Modeling and analysis of stochastic quantum magnetohydrodynamics equations with energy estimates

Abstract

This manuscript investigates the three-dimensional stochastic compressible quantum magnetohydrodynamic equation with density-dependent and stochastic external forces. First, we approximate the modelled system. In this work, the existence of a martingale solution within a given time interval is established by deriving energy estimates and applying the fixed-point argument. The proposed system is approximated using the Faedo-Galerkin method, the Jakubowski-Skorokhod theorem, and the compactness method. Furthermore, the result is analyzed in the limit $n \rightarrow \infty$ and $\eta \rightarrow 0$. Finally, numerical results for the quantum magnetohydrodynamic system are presented for the demonstrated theoretical findings.

Introduction

The Quantum Magnetohydrodynamics (QMHD) has gained significant attention in research^1,2,3,4. QMHD extends the classical Magnetohydrodynamic (MHD) by incorporating quantum mechanical effects, such as wavefunction coherence and tunneling^5,6. It distinguishes itself from classical MHD by combining quantum components^7,8,9,10. Specifically, trapped ions are governed by equations such as the Schrodinger equation for individual ions or the many-body quantum Hamiltonian that integrated into QMHD formulations¹¹. This method is a potent tool for many applications in physics, quantum mechanics and quantum information science since it enables scientists to work with and observe individual ions in extremely controlled system^{12,13,14,15,16}. It emerged as a cornerstone in the intersection of quantum mechanics and MHD^17,18. Their intrinsic properties and controllable quantum states make them invaluable for exploring the quantum mechanical underpinnings of macroscopic plasma phenomena^19,20.

The present study mainly focuses on the quantum counterpart of QMHD and the stochastic terms that influence the QMHD. To start with, the MHD equation provides the global properties of the plasma together with the QMHD model for charged particle systems and stochastic dynamics which play a role in the sudden changes in plasma behavior due to particle-wave interactions and fluctuations that happen at high-energy levels. To analyze the quantum phenomenon at microscopic scales, such as in dense astrophysical objects, neutron stars or in high-precision laboratory experiments involves superfluid helium or Bose-Einstein condensates²¹, the QMHD model is crucial for modeling and simulating electron transport. The pioneering work of F. Haas^22,23 introduced the Quantum Hydrodynamic (QHD) model for charged particle systems, extended it to cases with nonzero magnetic fields and applied it to describe the global properties of quantum plasmas. This approach starts with the QHD model with magnetic fields and leads to quantum corrections to MHD. The parameter H introduced by the author in²² indicates the significance of quantum corrections that are important in dense astrophysical plasmas.

On the otherhand, quantum effects play an important role in describing dense astrophysical plasmas, like those in white dwarfs and neutron stars. Particle densities are so high in extreme environments like white dwarfs, neutron stars, and quark-gluon plasmas that thermal and magnetic pressures are in opposition to quantum pressure. Bohm potential in system (2) represents quantum mechanical effects that become significant where quantum effects dominate classical hydrodynamic behavior. The Bohm potential, or quantum pressure term, accounts for quantum mechanical effects that influence the behavior of the fluid exerted by the probability waves²⁴. It essentially emerges from the wave-like nature of particles in quantum mechanics. It represents the influence of quantum effects on the motion of particles²⁵. Recent studies^26,27 on sampled-data $H_\infty$ attitude control of flexible spacecraft have addressed stochastic disturbances, missing measurements, and cyber-physical uncertainties using looped Lyapunov-Krasovskii functionals and LMI-based stability conditions. These robustness-oriented control approaches are conceptually relevant to the present stochastic QMHD setting, where random electromagnetic and plasma-flow interactions similarly influence system energy and stabilization behavior in space environments. The Bohm quantum potential contributes to the pressure by opposing density fluctuations. Due to interactions with the surrounding interstellar medium, random fluctuations arise in density and velocity. The quantum pressure term is crucial in maintaining equilibrium in the dynamical system.

Moreover, recently the authors in²⁸ studied the optimal decay and convergence rates for higher derivatives. Importantly, the authors in²⁸ considered the QMHD model for quantum plasmas and studied the optimal decay rates in the $L^p$-norm with $2 \le p \le 6$. In^28,29, the authors extended the works of long-time decay estimates for the compressible quantum model. With the help of Faedo-Galerkin method, the authors in³⁰ studied the global existence of solutions, and the large time behavior of solutions for the considered system is established. Specifically, the authors in³¹ investigated the global-in-time existence of weak solutions for the three-dimensional viscous QMHD system described by (1).

$$\begin{aligned}&d \texttt{m} + \operatorname {div}(\texttt{m} u) dt = 0, \nonumber \\&d(\texttt{m} u) + \left[ \operatorname {div}(\texttt{m} u \otimes u) - (\mu + \lambda )\nabla \operatorname {div} u - \mu \Delta u + \nabla p- \frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) \right] dt \nonumber \\&\quad = [(\nabla \times \mathscr {B}) \times \mathscr {B}]dt , \nonumber \\&d \mathscr {B}- \nu \Delta \mathscr {B}dt = [\nabla \times (u \times \mathscr {B})] dt, \ \operatorname {div}\mathscr {B}= 0. \end{aligned}$$

(1)

In the periodic setting, particularly on the three-dimensional torus, the global existence of weak solutions with large initial data has been established by combining energy estimates with compactness arguments. Furthermore, the authors in³⁰ reformulated the quantum Euler-Maxwell system. By employing Faedo-Galerkin method and standard fixed-point theory, the existence of solution is obtained³². By using a general energy method, authors in³³ proved decay rates of the compressible three-dimensional viscous QMHD model. Moreover, the authors in³⁴ studied the lower bound for the three-dimensional compressible viscous QMHD model and examined the solution converging to a constant equilibrium state (1, 0, 0) in both the spatial and time derivative. Moreover, the authors in³⁵ proved the existence and uniqueness of the time-periodic solution under suitable assumption. The authors in^36,37 examined the MHD instabilities in tokamaks by employing the Newton-Krylov method and a fourth-order precision difference scheme for the system. The authors in³⁸ investigated optimal decay rates for fourth and fifth-order spatial derivatives of density and velocity in the compressible viscous QMHD model. In terms of stochastic case, the authors in³⁹ investigated compressible QMHD with the external stochastic Brownian motion on a probability space. Moreover, the authors in⁴⁰ investigated the stochastic Navier-Stokes equation with the Brownian motion and multiplicative colored noise in a bounded domain of $R^d$ established under the condition that the adiabatic exponent $\gamma> \frac{d}{2}$ and $\eta \rightarrow \infty$ in the sequence of solutions. The works in⁴¹ established the existence, uniqueness, and long-time behavior of solutions under multiplicative noise. Recent studies on tamed and regularized fluid and MHD models have shown that taming techniques provide improved control over nonlinear growth conditions, enabling the analysis of strong solutions, Markov semigroups, and invariant measures in both whole-space and periodic settings. The work⁴² focused on the analysis of optimal feedback controls for the stochastic 3D MHD model and demonstrated the existence of $\epsilon$-optimal feedback controls through Galerkin approximations. Reserchers have focused on the control design of stochastic impulsive MHD models with their applicability to real-life systems such as MHD power plants, solar coronal MHD dynamics, and introducing the concept of trajectory controllability in infinite-dimensional stochastic MHD settings⁴³. Moreover⁴⁴ investigated the stability and large-time behavior of perturbations around a strong background magnetic field in a periodic channel, studying the 2D incompressible MHD system with no velocity dissipation and only horizontal magnetic diffusion, and established a sharp stability result using a careful constructed time-weighted energy functional that confirmed the stabilizing role of the background magnetic field. Recently, the works^38,43,45 have investigated stochastic MHD models from different perspectives such as solvability, stability analysis, optimal control, and trajectory controllability under various stochastic perturbations. These contributions are mainly concerned with the stochastic behavior of classical MHD flows, with applications ranging from magneto-fluid dynamics and plasma transport to MHD power systems and solar coronal models. Furthermore, the study³⁹ primarily addressed the existence of martingale solutions for compressible MHD systems subjected to stochastic external forces, where the emphasis is placed on probabilistic weak formulations and compactness-based construction of solutions. In contrast to the above works, the present study extends the stochastic MHD framework by incorporating quantum Bohm potential effects into the model, thereby bridging stochastic MHD with quantum fluid behavior. The inclusion of the quantum correction allows us to investigate the coupled interaction between quantum pressure, magnetic field dynamics, and stochastic fluctuations within the compressible QMHD system. This enables a deeper understanding of quantum–stochastic coupling mechanisms, which are not covered in the existing literature on classical stochastic MHD models.

Consider the following stochastic compressible QMHD model in the three-dimensional space:

$$\begin{aligned}&d \texttt{m} + \operatorname {div}(\texttt{m} u) dt = 0, \nonumber \\&d(\texttt{m} u) + \left[ \operatorname {div}(\texttt{m} u \otimes u) - (\mu + \lambda )\nabla \operatorname {div} u - \mu \Delta u + \nabla p- \frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) \right] dt \nonumber \\&\quad = [(\nabla \times \mathscr {B}) \times \mathscr {B}]dt + \sum \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u)d\beta _\varsigma ^1(t), \nonumber \\&d \mathscr {B}- \nu \Delta \mathscr {B}dt = [\nabla \times (u \times \mathscr {B})] dt + \sum \mathfrak {g}_\varsigma (x,\mathscr {B})d\beta _\varsigma ^2(t), \quad \operatorname {div}\mathscr {B}= 0. \end{aligned}$$

(2)

Consider the initial conditions

$$\begin{aligned} \texttt{m}|_{t=0}=\texttt{m}_0,\ \texttt{m} u|_{t=0}=y_0,\ \mathscr {B}|_{t_0}=\mathscr {B}_0,\ \operatorname {div}\mathscr {B}_0=0. \end{aligned}$$

(3)

The boundary conditions are

$$\begin{aligned} u|_{\partial \mathscr {Q}}=0, \mathscr {B}|_{\partial \mathscr {Q}}=0. \end{aligned}$$

(4)

Let $\mathscr {Q} \subset R^3$ be a bounded domain with smooth boundary $\partial \mathscr {Q}$. The unknowns $\texttt{m}$, $\nu$, u and $\mathscr {B}$ denote density, magnetic diffusion coefficient, the velocity field and the magnetic field, respectively. The two viscosity constants $\mu$ and $\lambda$ satisfy $2\mu +3\lambda \ge 0$. $\nu>0$ denotes the magnetic diffusivity of the magnetic field. Here scalar pressure p satisfies the condition $p=A\texttt{m}^{\gamma '}$ for $A>0$ with the adiabatic index $\gamma ' \ge 1$. The two sequences $\mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u)d\beta _\varsigma ^1$ and $\mathfrak {g}_\varsigma (x,\mathscr {B})d\beta _\varsigma ^2$ where $\beta _\varsigma ^1,\beta _\varsigma ^2, \varsigma =1,2, \ldots$ are stochastic external forces and independent one-dimensional R-valued Brownian motions. The Planck constant h is a positive constant. The term $\frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}}$ is considered as a quantum potential, thus so-called Bohm potential. System (2) models the dynamic behavior of conducting fluids and magnetic fields in the presence of external perturbations. Although the electric field E is not explicitly present in the system, (2)–(4), we note that the electric field is represented in terms of the density $\texttt{m}$, velocity u and magnetic field $\mathscr {B}$ as,

$$\begin{aligned} E= \nu \nabla \mathscr {B}-u \mathscr {B}+ \frac{e^2}{2}\nabla \frac{\Delta \sqrt{\texttt{m}}}{\sqrt{\texttt{m}}}, \end{aligned}$$

where e represents the elementary charge. The most important works done in this paper are as follows:

• The system (2) captures the complex dynamics of a compressible, magnetized fluid, incorporating mass and momentum conservation, viscous effects and the influence of a magnetic field.

• Firstly, we construct the approximation equation of the three-dimensional stochastic compressible QMHD model by using Faedo-Galerkin method.

• Withe help of a fixed-point argument, the existence of solution for the proposed model is established.

• The derivation of energy estimates is an essential part of comprehending complex systems. Energy estimations are fundamental for examining the interplay between dissipation, quantum effects and Brownian motion, assessing the well-posedness of the equations, and measuring the influence of stochastic forcing. The global Lipschitz continuity is guaranteed by the cut-off problem.

• From a physical point of view, the tightness property ensures that the sequence of approximations of the QMHD model, the dynamics of the system, and regularity and stability under stochastic external force.

• Due to the complexity involved in the QMHD system, it is difficult to handle the physical properties of the quantum effects. The most fundamental problem is to deal with the quantum pressure that is dependent on the density $\texttt{m}$, particularly in areas where the density fluctuates greatly. The main challenge is to deal with the density influenced by the Brownian motion in the system (2).

• The sustainable goals of compressible QMHD systems include ensuring effective and sustainable energy production, devising methods for providing these systems with energy stability, controllability and developing insightful predictions that help to lead experimental research in QMHD.

The organization of this paper is as follows: The basic definitions, functions and specific spaces are introduced in Section“Preliminaries”. The approximation problem is constructed in Section “Approximation scheme and a prior estimates” by using Galerkin approximation method and the compactness method. The fixed point argument establishes the existence and uniqueness of solution. An estimate of the tightness property is established in Section “Tightness property”. Finally, the passage of limit $n \rightarrow \infty$ and $\eta \rightarrow \infty$ is estimated in Section “’Estimates independent of n” and “Estimates independent of $\eta$”. In Section “Numerical results”, numerical results for QMHD system results is established and graphical representation of density, velocity and magnetic field is shown in three and two-dimensional field respectively.

Preliminaries

Let $(\Upsilon ,\mathscr {F},\mathbb {P})$ be the complete probablity space with the normal filtration $\{\mathscr {F}_t, t \ge 0\}$ of sub $\sigma -$fields of $\mathscr {F}$ such that $\mathscr {F}_0$ contains all $\mathbb {P}_0$-negligible subsets of $\Upsilon$. Let $C(\mathscr {I};H_w)$ consists of weakly continuous functions and also which is the subspace of $\mathscr {L}^\infty (\mathscr {I};H)$ where $\mathscr {I}=[0,T].$ Let $L_2(K,Z)$ be the collection of all Hilbert-Schmidt operators from K to Z. Let $\mathscr {L}^p(\mathscr {Q})$ denote the Banach space of Lebesgue measurable, $R^3$-valued functions that are integrable on $\mathscr {Q}$, equipped with standard norm $\Vert v\Vert _{\mathscr {L}^p(\mathscr {Q})}$, where $1 \le p < \infty$. Similarly, let $\mathscr {L}^\infty (\mathscr {Q})$ be the Banach space of Lebesgue measurable, essentially bounded, $R^3$-valued functions on $\mathscr {Q}$, equipped with the norm $\Vert v\Vert _{\mathscr {L}^\infty (\mathscr {Q})}$. When $p = 2$, $\mathscr {L}^2(\mathscr {Q})$ is a Hilbert space with the scalar product defined by

$$\begin{aligned} \langle v,w \rangle _{\mathscr {L}^2(\mathscr {Q})} = \int _\mathscr {Q} v(x)w(x)dx, \quad \text {for } v,w \in \mathscr {L}^2(\mathscr {Q}), \ x \in \mathscr {Q} \end{aligned}$$

and $\langle \cdot , \cdot \rangle$ is the inner product in $\mathscr {L}^2(\mathscr {Q})$. Let $H^1(\mathscr {Q})$ denotes the Sobolev space defined by

$$\begin{aligned} H^1(\mathscr {Q})=\{ v \in \mathscr {L}^2(\mathscr {Q}): \frac{\partial v}{\partial x_k} \in \mathscr {L}^2(\mathscr {Q}), k = 1, 2, 3 \} \end{aligned}$$

and also Hilbert space with the scalar product denoted by

$$\begin{aligned} \langle v,w \rangle _{H^1(\mathscr {Q})} = \langle v,w \rangle _{\mathscr {L}^2(\mathscr {Q})} + \langle \nabla v, \nabla w \rangle _{\mathscr {L}^2(\mathscr {Q})}, \ \text {for } v,w \in H^1(\mathscr {Q}). \end{aligned}$$

Let $\mathscr {L}^p(\Upsilon , \mathscr {L}^q(\mathscr {I};\mathscr {T}))$ denote the space of random functions defined on a probability space $(\Upsilon , \mathscr {F}, P)$ having its value in $\mathscr {L}^q(\mathscr {I};\mathscr {T})$ equipped with the norm:

$$\begin{aligned} \Vert v\Vert _{\mathscr {L}^p(\Upsilon , \mathscr {L}^q(\mathscr {I};\mathscr {T}))} = \left( \mathbb {E} \Vert v\Vert _{\mathscr {L}^q(\mathscr {I};\mathscr {T})}^p \right) ^{\frac{1}{p}}, \end{aligned}$$

where $\mathscr {T}$ is a Banach space and $1 \le p, q < \infty$. When $q = \infty$, we denote

$$\begin{aligned} \Vert v\Vert _{\mathscr {L}^p(\Upsilon , \mathscr {L}^\infty (\mathscr {I};\mathscr {L}^q(\mathscr {T})))} = \left( \, \textrm{ess} \sup _{0 \le t \le T} \mathbb {E} \Vert v\Vert _{\mathscr {L}^q(\mathscr {T})}^p \right) ^{\frac{1}{p}}. \end{aligned}$$

Denote

$$\begin{aligned} \mathfrak {F}(\texttt{m}, \texttt{m} u) dW_1 = \sum _{\varsigma =1}^\infty \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u) d \beta _\varsigma ^1 \quad \text {and} \quad \mathfrak {G}(\mathscr {B}) dW_2 = \sum _{\varsigma =1}^\infty \mathfrak {g}_\varsigma (x, B) d \beta _\varsigma ^2. \end{aligned}$$

The well-defined martingales $\int _0^T \mathfrak {F}(\texttt{m}, \texttt{m} u) dW_1$ and $\int _0^T \mathfrak {G}(\mathscr {B}) dW_2$ are Ito’s integrals in $\mathfrak {W}^{-l,2}(\mathscr {Q})$ and $\mathfrak {W}^{-1,2}(\mathscr {Q})$ respectively. Since $W_1$ and $W_2$ do not converge on K, we define $K_0 \supset K$ by $K_0 = \{c = \sum _{\varsigma \ge 1} \alpha _\varsigma e_\varsigma ; \, \sum _{\varsigma \ge 1} \frac{\alpha _\varsigma ^2}{\varsigma ^2} < \infty \}$ with the norm $\Vert c\Vert _{K_0}^2 = \sum _{\varsigma \ge 1} \frac{\alpha _\varsigma ^2}{\varsigma ^2},$ where $\{e_\varsigma \}_{\varsigma \ge 1}$ is a complete orthonormal basis of K. We see that $K \rightarrow K_0$ is Hilbert-Schmidt and we have $W_1,W_2 \in C(\mathscr {I};K_0) \ \mathscr {L}$ - a.s. Also, denote $\beta _\varsigma = (\beta _\varsigma ^1, \beta _\varsigma ^2)_{\varsigma \ge 1}$ with two given Brownian motions $\beta _\varsigma ^1$ and $\beta _\varsigma ^2$, $\varsigma \ge 1$ on $(\Upsilon , \mathscr {F}, P)$.

