $(\tilde{U}_{n_k}, \tilde{W}_{n_k})$ satisfies (4.12).

Let

$$\begin{aligned} {\tilde{V}}_{n_k} = e^{-\tau A_h} e^{-\gamma A_z} {\tilde{U}}_{n_k}, \ \ {\tilde{V}} = e^{-\tau A_h} e^{-\gamma A_z} {\tilde{U}}, \end{aligned}$$

and let $\tilde{{\mathcal {F}}}_t$ be the completion of $\sigma (\tilde{W}(s), \tilde{V}(s), 0 \le s \le t)$ and $\tilde{{\mathcal {S}}} = (\tilde{\Omega }, \tilde{{\mathcal {F}}}, ( \tilde{{\mathcal {F}}}_t )_{t \ge 0}, \tilde{{\mathbb {P}}})$. Then $(\tilde{{\mathcal {S}}}, \tilde{W}, \tilde{V},T)$ is a local martingale solution to the modified system (4.1) on the time interval [0, T]. Moreover, the solution $\tilde{V}$ satisfies

$$\begin{aligned} \begin{aligned}&\tilde{V} \in L^2\left( {\tilde{\Omega }}; C\left( [0, T]; {\mathcal {D}}_{\tau (t),r,\gamma (t),r} \right) \right) , \\&\quad \left( \Vert A_h^{\frac{1}{2}} A^r e^{\tau (t)A_h } e^{\gamma (t) A_z} \tilde{V} \Vert ^2 + \Vert A_z A^r e^{\tau (t)A_h } e^{\gamma (t) A_z} \tilde{V} \Vert ^2 \right) \Vert A^{r} e^{\tau (t)A_h } e^{\gamma (t) A_z} \tilde{V}\Vert ^{p-2} \\&\quad \in L^1\left( {\tilde{\Omega }}; L^1\left( 0, T \right) \right) . \end{aligned} \nonumber \\ \end{aligned}$$

(4.13)

Skecth of proof

The proof of $(\tilde{{\mathcal {S}}}, \tilde{W}, \tilde{V},T)$ being a local martingale solution to the modified system (4.1) is very similar to the proof in Proposition 3.4. Heuristically, the main differences between the inviscid case and the vertically viscous case are on the equipped norm and the additional vertical viscosity term. The difference on the norm is that in the viscous case the horizontal and vertical radii of analyticity are different. However, from Lemma 4.1 we see this will not give any trouble. The convergence of the vertical viscosity term is straightforward thanks to the regularity of the solution.

We will only provide a detailed proof of (4.13). By the growth condition of $\sigma$ assumed in (2.4) and the regularity of ${\tilde{V}}$ from Lemma 4.1, we have

$$\begin{aligned} \sigma ({\tilde{V}}) \in L^2\left( \tilde{\Omega }; L^2\left( 0,T; L_2\left( {\mathscr {U}}, {\mathcal {D}}_{\tau (t),r,\gamma (t),r}\right) \right) \right) . \end{aligned}$$

Therefore, the solution to

$$\begin{aligned} dZ - \nu _z \partial _{zz} Z = \sigma ({\tilde{V}}) \, d\tilde{W}, \qquad Z(0) = {\tilde{V}}_0, \end{aligned}$$

satisfies

$$\begin{aligned}&Z \in L^2\left( \tilde{\Omega }; C\left( [0,T]; {\mathcal {D}}_{\tau (t),r,\gamma (t),r}\right) \right) , \qquad \nonumber \\&\quad A^r e^{\tau A_h}e^{\gamma A_z} Z \in L^2\left( \tilde{\Omega }; L^2\left( 0,T; {\mathcal {D}}_{0,\frac{1}{2},0,1}\right) \right) . \end{aligned}$$

(4.14)

Similar to (3.18), we have the following regularity for $\tilde{V}$:

$$\begin{aligned} \begin{aligned}&\tilde{V} \in L^2 \left( {\tilde{\Omega }}; L^\infty \big ( 0, T; {\mathcal {D}}_{\tau (t),r,\gamma (t),r} \big ) \right) , \qquad \\&\quad A^r e^{\tau A_h}e^{\gamma A_z} \tilde{V} \in L^2\left( \tilde{\Omega }; L^2\left( 0,T; {\mathcal {D}}_{0,\frac{1}{2},0,1}\right) \right) . \end{aligned} \end{aligned}$$

