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 and . 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^1–4. 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^7–10. 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^12–16. 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 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 -norm with . 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).

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 established under the condition that the adiabatic exponent and 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 -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:

2

Consider the initial conditions

3

The boundary conditions are

4

Let be a bounded domain with smooth boundary . The unknowns , , u and denote density, magnetic diffusion coefficient, the velocity field and the magnetic field, respectively. The two viscosity constants and satisfy . denotes the magnetic diffusivity of the magnetic field. Here scalar pressure p satisfies the condition for with the adiabatic index . The two sequences and where are stochastic external forces and independent one-dimensional R-valued Brownian motions. The Planck constant h is a positive constant. The term 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 , velocity u and magnetic field as,

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 , 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 and is estimated in Section “’Estimates independent of n” and “Estimates independent of ”. 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 be the complete probablity space with the normal filtration of sub fields of such that contains all -negligible subsets of . Let consists of weakly continuous functions and also which is the subspace of where Let be the collection of all Hilbert-Schmidt operators from K to Z. Let denote the Banach space of Lebesgue measurable, -valued functions that are integrable on , equipped with standard norm , where . Similarly, let be the Banach space of Lebesgue measurable, essentially bounded, -valued functions on , equipped with the norm . When , is a Hilbert space with the scalar product defined by

and is the inner product in . Let denotes the Sobolev space defined by

and also Hilbert space with the scalar product denoted by

Let denote the space of random functions defined on a probability space having its value in equipped with the norm:

where is a Banach space and . When , we denote

Denote

The well-defined martingales and are Ito’s integrals in and respectively. Since and do not converge on K, we define by with the norm where is a complete orthonormal basis of K. We see that is Hilbert-Schmidt and we have - a.s. Also, denote with two given Brownian motions and , on .

Definition 2.1

A martingale solution of the model (2)–(4) is a system has the following properties:

1. For any , and are progressively measurable;

2. , , , , such that , and , it holds that

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

   

   holds in for any with for all large enough.

Next, let us impose the following conditions on the stochastic forces :

• (A)
  The functions are progressively measurable, also continuous in respectively and

  
• (B)
  The initial condition are satisfy the following conditions:

  

  
• (C)
  By using assumption (A), Holder’s inequality for we have

  

Let be an orthonormal basis of that also serves as an orthogonal basis of . Consider the eigenfunctions associated with the Dirichlet problem

Let Here is a finite dimensional Hilbert space. Let be the projection. Also to be a linear projection. We see that,

Now, we have consider the deterministic system (1) for the estimation of and :

Lemma 2.2

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

5

Now, we deduce space regularity for , , and in the following lemma.

Lemma 2.3

³¹ For some which is independent of n and , the following uniform estimate holds for deterministic system (1):

6

7

8

9

Lemma 2.4

³¹ For , the following uniform estimates holds for deterministic system (1):

10

11

12

13

Lemma 2.5

³⁹ For and , then a constant such that

where independent of .

Approximation scheme and a prior estimates

Let be fixed and , consider the following approximation problem:

14

with the initial conditions

15

with the boundary conditions:

16

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

almost surely and let us define

By using assumption (B), we have for all

As we know that is dense in so, one can with

Now, let us introduce a family of linear operators for a function with

We know that the operator is invertible and on and

where is the set of bounded linear mapping from to and denotes the inverse of . Furthermore, satisfies the condition

we can deduce that the map

is well-defined and also satisfies that

17

for any belonging to the set

Let us consider the sequence of pairs satisfying the integral equations :

18

where . For any function , where

Now, the integral (18) is reformulated as a stochastic ordinary differential equation on

19

Then we write the above integral (19) as

20

where

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

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

  21

  By [ Lemma 2.1, Lemma 2.2 in⁴⁹, we define the mapping with

  1. is the unique classical solution of (21);
  2. for all where . Here, and denotes the positive upper and lower bound of the density respectively.
  3. for any ,

  where . We have on for construct the approximate solutions for (14)–(16) by means of (20). We get the approximate solutions in the form of following integral equations:

  

• Step: 2 Cut-off problem.

  Let be a smooth cut-off function for any such that

  

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

  22

  where . For a fixed and n and by using the fixed point argument, we have the integral (22) at on the Banach space where and . With a slight abuse of notation, .

Theorem 3.1

For and for all fixed n and , then a with the system (22) has a unique solution .

Proof

Set

has the norm where .