Definition 2.1

A martingale solution of the model (2)–(4) is a system $(\texttt{m}, u,h, \mathscr {B})$ has the following properties:

• 1. For any $\psi , \varphi \in C_0^\infty (\mathscr {Q})$, $\langle \texttt{m}, \psi \rangle ,$ $\langle \texttt{m} u, \varphi \rangle$ and $\langle \mathscr {B}, \varphi \rangle$ are progressively measurable;

• 2. $\forall$ $1 \le p < \infty$, $\texttt{m} \ge 0$, $\texttt{m} \in \mathscr {L}^p(\Upsilon , \mathscr {L}^\infty (\mathscr {I}; \mathscr {L}^{\gamma '}(\mathscr {Q})))$, $u \in \mathscr {L}^p(\Upsilon , \mathscr {L}^2(\mathscr {I}; H_0^1(\mathscr {Q})))$, $\mathscr {B}\in \mathscr {L}^p(\Upsilon , \mathscr {L}^\infty (\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))) \cap \mathscr {L}^p(\Upsilon , \mathscr {L}^2(\mathscr {I}; H_0^1(\mathscr {Q})))$ such that $\forall$$t \in \mathscr {I}$, $\psi \in C_0^\infty (\mathscr {I} \times \mathscr {Q})$ and $\varphi \in C_0^\infty (\mathscr {I} \times \mathscr {Q})$, it holds that

  $$\begin{aligned}&\int _\mathscr {Q} \texttt{m}(t) \psi dx - \int _\mathscr {Q} \texttt{m}_0 \ \psi dx = \int _0^t \int _\mathscr {Q} \texttt{m} u \cdot \nabla \psi dx dt, \\&\int _\mathscr {Q} (\texttt{m} u)(t) \cdot \varphi dx =\int _0^t \int _\mathscr {Q} [\texttt{m} u u \cdot \nabla \varphi - (\mu + \lambda ) \nabla u \cdot \operatorname {div} \varphi - \mu \nabla u \cdot \operatorname {div} \varphi + A \texttt{m}^{\gamma '} \cdot \operatorname {div}\varphi ] dx dt \\&\quad = \int _\mathscr {Q} \texttt{m}_0 u_0 \cdot \varphi dx +\int _0^t \int _\mathscr {Q} \frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) \cdot \varphi dx dt + \int _0^t \int _\mathscr {Q} (\nabla \times \mathscr {B}) \cdot \mathscr {B}\cdot \varphi dx dt \\&\qquad + \sum _{\varsigma =1}^\infty \int _0^t \int _\mathscr {Q} \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u) \cdot \varphi dx d\beta _\varsigma ^1, \\&\int _\mathscr {Q} \mathscr {B}(t) \cdot \varphi dx - \int _0^t \int _\mathscr {Q} [\nabla \times (u \times \mathscr {B}) + \nu \Delta \mathscr {B}] \cdot \varphi dx dt = \int _\mathscr {Q} \mathscr {B}_0 \cdot \varphi dx + \sum _{\varsigma =1}^\infty \int _0^t \int _\mathscr {Q} \mathfrak {g}_\varsigma (x,\mathscr {B}) \cdot \varphi dx d\beta _\varsigma ^2. \end{aligned}$$

• 3. In the notion of renormalized solutions, the system (2) is satisfied, that is,

  $$\begin{aligned} r(\texttt{m})_t + \operatorname {div}(r(\texttt{m})u)+(r'(\texttt{m})\texttt{m}-r(\texttt{m}))\operatorname {div}u =0 \end{aligned}$$

  holds in $D'(\mathscr {I} \times R^3) \ \mathbb {P}-a.s.$ for any $r \in C^1(R)$ with $r'(w)\equiv 0$ for all $w \in R$ large enough.

Next, let us impose the following conditions on the stochastic forces $\mathfrak {f}_\varsigma , \mathfrak {g}_\varsigma$:

• (A) The functions $\mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u), \mathfrak {g}_\varsigma (x, \mathscr {B})$ are progressively measurable, also $C^1$ continuous in $x, \texttt{m}, \texttt{m} u; x,\mathscr {B},$ respectively and

  $$\begin{aligned} {\left\{ \begin{array}{ll} \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u) = \mathfrak {f}_\varsigma ^1(x, \texttt{m}) + \mathfrak {f}_\varsigma ^2(x) \texttt{m} u, \\ \sum _{\varsigma =1}^\infty |\mathfrak {f}_\varsigma ^1|^2 \le \upsilon |\texttt{m}|^{\gamma '+1}, \quad \sum _{\varsigma =1}^\infty |\partial _\texttt{m} \mathfrak {f}_\varsigma ^1|^2 \le \upsilon |\texttt{m}|^{\gamma -1}, \quad \sum _{\varsigma =1}^\infty |\mathfrak {f}_\varsigma ^2|^2 \le \upsilon , \\ \sum _{\varsigma =1}^\infty |\mathfrak {g}_\varsigma (x, \mathscr {B})|^2 \le \upsilon |\mathscr {B}|^2, \quad \sum _{\varsigma =1}^\infty |\partial _\mathscr {B}\mathfrak {g}_\varsigma | \le \upsilon . \end{array}\right. } \end{aligned}$$

• (B) The initial condition $\texttt{m}_0, u_0, \mathscr {B}_0$ are satisfy the following conditions:

  $$\texttt{m}_0 \in \mathscr {L}^{\gamma ' p}(\Upsilon , \mathscr {L}^{\gamma '}(\mathscr {Q})), \quad \texttt{m}_0 \ge 0; \quad y_0 = 0 \text { if } \texttt{m}_0 = 0,$$
  $$\frac{|y_0|^2}{\texttt{m}_0} \in \mathscr {L}^p(\Upsilon , \mathscr {L}^1(\mathscr {Q})); \quad \mathscr {B}_0 \in \mathscr {L}^{2p}(\Upsilon , \mathscr {L}^2(\mathscr {Q})), \ \textrm{div} \mathscr {B}_0 = 0.$$

• (C) By using assumption (A), Holder’s inequality for $l> \frac{3}{2},$ we have

  $$\begin{aligned} \Vert \mathfrak {F}(\texttt{m}, \texttt{m} u)\Vert _{\mathscr {L}^2(K, \mathfrak {W}^{-l,2}(\mathscr {Q}))}^2&= \sum _{\varsigma =1}^\infty \Vert \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u)\Vert _{ \mathfrak {W}^{-l,2}(\mathscr {Q}))}^2 \lesssim \sum _{\varsigma =1}^\infty \Vert \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u)\Vert _{ \mathscr {L}^1(\mathscr {Q})}^2 \\&\lesssim \left( \int _\mathscr {Q} \sqrt{\texttt{m}} \sum _{\varsigma =1}^\infty \frac{\mathfrak {f}^1_{\varsigma }(x,\texttt{m})}{\sqrt{\texttt{m}}} + \sqrt{\texttt{m}} \sqrt{\texttt{m}}u \sum _{\varsigma =1}^\infty \mathfrak {f}^2_{\varsigma }(x) dx \right) ^2 \\&\lesssim ||\texttt{m}||_{\mathscr {L}^1} (||\texttt{m}||^{\gamma '}_{\mathscr {L}^{\gamma '}(\mathscr {Q})} + ||\sqrt{\texttt{m}}u||^2_{\mathscr {L}^2(\mathscr {Q})}) \lesssim 1. \\ \Vert \mathfrak {G}(\mathscr {B})\Vert _{\mathscr {L}^2(K, \mathfrak {W}^{-1,2}(\mathscr {Q}))}^2&= \sum _{\varsigma =1}^\infty \Vert \mathfrak {g}_\varsigma (x, \mathscr {B})\Vert _{\mathfrak {W}^{-1,2}(\mathscr {Q})}^2 \le \sum _{\varsigma =1}^\infty \Vert \mathfrak {g}_\varsigma (x, \mathscr {B})\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \\ &\lesssim ||\mathscr {B}||_{\mathscr {L}^2(\mathscr {Q})}^2 \lesssim 1. \end{aligned}$$

Let $\{\rho _k\}$ be an orthonormal basis of $\mathscr {L}^2(\mathscr {Q})$ that also serves as an orthogonal basis of $H^1(\mathscr {Q})$. Consider the eigenfunctions associated with the Dirichlet problem

$$\begin{aligned} -\Delta \rho _j =\lambda _j \rho _j \ \text {on} \ \mathscr {Q},\ \rho _j|_{\partial \mathscr {Q}}=0. \end{aligned}$$

Let $\mathscr {T}_n = \textrm{span}\{\rho _j\}_{j=1}^n, \ n \in \mathfrak {N}.$ Here $\mathscr {T}_n$ is a finite dimensional Hilbert space. Let $\mathscr {P}:$ $\mathscr {L}^2(\mathscr {Q}) \rightarrow$ $\mathscr {T}_n$ be the projection. Also $\mathscr {P}:$ $H^l(\mathscr {Q})$ to $\mathscr {T}_n$ be a linear projection. We see that,

$$\begin{aligned} \mathscr {P}h = \sum _{r=1}^n \langle g, e_r \rangle _{H^{l}-H^{-l}} e_r, \quad g \in H^l(\mathscr {Q}), \ e_r \in \mathscr {T}_n. \end{aligned}$$

Now, we have consider the deterministic system (1) for the estimation of $\sqrt{\texttt{m}_{\texttt{N}}}$ and $\root 4 \of {\texttt{m}_\texttt{N}}$:

Lemma 2.2

^46,47 For any $\upsilon> 0$ that is independent of both n and $\eta$, the following uniform estimate is true for the deterministic system (1):

$$\begin{aligned} \Vert \sqrt{\texttt{m}_n}\Vert _{\mathscr {L}^2(\mathscr {I}; H^2(\mathscr {Q}))} + \Vert \root 4 \of {\texttt{m}_n}\Vert _{\mathscr {L}^4(\mathscr {I}; \mathfrak {W}^{1,4}(\mathscr {Q}))} \le \upsilon . \end{aligned}$$

(5)

Now, we deduce space regularity for $\texttt{m}_n$, $\texttt{m}_n u_n$, and $\mathscr {B}_n$ in the following lemma.

Lemma 2.3

³¹ For some $\upsilon>0$ which is independent of n and $\eta$, the following uniform estimate holds for deterministic system (1):

$$\begin{aligned} \Vert \texttt{m}_n u_n\Vert _{\mathscr {L}^2(\mathscr {I}; \mathfrak {W}^{1, \frac{3}{2}}(\mathscr {Q}))}&\le \upsilon , \end{aligned}$$

(6)

$$\begin{aligned} \Vert \texttt{m}_n\Vert _{\mathscr {L}^2\left( \mathscr {I}; \mathfrak {W}^{2, \frac{2\gamma '}{\gamma '+1}}(\mathscr {Q})\right) }&\le \upsilon , \end{aligned}$$

(7)

$$\begin{aligned} \Vert \texttt{m}_n\Vert _{\mathscr {L}^{\frac{4\gamma '+3}{3}}(\mathscr {I}; \mathscr {L}^{\frac{4\gamma '+3}{3}}(\mathscr {Q}))}&\le \upsilon , \end{aligned}$$

(8)

$$\begin{aligned} \Vert \mathscr {B}_n\Vert _{\mathscr {L}^{\frac{10}{3}}(\mathscr {I}; \mathscr {L}^{\frac{10}{3}}(\mathscr {Q}))}&\le \upsilon . \end{aligned}$$

(9)

Lemma 2.4

³¹ For $s> \frac{5}{2}$, the following uniform estimates holds for deterministic system (1):

$$\begin{aligned} \Vert \partial _t \texttt{m}_n\Vert _{\mathscr {L}^2(\mathscr {I}; \mathscr {L}^{\frac{3}{2}}(\mathscr {Q}))}&\le \upsilon , \end{aligned}$$

(10)

$$\begin{aligned} \Vert \partial _t (\texttt{m}_n u_n)\Vert _{\mathscr {L}^{\frac{4}{3}}(\mathscr {I}; (H^s(\mathscr {Q}))^*)}&\le \upsilon , \end{aligned}$$

(11)

$$\begin{aligned} \Vert \partial _t \sqrt{\texttt{m}_n}\Vert _{\mathscr {L}^2(\mathscr {I}; (H^1(\mathscr {Q}))^*)}&\le \upsilon , \end{aligned}$$

(12)

$$\begin{aligned} \Vert \partial _t \mathscr {B}_n\Vert _{\mathscr {L}^{\frac{4}{3}}(\mathscr {I}; (H^s(\mathscr {Q}))^*)}&\le \upsilon . \end{aligned}$$

(13)

Lemma 2.5

³⁹ For $\gamma '> \frac{3}{2}$ and $0< \sigma < \min \left\{ 1, \frac{\gamma '}{3}, \frac{2}{3} \gamma ' - 1 \right\}$, then $\exists$ a constant $\upsilon$ such that

$$\mathbb {E} \int _0^T \int _{\mathscr {Q}} \left( A \texttt{m}_\eta ^{\gamma ' + \sigma } + \eta \texttt{m}_\eta ^{\beta + \sigma } \right) \, dx \, dt \le \upsilon ,$$

where $\upsilon$ independent of $\eta> 0$.

Approximation scheme and a prior estimates

Let $\varepsilon , \eta> 0$ be fixed and $\beta> \max \{4, \gamma '\}$, consider the following approximation problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} d\texttt{m} + \textrm{div}(\texttt{m} u)dt = \varepsilon \Delta \texttt{m} dt \\ d(\texttt{m} u) + \left[ \textrm{div}(\texttt{m} u \otimes u) - \mu \Delta u + A \nabla \texttt{m}^{\gamma '} + \eta \nabla \texttt{m}^\beta + \varepsilon \nabla u \cdot \nabla \texttt{m} - (\mu + \lambda )\nabla \textrm{div}u - \frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) \right] dt \\ \qquad = (\nabla \times \mathscr {B}) \times \mathscr {B}dt + \sum _{\varsigma \ge 1} \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u)d\beta _\varsigma ^1(t) \\ d\mathscr {B} - \nu \Delta \mathscr {B}dt = \nabla \times (u \times \mathscr {B})dt + \sum _{\varsigma \ge 1} \mathfrak {g}_\varsigma ( x,\mathscr {B})d\beta _\varsigma ^2(t), \end{array}\right. } \end{aligned}$$

(14)

with the initial conditions

$$\begin{aligned} \left\{ \begin{array}{l} \texttt{m}|_{t=0} = \texttt{m}_{0, \eta } \in C^{2+\alpha }(\overline{\mathscr {Q}}), \quad \nabla \texttt{m}_{0, \eta } \cdot {\textbf {n}}|_{\partial \mathscr {Q}} = 0, \\ (\texttt{m} u)|_{t=0} = y_{0, \eta } \in C^2(\overline{\mathscr {Q}}), \\ \mathscr {B}|_{t=0} = \mathscr {B}_{0, \eta } \in C^2(\overline{\mathscr {Q}}), \quad \textrm{div} \mathscr {B}_{0, \eta } = 0, \end{array} \right. \end{aligned}$$

(15)

with the boundary conditions:

$$\begin{aligned} \nabla \texttt{m} \cdot {\textbf {n}}|_{\partial \mathscr {Q}} = 0, \quad u|_{\partial \mathscr {Q}} = 0, \quad \mathscr {B}|_{\partial \mathscr {Q}} = 0. \end{aligned}$$

(16)

Here n is the unit outer normal at the boundary. Consider the initial data which fulfills the conditions as follows:

$$\begin{aligned}&\texttt{m}_{0, \eta } \rightarrow \texttt{m}_0 \text { in } \mathscr {L}^{\gamma '}(\mathscr {Q}), \ \mathscr {B}_{0, \eta } \rightarrow \mathscr {B}_0 \text { in } \mathscr {L}^2(\mathscr {Q}) \text { as } \eta \rightarrow 0, \\&0 < \eta \le \texttt{m}_{0, \eta }(x) \le \eta ^{-\frac{1}{\beta }}, \ y_{0, \eta } = \texttt{y}_\eta \sqrt{\texttt{m}_{0, \eta }} \end{aligned}$$

almost surely and let us define

$$\tilde{y}_{0, \eta }(x) = {\left\{ \begin{array}{ll} y_0(x) \sqrt{\frac{\texttt{m}_{0, \eta }(x)}{\texttt{m}_0(x)}}, & \text {if } \texttt{m}_0(x)> 0, \\ 0, & \text {if } \texttt{m}_0(x) = 0. \end{array}\right. }$$

By using assumption (B), we have for all $\ p \in [1, \infty )$

$$\begin{aligned} \frac{|\tilde{y}_{0, \eta }|^2}{\texttt{m}_{0, \eta }} \text {is bounded in} \ \mathscr {L}^p(\Upsilon , \mathscr {L}^1(\mathscr {Q})) \ \text {independently of} \ \eta> 0. \end{aligned}$$

As we know that $C^2(\overline{\mathscr {Q}})$ is dense in $\mathscr {L}^2(\mathscr {Q})$ so, one can $\texttt{y}_\eta \in C^2(\overline{\mathscr {Q}})$ with

$$\left\| \frac{\tilde{y}_{0, \eta }}{\sqrt{\texttt{m}_{0, \eta }}} - \texttt{y}_\eta \right\| _{\mathscr {L}^p(\Upsilon , \mathscr {L}^2(\mathscr {Q}))} < \eta .$$

Now, let us introduce a family of linear operators for a function $\texttt{m} \in \mathscr {L}^1(\mathscr {Q})$ with $\texttt{m} \ge \underline{\texttt{m}}>0$

$$\mathbb {S}[\texttt{m}] : \mathscr {T}_n \rightarrow \mathscr {T}_n; \quad \langle \mathbb {S}[\texttt{m}] w, v \rangle = \int _\mathscr {Q} \texttt{m} w \cdot v dx, \quad w, v \in \mathscr {T}_n.$$

We know that the operator $\mathbb {S}$ is invertible and $\texttt{m}> 0$ on $\mathscr {Q}$ and

$$\begin{aligned} \Vert \mathbb {S}^{-1}[\texttt{m}]\Vert _{L(\mathscr {T}_n, \mathscr {T}_n)} \le \left( \inf _{x \in \mathscr {Q}} \texttt{m}(x) \right) ^{-1}, \ \inf _{x \in \mathscr {Q}} \texttt{m}> 0, \end{aligned}$$

where $L(\mathscr {T}_n,\mathscr {T}_n)$ is the set of bounded linear mapping from $\mathscr {T}_n$ to $\mathscr {T}_n$ and $\mathbb {S}^{-1}[\texttt{m}]$ denotes the inverse of $\mathbb {S}$. Furthermore, $\mathbb {S}^{-1}$ satisfies the condition

$$\begin{aligned} \mathbb {S}^{-1}[\texttt{m}_1] - \mathbb {S}^{-1}[\texttt{m}_2] = \mathbb {S}^{-1}[\texttt{m}_2](\mathbb {S}[\texttt{m}_2] - \mathbb {S}[\texttt{m}_1])\mathbb {S}^{-1}[\texttt{m}_1], \end{aligned}$$

we can deduce that the map

$$\begin{aligned} \texttt{m} \mapsto \mathbb {S}^{-1}[\texttt{m}] \text { mapping } \mathscr {L}^1(\mathscr {Q}) \text { into } L(\mathscr {T}_n, \mathscr {T}_n) \end{aligned}$$

is well-defined and also satisfies that

$$\begin{aligned} \Vert \mathbb {S}^{-1}[\texttt{m}_1] - \mathbb {S}^{-1}[\texttt{m}_2]\Vert _{L(\mathscr {T}_n, \mathscr {T}_n)} \le \upsilon (n,\theta )\Vert \texttt{m}_1 - \texttt{m}_2\Vert _{\mathscr {L}^1(\mathscr {Q})} \end{aligned}$$

(17)

for any $\texttt{m}_1, \texttt{m}_2$ belonging to the set $S_\theta = \{\texttt{m} \in \mathscr {L}^1 : \inf _{x \in \mathscr {Q}} \texttt{m} \ge \theta> 0\}.$

Let us consider the sequence of pairs $(u_n, \mathscr {B}_n) \in C(\mathscr {I}; \mathscr {T}_n)$ satisfying the integral equations $\forall t \in \mathscr {I}$:

$$\begin{aligned}&\int _\mathscr {Q} \texttt{m}(t) u_n(t) \cdot \psi dx + \int _0^t \int _\mathscr {Q} [\textrm{div}(\texttt{m} u_n \otimes u_n) - (\nabla \times \mathscr {B}_n) \times \mathscr {B}_n - \mu \Delta u_n] \cdot \psi \, dxd\phi \nonumber \\&\quad = \int _\mathscr {Q} y_0 \cdot \psi \, dx +\int _0^t \int _\mathscr {Q} \big [(\mu + \lambda ) \Delta u_n - A \nabla \texttt{m}^{\gamma '} - \eta \nabla \texttt{m}^\beta +\frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) - \varepsilon \nabla u_n \cdot \nabla \texttt{m} \big ] \cdot \psi \, dxd\phi \nonumber \\&\qquad + \int _0^t \langle \sum _{\varsigma \ge 1} \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u_n), \psi \rangle \, d\hat{\beta }_\varsigma ^1, \nonumber \\&\int _\mathscr {Q} \mathscr {B}_n(t) \cdot \psi \, dx = \int _\mathscr {Q} \mathscr {B}_0 \cdot \psi \, dx + \int _0^t \int _\mathscr {Q} \big [(\nabla \times (u_n \times \mathscr {B}_n)) + \nu \Delta \mathscr {B}_n\big ] \cdot \psi \, dxd\phi \nonumber \\&\qquad + \int _0^t \int _\mathscr {Q} \sum _{\varsigma \ge 1} \mathfrak {g}_\varsigma (x, \mathscr {B}_n) \cdot \psi \, dxd\hat{\beta }_\varsigma ^2, \end{aligned}$$