(4.15)

Defining ${\bar{V}} = {\tilde{V}} - Z$, by (4.1a) we have ${\mathbb {P}}$-almost surely

$$\begin{aligned} \tfrac{d}{dt} {\bar{V}} + \theta (\Vert {\tilde{V}} \Vert _{\tau ,r,\gamma ,r}) Q({\tilde{V}}, {\tilde{V}}) + F({\tilde{V}}) - \nu _z \partial _{zz} {\bar{V}}= 0, \qquad {\bar{V}}(0) = 0. \end{aligned}$$

Notice that

$$\begin{aligned} \frac{d}{dt} A^{r}e^{\tau (t)A_h} e^{\gamma (t) A_z}{\bar{V}}= & {} A^{r}e^{\tau (t)A_h} e^{\gamma (t) A_z} \frac{d}{dt} {\bar{V}} + \dot{\tau }A_h A^r e^{\tau (t)A_h} e^{\gamma (t) A_z} {\bar{V}}\\&+ \dot{\gamma }A_z A^r e^{\tau (t)A_h} e^{\gamma (t) A_z} {\bar{V}}, \end{aligned}$$

thanks to (4.14), (4.15), and Lemma A.4, for arbitrary $\phi \in {\mathcal {D}}_{0,\frac{1}{2},0,1}$, one has

$$\begin{aligned} \Big \langle \frac{d}{dt} A^{r}e^{\tau (t)A_h} e^{\gamma (t) A_z}{\bar{V}}, \phi \Big \rangle \in L^2\left( \tilde{\Omega }; L^2\left( 0,T\right) \right) . \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{d}{dt} A^{r}e^{\tau (t)A_h} e^{\gamma (t) A_z}{\bar{V}} \in L^2\left( \tilde{\Omega }; L^2\left( 0,T; {\mathcal {D}}_{0,\frac{1}{2},0,1}'\right) \right) , \end{aligned}$$

where ${\mathcal {D}}_{0,\frac{1}{2},0,1}'$ is the dual space of ${\mathcal {D}}_{0,\frac{1}{2},0,1}$. Since ${\mathcal {D}}_{0,\frac{1}{2},0,1} \subset {\mathcal {D}}_0 \equiv {\mathcal {D}}_0' \subset {\mathcal {D}}_{0,\frac{1}{2},0,1}'$, by the Lions-Magenes Lemma, see, e.g., [46, Lemma 1.2, Chapter 3], we infer that $A^{r}e^{\tau (t)A_h} e^{\gamma (t) A_z} {\bar{V}} \in L^2\left( {\tilde{\Omega }}; C\left( \left[ 0,T \right] ; {\mathcal {D}}_0\right) \right)$. Similarly, one can show that ${\bar{V}} \in L^2\left( {\tilde{\Omega }}; C\left( \left[ 0,T \right] ; {\mathcal {D}}_0\right) \right)$, and thus ${\bar{V}} \in L^2\left( {\tilde{\Omega }}; C\left( \left[ 0,T \right] ; {\mathcal {D}}_{\tau (t),r,\gamma (t),r}\right) \right)$. This together with (4.14) imply that $\tilde{V} \in L^2\left( {\tilde{\Omega }}; C\left( \left[ 0,T \right] ; {\mathcal {D}}_{\tau (t),r,\gamma (t),r}\right) \right)$. The second part of (4.13) follows directly from (4.15).

$\square$

Corollary 4.5

Suppose that $\mu _0$ satisfy (2.15) with constant $M>0$. Let $\rho \ge M$, and let $(\tilde{{\mathcal {S}}}, \tilde{W}, \tilde{V},T)$ be the local martingale solution to system (4.1) given in Proposition 4.4. Let

$$\begin{aligned} \eta = \inf \left\{ t \ge 0 \mid \Vert \tilde{V}\Vert _{\tau ,r,\gamma ,r} \ge \frac{\rho }{2} \right\} , \end{aligned}$$

(4.16)