A map defined as

where

By using the Burkholder-Davis-Gundy inequality, Holder’s inequality and similar to⁵⁰, the properties of , we conclude that maps into itself for some . Now we claim that is a contraction on for some . From^47,50, Section 7], the deterministic terms of (22) is proved. Hence, we consider the stochastic terms

For , , we have

Then

Since is a contraction map and by using the Burkholder-Davis-Gundy inequality, assumption (A), (17) the term becomes,

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

We know that , we have

Appling mean value theorem and definition of also in account of sufficiently small such that ,

Therefore, is a contraction on , maps into itself. Hence, admits a unique fixed point . The cut-off function guarantees the Lipschitz continuity globally.

Uniform prior estimates

Now we extend to T. From Theorem 3.1, let be the solution to system (20) on . Define stopping time:

Let . Now, let us compute the energy estimates.

Proposition 3.2

For , any satisfying (20), we have

where with

Proof

Define the function . Note that

where is the identity matrix. By using [Lemma 2.1 in³⁹ apply Ito’s formula into the function with . Again in the view of Ito’s formula with . From system (14), we obtain that

23

where and . Let

Integrating (23) on for all gives

24

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

25

By Holder inequality, assumption (A), the Burkholder-Davis-Gundy and Young inequality for small , we obtain for term

26

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

27

Also,

28

Now, we generalize , then (26) and (28) becomes

and

The identity

yields that

29

Substituting (25)–(29) in (24), for small enough ,

In view of Gronwall’s inequality yields that

Hence for any we get the following energy estimates

30

Thus, for , we get the energy estimates for

This completes the proof of the proposition.

Lemma 3.3

For any fixed n,

Proof

From (30), we get that

Since dimension of is finite, is bounded. Then, we have

31

32

where is independent of . From Proposition 3.2, (31) and (32), we get that

Tightness property

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

Lemma 4.1

³⁹ Let be the solution of (14)–(16) on defined above. For , we have,

33

where the constant is independent of .

Lemma 4.2

Define the set

equipped with its Borel -algebra. Consider the map be the probability on . For any ,

Then the family is tight.

Proof

Now, the following steps are need to verify that the family of is tight:

• Step 1 the tightness of

  First, let us start by examining the tightness of . It is enough to prove that for the compact subset with . Choose a constant , we consider the set

  

  By using Arzela -Ascoli’s Theorem, we conclude that is relatively compact in . Also, is closed and compact subset in . Now, we have to prove We know that, by Chebyshev’s inequality , we have

  

  So, we choose to obtain .

• Step 2 the tightness of

  Now, we have to find such that . Consider a function space X equipped with the norm

  

  Choose a closed ball in X. By Aubin-Lions Lemma implies that is compact in . From (33) in Lemma 4.1, we get

  

  Here, we choose , we get . Then .

• Step 3 the tightness of

  Next, we have to show that such that . Choose closed ball in . Then is compact in . From (33) in Lemma 4.1, we get

  

  Choosing , we get . Then

• Step 4 the tightness of

  Now, we have to find such that . In order to prove, we define a function space , , with the norm

  

  The Brownian motion cannot be differentiated in time because we cannot estimate directly. From (14), we have

  34

  where is a projection onto . From (33), we have

  35
  Similarly, by using Lemma 4.1 and , [Lemma 2.4⁵¹,, we have

  36

  From Lemma 4.1, for

  37

  By using Kolmogorov continuity theorem, we have that the stochastic terms is -Holder continuous for every almost surely. When , we have and . Choose in which is a closed ball. By [Corollary B.2⁵², and⁵³ we know that is compact in . In addition, from (34)–(), by the Chebyshev inequality, we get

  38

  Let in (38), we get that

  

  Then .
  We employ the tightness criterion. Let be a collection of -valued random variables defined on such that and for any , there exist an integer k, and such that for all . Then the sequence of the induced measures are tight on . By Lemma 4.1 and Holder’s inequality, we get that

  

  

  By applying , concludes the tightness of .

• Step 5 the tightness of

  Next, we have to find the subset in such that Finally, choose a closed ball in . Hence, is compact in . From Lemma 4.1, we have

  

  Here, we choose , then we have . Then . From (14), it follows that

  39

  where

  

  For the term , one has

  

  By Lemma 4.1 and Holder’s inequality , we get

  40

  Similarly, we have

  41

  By the assumption (A), Burkholder-Davis-Gundy inequality and Holder’s inequality, the term becomes

  42

  Combining (39)–(42) and by using Aubin-Simon Lemma in⁵³, we get that is compact in . Then

  From [Corollary1.3⁵⁴, the tightness property followed by the distribution of the joint processes . 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 converges weakly to , where is a probability on , a probability space . Then there exist random variables with distribution , with values in such that

43

Estimates independent of n

In this section, we first compute the limit , being fixed in the sequence in order to get a solution of the system (14)–(16). The limit is carried out in Section “Estimates independent of ”.