(18)

where $\psi \in \mathscr {T}_n$. For any function $\psi \in \mathscr {T}_n$, where

$$\sum _{\varsigma \ge 1} \mathfrak {f}^n_\varsigma (x, \texttt{m}, \texttt{m} u_n) = \mathbb {S}^{\frac{1}{2}}[\texttt{m}] \mathscr {P} \Bigg ( \sum _{\varsigma \ge 1} \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u_n) / \sqrt{\texttt{m}} \Bigg ) \ \text {with} \ \langle \mathbb {S}^{\frac{1}{2}}[\texttt{m}] w,v \rangle = \int _\mathscr {Q} \sqrt{\texttt{m}} w \cdot v \ dx.$$

Now, the integral (18) is reformulated as a stochastic ordinary differential equation on $\mathscr {T}_n:$

$$\begin{aligned}&\frac{d}{dt}(\mathbb {S}[\texttt{m}(t)]u_n(t)) = N_1[\texttt{m}, \texttt{m} u_n,\mathscr {B}_n]+N_2[u_n,\mathscr {B}_n] +\sum _{\varsigma \ge 1}\mathfrak {f}^n_\varsigma (x, \texttt{m}, \texttt{m} u_n)d\hat{\beta }_\varsigma ^1 \nonumber \\&\quad + \sum _{\varsigma \ge 1} \mathfrak {g}_k(x, \mathscr {B}_n)d\hat{\beta }_\varsigma ^2 ,\ t>0 \nonumber \\&\mathbb {S}[\texttt{m}_0]u_n(0)=\mathbb {S}[\texttt{m}_0]u_0. \end{aligned}$$

(19)

Then we write the above integral (19) as

$$\begin{aligned}&(u_n(t), \mathscr {B}_n(t)) = \Bigg ( \mathbb {S}^{-1}[\texttt{m}(t)] \Big [ y_0^* + \int _0^t N_1[\texttt{m}, \texttt{m} u_n(\phi ), \mathscr {B}_n(\phi )]d\phi + \int _0^t \sum _{\varsigma \ge 1} \mathfrak {f}^n_\varsigma (x, \texttt{m}, \texttt{m} u_n)d\hat{\beta }_\varsigma ^1 \Big ], \nonumber \\&\mathscr {B}_0^* + \int _0^t N_2[u_n(\phi ), \mathscr {B}_n(\phi )]d\phi + \int _0^t \sum _{\varsigma \ge 1} \mathfrak {g}_k(x, \mathscr {B}_n)d\hat{\beta }_\varsigma ^2 \Bigg ), \end{aligned}$$

(20)

where

$$\begin{aligned} &\langle y_0^*, \psi \rangle = \int _\mathscr {Q} y_0 \cdot \psi \ dx, \quad \langle \mathscr {B}_0^*, \psi \rangle = \int _\mathscr {Q} \mathscr {B}_0 \cdot \psi \ dx, \\ &\langle N_1[\texttt{m}, \texttt{m} u_n, \mathscr {B}_n], \psi \rangle = \int _\mathscr {Q} \big [ \mu \Delta u_n - \operatorname {div}(\texttt{m} u_n \otimes u_n) + (\mu + \lambda ) \Delta u_n - A \nabla \texttt{m}^{\gamma '} - \eta \nabla \texttt{m}^\beta +\frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) \\&\quad - \varepsilon \nabla u_n \cdot \nabla \texttt{m} + (\nabla \times \mathscr {B}_n) \times \mathscr {B}_n \big ] \cdot \psi \ dx, \\&\langle N_2[u_n, \mathscr {B}_n], \psi \rangle = \int _\mathscr {Q} \big [ \nabla \times (u_n \times \mathscr {B}_n) + \nu \Delta \mathscr {B}_n \big ] \cdot \psi \ dx. \end{aligned}$$

In order to solve system (20), we have the following steps:

• Step 1: Now, let us try to solve $\texttt{m}$ in terms of $u_n$. Let us observe that the following Neumann initial-boundary value problem has $\texttt{m}$ as its solution⁴⁸:

  $$\begin{aligned}&\texttt{m}_t + \operatorname {div}(\texttt{m} u) = \varepsilon \Delta \texttt{m}, \nonumber \\&\nabla \texttt{m} \cdot \textbf{n}|_{\partial \mathscr {Q}} = 0, \nonumber \\&\texttt{m}|_{t=0} = \texttt{m}_{0,\eta }(x). \end{aligned}$$

  (21)

  By [ Lemma 2.1, Lemma 2.2 in⁴⁹, we define the mapping $\mathfrak {M}: C(\mathscr {I}; C^2(\overline{\mathscr {Q}})) \rightarrow C(\mathscr {I}; C^{2+\alpha }(\overline{\mathscr {Q}}))$ with

  • 1. $\texttt{m} = \mathfrak {M}[u]$ is the unique classical solution of (21);

  • 2. $\underline{\texttt{m}} \exp \big ( - \int _{0}^{t} ||\nabla u(\phi )||_{\mathscr {L}^\infty (\mathscr {Q})} \, d\phi \big ) \le \mathfrak {M}[u](x,t) \le \overline{\texttt{m}} \exp \big ( - \int _{0}^{t} ||\nabla u(\phi )||_{\mathscr {L}^\infty (\mathscr {Q})} \, d\phi \big )$ for all $t \in \mathscr {I},$ where $0< \underline{\texttt{m}} < \texttt{m}_{0,\eta } \le \overline{\texttt{m}}$. Here, $\overline{\texttt{m}}$ and $\underline{\texttt{m}}$ denotes the positive upper and lower bound of the density respectively.

  • 3. $\Vert \mathfrak {M}[u_1] - \mathfrak {M}[u_2]\Vert _{C(\mathscr {I}; \mathfrak {W}^{1,2}(\mathscr {Q}))} \le T \upsilon (U, T)\Vert u_1 - u_2\Vert _{C(\mathscr {I}; \mathfrak {W}^{1,2}(\mathscr {Q}))}$ for any $u_1,u_2 \in M_R$,

  where $M_R=\{u \in C(\mathscr {I};\mathfrak {W}^{1,2}(\mathscr {Q})):\Vert u(t)\Vert _{\mathscr {L}^\infty (\mathscr {Q})}+\Vert \nabla u(t)\Vert _{\mathscr {L}^\infty (\mathscr {Q})} \le R \ \forall \ t \}$. We have $\texttt{m} = \mathfrak {M}[u_n]$ on $\mathscr {T}_n$ for construct the approximate solutions for (14)–(16) by means of (20). We get the approximate solutions in the form of following integral equations:

  $$\begin{aligned}&(u_n(t), \mathscr {B}_n(t)) = \Bigg ( \mathbb {S}^{-1}[\mathfrak {M}[u_n](t)] \Big [ y_0^* + \int _0^t N_1[\mathfrak {M}[u_n](\phi ), u_n(\phi ), \mathscr {B}_n(\phi )]d\phi \\ &\qquad + \int _0^t \sum _{\varsigma \ge 1} \mathfrak {f}^n_\varsigma (x,\mathfrak {M}[u_n], \mathfrak {M}[u_n]u_n)d\hat{\beta }_\varsigma ^1(\phi ) \Big ], \\&\mathscr {B}_0^* + \int _0^t N_2[u_n(\phi ), \mathscr {B}_n(\phi )]d\phi + \int _0^t \sum _{\varsigma \ge 1} \mathfrak {g}_\varsigma (x,\mathscr {B}_n)d\hat{\beta }_\varsigma ^2(\phi ) \Bigg ). \end{aligned}$$

• Step: 2 Cut-off problem.

  Let $\theta _{\texttt{N}}: [0, \infty ) \rightarrow [0, 1]$ be a $C^{\infty }$ smooth cut-off function for any $\texttt{N}> 0$ such that

  $$\theta _{ \texttt{N}}(x):= {\left\{ \begin{array}{ll} 1, & \text {for } |x| \le \texttt{N}, \\ 0, & \text {for } |x| \ge \texttt{N} + 1. \end{array}\right. }$$

  For a fixed n, let us consider (20) as the following cut-off problem :

  $$\begin{aligned}&(u_{n}^{ \texttt{N}}(t), \mathscr {B}_{n}^{ \texttt{N}}(t)) = \left( \mathbb {S}^{-1}[\mathfrak {M}[u_{n}^{ \texttt{N}}](t)] \left[ y_{0}^{*} + \int _{0}^{t} \theta _{ \texttt{N}}^{u_{n}^{ \texttt{N}}, \mathscr {B}_{n}^{ \texttt{N}}}(\phi )N_{1}[\mathfrak {M}[u_{n}^{ \texttt{N}}](\phi ), u_{n}^{ \texttt{N}}(\phi ), \mathscr {B}_{n}^{ \texttt{N}}(\phi )]d\phi \right. \right. \nonumber \\&\quad \left. + \int _{0}^{t} \sum _{\varsigma \ge 1}\theta _{ \texttt{N}}^{u_{n}^{ \texttt{N}}, \mathscr {B}_{n}^{ \texttt{N}}}(\phi ) \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}[u_{n}^{ \texttt{N}}], \mathfrak {M}[u_{n}^{ \texttt{N}}]u_{n}^{\texttt{N}})d\hat{\beta }_{\varsigma }^{1}(\phi ) \right] , \nonumber \\&B_{0}^{*} + \int _{0}^{t}\theta _{\texttt{N}}^{u_{n}^{ \texttt{N}}, \mathscr {B}_{n}^{ \texttt{N}}}(\phi )N_{2}[u_{n}^{ \texttt{N}}(\phi ), \mathscr {B}_{n}^{ \texttt{N}}(\phi )]d\phi + \int _{0}^{t} \sum _{\varsigma \ge 1}\theta _{ \texttt{N}}^{u_{n}^{\texttt{N}}, B_{n}^{\texttt{N}}}(\phi )\mathfrak {g}_{\varsigma }(x,\mathscr {B}_{n}^{ \texttt{N}})d\hat{\beta }_{\varsigma }^{2}(\phi ) \bigg ), \end{aligned}$$

  (22)

  where $\theta _{ \texttt{N}}^{u_{n}^{ \texttt{N}}, \mathscr {B}_{n}^{ \texttt{N}}}(\phi ) = \theta _{ \texttt{N}} \left( \max \{\Vert u_{n}^{ \texttt{N}}(\phi )\Vert _{\mathfrak {W}^{1,\infty }}, \Vert \mathscr {B}_{n}^{ \texttt{N}}(\phi )\Vert _{\mathfrak {W}^{1,\infty }}\} \right)$. For a fixed $\texttt{N}$ and n and by using the fixed point argument, we have the integral (22) at $\mathscr {I}_{n,\texttt{N}}$ on the Banach space $C(\mathscr {I}; \mathscr {T}_{n})$ where $\mathscr {I}_{n,\texttt{N}} = [\mathscr {I}_{n,\texttt{N}}]$ and $T_{n, \texttt{N}} \le T$. With a slight abuse of notation, $(u^{ \texttt{N}}, \mathscr {B}^{ \texttt{N}}):= (u_{n}^{ \texttt{N}}, \mathscr {B}_{n}^{ \texttt{N}})$.

Theorem 3.1

For $T> 0$ and for all fixed n and $\texttt{N}$, then $\exists$ a $T_{n,\texttt{N}} \in \mathscr {I}$ with the system (22) has a unique solution $(u^{\texttt{N}}, \mathscr {B}^{\texttt{N}}) \in \mathscr {L}^{2}(\Upsilon , C(\mathscr {I}_{n,\texttt{N}}; \mathscr {T}_{n}))^{2}$.

Proof

Set

$$Q_{\texttt{N},T_{n,\texttt{N}}} = \left\{ \mathscr {U}^{\texttt{N}} = (u^{\texttt{N}}, b^{\texttt{N}}) \in \mathscr {L}^{2}(\Upsilon ,C(\mathscr {I}_{n,\texttt{N}};\mathscr {T}_n))^2 : \Vert (u^{\texttt{N}}, b^{\texttt{N}})\Vert _{C(\mathscr {I}_{n,\texttt{N}}; \mathfrak {W}^{1,\infty }(\mathscr {Q}))} \le \texttt{N} \right\} ,$$

has the norm $\Vert \mathscr {U}^{\texttt{N}}\Vert _{\mathscr {B}_{\texttt{N},T_{n,\texttt{N}}}} = \sup _{\mathscr {I}_{n,\texttt{N}}} \mathbb {E}\Vert \mathscr {U}^{\texttt{N}}\Vert ^{2}_{\mathscr {T}_{n}}$ where $\Vert \mathscr {U}^{\texttt{N}}\Vert ^{2}_{\mathscr {T}_{n}} = \Vert u^{\texttt{N}}\Vert ^{2}_{\mathscr {T}_{n}} + \Vert b^{\texttt{N}}\Vert ^{2}_{\mathscr {T}_{n}}$.

A map $\mathfrak {R}: \mathscr {L}^{2}(\Upsilon ,C(\mathscr {I}_{n,\texttt{N}}; \mathscr {T}_{n}))^{2} \rightarrow \mathscr {L}^{2}(\Upsilon ,C(\mathscr {I}_{n,\texttt{N}}; \mathscr {T}_{n}))^{2}$ defined as

$$\begin{aligned} \mathfrak {R}(\mathscr {U}^{\texttt{N}}) = w_{0}^{*} + \int _{0}^{t}\theta _{\texttt{N}}^{\mathscr {U}^{\texttt{N}}}(\phi ) N d\phi + \int _{0}^{t} \sum _{\varsigma \ge 1}\theta _{\texttt{N}}^{\mathscr {U}^{\texttt{N}}} (\phi ) \mathfrak {F}_{\varsigma } \, d\beta _{k}^{1}(\phi ), \end{aligned}$$

where

$$\begin{aligned}&w_{0}^{*} = (\mathbb {S}^{-1}[\mathfrak {M}[u^{\texttt{N}}](t)]y_{0}^{*},\ \mathscr {B}_{0}^{*}), \\&N = (\mathbb {S}^{-1}[\mathfrak {M}(u^{\texttt{N}})(t)]N_1\left[ \mathfrak {M}(u^{\texttt{N}}),u^{\texttt{N}}, b^\texttt{N}\right] ,N_2\left[ u^{\texttt{N}}, b^\texttt{N}\right] ), \\&\mathfrak {F}_{\varsigma } = (\mathbb {S}^{-1}[\mathfrak {M}(u^{\texttt{N}})(t)] \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u^{\texttt{N}}), \mathfrak {M}(u^{\texttt{N}})u^{\texttt{N}}), \mathfrak {g}_\varsigma (x, b^{\texttt{N}})). \end{aligned}$$

By using the Burkholder-Davis-Gundy inequality, Holder’s inequality and similar to⁵⁰, the properties of $\mathbb {S}[\texttt{m}], \mathfrak {M}[u]$, we conclude that $\mathfrak {R}$ maps $Q_{\texttt{N},T_{n,\texttt{N}}}$ into itself for some $T_{n,\texttt{N}}$. Now we claim that $\mathfrak {R}$ is a contraction on $\mathscr {L}^{2}(\Upsilon , C(\mathscr {I}_{n,\texttt{N}}; \mathscr {T}_{n}))^{2}$ for some $T_{n,\texttt{N}}> 0$. From^47,50, Section 7], the deterministic terms of (22) is proved. Hence, we consider the stochastic terms

$$\begin{aligned} \mathfrak {R}^{\phi }(\mathscr {U}^{\texttt{N}}):= \int _{0}^{t}\sum _{\varsigma \ge 1}\theta _{\texttt{N}}^{\mathscr {U}^{\texttt{N}}}(\phi )\mathfrak {F}_{\varsigma } \, d\hat{\beta }_{\varsigma }(\phi ). \end{aligned}$$

For $\mathscr {U} = (u, b_1)$, $\mathscr {V} = (v, b_2)$, we have

$$\begin{aligned}&\mathfrak {R}^{\phi }(\mathscr {U}) - \mathfrak {R}^{\phi }(\mathscr {V}) = (\mathbb {S}^{-1}[\mathfrak {M}(u)(t)] - \mathbb {S}^{-1}[\mathfrak {M}(v)(t)]) \int _{0}^{t}\sum _{\varsigma \ge 1}\theta _{\texttt{N}}^{\mathscr {U}}(\phi )\mathfrak {f}^n_{\varsigma }(x,\mathfrak {M}(u), \mathfrak {M}(u)u)d\hat{\beta }_{\varsigma }^{1} \\&\quad + \mathbb {S}^{-1}[\mathfrak {M}(v)(t)]\int _{0}^{t}\sum _{\varsigma \ge 1}(\theta _{\texttt{N}}^{\mathscr {U}}(\phi )\mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u) - \theta _{\texttt{N}}^{\mathscr {V}}(\phi )\mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(v), \mathfrak {M}(v)v)) d\hat{\beta }_{\varsigma }^{1} \\&\quad + \int _{0}^{t}\sum _{\varsigma \ge 1}(\theta _{\texttt{N}}^{\mathscr {U}}\mathfrak {g}_{\varsigma }(x, b_1) - \theta _{\texttt{N}}^{\mathscr {V}}\mathfrak {g}_{\varsigma }(x, b_2)) d\hat{\beta }_{\varsigma }^{2}. \end{aligned}$$

Then

$$\begin{aligned}&\mathbb {E}\left\| \mathfrak {R}^{\phi }(\mathscr {U}) - \mathfrak {R}^{\phi }(\mathscr {V})\right\| ^{2}_{\mathscr {T}_{n}} \le \, \mathbb {E}\sup _{0 \le t \le T_{n,\texttt{N}}}\left\| \int _{0}^{t}\left( \mathbb {S}^{-1}[\mathfrak {M}(u)(t)] - \mathbb {S}^{-1}[\mathfrak {M}(v)(t)] \right) \sum _{\varsigma \ge 1}\mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u)d\hat{\beta }_{\varsigma }^{1} \right\| ^{2}_{\mathscr {T}_{n}} \\&\qquad + \mathbb {E}\sup _{0 \le t \le T_{n,\texttt{N}}} \left\| \int _{0}^{t} \mathbb {S}^{-1}[\mathfrak {M}(v)(t)]\sum _{\varsigma \ge 1}[\mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u) - \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(v), \mathfrak {M}(v)v)] d\hat{\beta }_{\varsigma }^{1}\right\| ^{2}_{\mathscr {T}_{n}} \\&\qquad + \mathbb {E}\sup _{0 \le t \le T_{n,\texttt{N}}} \left\| \int _{0}^{t}\sum _{\varsigma \ge 1}(\mathfrak {g}_{\varsigma }(x, b_1) - \mathfrak {g}_{\varsigma }(x, b_2)) d\hat{\beta }_{\varsigma }^{2}\right\| ^{2}_{\mathscr {T}_{n}} \\&\quad =: I_{1} + I_{2} + I_{3}. \end{aligned}$$

Since $\mathfrak {M}(u)$ is a contraction map and by using the Burkholder-Davis-Gundy inequality, assumption (A), (17) the term $I_{1}$ becomes,

$$\begin{aligned} I_{1}&\le \mathbb {E}\left( \int _{0}^{T_{n,\texttt{N}}}\left\| \left( \mathbb {S}^{-1}[\mathfrak {M}(u)(t)] - \mathbb {S}^{-1}[\mathfrak {M}(v)(t)] \right) \sum _{\varsigma \ge 1}\mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u)\right\| ^2_{\mathscr {T}_{n}} \, d\phi \right) \\&\le \mathbb {E}\left( \Vert \mathbb {S}^{-1}[\mathfrak {M}(u)(t)] - \mathbb {S}^{-1}[\mathfrak {M}(v)(t)]\Vert _{L(\mathscr {T}_{n}, \mathscr {T}_{n})}^{2} \int _{0}^{T_{n,\texttt{N}}} \sum _{\varsigma \ge 1}\Vert \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u)\Vert ^{2}_{\mathscr {T}_{n}} \, d\phi \right) \\&\le \mathbb {E}\left( \Vert \mathfrak {M}(u) - \mathfrak {M}(v)\Vert ^2_{\mathscr {L}^{1}(\mathscr {Q})} \int _{0}^{T_{n,\texttt{N}}} \sum _{\varsigma \ge 1}\Vert \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u)\Vert ^{2}_{\mathscr {T}_{n}} \, d\phi \right) \\&\le \upsilon T_{n,\texttt{N}} \mathbb {E}\left( \Vert u - v\Vert ^{2}_{\mathscr {T}_{n}}\right) \\&\le \upsilon T_{n,\texttt{N}} \mathbb {E}\left( \Vert \mathscr {U} - \mathscr {V}\Vert ^{2}_{\mathscr {T}_{n}}\right) . \end{aligned}$$