Then $(\tilde{{\mathcal {S}}}, \tilde{W}, \tilde{V}, \eta \wedge T)$ is a local martingale solution to the problem (2.14). Moreover,

$$\begin{aligned} \begin{aligned}&\tilde{V}\left( \cdot \wedge \eta \right) \in L^2 \left( \Omega ; C\left( [0, T], {\mathcal {D}}_{\tau (t),r,\gamma (t),r} \right) \right) , \\&\quad \mathbbm {1}_{[0, \eta ]} \left( \Vert A_h^{\frac{1}{2}} A^r e^{\tau (t)A_h } e^{\gamma (t) A_z} \tilde{V} \Vert ^2 + \Vert A_z A^r e^{\tau (t)A_h } e^{\gamma (t) A_z} \tilde{V} \Vert ^2 \right) \\&\qquad \times \Vert A^{r} e^{\tau (t)A_h } e^{\gamma (t) A_z} \tilde{V}\Vert ^{p-2}\in L^1\left( {\tilde{\Omega }}; L^1\left( 0, T \right) \right) . \end{aligned} \end{aligned}$$

Finally, we state the pathwise uniqueness for the original system (2.14).

Proposition 4.6

Let $\tau (t), \gamma (t)$ and T defined as in (4.6) and (4.8), respectively. Suppose that $\sigma$ and f satisfy (2.4) and (2.5), respectively, with $\delta$ satisfying a similar smallness condition as (4.5). Let ${\mathcal {S}}= \left( \Omega , {\mathcal {F}}, \left( {\mathcal {F}}_t \right) _{t \ge 0}, {\mathbb {P}}\right)$ and W be fixed. Suppose that there exist two local martingale solutions $\left( {\mathcal {S}}, W, V^1,T\right)$ and $\left( {\mathcal {S}}, W, V^2,T\right)$ to the modified system (4.1). Correspondingly, $\left( {\mathcal {S}}, W, V^1, \eta _1\wedge T\right)$ and $\left( {\mathcal {S}}, W, V^2, \eta _2\wedge T\right)$ are the two local martingale solutions to the original system (2.14). Denote $\Omega _0 = \left\{ V^1(0) = V^2(0) \right\} \subseteq \Omega$, and $\eta = \eta _1 \wedge \eta _2$. Then

$$\begin{aligned} {\mathbb {P}}\left( \left\{ \mathbbm {1}_{\Omega _0}\left( V^1(t\wedge \eta ) - V^2(t \wedge \eta )\right) = 0 \ \text {for all} \ t \in [0,T] \right\} \right) = 1. \end{aligned}$$

The proof is very similar to that of Proposition 3.6, and thus we omit it.

5 Conclusive remarks

We prove the local existence of martingale solutions and pathwise uniqueness for the 3D stochastic inviscid primitive equations (PEs, or hydrostatic Euler equations), where we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. By adding vertical viscosity, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. Moreover, the solution becomes analytic in z instantaneously as $t>0$ and the analytic radius in z increases as long as the solutions exist. The global in time existence of solutions is not achieved, as the parallel results in the deterministic case are still open in the vertically viscous case, and not true in the inviscid case. The existence of pathwise solutions still remains open, since the nonlinear estimates in the analytic function space are not as good as the horizontal viscosity case in Sobolev spaces. We shall leave it as future work.

Appendix A. Estimates of nonlinear terms

In this appendix, we provide estimates of nonlinear terms in the analytic function space which has been used throughout the paper. Recall that $Q(f,g) = f\cdot \nabla g -\int _0^z (\nabla \cdot f) ({\varvec{x}}', {\tilde{z}}) d{\tilde{z}} \, \partial _z g$, and for $r,\tau ,s,\gamma \ge 0$,

$$\begin{aligned} \begin{aligned} A^r f&:= \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |{\varvec{k}}|^r {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}, \qquad A_h^r f := \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |{\varvec{k}}'|^r {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}, \qquad \\ A_z^s f&:= \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} |k_3|^s {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}, \\ e^{\tau A} f&:= \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} e^{\tau (t)|{\varvec{k}}|} {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}, \qquad e^{\tau A_h} e^{\gamma A_z} f := \sum \limits _{{\varvec{k}}\in 2\pi {\mathbb {Z}}^3} e^{\tau (t)|{\varvec{k}}'|} e^{\gamma (t)|k_3|} {\hat{f}}_{{\varvec{k}}} e^{ i {\varvec{k}}\cdot {\varvec{x}}}, \end{aligned} \end{aligned}$$

where ${\varvec{k}}= ({\varvec{k}}', k_3)$.

The first lemma is used in the inviscid case.