Let with

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

44

45

and

46

where is a positive constant. Since embedes continuously into , the estimate (45) proves the bound only if ³⁸. From the Aubin lemma, we draw a conclusion and taking into account the regularity (7) and (10) for , the regularity (5) and (11) for , and the regularity (6) and (11) for , that there exist subsequences of , which are not relabeled, such that, for some functions and B as ,

By the estimates of Lemma 4.1 and the fact that satisfies (14), we use (43) such that there exists a function such that

47

Also, (43) implies that For , since and using the interpolation of and , we have (47). In addition to (47) and in view of Lemma 4.1, we have

Let us pass to the limit in the products and . By using Lemma 4.1, we see that

Since in and in we have

By⁴⁹, Lemma 2.4], we get . Then we have

Since and by Aubin-Lions Lemma, we obtain that

48

Since in -a.s. and by (48), we obtain that

Moreover, if we pass the limit of

satisfying for sufficiently smooth test functions, we get

By (43), we get

49

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

Also, we have

From (43),(49) and since the convergence of the sequences and

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

where is the Lebesgue measure in time. By using Assumption (A) and Lemma 4.1, we get

51

Thus, we obtain the boundedness of Since

we get that

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

This completes the proof.

Estimates independent of

Let be the solutions to (14), and similarly we apply the test functions and to (14), we obtain

52

First, we apply Lemma 2.5 to obtain

In addition, we obtain that

53

Since , then [Proposition 4.6³⁹,. By the (53), we have

By using (43), we see that

54

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

Also, we have

From (53) and (54), we have

The Aubin lemma and regularity results allow us to obtain subsequences such that for some functions and B

By using (46) and Fatou’s lemma yields

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

for some function . Hence, converges strongly to in . Now, we pass the limit in the (52) term by term to obtain that

This implies that

Moreover, we have

It holds that , since we have . From the above convergence, we conclude that

Here, we have to assume that if that allows to yield compactness of in with . Also, weakly converges in and converge strongly in , we obtain weakly in .

Since

By the above proof, there exists a martingale solution to the system (14)–(16) with the initial condition 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:

55

Let us consider the three-dimensional computational grid with . Spatial resolution is about the axis (dx, dy, dz), and temporal parameters (dt, T) are set. The time step is . The magnetic field components are defined as divergence-free conditions. The two viscosity coefficients ( and ), magnetic coefficients, quantum potential, and Brownian motion are defined as , and .

Numerical Steps for Stochastic QMHD System:

1. Mesh and Time Setup:

   • Define spatial grid: , , .
   • Define time steps: .
2. Initialize Fields:

   • Set initial density .
   • Set initial velocity .
   • Set initial magnetic field and enforce .
3. Compute Deterministic Operators:

   • Convective term , viscous term , pressure gradient , quantum Bohm potential , Lorentz force , magnetic diffusion , and induction term .
4. Discretization of the Bohm Potential: In two spatial dimensions, the Bohm potential is defined by

   

   The density is discretized on a uniform cartesian grid and all spatial derivatives are approximated using second-order central finite differences.

   • Step 1: Discretization of . At each time level , we compute
   • Step 2: Discretization of the Laplacian. The Laplacian of is approximated by

     

     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

     
5. Stochastic Update for Velocity:

   • Generate Gaussian increments: , .
   • Update velocity using Euler-Maruyama:

     

     where

     
6. Stochastic Update for Magnetic Field:

   • Generate Gaussian increments: , .
   • Update magnetic field:

     
7. Update Density:

   

8. Iterate in Time: Repeat steps 3-6 for .

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.

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. 1.

3D Density field distribution in QMHD system.

The Fig. 2 represents the spatial distribution of plasma density 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. 2.

Contour plot of density 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. 3.

2D velocity field distribution in QMHD system.

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 and 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.

Author contributions

Divyabala K: Writing-original draft; visualization; methodology; software; investigation; conceptualization. Durga N: Conceptualization; formal analysis; methodology; supervision; validation; writing-review and editing.

Funding

Open access funding provided by Vellore Institute of Technology. No funding.

Data availability

All data generated or analysed during this study are included in this published article and its supplementary information files.

Code availability

The MATLAB code used in this study is available at the following GitHub repository: https://github.com/divyabalak2023-create/stochastic-mhd-matlab.git.

Declarations

Competing interests

The authors declare no competing interests.

Ethical approval

The authors declare that they comply with ethical standards.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-43494-9.