Similarly, by using the assumption (A), we obtain

$$\begin{aligned} I_{2}&\le \mathbb {E}\left( \int _{0}^{T_{n,\texttt{N}}}\left\| \mathbb {S}^{-1}[\mathfrak {M}(u)(t)]\right\| ^2_{L(\mathscr {T}_{n},\mathscr {T}_{n})}\left\| \sum _{\varsigma \ge 1}(\mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u) - \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(v), \mathfrak {M}(v)v)) \right\| ^{2}_{\mathscr {T}_{n}}d\phi \right) \\&\le \mathbb {E} \left( [\inf \mathfrak {M}(u)(t)]^{-1}\int _{0}^{T_{n,\texttt{N}}}\sum _{\varsigma \ge 1}\left\| \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u) - \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(v), \mathfrak {M}(v)v)\right\| ^{2}_{\mathscr {T}_{n}} d\phi \right) \\&\le [\inf \mathfrak {M}(u)(t)]^{-1} \mathbb {E}\left( \int _{0}^{T_{n,\texttt{N}}}\sum _{\varsigma \ge 1}\left\| \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(u), \mathfrak {M}(u)u) - \mathfrak {f}^n_{\varsigma }(x, \mathfrak {M}(v), \mathfrak {M}(v)v)\right\| ^{2}_{\mathscr {T}_{n}} d\phi \right) \\&\le \upsilon T_{n,\texttt{N}}\mathbb {E}\left( \Vert \mathfrak {M}(u) - \mathfrak {M}(v)\Vert ^{2}_{\mathscr {T}_{n}} + \Vert u - v\Vert ^{2}_{\mathscr {T}_{n}}\right) \\&\le \upsilon T_{n,\texttt{N}}\mathbb {E}\left( \Vert u - v\Vert ^{2}_{\mathscr {T}_{n}} + \Vert b_1 - b_2\Vert ^{2}_{\mathscr {T}_{n}}\right) \\&\le \upsilon T_{n,\texttt{N}}\mathbb {E}\left( \Vert \mathscr {U} - \mathscr {V}\Vert ^{2}_{\mathscr {T}_{n}}\right) . \end{aligned}$$

We know that $\Vert \mathbb {S}^{\frac{1}{2}}[\mathfrak {M}(u)] - \mathbb {S}^{\frac{1}{2}}[\mathfrak {M}(v)]\Vert _{L(\mathscr {T}_{n}, \mathscr {T}_{n})} \le \upsilon \Vert \mathfrak {M}(u) - \mathfrak {M}(v)\Vert _{\mathscr {L}^{2}}$, we have

$$\begin{aligned} I_{3} \le \mathbb {E}\left( \int _{0}^{T_{n,\texttt{N}}}\left\| \sum _{\varsigma \ge 1}(\mathfrak {g}_{\varsigma }(x, b_1) - \mathfrak {g}_{\varsigma }(x, b_2))\right\| ^{2}_{\mathscr {T}_{n}} dt \right) \le \upsilon T_{n,\texttt{N}}\mathbb {E}\left( \Vert \mathscr {U} - \mathscr {V}\Vert ^{2}_{\mathscr {T}_{n}}\right) . \end{aligned}$$

Appling mean value theorem and definition of $\mathbb {S}$ also in account of sufficiently small $T_{n,\texttt{N}}$ such that $m=\upsilon T_{n,\texttt{N}} <1$,

$$\mathbb {E}\left\| \mathfrak {R}(\mathscr {U}) - \mathfrak {R}(\mathscr {V})\right\| ^{2}_{\mathscr {T}_{n}} \le m \mathbb {E}\left( \Vert \mathscr {U} - \mathscr {V}\Vert ^{2}_{\mathscr {T}_{n}}\right) .$$

Therefore, $\mathfrak {R}$ is a contraction on $T_{n,\texttt{N}}$, $\mathfrak {R}$ maps $Q_{\texttt{N},T_{n,\texttt{N}}}$ into itself. Hence, $\mathfrak {R}$ admits a unique fixed point $(u^\texttt{N}, \mathscr {B}^\texttt{N})$. The cut-off function $\theta _\texttt{N}(x)$ guarantees the Lipschitz continuity globally. $\square$

Uniform prior estimates

Now we extend $\tau _{n,\texttt{N}}$ to T. From Theorem 3.1, let $\mathscr {U}_n^{\texttt{N}} = (u^{\texttt{N}}, \mathscr {B}^{\texttt{N}})$ be the solution to system (20) on $[0, \tau _{n,\texttt{N}})$. Define stopping time:

$$\tau _{n,\texttt{N}} = {\left\{ \begin{array}{ll} \inf \left\{ t \ge 0 : \Vert \mathscr {U}_n^{\texttt{N}}(t)\Vert ^{2}_{\mathscr {T}_{n}} \ge \texttt{N} \right\} \wedge \inf \left\{ t \ge 0 : \int _{0}^{t}\Vert \mathscr {F}_n^{\texttt{N}}dW\Vert _{\mathscr {L}^{2}(\mathscr {Q})} d\phi \ge \texttt{N}\right\} , T,\ Otherwise. \end{array}\right. }$$

Let $\texttt{m}_\texttt{N}:= \mathfrak {M}(u^\texttt{N})$. Now, let us compute the energy estimates.

Proposition 3.2

For $1 \le p < \infty$, any $(u_{\texttt{N}}, \mathscr {B}_{\texttt{N}})$ satisfying (20), we have

$$\begin{aligned}&\sup _{0 \le t \le T} E_\eta + \mathbb {E} \Bigg [\int _{0}^{T} \left( \mu \Vert \nabla u_{\texttt{N}}(t)\Vert _{\mathscr {L}^{2}(\mathscr {Q})}^{2} + (\mu +\lambda )\Vert \textrm{div}(u_{\texttt{N}}(t))\Vert _{\mathscr {L}^{2}(\mathscr {Q})}^{2} + \nu \Vert \nabla \mathscr {B}_{\texttt{N}}(t)\Vert _{\mathscr {L}^{2}(\mathscr {Q})}^{2} + \mu H''(\texttt{m}_{\texttt{N}})\Vert \nabla \texttt{m}_{\texttt{N}}\Vert ^{2} \right. \\&\qquad \left. + \frac{\mu h^2}{4}\Vert \nabla ^2 \ln \texttt{m}_{\texttt{N}}\Vert ^{2} \right) dt + \varepsilon \mathbb {E} \int _{0}^{T} \int _{\mathscr {Q}} \left( A \gamma ' \texttt{m}_{\texttt{N}}^{\gamma '-2} + \eta \beta _{\texttt{m}_{\texttt{N}}}^{\beta -2} \right) |\nabla \texttt{m}_{\texttt{N}}(t)|^{2} \, dx \, dt \Bigg ]^p \\&\quad \le \upsilon \mathbb {E}(E_{0,\eta })^{p}, \end{aligned}$$

where $H(\texttt{m})=\frac{A \texttt{m}^{\gamma '}}{\gamma '-1}$ with $\gamma '>1.$

Proof

Define the function $\Psi (\texttt{m}, y):= \int _{\mathscr {Q}} \frac{|y|^2}{\texttt{m}} dx$. Note that

$$\nabla _y \Psi (\texttt{m}, y) = \int _{\mathscr {Q}} \frac{2y}{\texttt{m}} dx, \quad \nabla ^2_y \Psi (\texttt{m}, y) = \int _{\mathscr {Q}} \frac{2}{\texttt{m}} \texttt{I} dx, \quad \partial _\texttt{m} \Psi (\texttt{m}, y) = -\int _{\mathscr {Q}} \frac{|y|^2}{\texttt{m}^2} dx,$$

where $\texttt{I}$ is the identity matrix. By using [Lemma 2.1 in³⁹ apply Ito’s formula into the function $\Psi$ with $(\texttt{m}, y) = (\texttt{m}_{\texttt{N}}, \texttt{m}_{\texttt{N}} u_{\texttt{N}})$. Again in the view of Ito’s formula with $\Psi (\mathscr {B}):= \int _{\mathscr {Q}} |\mathscr {B}|^2 dx$. From system (14), we obtain that

$$\begin{aligned}&d\int _{\mathscr {Q}} \left( \frac{1}{2} | \sqrt{\texttt{m}_{\texttt{N}}} u_{\texttt{N}}|^2 + \frac{A}{\gamma '-1} \texttt{m}_{\texttt{N}}^{\gamma '} + \frac{\eta }{\beta -1} \texttt{m}_{\texttt{N}}^{\beta } + \frac{1}{2} |\mathscr {B}_{\texttt{N}}|^2 \right) dx + \int _{\mathscr {Q}} \left( \mu |\nabla u_{\texttt{N}}|^2 + (\mu + \lambda ) |\textrm{div} u_{\texttt{N}}|^2 \right) dxd\phi \nonumber \\&\qquad + \mu H''(\texttt{m}_\texttt{N})|\nabla \texttt{m}_\texttt{N}|^2 + \int _{\mathscr {Q}} \left( H(\texttt{m}_{\texttt{N}}) +\frac{h^2}{2}|\nabla \sqrt{\texttt{m}_\texttt{N}}|^2 \right) dxd\phi + \frac{\mu h^2}{2} \frac{\Delta \sqrt{\texttt{m}_\texttt{N}}}{\sqrt{\texttt{m}_\texttt{N}}} \Delta \texttt{m}_\texttt{N} +\nu \int _{\mathscr {Q}} |\nabla \mathscr {B}_{\texttt{N}}|^2 dxd\phi \nonumber \\&\qquad + \varepsilon \int _{\mathscr {Q}} \left( A \gamma ' \texttt{m}_{\texttt{N}}^{\gamma '-2} + \eta \beta \texttt{m}_{\texttt{N}}^{\beta -2} \right) |\nabla \texttt{m}_{\texttt{N}}|^{2} \, dx \, d\phi \nonumber \\&\quad = \int _{\mathscr {Q}} \sum _{\varsigma =1}^{\infty } \mathfrak {f}_{\varsigma }^{n} (x, \texttt{m}_{\texttt{N}}, \texttt{m}_{\texttt{N}} u_{\texttt{N}}) \cdot u_{\texttt{N}} dx d\hat{\beta }_{\varsigma }^{1} + \frac{1}{2} \int _{\mathscr {Q}} \sum _{\varsigma \ge 1} \Big |\frac{\mathfrak {f}_{\varsigma }(x, \texttt{m}_{\texttt{N}}, \texttt{m}_{\texttt{N}} u_{\texttt{N}})}{\sqrt{\texttt{m}_{\texttt{N}}}}\Big |^2 dxd\phi \nonumber \\&\qquad + \int _{\mathscr {Q}} \sum _{\varsigma \ge 1} \mathfrak {g}_{\varsigma }(x, \mathscr {B}_{\texttt{N}}) \cdot \mathscr {B}_{\texttt{N}} dx d\hat{\beta }_{\varsigma }^{2} + \frac{1}{2} \int _{\mathscr {Q}} \sum _{\varsigma \ge 1} |\mathfrak {g}_{\varsigma }(x, \mathscr {B}_{\texttt{N}})|^2 dxd\phi , \end{aligned}$$

(23)

where $\phi \in [0,t \wedge \tau _{n,\texttt{N}}], \, t \wedge \tau _{n,\texttt{N}} = \min \{t, \tau _{n,\texttt{N}}\}$ and $t \in \mathscr {I}$. Let

$$\begin{aligned}&E_{\eta } = \int _{\mathscr {Q}} \left( \frac{1}{2} \texttt{m}_{\texttt{N}} |u_{\texttt{N}}|^2 + \frac{A}{\gamma '-1} \texttt{m}_{\texttt{N}}^{\gamma '} + \frac{\eta }{\beta -1} \texttt{m}_{\texttt{N}}^{\beta } + H(\texttt{m}_{\texttt{N}})+ \frac{h^2}{2}|\nabla \sqrt{\texttt{m}_\texttt{N}}|^2 + \frac{1}{2} |\mathscr {B}_{\texttt{N}}|^2 \right) dx, \\&E_{0, \eta } = \int _{\mathscr {Q}} \left( \frac{1}{2} \texttt{m}_{0,\eta } |u_{0,\eta }|^2 + \frac{A}{\gamma '-1} \texttt{m}_{0,\eta }^{\gamma '} + \frac{\eta }{\beta -1} \texttt{m}_{0,\eta }^{\beta } +H(\texttt{m}_{0,\eta }) +\frac{h^2}{2}|\nabla \sqrt{\texttt{m}_{0,\eta }}|^2 + \frac{1}{2} |\mathscr {B}_{0,\eta }|^2 \right) dx. \end{aligned}$$

Integrating (23) on $[0,\phi ]$ for all $[0, t \wedge \tau _{n, \texttt{N}}]$ gives

$$\begin{aligned}&E_\eta + \int _{0}^{\phi } \left[ \mu \Vert \nabla u_{\texttt{N}}(\zeta )\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + (\mu + \lambda ) \Vert \textrm{div} u_{\texttt{N}}(\zeta )\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \nu \Vert \nabla \mathscr {B}_{\texttt{N}}(\zeta )\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \right] \, d\zeta + \frac{\mu h^2}{2} \frac{\Delta \sqrt{\texttt{m}_\texttt{N}}}{\sqrt{\texttt{m}_\texttt{N}}} \Delta \texttt{m}_\texttt{N}\\&\qquad + \mu H''(\texttt{m}_\texttt{N})|\Delta \texttt{m}_\texttt{N}|^2 + \varepsilon \int _{0}^{\phi } \int _{\mathscr {Q}} \left( A \gamma ' \texttt{m}_{\texttt{N}}^{\gamma '-2} + \eta \beta \texttt{m}_{\texttt{N}}^{\beta -2} \right) |\nabla \texttt{m}_{\texttt{N}}|^2 \, dx \, d\zeta \\&\quad \le E_{0,\eta } + \frac{1}{2} \int _{0}^{\phi } \sum _{\varsigma \ge 1} \left\| \frac{f_{\varsigma }^{n}(x, \texttt{m}_{\texttt{N}}, \texttt{m}_{\texttt{N}} u_{\texttt{N}})}{\sqrt{\texttt{m}_{\texttt{N}}(\zeta )}}\right\| ^2_{\mathscr {L}^2(\mathscr {Q})} \, d\zeta + \left| \int _{0}^{s} \langle u_{\texttt{N}}(\zeta ), \sum _{\varsigma \ge 1} \mathfrak {f}_{\varsigma }^{n}(x, \texttt{m}_{\texttt{N}}, \texttt{m}_{\texttt{N}} u_{\texttt{N}}) \rangle \, d\hat{\beta }_{\varsigma }^1 \right| \\&\qquad + \frac{1}{2} \int _{0}^{\phi } \sum _{\varsigma \ge 1} \Vert \mathfrak {g}_{\varsigma }(x,\mathscr {B}_{\texttt{N}})\Vert ^2_{\mathscr {L}^2(\mathscr {Q})} \, d\zeta + \left| \int _{0}^{\phi } \langle \mathscr {B}_{\texttt{N}}(\zeta ), \sum _{\varsigma \ge 1} \mathfrak {g}_{\varsigma }(x, \mathscr {B}_{\texttt{N}}) \rangle \, d\hat{\beta }_{\varsigma }^2 \right| . \end{aligned}$$

$$\begin{aligned}&\sup _{0 \le \phi \le t \wedge \tau _{\texttt{N}}} E_\eta + \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \left[ \mu \Vert \nabla u_{\texttt{N}}(\phi )\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + (\mu + \lambda ) \Vert \textrm{div} u_{\texttt{N}}(\phi )\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \nu \Vert \nabla \mathscr {B}_{\texttt{N}}(\phi )\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \right] \, d\phi \nonumber \\&\qquad + \varepsilon \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \left( A \gamma ' \texttt{m}_{\texttt{N}}^{\gamma '-2} + \eta \texttt{m}_{\texttt{N}}^{\beta -2} \right) |\nabla \texttt{m}_{\texttt{N}}|^2 \, dx \, d\phi + \mu H''(\texttt{m}_\texttt{N})|\Delta \texttt{m}_\texttt{N}|^2 + \frac{\mu h^2}{2} \frac{\Delta \sqrt{\texttt{m}_\texttt{N}}}{\sqrt{\texttt{m}_\texttt{N}}} \Delta \texttt{m}_\texttt{N} \nonumber \\&\quad \le E_{0,\eta } + \frac{1}{2} \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \sum _{\varsigma \ge 1} \left\| \frac{\mathfrak {f}_{\varsigma }^{n}(x, \texttt{m}_{\texttt{N}}, \texttt{m}_{\texttt{N}} u_{N})}{\texttt{m}_{\texttt{N}}(\zeta )}\right\| ^2_{\mathscr {L}^2(\mathscr {Q})} \, d\phi + \sup _{\phi } \mathbb {E}\left| \int _{0}^{\phi } \langle u_{\texttt{N}}, \sum _{\varsigma \ge 1} \mathfrak {f}_{\varsigma }^{n}(x, \texttt{m}_{\texttt{N}}, \texttt{m}_{\texttt{N}} u_{\texttt{N}}) \rangle \, d\hat{\beta }_{\varsigma }^1 \right| \nonumber \\&\qquad + \frac{1}{2} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \sum _{\varsigma \ge 1} \Vert \mathfrak {g}_{\varsigma }(x,\mathscr {B}_{\texttt{N}})\Vert ^2_{\mathscr {L}^2(\mathscr {Q})} + \sup _{\phi } \mathbb {E} \left| \int _{0}^{\phi } \langle \mathscr {B}_{\texttt{N}}, \sum _{\varsigma \ge 1} \mathfrak {g}_{\varsigma }(x, \mathscr {B}_{\texttt{N}}) \rangle \, d\hat{\beta }_{\varsigma }^2 \right| \nonumber \\&\quad = E_{0,\eta } + J_1 + J_2 +J_3 +J_4. \end{aligned}$$

(24)

By the Cauchy-Schwarz inequality and the assumption (A), we obtain

$$\begin{aligned} J_1&\lesssim \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \int _{\mathscr {Q}} \left( |\sqrt{\texttt{m}_{\texttt{N}}} u_{\texttt{N}}|^2 + |\texttt{m}_{\texttt{N}}|^{\gamma '} \right) \, dx \, d\phi , \nonumber \\&\lesssim \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \left( \Vert \sqrt{\texttt{m}_{\texttt{N}}} u_{\texttt{N}}\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \Vert \texttt{m}_{\texttt{N}}\Vert _{\mathscr {L}^{\gamma '}(\mathscr {Q})}^{\gamma '} \right) \, d\phi . \end{aligned}$$

(25)

By Holder inequality, assumption (A), the Burkholder-Davis-Gundy and Young inequality for small $\varrho> 0$, we obtain for term $J_2$

$$\begin{aligned} J_2&\lesssim \mathbb {E} \left[ \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \sum _{\varsigma \ge 1} \langle \mathfrak {f}_{\varsigma }^n(x, \texttt{m}_{\texttt{N}}, \texttt{m}_{\texttt{N}} u_{\texttt{N}}), u_{\texttt{N}} \rangle ^2 \, d\phi \right] ^{\frac{1}{2}} \nonumber \\&\lesssim \mathbb {E} \left[ \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \Vert \sqrt{\texttt{m}_{\texttt{N}}} u_{\texttt{N}}\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \left( \Vert \sqrt{\texttt{m}_{\texttt{N}}} u_{\texttt{N}}\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \Vert \texttt{m}_{\texttt{N}}\Vert _{\mathscr {L}^\gamma (\mathscr {Q})}^{\gamma '} \right) \, d\phi \right] ^{\frac{1}{2}} \nonumber \\&\le \varrho \ \mathbb {E} \sup _{0 \le \phi \le t \wedge \tau _{n, \texttt{N}}} \Vert \sqrt{\texttt{m}_{\texttt{N}}} u_{\texttt{N}}(\phi )\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \upsilon _\varrho \ \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}}\int _{\mathscr {Q}} \left( |\sqrt{\texttt{m}_{\texttt{N}}} u_{\texttt{N}}|^2 + |\texttt{m}_{\texttt{N}}|^{\gamma '} \right) \, dx \, d\phi . \end{aligned}$$