Lemma A.1

([22, Lemmas A.1 and A.3]) For $f, g, h\in {\mathcal {D}}_{\tau ,r+\frac{1}{2}}$, where $r>2$ and $\tau \ge 0$, one has

$$\begin{aligned} \begin{aligned} \Big |\Big \langle A^r e^{\tau A} Q(f,g), A^r e^{\tau A} h \Big \rangle \Big | \le&C_r\Big ( \Vert f\Vert _{\tau ,r} \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \\&+ \Vert g\Vert _{\tau ,r} \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} h\Vert \\&+ \Vert h\Vert _{\tau ,r} \Vert A^{r+\frac{1}{2}} e^{\tau A} f\Vert \Vert A^{r+\frac{1}{2}} e^{\tau A} g\Vert \Big ). \end{aligned} \end{aligned}$$

The next lemma is used in the vertically viscous case.

Lemma A.2

For $f, g, h\in {\mathcal {D}}_{\tau ,r+\frac{1}{2},\gamma ,r} \cap {\mathcal {D}}_{\tau ,r,\gamma ,r+1}$, where $r>2$, $\tau ,\gamma \ge 0$, and $\int _0^1 \nabla \cdot f({\varvec{x}}', z)dz = 0$, one has

$$\begin{aligned} \begin{aligned}&\Big |\Big \langle A^r e^{\tau A_h} e^{\gamma A_z} Q(f,g), A^r e^{\tau A_h} e^{\gamma A_z} h \Big \rangle \Big | \\&\quad \le C_r\Big [\Vert f\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} h\Vert \\&\qquad + \Vert g\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} h\Vert \\&\qquad + \Vert h\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} g\Vert \Big ] \\&\qquad + C_r \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \left( \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right. \\&\qquad \left. +\Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right) . \end{aligned} \end{aligned}$$

Proof

The proof is similar to [22, Lemma A.1 and Lemma A.3] with some modifications. For completeness, we present it below but provide only necessary details. First, since $Q(f,g) = f\cdot \nabla g -\int _0^z (\nabla \cdot f) ({\tilde{z}}) d{\tilde{z}} \, \partial _z g$, one has

$$\begin{aligned} \begin{aligned}&\Big |\Big \langle A^r e^{\tau A_h} e^{\gamma A_z} Q(f,g), A^r e^{\tau A_h} e^{\gamma A_z} h \Big \rangle \Big | \\&\quad \le \Big |\Big \langle A^r e^{\tau A_h} e^{\gamma A_z} \left( f\cdot \nabla g\right) , A^r e^{\tau A_h} e^{\gamma A_z} h \Big \rangle \Big |\\&\qquad + \Big |\Big \langle A^r e^{\tau A_h} e^{\gamma A_z} \left( \int _0^z (\nabla \cdot f) ({\tilde{z}}) d{\tilde{z}} \, \partial _z g\right) , A^r e^{\tau A_h} e^{\gamma A_z} h \Big \rangle \Big | \\&\quad := I_1 + I_2. \end{aligned} \end{aligned}$$

For $I_1$, follows almost exactly the proof of [22, Lemma A.1], we have

$$\begin{aligned} \begin{aligned} I_1 \le&C_r\Big (\Vert f\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} h\Vert \\&\quad + \Vert g\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} h\Vert \\&\quad + \Vert h\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} g\Vert \Big ). \end{aligned} \end{aligned}$$

For $I_2$, we use Fourier representation of f, g and $H := A^r e^{\tau A_h} e^{\gamma A_z} h$, in which we can write

$$\begin{aligned}&f({\varvec{x}}) = \sum \limits _{{\varvec{j}}\in 2\pi {\mathbb {Z}}^3} {\hat{f}}_{{\varvec{j}}} e^{ i{\varvec{j}}\cdot {\varvec{x}}}, \quad g({\varvec{x}}) = \sum \limits _{2\pi {\varvec{k}}\in {\mathbb {Z}}^3} {\hat{g}}_{{\varvec{k}}} e^{ i{\varvec{k}}\cdot {\varvec{x}}}, \quad H({\varvec{x}}) = \sum \limits _{2\pi {\varvec{l}}\in {\mathbb {Z}}^3} |{\varvec{l}}|^r e^{\tau |{\varvec{l}}'|} e^{\gamma |l_3|}{\hat{h}}_{{\varvec{l}}} e^{ i{\varvec{l}}\cdot {\varvec{x}}}. \end{aligned}$$

Since $\int _0^1 \nabla \cdot f({\varvec{x}}', z)dz = 0$, one has $f({\varvec{x}}) = \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0 \end{array}} {\hat{f}}_{{\varvec{j}}} e^{2\pi ( i{\varvec{j}}' \cdot {\varvec{x}}' + ij_3 z)}$, and

$$\begin{aligned} \int _0^z \nabla \cdot f({\varvec{x}}',{\tilde{z}}) d{\tilde{z}} = \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi ( i{\varvec{j}}' \cdot {\varvec{x}}' + ij_3 z)} - \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi i{\varvec{j}}'\cdot {\varvec{x}}'}. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} I_2 =&\Big |\Big \langle A^r e^{\tau A_h} e^{\gamma A_z} \left( \int _0^z (\nabla \cdot f) ({\tilde{z}}) d{\tilde{z}} \, \partial _z g\right) , H \Big \rangle \Big | \\ \le&\Big |\Big \langle A^r e^{\tau A_h} e^{\gamma A_z} \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi ( i{\varvec{j}}' \cdot {\varvec{x}}' + ij_3 z)} \, \partial _z g\Big ), H \Big \rangle \Big | \\&+ \Big |\Big \langle A^r e^{\tau A_h} e^{\gamma A_z} \Big (\sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3 \ne 0, {\varvec{j}}' \ne 0 \end{array}} \frac{1}{j_3}{\varvec{j}}'\cdot {\hat{f}}_{{\varvec{j}}} e^{2\pi i{\varvec{j}}'\cdot {\varvec{x}}'}\, \partial _z g\Big ), H \Big \rangle \Big | := I_{21} + I_{22}. \end{aligned} \end{aligned}$$

We estimate $I_{22}$ first. For ${\varvec{l}} = ({\varvec{l}}', l_3) = (-{\varvec{j}}'-{\varvec{k}}', -k_3)$, by using the inequalities

$$\begin{aligned} |{\varvec{j}}'|^{\frac{1}{2}} \le C(|{\varvec{k}}'|^{\frac{1}{2}} + |{\varvec{l}}'|^{\frac{1}{2}}), \;\; |{\varvec{l}}|^r \le C_r (|{\varvec{j}}|^r+|{\varvec{k}}|^r), \end{aligned}$$

one has

$$\begin{aligned} I_{22}\le & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r\frac{1}{|j_3|}|{\varvec{j}}'||k_3|e^{\tau |{\varvec{j}}'|} e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} |{\hat{f}}_{{\varvec{j}}}||{\hat{g}}_{{\varvec{k}}}|(|{\varvec{j}}|^r+|{\varvec{k}}|^r)|{\varvec{l}}|^r e^{\tau |{\varvec{l}}'|} e^{\gamma |l_3|}|{\hat{h}}_{{\varvec{l}}}|\\\le & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{|k_3|}{|j_3|}\Big (|{\varvec{k}}'|^{\frac{1}{2}} |{\varvec{j}}'|^{\frac{1}{2}}|{\varvec{j}}|^{r} |{\varvec{l}}|^{r} + |{\varvec{j}}'|^{\frac{1}{2}}|{\varvec{j}}|^{r} |{\varvec{l}}'|^{\frac{1}{2}} |{\varvec{l}}|^{r} + |{\varvec{j}}'|^{\frac{1}{2}} |{\varvec{k}}'|^{\frac{1}{2}}|{\varvec{k}}|^{r} |{\varvec{l}}|^{r}\\&+ |{\varvec{j}}'|^{\frac{1}{2}}|{\varvec{k}}|^{r} |{\varvec{l}}'|^{\frac{1}{2}}|{\varvec{l}}|^{r} \Big ) e^{\tau |{\varvec{j}}'|} e^{\gamma |j_3|}e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} e^{\tau |{\varvec{l}}'|} e^{\gamma |l_3|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| =: B_1\\&+ B_2 + B_3 + B_4. \end{aligned}$$