(26)

Again, by using the assumption (A), we have

$$\begin{aligned} J_3&\le \upsilon \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \Vert \mathscr {B}_{\texttt{N}}\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \, d\phi . \end{aligned}$$

(27)

Also,

$$\begin{aligned} J_4 \le \varrho \sup _{0 \le \phi \le t \wedge \tau _{n, \texttt{N}}} \mathbb {E}\Vert \mathscr {B}_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \upsilon _\varrho \ \int _{0}^{t \wedge \tau _{n, \texttt{N}}}\mathbb {E} \Vert \mathscr {B}_\texttt{N} \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \, d\phi . \end{aligned}$$

(28)

Now, we generalize $p> 2$, then (26) and (28) becomes

$$\begin{aligned} |J_2|^p&\le \upsilon \mathbb {E} \left[ \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \sum _{\varsigma \ge 1} \langle \mathfrak {f}_\varsigma (x, \texttt{m}_\texttt{N}, \texttt{m}_\texttt{N} u_\texttt{N}), u_\texttt{N} \rangle ^2 \, d\phi \right] ^{\frac{p}{2}} \\&\le \varrho \mathbb {E} \left( \sup _{0 \le \phi \le t \wedge \tau _{n, \texttt{N}}} \Vert \sqrt{\texttt{m}_\texttt{N}} u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^{2p} \right) + \upsilon _\varrho \ \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \left( \int _{\mathscr {Q}} |\sqrt{\texttt{m}_\texttt{N}} u_\texttt{N}|^2 + |\texttt{m}_\texttt{N}|^{\gamma '} \, dx \right) ^p d\phi \end{aligned}$$

and

$$\begin{aligned} |J_4|^p&\le \upsilon \mathbb {E} \left[ \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \sum _{\varsigma \ge 1} \langle \mathfrak {g}_\varsigma (x, \mathscr {B}_\texttt{N}), \mathscr {B}_\texttt{N} \rangle ^2 \, d\phi \right] ^{p/2} \\&\le \varrho \mathbb {E} \left( \sup _{0 \le s \le t \wedge \tau _{n, \texttt{N}}} \Vert \mathscr {B}_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^{2p} \right) + \upsilon _\varrho \ \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \left( \int _\mathscr {Q} |\mathscr {B}_\texttt{N}|^2 \, dx \right) ^p d\phi . \end{aligned}$$

The identity

$$\begin{aligned} \texttt{m}_{\texttt{N}} \nabla \big (\frac{\Delta \sqrt{\texttt{m}_\texttt{N}}}{\sqrt{\texttt{m}_\texttt{N}}} \big ) = \operatorname {div}(\texttt{m}_{\texttt{N}} \nabla ^2 \text {In} \texttt{m}_{\texttt{N}}) \end{aligned}$$

yields that

$$\begin{aligned} \int _{\mathscr {Q}} \frac{\Delta \sqrt{\texttt{m}_\texttt{N}}}{\sqrt{\texttt{m}_\texttt{N}}} \Delta \texttt{m}_\texttt{N} dx =\frac{1}{2} \int _{\mathscr {Q}} |\nabla ^2 \text {In} \texttt{m}_{\texttt{N}}|^2 dx. \end{aligned}$$

(29)

Substituting (25)–(29) in (24), for small enough $\varrho> 0$,

$$\begin{aligned}&\sup _{0 \le \phi \le t \wedge \tau _{n, \texttt{N}}} E_\eta + \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \left( \mu \Vert \nabla u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + (\mu + \lambda ) \Vert \operatorname {div} u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \nu \Vert \nabla \mathscr {B}_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \right) d\phi \\&\qquad + \varepsilon \ \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \int _\mathscr {Q} \left( A \gamma ' \texttt{m}_\texttt{N}^{\gamma ' - 2} + \delta \beta \texttt{m}_\texttt{N}^{\beta - 2} \right) | \nabla \texttt{m}_\texttt{N}(\phi ) |^2 \, dx \, d\phi \\ \\ &\qquad + \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \left( \mu H''(\texttt{m}_{\texttt{N}})\Vert \nabla \texttt{m}_{\texttt{N}}\Vert ^{2}+\frac{\mu h^2}{4}|\nabla ^2 \ln \texttt{m}_{\texttt{N}}|^2\right) d\phi \\&\quad \le \mathbb {E} E_{0,\eta } + \upsilon _\varrho \ \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \int _\mathscr {Q} \left( |\sqrt{\texttt{m}_\texttt{N}} u_\texttt{N}|^2 + |\mathscr {B}_\texttt{N}|^2 \right) dx d\phi . \end{aligned}$$

In view of Gronwall’s inequality yields that

$$\begin{aligned}&\sup _{0 \le \phi \le t \wedge \tau _{n, \texttt{N}}} E_\eta + \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}}\left( \mu \Vert \nabla u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + (\mu + \lambda ) \Vert \operatorname {div} u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \nu \Vert \nabla \mathscr {B}_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \right) d\phi \\&\qquad + \varepsilon \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \int _\mathscr {Q} \left( A \gamma ' \texttt{m}_\texttt{N}^{\gamma ' - 2} + \delta \beta \texttt{m}_\texttt{N}^{\beta - 2} \right) | \nabla \texttt{m}_\texttt{N}(\phi ) |^2 \, dx \, d\phi \\&\qquad + \mathbb {E} \int _{0}^{t \wedge \tau _{n, \texttt{N}}} \left( \mu H''(\texttt{m}_{\texttt{N}})\Vert \nabla \texttt{m}_{\texttt{N}}\Vert ^{2}+\frac{\mu h^2}{4}|\nabla ^2 \ln \texttt{m}_{\texttt{N}}|^2\right) d\phi \\&\quad \le \upsilon \mathbb {E}(E_{0,\eta }). \end{aligned}$$

Hence for any $t \in \mathscr {I}$ we get the following $\mathscr {L}^2$ energy estimates

$$\begin{aligned}&\sup _{0 \le \phi \le t} E_\eta + \mathbb {E} \int _0^t \left( \mu \Vert \nabla u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + (\mu + \lambda ) \Vert \operatorname {div} u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \nu \Vert \nabla \mathscr {B}_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \right) d\phi \nonumber \\&\qquad + \varepsilon \mathbb {E} \int _0^t \int _\mathscr {Q} \left( A \gamma ' \texttt{m}_\texttt{N}^{\gamma ' - 2} + \delta \beta \texttt{m}_\texttt{N}^{\beta - 2} \right) | \nabla \texttt{m}_\texttt{N}(\phi ) |^2 \, dx \, d\phi \nonumber \\&\qquad + \mathbb {E} \int _0^t \left( \mu H''(\texttt{m}_{\texttt{N}})\Vert \nabla \texttt{m}_{\texttt{N}}\Vert ^{2}+\frac{\mu h^2}{4}|\nabla ^2 \ln \texttt{m}_{\texttt{N}}|^2\right) d\phi \nonumber \\&\quad \le \upsilon \mathbb {E}(E_{0,\eta }). \end{aligned}$$

(30)

Thus, for $p> 2$, we get the $\mathscr {L}^p$ energy estimates for $t \in \mathscr {I}$

$$\begin{aligned}&\sup _{0 \le \phi \le t} E_\eta + \mathbb {E}\Bigg [ \int _0^t \bigg ( \mu \Vert \nabla u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + (\mu + \lambda ) \Vert \operatorname {div} \, u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 + \nu \Vert \nabla \mathscr {B}_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \bigg ) \, d\phi \nonumber \\&\qquad + \varepsilon \int _0^t \int _\mathscr {Q} \big ( A \gamma ' \texttt{m}_\texttt{N}^{\gamma ' - 2} + \delta \beta \texttt{m}_\texttt{N}^{\beta - 2} \big ) | \nabla \texttt{m}_\texttt{N}(\phi ) |^2 \, dx \, d\phi \\&\qquad + \mathbb {E} \int _0^t \left( \mu H''(\texttt{m}_{\texttt{N}})\Vert \nabla \texttt{m}_{\texttt{N}}\Vert ^{2}+\frac{\mu h^2}{4}|\nabla ^2 \ln \texttt{m}_{\texttt{N}}|^2\right) d\phi \Bigg ]^p\nonumber \\&\quad \le \upsilon \mathbb {E}(E_{0,\eta })^p. \end{aligned}$$

This completes the proof of the proposition. $\square$

Lemma 3.3

For any fixed n,

$$\lim _{\texttt{N} \rightarrow \infty } \mathbb {P}(\tau _{n,\texttt{N}} = T) = 1.$$

Proof

From (30), we get that

$$\begin{aligned} \mathbb {E} \int _{0}^{T} \Vert \nabla u_\texttt{N}(\phi ) \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \le \upsilon \mathbb {E}(E_{0,\eta }) , \ \ \underset{t \in \mathscr {I}}{\sup } \mathbb {E}\Vert \sqrt{\texttt{m}_\texttt{N}}u_\texttt{N}(t)\Vert ^2_{\mathscr {L}^2(\mathscr {Q})} \le \upsilon \mathbb {E}(E_{0,\eta }). \end{aligned}$$

Since dimension of $\mathscr {T}_n$ is finite, $\texttt{m}=\mathfrak {M}[u]$ is bounded. Then, we have

$$\begin{aligned} \underline{\texttt{m}} \ \exp \big ( - \int _{0}^{T}&||\nabla u_\texttt{N}(\phi )||^2_{\mathscr {L}^2(\mathscr {Q})} \, d\phi \big ) \lesssim \texttt{m}_\texttt{N}(x,t) \lesssim \overline{\texttt{m}} \ \exp \big ( - \int _{0}^{T} ||\nabla u_{\texttt{N}}(\phi )||^2_{\mathscr {L}^2(\mathscr {Q})} \, d\phi \big ), \end{aligned}$$

(31)

$$\begin{aligned}&\mathbb {E} \big [\exp \big ( -\int _{0}^{T} ||\nabla u_\texttt{N}(\phi )||_{\mathscr {L}^2(\mathscr {Q})} d\phi \big ) \sup _{0 \le \phi \le t} \Vert u_\texttt{N}\Vert _{\mathscr {L}^2(\mathscr {Q})}\big ] \le \upsilon , \end{aligned}$$

(32)

where $\upsilon$ is independent of $\texttt{N}$. From Proposition 3.2, (31) and (32), we get that

$$\begin{aligned} \lim _{\texttt{N} \rightarrow \infty } \mathbb {P}(\tau _{n,\texttt{N}}=T)=1. \end{aligned}$$

$\square$

Tightness property

In this section we demonstrate the tightness property for each term in the approximation solution.

Lemma 4.1

³⁹ Let $(\texttt{m}_n, u_n, \mathscr {B}_n)$ be the solution of (14)–(16) on $\Upsilon \times \mathscr {I} \times \mathscr {Q}$ defined above. For $\beta \ge 4$, we have,

$$\begin{aligned}&\mathbb {E} \left( \sup _{t \in \mathscr {I}} \Vert \texttt{m}_n(t)\Vert ^{\gamma '}_{\mathscr {L}^{\gamma '}(\mathscr {Q})} \right) ^p \le \upsilon , \quad \mathbb {E} \left( \eta \sup _{t \in \mathscr {I}} \Vert \texttt{m}_n(t)\Vert ^\beta _{\mathscr {L}^\beta (\mathscr {Q})} \right) ^p \le \upsilon , \nonumber \\&\mathbb {E} \left( \sup _{t \in \mathscr {I}} \Vert \sqrt{\texttt{m}_n(t)} u_n(t)\Vert ^2_{\mathscr {L}^2(\mathscr {Q})} \right) ^p \le \upsilon , \quad \mathbb {E} \left( \Vert u_n(t)\Vert ^2_{\mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q}))} \right) ^p \le \upsilon , \nonumber \\&\mathbb {E} \left( \sup _{t \in \mathscr {I}} \Vert \mathscr {B}_n(t)\Vert ^2_{\mathscr {L}^2(\mathscr {Q})} \right) ^p \le \upsilon , \quad \mathbb {E} \left( \Vert \mathscr {B}_n(t)\Vert ^2_{\mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q}))} \right) ^p \le \upsilon , \nonumber \\&\mathbb {E} \left( \varepsilon \Vert \nabla \texttt{m}_n(t)\Vert ^2_{\mathscr {L}^2(\mathscr {I}\times \mathscr {Q})} \right) ^p \le \upsilon , \quad \mathbb {E} \left( \Vert \texttt{m}_n\Vert _{\mathscr {L}^{\beta +1}(\mathscr {I}\times \mathscr {Q})} \right) ^p \le \upsilon , \end{aligned}$$

(33)

where the constant $\upsilon$ is independent of $n$.

Lemma 4.2

Define the set

$$\begin{aligned} \mathscr {S}&:= C(\mathscr {I}; R) \times \Bigg ( C(\mathscr {I}; \mathscr {L}_w^\beta (\mathscr {Q})) \cap \mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q})) \cap \mathscr {L}^2(\mathscr {I}; H_w^1(\mathscr {Q}))\Bigg ) \\ &\quad \times \mathscr {L}^2(\mathscr {I}; H_w^1(\mathscr {Q})) \times C(\mathscr {I}; \mathscr {L}_w^{\frac{2\beta }{\beta +1}}(\mathscr {Q})) \times (\mathscr {L}^2(\mathscr {I}; H_w^1(\mathscr {Q})) \cap \mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))) \end{aligned}$$

equipped with its Borel $\sigma$-algebra. Consider the map $\Gamma _n: \Upsilon \mapsto (\beta _\varsigma ( \vartheta , \cdot ), \texttt{m}_n( \vartheta , \cdot ), u_n( \vartheta , \cdot ), \texttt{m}_n u_n( \vartheta , \cdot ), \mathscr {B}_n( \vartheta , \cdot ))$ be the probability on $\mathscr {S}$. For any $F \subseteq \mathscr {S}$,

$$\Gamma _n(F) = \mathbb {P}\{ \vartheta \in \Upsilon : (\hat{\beta }_\varsigma ( \vartheta , \cdot ), \texttt{m}_n( \vartheta , \cdot ), u_n( \vartheta , \cdot ), \texttt{m}_n u_n( \vartheta , \cdot ), \mathscr {B}_n( \vartheta , \cdot )) \in F\}.$$

Then the family $\Gamma _n$ is tight.

Proof

Now, the following steps are need to verify that the family of $\Gamma _n$ is tight: $.$

• Step 1 the tightness of $\hat{\beta }_\varsigma$

  First, let us start by examining the tightness of $\hat{\beta }_\varsigma$. It is enough to prove that for $\varepsilon> 0$ the compact subset $\Xi _\varepsilon \subset C(\mathscr {I}; R)$ with $\mathbb {P}(\beta _\varsigma \notin \Xi _\varepsilon ) \le \frac{\varepsilon }{5}$. Choose a constant $M_\varepsilon>0$, we consider the set

  $$\begin{aligned} \Xi _\varepsilon = \left\{ \beta _\varsigma (\cdot ) \in C(\mathscr {I}; R) : \sup _{t_1, t_2 \in \mathscr {I}, \, |t_1 - t_2| < \frac{1}{j^6}} j |\hat{\beta }_\varsigma (t_2) - \hat{\beta }_\varsigma (t_1)| \le M_\varepsilon , \, \forall j \in \mathfrak {N} \right\} . \end{aligned}$$

  By using Arzela -Ascoli’s Theorem, we conclude that $\Xi _\varepsilon$ is relatively compact in $C(\mathscr {I}; R)$. Also, $\Xi _\varepsilon$ is closed and compact subset in $C(\mathscr {I}; R)$. Now, we have to prove $\mathbb {P}(\hat{\beta }_\varsigma \notin \Xi _\varepsilon ) \le \frac{\upsilon }{M_\varepsilon ^4}.$ We know that, by Chebyshev’s inequality $\mathbb {P}\{\vartheta : \xi (\vartheta ) \ge q\} \le \frac{1}{q^r}\mathbb {E}[|\xi (\vartheta )|^r]$, we have

  $$\begin{aligned} \mathbb {P}\{ \vartheta : \hat{\beta }_\varsigma ( \vartheta , \cdot ) \notin \Xi _\varepsilon \}&\le \mathbb {P}\left[ \bigcup _{j=1}^\infty \left\{ \vartheta : \sup _{t_1, t_2 \in \mathscr {I}, \, |t_1 - t_2| < j^{-6}} |\hat{\beta }_\varsigma (t_1) - \hat{\beta }_\varsigma (t_2)|> \frac{M_\varepsilon }{j} \right\} \right] \\&\le \sum _{j=1}^\infty \sum _{i=0}^{j^6 - 1} \left( \frac{j}{M_\varepsilon }\right) ^4 \mathbb {E} \left[ \sup _{iT j^{-6} \le t \le (i+1)T j^{-6}} |\hat{\beta }_\varsigma (t) - \hat{\beta }_\varsigma (iT j^{-6})|^4 \right] \\&\le \upsilon \sum _{j=1}^\infty \left( \frac{j}{M_\varepsilon } \right) ^4 (Tj^{-6}) j^6 = \frac{\upsilon }{M_\varepsilon ^4} \sum _{j=1}^\infty \frac{1}{j^2}. \end{aligned}$$

  So, we choose $M_\varepsilon ^4 = \frac{1}{5 \upsilon \varepsilon } \left( \sum _{j=1}^\infty \frac{1}{j^2}\right) ^{-1}$ to obtain $\mathbb {P}(\hat{\beta }_\varsigma \notin \Xi _\varepsilon ) \le \frac{\varepsilon }{5}$.

• Step 2 the tightness of $\texttt{m}_n$

  Now, we have to find $\Lambda _\varepsilon \subset C(\mathscr {I}; \mathscr {L}_w^\beta (\mathscr {Q})) \cap \mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q})) \cap \mathscr {L}^2(\mathscr {I}; H_w^1(\mathscr {Q}))$ such that $\mathbb {P}(\texttt{m}_n \notin \Lambda _\varepsilon ) \le \frac{\varepsilon }{5}$. Consider a function space X equipped with the norm

  $$\begin{aligned} \Vert \mathfrak {f}\Vert _{X} = \sup _{0 \le t \le T} \Vert \mathfrak {f}(t)\Vert _{\mathscr {L}^\beta (\mathscr {Q})} + \Vert \mathfrak {f}(t)\Vert _{\mathscr {L}^2(\mathscr {I};H^1(\mathscr {Q}))} + \sup _{0 \le t \le T} \Vert \partial _t \mathfrak {f}\Vert _{\mathfrak {W}^{-1, \frac{2\beta }{\beta +1}}(\mathscr {Q})} + \Vert \partial _t \mathfrak {f}\Vert _{\mathscr {L}^2(\mathscr {I}; H^{-1}(\mathscr {Q}))}. \end{aligned}$$

  Choose a closed ball $\Lambda _\varepsilon (0,\texttt{r}_\varepsilon )$ in X. By Aubin-Lions Lemma implies that $\Lambda _\varepsilon$ is compact in $C(\mathscr {I};L_w^\beta (\mathscr {Q})) \cap \mathscr {L}^2(\mathscr {I}; H_w^1(\mathscr {Q})) \cap \mathscr {L}^2(\mathscr {I};\mathscr {L}^2(\mathscr {Q}))$. From (33) in Lemma 4.1, we get

  $$\mathbb {P}(\texttt{m}_n \notin \Lambda _\varepsilon ) = \mathbb {P}(\Vert \texttt{m}_n\Vert _{X}> \texttt{r}_\varepsilon ) \le \frac{1}{\texttt{r}_\varepsilon } \mathbb {E}(\Vert \texttt{m}_n\Vert _{X}) \le \frac{\upsilon }{\texttt{r}_\varepsilon }.$$

  Here, we choose $\texttt{r}_\varepsilon = 5\upsilon \varepsilon ^{-1}$, we get $\mathbb {P}(\texttt{m}_n \notin \Lambda _\varepsilon ) \le \frac{\varepsilon }{5}$. Then $\mathbb {P}(\{ \vartheta : \texttt{m}_n( \vartheta , \cdot ) \in \Lambda _\varepsilon \}) \ge 1 - \frac{\varepsilon }{5}$.

• Step 3 the tightness of $u_n$

  Next, we have to show that $\Phi _\varepsilon \subset \mathscr {L}^2(\mathscr {I}; H_w^1(\mathscr {Q}))$ such that $\mathbb {P}(u_n \notin \Phi _\varepsilon ) \le \frac{\varepsilon }{5}$. Choose closed ball $\Phi _\varepsilon (0;\tilde{\texttt{r}}_\varepsilon )$ in $\mathscr {L}^2(\mathscr {I};H_w^1(\mathscr {Q}))$. Then $\Phi _\varepsilon$ is compact in $\mathscr {L}^2(\mathscr {I};H_w^1(\mathscr {Q}))$. From (33) in Lemma 4.1, we get

  $$\begin{aligned} \mathbb {P}(u_n \notin \Phi _\varepsilon ) = \mathbb {P}(\Vert u_n\Vert _{\mathscr {L}^2(\mathscr {I};H^1(\mathscr {Q}))}> \tilde{\texttt{r}}_\varepsilon ) \le \frac{1}{\tilde{\texttt{r}}_\varepsilon } \mathbb {E}(\Vert u_n\Vert _{\mathscr {L}^2(\mathscr {I};H^1(\mathscr {Q}))}) \le \frac{\upsilon }{\tilde{\texttt{r}}_\varepsilon }. \end{aligned}$$

  Choosing $\tilde{\texttt{r}}_\varepsilon = 5\upsilon \varepsilon ^{-1}$, we get $\mathbb {P}(u_n \notin \Phi _\varepsilon ) \le \frac{\varepsilon }{5}$. Then $\mathbb {P}\{\vartheta : u_n( \vartheta , \cdot ) \in \Phi _\varepsilon \} \ge 1 - \frac{\varepsilon }{5}.$

• Step 4 the tightness of $\texttt{m}_n u_n$

  Now, we have to find $\Pi _\varepsilon \subset C(\mathscr {I}; L_w^{\frac{2\beta }{\beta +1}}(\mathscr {Q}))$ such that $\mathbb {P}(\texttt{m}_n u_n \notin \Pi _\varepsilon ) \le \frac{\varepsilon }{5}$. In order to prove, we define a function space $\Pi = \mathscr {L}^\infty (\mathscr {I}; \mathscr {L}_w^{\frac{2\beta }{\beta +1}}(\mathscr {Q})) \cap C^{0, \alpha }(\mathscr {I}; \mathfrak {W}^{-l, 2}(\mathscr {Q}))$, $l> 3$, $0< \alpha < \frac{1}{2}$ with the norm

  $$\Vert \mathfrak {f}\Vert _\Pi = \sup _{0 \le t \le T} \Vert \mathfrak {f}(t)\Vert _{\mathscr {L}^{\frac{2\beta }{\beta +1}}(\mathscr {Q})} + \Vert \mathfrak {f}(t)\Vert _{C^{0, \alpha }(\mathscr {I}; \mathfrak {W}^{-l, 2}(\mathscr {Q}))}.$$

  The Brownian motion cannot be differentiated in time because we cannot estimate $d (\texttt{m}_n u_n)$ directly. From (14), we have

  $$\begin{aligned} \mathscr {P}[\texttt{m}_n u_n(t)]&=\mathscr {P}[\texttt{m}_0 u_0] - \int _0^t \mathscr {P}\left[ \operatorname {div}(\texttt{m}_n u_n \otimes u_n) - \mu \Delta u_n - (\mu + \lambda ) \operatorname {div} u_n - A \nabla \texttt{m}_n^{\gamma '} - \eta \nabla \texttt{m}_n^\beta \right] d\phi \nonumber \\&\quad + \int _0^t \mathscr {P}\left[ (\nabla \times \mathscr {B}_n) \times \mathscr {B}_n - \varepsilon \nabla u_n \cdot \nabla \texttt{m}_n d\phi + \frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) \right] \nonumber \\&\quad - \int _0^t \sum _{\varsigma \ge 1} \mathfrak {f}_\varsigma ^n(x, \texttt{m}_n, \texttt{m}_n u_n) d\hat{\beta }_\varsigma ^1, \end{aligned}$$

  (34)

  where $\mathscr {P}: \mathscr {L}^2(\mathscr {Q}) \rightarrow \mathscr {T}_n$ is a projection onto $\mathscr {T}_n$. From (33), we have

  $$\begin{aligned} \mathbb {E} \left[ \int _0^T \Vert \mathscr {P}[\operatorname {div}(\texttt{m}_n u_n \otimes u_n)]\Vert _{\mathfrak {W}^{-1, \frac{6 \beta }{4 \beta + 3}}(\mathscr {Q})}^2 d\phi \right] \le \mathbb {E} \left[ \int _0^T \Vert \texttt{m}_n u_n \otimes u_n\Vert _{\mathscr {L}^{\frac{6 \beta }{4 \beta + 3}}(\mathscr {Q})}^2 d\phi \right] \le \upsilon . \end{aligned}$$

  (35)

  Similarly, by using Lemma 4.1 and $\nabla \texttt{m}_n \in \mathscr {L}^q(\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))$, $q> 2$[Lemma 2.4⁵¹,, we have

  $$\begin{aligned}&\mathbb {E} \int _0^T \Vert \mathscr {P}[\mu \Delta u_n + (\mu + \lambda ) \nabla \operatorname {div} u_n]\Vert ^2_{\mathfrak {W}^{-1,2}(\mathscr {Q})} \, d\phi \le \upsilon , \nonumber \\&\mathbb {E} \int _0^T \Vert \mathscr {P} (\varepsilon \nabla u_n \cdot \nabla \texttt{m}_n)\Vert ^{\frac{2q}{q+2}}_{\mathfrak {W}^{-l,2}(\mathscr {Q})} \, d\phi \le \upsilon , \nonumber \\&\mathbb {E} \int _0^T \Vert \mathscr {P} \big (A \nabla \texttt{m}^\beta _n + \eta \nabla \texttt{m}^\beta _n \big )\Vert ^{\frac{\beta +1}{\beta }}_{\mathfrak {W}^{-1,\frac{\beta +1}{\beta }}(\mathscr {Q})} \, d\phi \le \upsilon , \nonumber \\&\mathbb {E} \int _0^T \Vert \mathscr {P} \left( \frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) \right) \Vert ^2 \, d\phi \le \upsilon , \nonumber \\&\mathbb {E} \int _0^T \Vert \mathscr {P} \big [(\nabla \times \mathscr {B}_n) \times \mathscr {B}_n\Big ]\Vert ^2_{\mathfrak {W}^{-l,2}(\mathscr {Q})} \, d\phi \le \upsilon . \end{aligned}$$

  (36)

  From Lemma 4.1, for $\varpi> 1$

  $$\begin{aligned}&\mathbb {E} \left( \int _\phi ^t \sum _{\varsigma =1}^\infty \Vert \mathfrak {f}^n_\varsigma (x, \texttt{m}_n, \texttt{m}_n)\Vert _{\mathfrak {W}^{-l,2}(\mathscr {Q})} \, d\hat{\beta }^1_\varsigma (\phi ) \right) ^{2\varpi } \le \mathbb {E} \left[ \int _\phi ^t \sum _{\varsigma =1}\Vert \mathfrak {f}^n_\varsigma (x, \texttt{m}_n, \texttt{m}_n)\Vert ^2_{\mathfrak {W}^{-l,2}(\mathscr {Q})} \, d\phi \right] ^\varpi \nonumber \\&\quad \le \mathbb {E} \left[ \int _s^t \sum _{\varsigma =1} \Vert \mathscr {P} \mathfrak {f}_\varsigma (x, \texttt{m}_n, \texttt{m}_n)/\sqrt{\texttt{m}_n}\Vert ^2_{\mathscr {L}^2(\mathscr {Q})} \sup _{\Vert \psi \Vert _{\mathfrak {W}^{l,2}(\mathscr {Q})}=1} \Vert \mathbb {S}_{\texttt{m}_n}^{1/2}[\texttt{m}_n] \psi \Vert ^2_{\mathscr {L}^2(\mathscr {Q})} \, d\phi \right] ^\varpi \le \upsilon (t-\phi )^\varpi . \end{aligned}$$

  (37)

  By using Kolmogorov continuity theorem, we have that the stochastic terms $\mathfrak {f}_\varsigma (x, \texttt{m}_n, \texttt{m}_n)$ is $\alpha$-Holder continuous for every $0< \alpha < \frac{1}{2} - \frac{1}{2\varpi }$ almost surely. When $l> 3$, we have $\mathfrak {W}^{-1,\frac{6\beta }{4\beta +3}}(\mathscr {Q}) \hookrightarrow \mathfrak {W}^{-l,2}(\mathscr {Q})$ and $\mathfrak {W}^{-1,\frac{\beta +1}{\beta }}(\mathscr {Q}) \hookrightarrow \mathfrak {W}^{-l,2}(\mathscr {Q})$. Choose $\Pi _\varepsilon (0,\texttt{r}'_\varepsilon )$ in $\tilde{\Pi }$which is a closed ball. By [Corollary B.2⁵², and⁵³ we know that $\Pi _\varepsilon$ is compact in $C(\mathscr {I}; L_w^{\frac{2\beta +1}{\beta }}(\mathscr {Q}))$. In addition, from (34)–(), by the Chebyshev inequality, we get

  $$\begin{aligned} \mathbb {P}(\texttt{m}_n u_n \notin \Pi _\varepsilon ) = \mathbb {P} \left( \Vert \texttt{m}_n u_n\Vert _{\tilde{\Pi }}> \texttt{r}'_\varepsilon \right) \le \frac{1}{\texttt{r}'_\varepsilon } \mathbb {E}(\Vert \texttt{m}_n u_n\Vert _Z) \le \frac{\upsilon }{\texttt{r}'_\varepsilon }. \end{aligned}$$

  (38)

  Let $\texttt{r}'_\varepsilon = 5\upsilon \varepsilon ^{-1}$ in (38), we get that

  $$\mathbb {P}(\texttt{m}_n u_n \notin \Pi _\varepsilon ) \le \frac{\varepsilon }{5}.$$

  Then $\mathbb {P}(\{\vartheta : \vartheta _n u_n( \vartheta , \cdot ) \in \Pi _\varepsilon \}) \ge 1 - \frac{\varepsilon }{5}$.

  We employ the tightness criterion. Let $\{\mathfrak {f}_n\}_{n=1}^\infty$ be a collection of $C(\mathscr {I},\mathscr {L}_w^p(\mathscr {Q}))$-valued random variables defined on $(\Upsilon , \mathscr {F}_n, P_n)$ such that $\sup _n \mathbb {E}(\Vert \mathfrak {f}_n\Vert _{\mathscr {L}^\infty (\mathscr {I}, \mathscr {L}^p(\mathscr {Q}))}) < \infty$ and for any $\psi \in C_0^\infty (\mathscr {Q})$, there exist an integer k, $\varpi> 0$ and $q> \frac{1}{\varpi }$ such that $\sup _n \mathbb {E}[|\langle \mathfrak {f}_n(t) - \mathfrak {f}_n(\phi ), \varphi \rangle |^q] \le \Vert \psi \Vert _{C^k}^q |t-\phi |^{\varpi q}$ for all $0 \le \phi , t \le T$. Then the sequence of the induced measures $P \circ \mathfrak {f}_n^{-1}$ are tight on $C(\mathscr {I}, \mathscr {L}_w^p(\mathscr {Q}))$. By Lemma 4.1 and Holder’s inequality, we get that

  $$\begin{aligned} \mathbb {E} \Bigg |\int _0^t&\left( \operatorname {div} (\texttt{m}_n u_n \otimes u_n), \psi \right) d\zeta \Bigg | \le \mathbb {E} \int _\phi ^t \Vert \nabla \psi \Vert _{\mathscr {L}^\infty } \Vert \sqrt{\texttt{m}_n} u_n\Vert _{\mathscr {L}^2(\mathscr {Q})} \Vert \nabla u_n\Vert _{\mathscr {L}^2(\mathscr {Q})} \Vert \texttt{m}_n\Vert ^{\frac{1}{2 }}_{\mathscr {L}^{\gamma '}(\mathscr {Q})} \, d\zeta \\&\le \upsilon (t-\phi )^\frac{1}{2} \mathbb {E} \left( \Vert \sqrt{\texttt{m}_n}u_n\Vert _{\mathscr {L}^\infty (\mathscr {I};\mathscr {L}^2(\mathscr {Q}))} \Vert \nabla u_n\Vert _{\mathscr {L}^2(\mathscr {I};\mathscr {L}^2(\mathscr {Q}))} \Vert \texttt{m}_n\Vert ^{\frac{1}{2}}_{\mathscr {L}^{\infty }(\mathscr {I};\mathscr {L}^{\gamma '}(\mathscr {Q}))} \right) \\&\le \upsilon (t-\phi )^\frac{1}{2}, \end{aligned}$$
  $$\begin{aligned} \mathbb {E} \left| \int _\phi ^t \langle (\nabla \times \mathscr {B}_n) \times \mathscr {B}_n, \psi \rangle d\zeta \right|&\le \upsilon (t-\phi )^\frac{1}{2} \Vert \psi \Vert _{\mathscr {L}^\infty } \mathbb {E}(\Vert \nabla \mathscr {B}_n\Vert _{\mathscr {L}^2(\mathscr {I};\mathscr {L}^2(\mathscr {Q}))} \Vert \mathscr {B}_n\Vert _{\mathscr {L}^\infty (\mathscr {I},\mathscr {L}^2(\mathscr {Q}))}) \\&\le \upsilon (t-\phi )^\frac{1}{2}, \\ \mathbb {E} \left| \int _\phi ^t \langle \mu \Delta u_n + (\mu + \lambda ) \nabla \operatorname {div} u_n, \psi \rangle d\zeta \right|&\le \upsilon (t-\phi )^\frac{1}{2} \Vert \nabla \psi \Vert _{\mathscr {L}^\infty } \mathbb {E}(\Vert \nabla u_n\Vert _{\mathscr {L}^2(\mathscr {I};\mathscr {L}^2(\mathscr {Q}))}) \\&\le \upsilon (t-\phi )^\frac{1}{2}, \\ \mathbb {E} \left| \int _\phi ^t \langle A \nabla \texttt{m}_n^{\gamma '} - \eta \nabla \texttt{m}_n^\beta , \psi \rangle d\zeta \right|&\le \upsilon \mathbb {E} \int _\phi ^t \Vert \nabla \psi \Vert _{\mathscr {L}^\infty } \Vert \texttt{m}_n\Vert ^{\gamma '}_{\mathscr {L}^\gamma (\mathscr {Q})} \Vert \texttt{m}_n\Vert ^\beta _{\mathscr {L}^\beta (\mathscr {Q})} d\zeta \le \upsilon (t-\phi )^{\frac{1}{2} }\\ \mathbb {E} \left| \int _\phi ^t \left\langle \texttt{m}_n \nabla \left( \frac{\Delta \sqrt{\texttt{m}_n}}{\sqrt{\texttt{m}_n}} \right) , \psi \right\rangle d\zeta \right|&\le \mathbb {E} \int _\phi ^t \Vert \Delta \sqrt{\texttt{m}_n}\Vert _{\mathscr {L}^2(\mathscr {I};\mathscr {L}^2(\mathscr {Q}))} \mathbb {E}\big ( 2 \Vert \sqrt{\texttt{m}_n}\Vert _{\mathscr {L}^4(\mathscr {I};\mathfrak {W}^{1,3}(\mathscr {Q}))} \Vert \psi \Vert _{\mathscr {L}^4(\mathscr {I};\mathscr {L}^6(\mathscr {Q}))} \\&\quad + \Vert \sqrt{\texttt{m}_n}\Vert _{\mathscr {L}^\infty (\mathscr {I};\mathscr {L}^6(\mathscr {Q}))} \Vert \psi \Vert _{\mathscr {L}^2(\mathscr {I};\mathfrak {W}^{1,3}(\mathscr {Q}))} \big ) \\&\le \upsilon (t-\phi )^{\frac{1}{2}}, \\ \mathbb {E} \left| \int _\phi ^t \langle \varepsilon \nabla u_n \cdot \nabla \texttt{m}_n, \psi \rangle d\zeta \right|&\le \mathbb {E} \int _\phi ^t \varepsilon \Vert \nabla u_n\Vert _{\mathscr {L}^2(\mathscr {Q})} \Vert \nabla \texttt{m}_n\Vert _{\mathscr {L}^2(\mathscr {Q})} \Vert \psi \Vert _{\mathscr {L}^\infty (\mathscr {Q})} d\zeta \\&\le (t-\phi )^{\frac{1}{2}-\frac{1}{q}} \mathbb {E}(\Vert \nabla u_n\Vert _{\mathscr {L}^2(\mathscr {I};\mathscr {L}^2(\mathscr {Q}))} \Vert \nabla \texttt{m}_n\Vert _{\mathscr {L}^q(\mathscr {I};\mathscr {L}^2(\mathscr {Q}))}) \\&\le \upsilon (t-\phi )^{\frac{1}{2} - \frac{1}{q}}. \end{aligned}$$

  By applying $\nabla \texttt{m}_n \in \mathscr {L}^q(\mathscr {I};\mathscr {L}^2(\mathscr {Q})), q> 2$, concludes the tightness of $\texttt{m}_n u_n$.