When ${\varvec{k}}'\ne 0$, $k_3 \ne 0$ and $r>2$, thanks to the Cauchy–Schwarz inequality, we have

$$\begin{aligned} B_1= & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{|k_3|}{|j_3|} |{\varvec{k}}'|^{\frac{1}{2}} |{\varvec{j}}'|^{\frac{1}{2}}|{\varvec{j}}|^{r} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}'|} e^{\gamma |j_3|}e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} e^{\tau |{\varvec{l}}'|} e^{\gamma |l_3|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}}',k_3\ne 0 \end{array} } |{\varvec{k}}'|^{\frac{1}{2}} |k_3| e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} |{\hat{g}}_{{\varvec{k}}}| \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}' \ne 0 \end{array} } \frac{1}{|j_3|} |{\varvec{j}}'|^{\frac{1}{2}} |{\varvec{j}}|^{r} e^{\tau |{\varvec{j}}'|} e^{\gamma |j_3|} |{\hat{f}}_{{\varvec{j}}}| \\&\times |({\varvec{j}}'+{\varvec{k}}',k_3)|^{r}e^{\tau |{\varvec{j}}'+{\varvec{k}}'|} e^{\gamma |k_3|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}| \\\le & {} C_r \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |{\varvec{k}}'|^{1-r} \sum \limits _{k_3\in {{\mathbb {Z}}}} |{\varvec{k}}'|^{r-\frac{1}{2}} |k_3| |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} \Big ( \sum \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} |{\varvec{j}}'||{\varvec{j}}|^{2r} e^{2\tau |{\varvec{j}}'|} e^{2\gamma |j_3|}|{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{j_3\ne 0} \frac{1}{|j_3|^2} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}e^{2\tau |{\varvec{j}}'+{\varvec{k}}'|} e^{2\gamma |k_3|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |{\varvec{k}}'|^{1-r} \Big (\sum \limits _{k_3\in {{\mathbb {Z}}}} |{\varvec{k}}'| |{\varvec{k}}|^{2r} |{\hat{g}}_{{\varvec{k}}}|^2 e^{2\tau |{\varvec{k}}'|} e^{2\gamma |k_3|} \Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{k_3\in {{\mathbb {Z}}}} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}e^{2\tau |{\varvec{j}}'+{\varvec{k}}'|} e^{2\gamma |k_3|} |{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |{\varvec{k}}'|^{2-2r}\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} \sum \limits _{k_3\in {{\mathbb {Z}}}} |{\varvec{k}}'||{\varvec{k}}|^{2r} |{\hat{g}}_{{\varvec{k}}}|^2 e^{2\tau |{\varvec{k}}'|} e^{2\gamma |k_3|}\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert . \end{aligned}$$

For $B_2$, since $r >2$ and $k_3\ne 0$, thanks to the Cauchy–Schwarz inequality, we have

$$\begin{aligned} B_2= & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{|k_3|}{|j_3|} |{\varvec{j}}'|^{\frac{1}{2}} |{\varvec{l}}'|^{\frac{1}{2}}|{\varvec{j}}|^{r} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}'|} e^{\gamma |j_3|}e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} e^{\tau |{\varvec{l}}'|} e^{\gamma |l_3|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\\le & {} C_r \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ {\varvec{k}}',k_3\ne 0 \end{array} } |{\varvec{k}}| e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} |{\hat{g}}_{{\varvec{k}}}| \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}' \ne 0 \end{array} } \frac{1}{|j_3|} |{\varvec{j}}'|^{\frac{1}{2}} |{\varvec{j}}|^{r} e^{\tau |{\varvec{j}}'|} e^{\gamma |j_3|} |{\hat{f}}_{{\varvec{j}}}| \\&\times |{\varvec{j}}'+{\varvec{k}}'|^{\frac{1}{2}} |({\varvec{j}}'+{\varvec{k}}',k_3)|^{r}e^{\tau |{\varvec{j}}'+{\varvec{k}}'|} e^{\gamma |k_3|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}| \\\le & {} C_r \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \sum \limits _{k_3\in {{\mathbb {Z}}}} |{\varvec{k}}|^{r} | |{\hat{g}}_{{\varvec{k}}}| e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} \Big ( \sum \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} |{\varvec{j}}'||{\varvec{j}}|^{2r} e^{2\tau |{\varvec{j}}'|} e^{2\gamma |j_3|}|{\hat{f}}_{{\varvec{j}}}|^2\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{j_3\ne 0} \frac{1}{|j_3|^2} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |{\varvec{j}}'+{\varvec{k}}'| |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}e^{2\tau |{\varvec{j}}'+{\varvec{k}}'|} e^{2\gamma |k_3|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{1-r} \Big (\sum \limits _{k_3\in {{\mathbb {Z}}}} |{\varvec{k}}|^{2r} |{\hat{g}}_{{\varvec{k}}}|^2 e^{2\tau |{\varvec{k}}'|} e^{2\gamma |k_3|} \Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{k_3\in {{\mathbb {Z}}}} \sum \limits _{{\varvec{j}}'\in {\mathbb {Z}}^2 } |{\varvec{j}}'+{\varvec{k}}'| |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}e^{2\tau |{\varvec{j}}'+{\varvec{k}}'|} e^{2\gamma |k_3|} |{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \Big (\sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} |({\varvec{k}}',\pm 1)|^{2-2r}\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{{\varvec{k}}'\in {\mathbb {Z}}^2} \sum \limits _{k_3\in {{\mathbb {Z}}}} |{\varvec{k}}|^{2r} |{\hat{g}}_{{\varvec{k}}}|^2 e^{2\tau |{\varvec{k}}'|} e^{2\gamma |k_3|}\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert , \end{aligned}$$