• Step 5 the tightness of $\mathscr {B}_n$

  Next, we have to find the subset $Y_\varepsilon$ in $\mathscr {L}^2(\mathscr {I};H^1_w(\mathscr {Q}))$ such that $\mathbb {P}(\mathscr {B}_n \notin R_\varepsilon ) \le \frac{\varepsilon }{5}.$ Finally, choose a closed ball $Y_\varepsilon (0,\ddot{\texttt{r}}_\varepsilon )$ in $\mathscr {L}^2(\mathscr {I};H^1(\mathscr {Q}))$. Hence, $Y_\varepsilon$ is compact in $\mathscr {L}^2(\mathscr {I};H^1_w(\mathscr {Q}))$. From Lemma 4.1, we have

  $$\mathbb {P}(\mathscr {B}_n \notin Y_\varepsilon ) = \mathbb {P}(\Vert \mathscr {B}_n\Vert _{\mathscr {L}^2(\mathscr {I};H^1(\mathscr {Q}))}> \ddot{\texttt{r}}_\varepsilon ) \le \frac{1}{\ddot{\texttt{r}}_\varepsilon } \mathbb {E}\left( \Vert \mathscr {B}_n\Vert _{\mathscr {L}^2(\mathscr {I};H^1(\mathscr {Q}))} \right) \le \frac{\upsilon }{\ddot{\texttt{r}}_\varepsilon }.$$

  Here, we choose $\ddot{\texttt{r}}_\varepsilon = 5\upsilon \varepsilon ^{-1}$, then we have $\mathbb {P}(\mathscr {B}_n \notin Y_\varepsilon ) \le \frac{\varepsilon }{5}$. Then $\mathbb {P}\{\varepsilon : \mathscr {B}_n( \vartheta , \cdot ) \in Y_\varepsilon \} \ge 1 - \frac{\varepsilon }{5}$. From (14), it follows that

  $$\begin{aligned} \mathbb {E} \int _0^{T-\sigma } \Vert \mathscr {B}_n(t+\sigma ) - \mathscr {B}_n(t)\Vert ^2_{H^{-1}(\mathscr {Q})} dt&= \mathbb {E} \int _0^{T-\sigma } \left\| \int _t^{t+\theta } d\mathscr {B}_n(\phi ) \right\| ^2_{H^{-1}(\mathscr {Q})} dt \nonumber \\&\le \mathbb {E} \int _0^{T-\sigma } (\mathscr {J}_1 + \mathscr {J}_2 + \mathscr {J}_3) dt, \end{aligned}$$

  (39)

  where

  $$\begin{aligned}&\mathscr {J}_1(t) = \left\| \int _t^{t+\sigma } \nabla \times (u_n \times \mathscr {B}_n) d\phi \right\| ^2_{H^{-1}(\mathscr {Q})}, \quad \mathscr {J}_2(t) = \left\| \int _t^{t+\sigma } \nu \Delta \mathscr {B}_n d\phi \right\| ^2_{H^{-1}(\mathscr {Q})}, \\&\mathscr {J}_3(t) = \left\Vert \int _t^{t+\sigma } \sum _{\varsigma \ge 1} \mathfrak {g}_\varsigma (x, \mathscr {B}_n) d\hat{\beta }_\varsigma ^2 \right\Vert _{H^{-1}(\mathscr {Q})}^2. \end{aligned}$$

  For the term $\mathscr {J}_1(t)$, one has

  $$\begin{aligned} \mathscr {J}_1^{1/2}&= \sup _{\psi \in H_0^1(\mathscr {Q}), \Vert \psi \Vert _{H^1(\mathscr {Q})} = 1} \left\{ \int _\mathscr {Q} \left( \int _t^{t+\sigma } \nabla \cdot (u_n \times \mathscr {B}_n) d\phi \right) \psi (x) dx \right\} \\&\le \upsilon \int _t^{t+\sigma } \Vert u_n\Vert _{\mathscr {L}^4(\mathscr {Q})} \Vert \mathscr {B}_n\Vert _{\mathscr {L}^4(\mathscr {Q})} d\phi . \end{aligned}$$

  By Lemma 4.1 and Holder’s inequality , we get

  $$\begin{aligned} \mathbb {E} \int _0^{T-\sigma } \mathscr {J}_1(t) dt \lesssim \sigma ^{\frac{1}{4}} \mathbb {E} \left( \Vert u_n\Vert ^2_{\mathscr {L}^2(\mathscr {I};H^1(\mathscr {Q}))} \Vert \mathscr {B}_n\Vert ^2_{\mathscr {L}^{\frac{8}{3}}(\mathscr {I};\mathscr {L}^4(\mathscr {Q}))} \right) \lesssim \sigma ^{\frac{1}{4}}. \end{aligned}$$

  (40)

  Similarly, we have

  $$\begin{aligned} \mathbb {E} \int _0^{T-\sigma } \mathscr {J}_2(t) dt&\lesssim \mathbb {E} \int _0^{T-\sigma } \left( \int _t^{t+\sigma } \Vert \nabla \mathscr {B}_n\Vert _{\mathscr {L}^2(\mathscr {Q})} d\phi \right) ^2 dt \nonumber \\&\lesssim \sigma \mathbb {E} \int _0^T \Vert \nabla \mathscr {B}_n \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 dt \lesssim \sigma . \end{aligned}$$

  (41)

  By the assumption (A), Burkholder-Davis-Gundy inequality and Holder’s inequality, the term $\mathscr {J}_3$ becomes

  $$\begin{aligned} \mathbb {E} \int _0^{T-\sigma } \mathscr {J}_3(t) dt&\le \int _0^T \mathbb {E} \left( \sup _{\psi \in H_0^1(\mathscr {Q}), \Vert \psi \Vert _{H^1(\mathscr {Q})}=1} \int _t^{t+\sigma } \int _\mathscr {Q} \sum _{\varsigma \ge 1} \mathfrak {g}_\varsigma (x, \mathscr {B}_n) \psi dx d\hat{\beta }_\varsigma ^2 \right) ^2 dt \nonumber \\&\le \int _0^T \left( \mathbb {E} \sup _{\psi \in H_0^1(\mathscr {Q}), \Vert \psi \Vert _{H^1(\mathscr {Q})}=1} \int _t^{t+\sigma } \left( \sum _{\varsigma \ge 1} \int _\mathscr {Q} \mathfrak {g}_\varsigma (x, \mathscr {B}_n)\psi dx \right) ^2 d\phi \right) dt \nonumber \\&\le \int _0^T \left( \mathbb {E} \int _t^{t+\sigma } \Vert \psi \Vert _{\mathscr {L}^2(\mathscr {Q})}^2 \sum _{\varsigma =1} \Vert \mathfrak {g}_\varsigma (x, \mathscr {B}_n)\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 d\phi \right) dt \nonumber \\&\le \int _0^T \left( \mathbb {E} \int _t^{t+\sigma } \Vert \mathscr {B}_n\Vert _{\mathscr {L}^2(\mathscr {Q})}^2 d\phi \right) dt \nonumber \\&\lesssim \sigma . \end{aligned}$$

  (42)

  Combining (39)–(42) and by using Aubin-Simon Lemma in⁵³, we get that $\mathscr {B}_n$ is compact in $\mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))$. Then $\mathbb {P}(\mathscr {B}_n \notin Y_\varepsilon ) \le \frac{\upsilon }{\ddot{\texttt{r}}_\varepsilon }.$

  From [Corollary1.3⁵⁴, the tightness property followed by the distribution of the joint processes $(\hat{\beta }_\varsigma , \texttt{m}_n, u_n, \texttt{m}_n u_n, \mathscr {B}_n)$. This completes the proof of lemma. 

Remark 4.3

By employing Jakubowski-Skorokhod Theorem, also from the tightness property, there exists a subsequence such that $\Gamma _n$ converges weakly to $\Gamma$, where $\Gamma$ is a probability on $\mathscr {S}$, a probability space $(\tilde{\Upsilon }, \tilde{\mathscr {F}}, \tilde{\mathbb {P}})$. Then there exist random variables $(\tilde{\beta }_\varsigma , \tilde{\texttt{m}}_n, \tilde{u}_n, \tilde{\texttt{m}}_n \tilde{u}_n, \tilde{\mathscr {B}}_n)$ with distribution $\Gamma _n$, $(\beta _\varsigma , \texttt{m}, u, h, \mathscr {B})$ with values in $\mathscr {S}$ such that

$$\begin{aligned} (\tilde{\beta }_{\varsigma ,n}, \tilde{\texttt{m}}_n, \tilde{u}_n, \tilde{\texttt{m}}_n \tilde{u}_n, \tilde{\mathscr {B}}_n) \rightarrow (\beta _\varsigma , \texttt{m}, u, h, \mathscr {B}) \quad \text {in} \ \mathscr {S} \quad \tilde{\mathbb {P}} \text {-a.s.}. \end{aligned}$$

(43)

Estimates independent of n

In this section, we first compute the limit $n \rightarrow \infty$, $\eta> 0$ being fixed in the sequence $(\texttt{m}_n,u_n,\mathscr {B}_n)$ in order to get a solution of the system (14)–(16). The limit $\eta \rightarrow 0$ is carried out in Section “Estimates independent of $\eta$”.

Let $(\texttt{m}_n, u_n, \mathscr {B}_n) \in C^1(\mathscr {I}; C^3(\mathscr {Q}) \times C^1(\mathscr {I}; \mathscr {T}_n) \times \mathscr {Y}$ with

$$\mathscr {Y} := \mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q})) \cap \mathscr {L}^\infty (\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))$$

be a solution to system (14), (18) and (19). It is easy to obtain from the energy estimate of Proposition 3.2 to get the uniform bounds

$$\begin{aligned} \Vert \sqrt{\texttt{m}_n} \Vert _{\mathscr {L}^\infty (\mathscr {I}; H^1(\mathscr {Q}))}&\le \upsilon , \end{aligned}$$

(44)

$$\begin{aligned} \Vert \texttt{m}_n \Vert _{\mathscr {L}^\infty (\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))}&\le \upsilon , \end{aligned}$$

(45)

and

$$\begin{aligned}&\Vert \sqrt{\texttt{m}_n} u_n \Vert _{\mathscr {L}^\infty (\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))} + \Vert \sqrt{\texttt{m}_n} \nabla u_n \Vert _{\mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))}&\le \upsilon , \nonumber \\&\Vert \mathscr {B}_n \Vert _{\mathscr {L}^\infty (\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))} + \Vert \mathscr {B}_n \Vert _{\mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q}))}&\le \upsilon , \end{aligned}$$

(46)

where $\upsilon$ is a positive constant. Since $H^1(\mathscr {Q})$ embedes continuously into $\mathscr {L}^6(\mathscr {Q})$, the estimate (45) proves the bound only if $\gamma> 3$³⁸. From the Aubin lemma, we draw a conclusion and taking into account the regularity (7) and (10) for $\texttt{m}_n$, the regularity (5) and (11) for $\sqrt{\texttt{m}_n}$, and the regularity (6) and (11) for $\texttt{m}_n u_n$, that there exist subsequences of $(\texttt{m}_n),\sqrt{\texttt{m}_n},\texttt{m}_n u_n)$, which are not relabeled, such that, for some functions $\texttt{m}, y$ and B as $n \rightarrow \infty$,

$$\texttt{m}_n \rightarrow \texttt{m} \quad \text {strongly in } \mathscr {L}^2(\mathscr {I};\mathscr {L}^\infty (\mathscr {Q})),$$

$$\sqrt{\texttt{m}_n} \rightharpoonup \sqrt{\texttt{m}} \quad \text {weakly in } \mathscr {L}^2(\mathscr {I};H^2(\mathscr {Q})),$$

$$\sqrt{\texttt{m}_n} \rightarrow \sqrt{\texttt{m}} \quad \text {strongly in } \mathscr {L}^2(\mathscr {I};H^1(\mathscr {Q})),$$

$$\texttt{m}_n u_n \rightarrow y \quad \text {strongly in } \mathscr {L}^2(\mathscr {I};\mathscr {L}^2(\mathscr {Q})).$$

By the estimates of Lemma 4.1 and the fact that $d \texttt{m}_n$ satisfies (14), we use (43) such that there exists a function $\texttt{m}$ such that

$$\begin{aligned} \texttt{m}_n \rightarrow \texttt{m} \quad \text {in } \mathscr {L}^4(\mathscr {I} \times \mathscr {Q}) \quad \widetilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

(47)

Also, (43) implies that $\texttt{m}_n \rightarrow \texttt{m} \quad \text {in } \mathscr {L}^2(\mathscr {I} \times \mathscr {Q}) \quad \widetilde{\mathbb {P}}\text {-a.s.}.$ For $\beta> 4$, since $\texttt{m}_n \in \mathscr {L}^\infty (\mathscr {I};\mathscr {L}^\beta (\mathscr {Q}))$ and using the interpolation of $\mathscr {L}^2(\mathscr {I}\times \mathscr {Q})$ and $\mathscr {L}^\infty (\mathscr {I};\mathscr {L}^\beta (\mathscr {Q}))$, we have (47). In addition to (47) and in view of Lemma 4.1, we have

$$\texttt{m}_n^{\gamma '} \rightarrow \texttt{m}^{\gamma '},\ \texttt{m}_n^\beta \rightarrow \texttt{m}^\beta \quad \text {in } \mathscr {L}^1(\mathscr {I} \times \mathscr {Q}) \quad \widetilde{\mathbb {P}}\text {-a.s. for } \beta> \gamma '.$$

Let us pass to the limit in the products $\texttt{m}_n u_n$ and $\texttt{m}_n u_n \otimes u_n$. By using Lemma 4.1, we see that

$$\texttt{m}_n u_n \rightarrow y \quad \text {in } C(\mathscr {I}; \mathscr {L}^\frac{2\beta }{\beta +1}(\mathscr {Q})) \quad \widetilde{\mathbb {P}}\text {-a.s.}.$$

Since $\texttt{m}_n \rightarrow \texttt{m}$ in $\mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q}))$ and $u_n \rightarrow u$ in $\mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q})) \quad \widetilde{\mathbb {P}}\text {-a.s.},$ we have

$$\texttt{m}_n u_n \rightarrow \texttt{m} u \quad \text {in } \mathscr {L}^1(\mathscr {I}; \mathscr {L}^1(\mathscr {Q})) \quad \widetilde{\mathbb {P}}\text {-a.s.}.$$

By⁴⁹, Lemma 2.4], we get $y = \texttt{m} u$. Then we have

$$\texttt{m}_n u_n \rightarrow \texttt{m} u \quad \text {in } C(\mathscr {I}; \mathscr {L}^\frac{2\beta }{\beta +1}(\mathscr {Q})) \quad \widetilde{\mathbb {P}}\text {-a.s.}.$$

Since $\mathscr {L}^\frac{2\beta }{\beta +1}(\mathscr {Q}) \hookrightarrow \hookrightarrow H^{-1}(\mathscr {Q})$ and by Aubin-Lions Lemma, we obtain that

$$\begin{aligned} \texttt{m}_n u_n \rightarrow \texttt{m} u \quad \text {in } C(\mathscr {I}; H^{-1}(\mathscr {Q})) \quad \widetilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

(48)

Since $u_n \rightarrow u$ in $\mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q}))$ $\widetilde{\mathbb {P}}$-a.s. and by (48), we obtain that

$$\texttt{m}_n u_n \otimes u_n \rightarrow \texttt{m} u \otimes u \quad \text {in } D'(\mathscr {I} \times \mathscr {Q}) \quad \widetilde{\mathbb {P}}\text {-a.s.}.$$

Moreover, if we pass the limit $n \rightarrow \infty$ of

$$\int _\mathscr {Q} \frac{\Delta \sqrt{\texttt{m}_n}}{\sqrt{\texttt{m}_n}} \operatorname {div} (\texttt{m}_n \psi ) dx = \int _\mathscr {Q} \Delta \sqrt{\texttt{m}_n} (2 \nabla \sqrt{\texttt{m}_n} \cdot \psi + \sqrt{\texttt{m}_n} \operatorname {div} \psi ) \, dx,$$

satisfying for sufficiently smooth test functions, we get

$$\int _\mathscr {Q} \Delta \sqrt{\texttt{m}} (2 \nabla \sqrt{\texttt{m}} \cdot \psi + \sqrt{\texttt{m}} \operatorname {div} \psi ) \, dx.$$

By (43), we get

$$\begin{aligned} \mathscr {B}_n \rightarrow \mathscr {B}\quad \text {in } \mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q})) \quad \text {and } \quad \mathscr {B}_n \rightarrow \mathscr {B}\quad \text {in } \mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q})) \quad \widetilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

(49)

Furthermore, from (15) and Lemma 4.1, and by the Aubin-Lions lemma, we have

$$\mathscr {B}_n \rightarrow \mathscr {B}\quad \text {in } C(\mathscr {I}; H^{-1}(\mathscr {Q})) \quad \widetilde{\mathbb {P}}\text {-a.s.}.$$

Also, we have

$$(\nabla \times \mathscr {B}_n) \times \mathscr {B}_n \rightarrow (\nabla \times \mathscr {B}) \times \mathscr {B}\quad \text {in } D'(\mathscr {I} \times \mathscr {Q}) \quad \widetilde{\mathbb {P}}\text {-a.s.} .$$

From (43),(49) and since the convergence of the sequences $u_n \rightarrow u \ \text {in} \ \mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q})) \ \widetilde{\mathbb {P}}\text {-a.s.}$ and

$$\begin{aligned} \nabla \times (u_n \times \mathscr {B}_n) \rightarrow \nabla \times (u \times \mathscr {B}) \quad \text {in } D'(\mathscr {I} \times \mathscr {Q}) \quad \widetilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

(50)

Next, we focus on the stochastic force terms. By using Theorem 3.1, Lemma 4.1, Assumption (A), and the strong convergence in (43), (47) and (48), we get that

$$\begin{aligned} \frac{\mathfrak {f}_\varsigma (x, \texttt{m}_n, \texttt{m}_n u_n)}{\sqrt{\texttt{m}_n}} \rightarrow \frac{\mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u)}{\sqrt{\texttt{m}}} \quad \text {in } \mathscr {L}^2(\mathscr {Q}) \quad \widetilde{\mathbb {P}} \otimes \tilde{L}\text {-a.s.}, \end{aligned}$$

where $\tilde{L}$ is the Lebesgue measure in time. By using Assumption (A) and Lemma 4.1, we get

$$\begin{aligned} \sum _{\varsigma =1}^\infty \left| \langle \mathfrak {f}_\varsigma (x, \texttt{m}_n, \texttt{m}_n u_n), \varphi \rangle \right| \le \sum _{k\ge 1}^\infty \Vert \mathfrak {f}_\varsigma \Vert _{\mathscr {L}^2} \le \Vert \texttt{m}_n \Vert ^{\frac{\gamma '+1}{2}}_{\mathscr {L}^{\gamma '+1}} + \Vert \texttt{m}_n u_n \Vert _{\mathscr {L}^2} \le \upsilon . \end{aligned}$$

(51)

Thus, we obtain the boundedness of $\sum _{\varsigma =1}^\infty \left| \langle \mathfrak {f}_\varsigma (x, \texttt{m}_n, \texttt{m}_n u_n), \mathscr {P} \varphi \rangle \right| .$ Since

$$\begin{aligned}\langle \mathfrak {f}^n_\varsigma (x, \texttt{m}_n, \texttt{m}_n u_n), \mathscr {P} \varphi \rangle \rightarrow \langle \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u), \varphi \rangle \quad \text {in} \ \mathscr {L}^1(\mathscr {I}) \ \widetilde{\mathbb {P}}\text {-a.s.},\end{aligned}$$

we get that

$$\sum _{\varsigma \ge 1} \langle \mathfrak {f}^n_\varsigma (x, \texttt{m}_n, \texttt{m}_n u_n), \mathscr {P} \varphi \rangle ^2 \rightarrow \langle \sum _{\varsigma \ge 1} \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u), \varphi \rangle ^2 \quad \text {in } \mathscr {L}^1(\mathscr {I}) \ \widetilde{\mathbb {P}}\text {-a.s.}.$$

Similarly in the (51), with the help of Theorem 3.1, Assumption (A) and (50), we obtain that

$$\begin{aligned}&\langle \mathfrak {g}_\varsigma (x, \mathscr {B}_n), \varphi \rangle \rightarrow \langle \mathfrak {g}_\varsigma (x, \mathscr {B}), \varphi \rangle \quad \text {in } \mathscr {L}^1(\mathscr {I}) \ \widetilde{\mathbb {P}}\text {-a.s.}, \\&\left\langle \sum _{\varsigma \ge 1} \mathfrak {g}_\varsigma (x, \mathscr {B}_n), \varphi \right\rangle ^2 \rightarrow \sum _{\varsigma \ge 1} \langle \mathfrak {g}_\varsigma (x, \mathscr {B}), \varphi \rangle ^2 \quad \text {in } \mathscr {L}^1(\mathscr {I}) \ \widetilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

This completes the proof.