where we have used that $|{\varvec{k}}|^{1-r} \le |({\varvec{k}}',\pm 1)|^{1-r}$ when $k_3\ne 0$. For $B_3$, since $r>2$, thanks to the Cauchy–Schwarz inequality, we have

$$\begin{aligned} B_3= & {} \sum \limits _{\begin{array}{c} {\varvec{j}}'+{\varvec{k}}'+{\varvec{l}}'=0\\ k_3+l_3 = 0 \\ j_3, k_3, {\varvec{j}}' \ne 0 \end{array}} C_r \frac{|k_3|}{|j_3|} |{\varvec{k}}'|^{\frac{1}{2}} |{\varvec{j}}'|^{\frac{1}{2}}|{\varvec{k}}|^{r} |{\varvec{l}}|^{r} e^{\tau |{\varvec{j}}'|} e^{\gamma |j_3|}e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} e^{\tau |{\varvec{l}}'|} e^{\gamma |l_3|} |{\hat{f}}_{{\varvec{j}}}| |{\hat{g}}_{{\varvec{k}}}| |{\hat{h}}_{{\varvec{l}}}| \\= & {} C_r \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } \frac{1}{|j_3|} |{\varvec{j}}'|^{\frac{1}{2}} e^{\tau |{\varvec{j}}'|} e^{\gamma |j_3|} |{\hat{f}}_{{\varvec{j}}}| \sum \limits _{\begin{array}{c} {\varvec{k}}\in {\mathbb {Z}}^3 \\ k_3, {\varvec{k}}' \ne 0 \end{array} } |{\varvec{k}}'|^{\frac{1}{2}} |{\varvec{k}}|^{r} e^{\tau |{\varvec{k}}'|} e^{\gamma |k_3|} |{\hat{g}}_{{\varvec{k}}}| \\&\times |({\varvec{j}}'+{\varvec{k}}',k_3)|^{r} |k_3| e^{\tau |{\varvec{j}}'+{\varvec{k}}'|} e^{\gamma |k_3|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}| \\\le & {} C_r \sum \limits _{\begin{array}{c} {\varvec{j}}\in {\mathbb {Z}}^3 \\ j_3, {\varvec{j}}'\ne 0 \end{array} } |{\varvec{j}}'|^{\frac{1}{2}-r} \frac{1}{|j_3|} |{\varvec{j}}|^{r} |{\hat{f}}_{{\varvec{j}}}| e^{\tau |{\varvec{j}}'|} e^{\gamma |j_3|} \Big ( \sum \limits _{{\varvec{k}}\in {\mathbb {Z}}^3} |{\varvec{k}}'||{\varvec{k}}|^{2r} e^{2\tau |{\varvec{k}}'|} e^{2\gamma |k_3|}|{\hat{g}}_{{\varvec{k}}}|^2\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{{\varvec{k}}\in {\mathbb {Z}}^3 } |k_3|^2 |({\varvec{j}}'+{\varvec{k}}',k_3)|^{2r}e^{2\tau |{\varvec{j}}'+{\varvec{k}}'|} e^{2\gamma |k_3|}|{\hat{h}}_{-({\varvec{j}}'+{\varvec{k}}',k_3)}|^2\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \Big (\sum \limits _{{\varvec{j}}'\ne 0} \sum \limits _{j_3 \ne 0} |{\varvec{j}}'|^{1-2r} \frac{1}{|j_3|^2}\Big )^{\frac{1}{2}} \\&\times \Big ( \sum \limits _{{\varvec{j}}\in {\mathbb {Z}}^3} |{\varvec{j}}|^{2r} |{\hat{f}}_{{\varvec{j}}}|^2 e^{2\tau |{\varvec{j}}'|} e^{2\gamma |j_3|}\Big )^{\frac{1}{2}} \\\le & {} C_r \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert . \end{aligned}$$

The estimate of $B_4$ is similar to that of $B_3$, and one can get

$$\begin{aligned} B_4 \le C_r \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert . \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned} I_{22} \le&C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \left( \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right. \\&\left. + \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right) \\&+ C_r \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \left( \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right. \\&\left. +\Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right) . \end{aligned} \end{aligned}$$

The estimates of $I_{21}$ follow similarly to those of $I_{22}$, and the result is the same. Thus,

$$\begin{aligned} \begin{aligned} I_{2} \le&C_r \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \left( \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right. \\&\left. + \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right) \\&+ C_r \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \left( \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right. \\&\left. +\Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right) . \end{aligned} \end{aligned}$$

Combining the estimates of $I_1$ and $I_2$ yields

$$\begin{aligned} \begin{aligned}&\Big |\Big \langle A^r e^{\tau A_h} e^{\gamma A_z} Q(f,g), A^r e^{\tau A_h} e^{\gamma A_z} h \Big \rangle \Big | \\&\quad \le C_r\Big [\Vert f\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} h\Vert \\&\qquad + \Vert g\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} h\Vert \\&\qquad + \Vert h\Vert _{\tau ,r,\gamma ,r} \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} f\Vert \Vert A_h^{\frac{1}{2}} A^r e^{\tau A_h} e^{\gamma A_z} g\Vert \Big ] \\&\qquad + C_r \Vert A^{r} e^{\tau A_h} e^{\gamma A_z} f\Vert \left( \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right. \\&\qquad \left. +\Vert A_z A^{r} e^{\tau A_h} e^{\gamma A_z} g\Vert \Vert A_h^{\frac{1}{2}} A^{r} e^{\tau A_h} e^{\gamma A_z} h\Vert \right) . \end{aligned} \end{aligned}$$

$\square$

The next lemma comes from [20, Lemma 2.1], addressing an important property of the space ${\mathcal {D}}_{\tau ,r}$.

Lemma A.3

If $\tau \ge 0$ and $r>\frac{3}{2}$, then ${\mathcal {D}}_{\tau ,r}$ is a Banach algebra, and for any $f, g\in {\mathcal {D}}_{\tau ,r}$, we have

$$\begin{aligned} \Vert fg\Vert _{\tau ,r} \le C_{r}\Vert f\Vert _{\tau ,r} \Vert g\Vert _{\tau ,r}. \end{aligned}$$

For the semi-norm, we also have a similar estimate

$$\begin{aligned} \Vert A^r e^{\tau A} (fg)\Vert \le C_{r} \Big (|{\hat{f}}_0|+\Vert A^r e^{\tau A} f\Vert \Big ) \Big (|{\hat{g}}_0|+ \Vert A^r e^{\tau A} g\Vert \Big ). \end{aligned}$$

Similar to Lemma A.3, one has the follwing lemma concerning the space ${\mathcal {D}}_{\tau ,r,\gamma ,r}$.

Lemma A.4

If $\tau ,\gamma \ge 0$ and $r>\frac{3}{2}$, then ${\mathcal {D}}_{\tau ,r,\gamma ,r}$ is a Banach algebra, and for any $f, g\in {\mathcal {D}}_{\tau ,r,\gamma ,r}$, we have

$$\begin{aligned} \Vert fg\Vert _{\tau ,r,\gamma ,r} \le C_{r}\Vert f\Vert _{\tau ,r,\gamma ,r} \Vert g\Vert _{\tau ,r,\gamma ,r}. \end{aligned}$$

For the semi-norm, we also have a similar estimate

$$\begin{aligned} \Vert A^r e^{\tau A_h} e^{\gamma A_z} (fg)\Vert \le C_{r} \Big (|{\hat{f}}_0|+\Vert A^r e^{\tau A_h} e^{\gamma A_z} f\Vert \Big ) \Big (|{\hat{g}}_0|+ \Vert A^r e^{\tau A_h} e^{\gamma A_z} g\Vert \Big ). \end{aligned}$$