Estimates independent of $\eta$

Let $(\texttt{m}_\eta , u_\eta , \mathscr {B}_\eta )$ be the solutions to (14), and similarly we apply the test functions $\texttt{m}_\eta \psi$ and $\psi$ to (14), we obtain

$$\begin{aligned}&\int _\mathscr {Q} \texttt{m}_\eta (t) u_\eta (t) \cdot \psi dx + \int _0^t \int _\mathscr {Q} [\textrm{div}(\texttt{m}_\eta u_\eta \otimes u_\eta ) - (\nabla \times \mathscr {B}_\eta ) \times \mathscr {B}_\eta - \mu \Delta u_\eta ] \cdot \psi \, dxd\phi \nonumber \\&\quad = \int _0^t \int _\mathscr {Q} \big [(\mu + \lambda ) \Delta u - A \nabla \texttt{m}^{\gamma '} - \eta \nabla \texttt{m}^\beta +\frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) - \varepsilon \nabla u \cdot \texttt{m} \big ] \cdot \psi \, dxd\phi \nonumber \\&\qquad + \int _0^t \int _\mathscr {Q} \sum _{\varsigma \ge 1} \mathfrak {f}_\varsigma (x, \texttt{m}_\eta , \texttt{m}_\eta u_\eta ) \psi \, dxd\beta _\varsigma ^1, \nonumber \\&\int _\mathscr {Q}\mathscr {B}(t) \cdot \psi \, dx = \int _\mathscr {Q} \mathscr {B}_0 \cdot \psi \, dx + \int _0^t \int _\mathscr {Q} \big [(\nabla \times (u_\eta \times \mathscr {B}_\eta )) + \nu \Delta \mathscr {B}_\eta \big ] \cdot \psi \, dxd\phi \nonumber \\&\qquad + \int _0^t \int _\mathscr {Q} \sum _{\varsigma \ge 1} \mathfrak {g}_\varsigma (x, \mathscr {B}_\eta ) \cdot \psi \, dxd\beta _\varsigma ^2. \end{aligned}$$

(52)

First, we apply Lemma 2.5 to obtain

$$\begin{aligned}&\eta \texttt{m}^\beta _\eta \rightarrow 0 \quad \text {in} \quad \mathscr {L}^{\frac{\beta + \sigma }{\beta }}(\tilde{\Upsilon } \times \mathscr {I} \times \mathscr {Q}), \\&\texttt{m}^{\gamma '}_\eta \rightarrow \tilde{\texttt{m}^{\gamma '}} \quad \text {in} \quad \mathscr {L}^{\frac{\gamma + \sigma }{\gamma }}(\tilde{\Upsilon } \times \mathscr {I} \times \mathscr {Q}). \end{aligned}$$

In addition, we obtain that

$$\begin{aligned}&\texttt{m}_\eta \rightarrow \texttt{m} \quad \text {in} \quad C(\mathscr {I}; \mathscr {L}^{\gamma '}_w(\mathscr {Q})) \cap C(\mathscr {I}; H^{-1}(\mathscr {Q})) \quad \tilde{\mathbb {P}}\text {-a.s.}, \nonumber \\&u_\eta \rightarrow u \quad \text {in} \quad \mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q})) \quad \tilde{\mathbb {P}}\text {-a.s.}, \nonumber \\&\texttt{m}_\eta u_\eta \rightarrow \texttt{m} u \quad \text {in} \quad C(\mathscr {I}; \mathscr {L}^{\frac{2 \gamma '}{\gamma '+1}}(\mathscr {Q})) \quad \tilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

(53)

Since $\gamma '> \frac{3}{2}$, then $\frac{2\gamma '}{\gamma '+1}> \frac{6}{5}$[Proposition 4.6³⁹,. By the (53), we have

$$\texttt{m}_\eta u_\eta ^i u_\eta ^j \rightarrow \texttt{m} u^i u^j \quad \text {in} \quad D'(\mathscr {I} \times \mathscr {Q}) \quad \tilde{\mathbb {P}}\text {-a.s.}.$$

By using (43), we see that

$$\begin{aligned} \mathscr {B}_\eta \rightarrow \mathscr {B}\quad \text {in} \quad \mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q})) \quad \text {and} \quad \mathscr {B}_\eta \rightarrow \mathscr {B}\quad \text {in} \quad \mathscr {L}^2(\mathscr {I}; H^1(\mathscr {Q})) \quad \tilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

(54)

By using the (15) and [Proposition 4.6³⁹,, by Aubin-Lions lemma gives

$$\begin{aligned} \mathscr {B}_\eta \rightarrow \mathscr {B}\quad \text {in} \quad C(\mathscr {I}; H^{-1}(\mathscr {Q})) \quad \tilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

Also, we have

$$\begin{aligned} (\nabla \times \mathscr {B}_\eta ) \times \mathscr {B}_\eta \rightarrow (\nabla \times \mathscr {B}) \times \mathscr {B}\quad \text {in} \quad D'(\mathscr {I} \times \mathscr {Q}) \quad \tilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

From (53) and (54), we have

$$\begin{aligned} \nabla \times (u_\eta \times \mathscr {B}_\eta ) \rightarrow \nabla \times (u \times \mathscr {B}) \quad \text {in} \quad D'(\mathscr {I} \times \mathscr {Q}) \quad \tilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

The Aubin lemma and regularity results allow us to obtain subsequences such that $\eta \rightarrow 0$ for some functions $\texttt{m}, y$ and B

$$\begin{aligned}&\texttt{m}_\eta \rightarrow \texttt{m} \quad \text {strongly in} \quad \mathscr {L}^2(\mathscr {I}; W^{1,p}(\mathscr {Q})), \quad 3< p< \frac{6\gamma '}{\gamma '+3}, \\&\texttt{m}_\eta u_\eta \rightarrow y \quad \text {strongly in} \quad \mathscr {L}^2(\mathscr {I}; \mathscr {L}^q(\mathscr {Q})), \quad 1 \le q <3 , \\&\sqrt{\texttt{m}_\eta } \rightarrow \sqrt{\texttt{m}} \quad \text {strongly in} \quad \mathscr {L}^\infty (\mathscr {I}; \mathscr {L}^r(\mathscr {Q})), \quad 1 \le r \le 6. \end{aligned}$$

By using (46) and Fatou’s lemma yields

$$\int _\mathscr {Q} \liminf _{\eta \rightarrow 0} \frac{\left| \texttt{m}_\eta u_\eta \right| ^2}{\texttt{m}_\eta } dx < \infty .$$

Again, by using (46), there exists a subsequence such that

$$\sqrt{\texttt{m}_\eta } u_\eta \rightharpoonup g \quad \text {weakly}^* \ \text {in} \ \mathscr {L}^{\infty }(\mathscr {I}; \mathscr {L}^2(\mathscr {Q})),$$

for some function $g$. Hence, $\sqrt{\texttt{m}_\eta }$ converges strongly to $\sqrt{\texttt{m}}$ in $\mathscr {L}^2(\mathscr {I}; \mathscr {L}^\infty (\mathscr {Q}))$. Now, we pass the limit $\eta \rightarrow \infty$ in the (52) term by term to obtain that

$$\begin{aligned}&\texttt{m}^2_\eta u_\eta \rightarrow \texttt{m}^2 u \quad \text {strongly in} \quad \mathscr {L}^1(\mathscr {I}; \mathscr {L}^q(\mathscr {Q})),\quad q <3, \\&\texttt{m}_\eta u_\eta \otimes \nabla \texttt{m}_\eta \rightarrow \texttt{m} u \otimes \nabla \texttt{m} \quad \text {strongly in} \quad \mathscr {L}^1(\mathscr {I}; \mathscr {L}^{\frac{3}{2}}(\mathscr {Q})). \end{aligned}$$

This implies that

$$\texttt{m}_\eta u_\eta \otimes \texttt{m}_\eta u_\eta \rightarrow \texttt{m} u \otimes \texttt{m} u \quad \text {strongly in} \quad \mathscr {L}^1(\mathscr {I}; \mathscr {L}^{\frac{q}{2}}(\mathscr {Q})), \quad q < 3.$$

Moreover, we have

$$\begin{aligned}&\nabla \texttt{m}_\eta \rightarrow \nabla \texttt{m} \quad \text {strongly in} \quad \mathscr {L}^2(\mathscr {I}; \mathscr {L}^p(\mathscr {Q})), \quad p> 3, \\&\sqrt{\texttt{m}_\eta } \rightarrow \sqrt{\texttt{m}} \quad \text {strongly in} \quad \mathscr {L}^{\infty }(\mathscr {I}; \mathscr {L}^r(\mathscr {Q})) \quad \text {with} \quad r = \frac{2p}{p-2}, \\&\Delta \sqrt{\texttt{m}_\eta } \rightarrow \Delta \sqrt{\texttt{m}} \quad \text {strongly in} \quad \mathscr {L}^2(\mathscr {I}; \mathscr {L}^2(\mathscr {Q})). \end{aligned}$$

It holds that $r<6$, since we have $p>3$. From the above convergence, we conclude that

$$\Delta \sqrt{\texttt{m}_\eta } \sqrt{\texttt{m}_\eta } \nabla \texttt{m}_\eta \rightarrow \Delta \sqrt{\texttt{m}} \sqrt{\texttt{m}} \nabla \texttt{m}_\eta \quad \text {weakly in} \quad \mathscr {L}^1(\mathscr {I}; \mathscr {L}^1(\mathscr {Q})).$$

Here, we have to assume that $\gamma '> 3$ if $d = 3$ that allows to yield compactness of $(\texttt{m}_\eta )$ in $\mathfrak {W}^{1,p}(\mathscr {Q})$ with $p> 3$. Also, $\nabla (\texttt{m}_\eta u_\eta )$ weakly converges in $\mathscr {L}^2(\mathscr {I}$ $\mathscr {L}^{\frac{3}{2}}(\mathscr {Q}))$ and $\nabla \texttt{m}_\eta$ converge strongly in $\mathscr {L}^2(\mathscr {I};\mathscr {L}^3(\mathscr {Q}))$, we obtain $\nabla (\texttt{m}_\eta u_\eta ) \cdot \nabla u_\eta \rightarrow \nabla (\texttt{m} u) \cdot \nabla \texttt{m}$ weakly in $\mathscr {L}^1(\mathscr {I};\mathscr {L}^1(\mathscr {Q}))$.

Since

$$\begin{aligned}&\langle \mathfrak {f}_\varsigma (x, \texttt{m}_\eta , \texttt{m}_\eta u_\eta ), \varphi \rangle \rightarrow \langle \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u), \varphi \rangle \quad \text {in} \ \mathscr {L}^1(\mathscr {I}) \ \widetilde{\mathbb {P}}\text {-a.s.},\\&\langle \mathfrak {g}_\varsigma (x, \mathscr {B}_n), \varphi \rangle \rightarrow \langle \mathfrak {g}_\varsigma (x, B), \varphi \rangle \quad \text {in } \mathscr {L}^1(\mathscr {I}) \ \widetilde{\mathbb {P}}\text {-a.s.}. \end{aligned}$$

By the above proof, there exists a martingale solution $(\texttt{m}, u,\mathscr {B})$ to the system (14)–(16) with the initial condition $(\texttt{m}_0, y_0, \mathscr {B}_0)$ ensures the boundeness property. The proof is completed.

Numerical results

In this section, let us present a numerical simulation of the QMHD equation with Brownian motion.

Consider the following stochastic compressible QMHD model in the three-dimensional space:

$$\begin{aligned}&d \texttt{m} + \operatorname {div}(\texttt{m} u) dt = 0, \nonumber \\&d(\texttt{m} u) + \left[ \operatorname {div}(\texttt{m} u \otimes u) - (\mu + \lambda )\nabla \operatorname {div} u - \mu \Delta u + \nabla p- \frac{h^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{ \sqrt{\texttt{m}}} \right) \right] dt \nonumber \\&\quad = [(\nabla \times \mathscr {B}) \times \mathscr {B}]dt + \sum \mathfrak {f}_\varsigma (x, \texttt{m}, \texttt{m} u)d\beta _\varsigma ^1(t), \nonumber \\&d \mathscr {B}- \nu \Delta \mathscr {B}dt = [\nabla \times (u \times \mathscr {B})] dt + \sum \mathfrak {g}_\varsigma (x,\mathscr {B})d\beta _\varsigma ^2(t), \quad \operatorname {div}\mathscr {B}= 0. \end{aligned}$$

(55)

Let us consider the three-dimensional computational grid with $30 \times 30 \times 30$. Spatial resolution is about the axis (dx, dy, dz), and temporal parameters (dt, T) are set. The time step is $dt=0.01$. The magnetic field components $\mathscr {B}_x, \mathscr {B}_y, \mathscr {B}_z$ are defined as divergence-free conditions. The two viscosity coefficients ($\mu$ and $\lambda$), magnetic coefficients, quantum potential, and Brownian motion are defined as $\mu = 0.1, \lambda = 1, \nu = 0.1, h = 1$, and $\sigma = 0.05$.

Numerical Steps for Stochastic QMHD System:

• 1. Mesh and Time Setup:

  • Define spatial grid: $\Delta x = 1/N_x$, $\Delta y = 1/N_y$, $\Delta z = 1/N_z$.

  • Define time steps: $N_t = T/\Delta t$.

• 2. Initialize Fields:

  • Set initial density $\texttt{m}^0 = \texttt{m}_0(x,y,z)$.

  • Set initial velocity $u^0 = (u,v,w) = 0$.

  • Set initial magnetic field $\mathscr {B}^0 = \mathscr {B}_0(x,y,z)$ and enforce $\operatorname {div} \mathscr {B}^0 = 0$.

• 3. Compute Deterministic Operators:

  • Convective term $\operatorname {div}(\texttt{m} u \otimes u)$, viscous term $-(\mu +\lambda )\nabla \operatorname {div} u - \mu \Delta u$, pressure gradient $\nabla p$, quantum Bohm potential $\frac{\hbar ^2}{2} \texttt{m} \nabla \left( \frac{\Delta \sqrt{\texttt{m}}}{\sqrt{\texttt{m}}} \right)$, Lorentz force $(\nabla \times \mathscr {B}) \times \mathscr {B}$, magnetic diffusion $\nu \Delta \mathscr {B}$, and induction term $\nabla \times (u \times \mathscr {B})$.

• 4. Discretization of the Bohm Potential: In two spatial dimensions, the Bohm potential is defined by

  $$Q(\texttt{m}) = \frac{h ^2}{2}\, \texttt{m} \, \nabla \!\left( \frac{\Delta \sqrt{\texttt{m}}}{\sqrt{\texttt{m}}}\right) , \qquad \texttt{m} =\texttt{m}(x,y,t).$$

  The density $\texttt{m}$ is discretized on a uniform cartesian grid $(x_i,y_j) = (i\,\Delta x,\; j\,\Delta y),$ and all spatial derivatives are approximated using second-order central finite differences.

  • Step 1: Discretization of $\sqrt{\texttt{m}}$. At each time level $t^n$, we compute $(\sqrt{\texttt{m}})_{i,j}^n = \sqrt{\texttt{m}_{i,j}^n}.$

  • Step 2: Discretization of the Laplacian. The Laplacian of $\sqrt{\texttt{m}}$ is approximated by

    $$\begin{aligned} (\Delta \sqrt{\texttt{m}})_{i,j}^n&\approx \frac{\sqrt{\texttt{m}}_{i+1,j}^n - 2\sqrt{\texttt{m}}_{i,j}^n + \sqrt{\texttt{m}}_{i-1,j}^n}{(\Delta x)^2}+ \frac{\sqrt{\texttt{m}}_{i,j+1}^n - 2\sqrt{\texttt{m}}_{i,j}^n + \sqrt{\texttt{m}}_{i,j-1}^n}{(\Delta y)^2}. \end{aligned}$$

    Periodic boundary conditions are imposed in both spatial directions.

  • Step 3: Discretization of the Bohm potential. The discrete Bohm potential is then evaluated pointwise as

    $$Q_{i,j}^n = \frac{h ^2}{2}\, \texttt{m}_{i,j}^n \frac{(\Delta \sqrt{\texttt{m}})_{i,j}^n}{(\sqrt{\texttt{m}})_{i,j}^n}.$$

• 5. Stochastic Update for Velocity:

  • Generate Gaussian increments: $\Delta \beta _\varsigma ^n = \sigma \sqrt{\Delta t} \, \xi _\varsigma ^n$, $\xi _\varsigma ^n \sim \mathscr {N}(0,1)$.

  • Update velocity using Euler-Maruyama:

    $$u^{n+1} = u^n + \Delta t \, F(u^n, \texttt{m}^n, \mathscr {B}^n) + \sum _\varsigma f_\varsigma (x, \texttt{m}^n, \texttt{m}^n u^n) \, \Delta \beta _\varsigma ^n,$$

    where

    $$F(u^n, \texttt{m}^n, \mathscr {B}^n) = - \operatorname {div}(\texttt{m}^n u^n \otimes u^n) + (\mu + \lambda ) \nabla \operatorname {div} u^n + \mu \Delta u^n - \nabla p^n + \frac{\hbar ^2}{2} \texttt{m}^n \nabla \left( \frac{\Delta \sqrt{\texttt{m}^n}}{\sqrt{\texttt{m}^n}} \right) + (\nabla \times \mathscr {B}^n) \times \mathscr {B}^n.$$

• 6. Stochastic Update for Magnetic Field:

  • Generate Gaussian increments: $\Delta \alpha _\varsigma ^n = \sigma \sqrt{\Delta t} \, \eta _\varsigma ^n$, $\eta _\varsigma ^n \sim \mathscr {N}(0,1)$.

  • Update magnetic field:

    $$\mathscr {B}^{n+1} = \mathscr {B}^n + \Delta t (\nu \Delta \mathscr {B}^n + \nabla \times (u^n \times \mathscr {B}^n)) + \sum _\varsigma g_\varsigma (x, \mathscr {B}^n) \, \Delta \alpha _\varsigma ^n.$$

• 7. Update Density:

  $$\texttt{m}^{n+1} = \texttt{m}^n - \Delta t \, \operatorname {div} (\texttt{m}^n u^n).$$

• 8. Iterate in Time: Repeat steps 3-6 for $n = 0,1,\dots ,N_t-1$.

The deterministic spatial operators are discretized using second-order central finite differences, while the stochastic forcing terms are incorporated through an Euler-Maruyama time stepping scheme with Wiener increments.

Fig. 1

3D Density field distribution in QMHD system.

The three-dimensional Fig. 1 indicates the probability density of quantum particles, where the central yellow peak regions represent a higher probability of electrons. In addition, the central peak region, also interpreted as the influence of Bohm potential or pressure, causes the smooth transition of particles under the influence of stochastic forces. It also represents the energy density distribution, indicating a region of higher energy concentration due to stochastic and quantum effects.

Fig. 2

Contour plot of density in QMHD system.

The Fig. 2 represents the spatial distribution of plasma density $\texttt{m}(x,y)$ of the QMHD system given in the contour plot. A radially symmetric distribution is provided by the concentric rings, with the maximum density in the center (yellow region) and decreasing density outward (black regions).

Fig. 3

2D velocity field distribution in QMHD system.

The Figs. 3 and 4 show the direction of the velocity field and magnetic field in two-dimensional space. This illustrates the motion of plasma particles in the QMHD system, the Fig. 3 shows the velocity field’s magnitude and direction. The shorter arrows highlight the minimal velocities of motion, whereas the longer arrows point out the greatest velocities. The uniformity and direction of the arrows indicate consistent and laminar flow without stochastic turbulence. This implies that the plasma is in a stable state with well-defined. The velocity field in QMHD is influenced by magnetic, stochastic, and quantum forces. The presence of a magnetic field is indicated by the diagonal alignment of the arrows. The arrow pattern near the boundaries is aligned smoothly, indicating no-slip conditions, which satisfy the boundary conditions of the QMHD system (2).

Fig. 4

2D magnetic field distribution in QMHD system.

The Fig. 4 implies confined magnetic force, which helps to trap plasma particles effectively. The desirable feature in QMHD systems is to prevent energy loss. The arrows converge towards the center, indicating a magnetic null point where the magnetic field strength reduces significantly. Magnetic shear is indicated by the change in arrow direction across the plot. This shear helps to stabilize the plasma against specific instabilities and improve energy flow.

In the theoretical analysis, the stochastic QMHD system is treated by using the Faedo-Galerkin method, the Jakubowski-Skorokhod theorem, and the compactness method to rigorously establish the existence, uniqueness, and convergence of solutions. The Faedo-Galerkin method provides a finite-dimensional approximation for the infinite-dimensional system, ensuring that the sequence of approximate solutions satisfies the necessary priori estimates. The Jakubowski-Skorokhod theorem allowed the solution to pass to the limit in probability for the stochastic components, while the compactness method guaranteed convergence of the nonlinear and coupled terms. To implement these theoretical results numerically, a discrete scheme is constructed using finite differences for the spatial derivatives and the Euler-Maruyama method for the stochastic time integration. The deterministic operators - convective, viscous, pressure, quantum Bohm potential, Lorentz force, magnetic diffusion, and induction terms are discretized consistently with the theoretical operators used in the Faedo-Galerkin approximation. The stochastic forcing terms are incorporated as Wiener increments, reflecting the multiplicative Wiener noise, thus providing a direct numerical realization of the probabilistic elements treated in the Jakubowski-Skorokhod framework. This numerical scheme is allowed to simulate the system in three dimensions and observe the evolution of density, velocity, and magnetic fields over time. Specifically, the simulations produced the three-dimensional density field distribution, contour plots of density, and two-dimensional slices of velocity and magnetic field structures, which illustrate the interaction between stochastic fluctuations and quantum effects. The agreement of these numerical results with the theoretical predictions confirms the validity of the Faedo-Galerkin approximations and convergence analysis and provides a tangible visualization of the QMHD dynamics that are otherwise difficult to analyze analytically.

Conclusion

A theoretical approach to the compressible QMHD equation with Brownian motion has been established. The proposed model is reformulated to approximate the equation. With the help of the Faedo-Galerkin method, the Jakubowski-Skorokhod theorem, and the compactness method, the approximation solution has been derived. Moreover, the energy estimation is carried out for the sequence of solutions. Finally, we pass the limit $n \rightarrow \infty$ and $\eta \rightarrow 0$ in the approximation solution. Further, numerical results and graphical representations of density, velocity, and magnetic field are shown. The future work involves numerical simulation to capture the quantum effects or Bohm potential. The presented system will be investigated with the help of fractional differential equations and semigroup theory. In the future, the tightness property will be verified by testing the convergence of numerical solutions using simulations. For higher-dimensional QMHD systems, future work will focus on incorporating machine-learning methods into the analysis framework.