Stochastic resistive Hall--MHD with current fluctuations: martingale weak solutions via a convergent structure-preserving finite element method

Abstract: We study the stochastic resistive Hall--magnetohydrodynamic (Hall--MHD) system on bounded convex polyhedral domains, subject to nonlinear perfectly conducting boundary conditions. The system is driven by multiplicative Gaussian forcing in the momentum equation together with structured curl-type noise in the induction equation arising from stochastic perturbations of the resistive and Hall parts in the generalised Ohm law. We develop a fully discrete, linearly implicit, structure-preserving mixed finite element method based on a compatible discrete de Rham complex, which preserves the magnetic Gauss law exactly. We prove subsequential convergence of the discrete solutions to a finite-energy weak martingale solution, thereby obtaining a constructive existence result for the stochastic Hall--MHD system with curl-type noise. Numerical experiments illustrate the stochastic dynamics, exact divergence preservation, and convergence of the method.

• The paper establishes subsequential convergence of a fully discrete compatible finite element scheme to martingale weak solutions for three-dimensional stochastic resistive Hall–MHD with current-type Gaussian noise.
• The method preserves the magnetic Gauss law exactly at every discrete time step and achieves uniform energy and dissipation moment bounds when Hall-noise intensity is sufficiently small relative to resistivity.
• Numerical tests confirm machine-precision divergence control and show empirical near-first-order temporal self-convergence, while uniqueness, arbitrary noise amplitudes, and rigorous convergence rates remain open.

Problem setting and motivation

This paper studies the incompressible stochastic resistive Hall–magnetohydrodynamic (Hall–MHD) system on a bounded convex polyhedral domain $\mathscr{D}\subset\mathbb{R}^3$ over a finite time interval $[0,T]$, subject to perfectly conducting boundary conditions. The governing system couples the Navier–Stokes equations for the velocity $\boldsymbol{u}$ with the induction equation for the magnetic field $\boldsymbol{B}$, closed by a generalised Ohm law

$$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$

where $\sigma>0$ is the resistivity and $\eta\ge 0$ the Hall coefficient. Randomness enters through two independent channels: multiplicative Gaussian forcing $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$ in the momentum equation, and structured $curl$-type noise in the induction equation,

$$-\sum_{i=1}^\infty curl\bigl(\sigma\boldsymbol{\chi}_i+\eta\,\boldsymbol{\chi}_i\times\boldsymbol{B}\bigr)\,\mathrm{d}\beta_i^B,$$

interpreted as stochastic perturbations of the resistive and Hall (electron-drift) parts of the Ohm law. The fields $[0,T]$0 are prescribed current-like spatial modes with $[0,T]$1. Because both deterministic and stochastic electromotive contributions enter the induction equation under $[0,T]$2, magnetic solenoidality is propagated formally from divergence-free initial data — a property the numerical method is designed to preserve exactly.

The Hall term is significant at length scales below the ion inertial length, relevant to magnetic reconnection and solar plasmas. Mathematically it is a strongly nonlinear term of the same differential order as magnetic diffusion, making Hall–MHD a quasilinear parabolic-dispersive system with geometric constraints.

Position within the literature

The mathematical literature on Hall–MHD is comparatively sparse: global weak solutions were established by Acheritogaray et al. (2011) and extended by Tan (2021), while local well-posedness results are due to Chae–Degond–Liu (2014) and Dai (2021). In the stochastic setting, Yamazaki (2017) obtained global martingale solutions for three- and two-and-a-half-dimensional Hall–MHD under Gaussian multiplicative noise, and Motyl (2023) treated $[0,T]$3 via stochastic compactness. The author states that the combination studied here — $[0,T]$4-type multiplicative noise together with nonlinear perfectly conducting boundary conditions — had not previously been analysed.

On the numerical side, compatible mixed finite element methods preserving $[0,T]$5 are well developed for deterministic MHD without the Hall term; rigorous analysis for deterministic Hall–MHD exists only for the stationary case (Laakmann–Hu–Farrell, 2023) and a recent non-stationary scheme (Goldys–Soenjaya–Tran). For stochastic MHD, the only prior finite element convergence result (Deugoue et al., 2025) uses a nonlinear scheme that does not preserve the magnetic Gauss law exactly. The paper accordingly claims to be the first to establish convergence of a fully discrete structure-preserving finite element method for stochastic MHD, and the first such result for stochastic Hall–MHD with current-type stochastic electromotive forcing.

The fully discrete structure-preserving scheme

The spatial discretisation combines the MINI element pair $[0,T]$6 for velocity–pressure with lowest-order Nédélec ($[0,T]$7), Raviart–Thomas ($[0,T]$8), and piecewise constant ($[0,T]$9) spaces for electric field, magnetic induction, and Lagrange multiplier variables. These spaces form an exact discrete de Rham complex commuting with the canonical interpolators, which is the structural basis for exact preservation of the magnetic Gauss law. A discretely solenoidal velocity space $\boldsymbol{u}$0 enforces $\boldsymbol{u}$1 weakly.

The time stepping is linearly implicit (implicit Euler in the dissipative terms, explicit treatment of convective and Lorentz/Hall couplings through lagged fields), with Brownian increments evaluated at the previous time level. Two auxiliary operators are central:

• Discrete curl: $\boldsymbol{u}$2, defined dually so that $\boldsymbol{u}$3 and $\boldsymbol{u}$4 commute with $\boldsymbol{u}$5.
• Maxwell reconstruction $\boldsymbol{u}$6: an $\boldsymbol{u}$7/$\boldsymbol{u}$8 elliptic projection used to evaluate the state-dependent Hall noise coefficient $\boldsymbol{u}$9. The reconstruction is shown to be stable in the Maxwell graph norm and asymptotically $\boldsymbol{B}$0-consistent, with defect bounded by a modulus $\boldsymbol{B}$1.

The scheme is well-posed by arguments analogous to the deterministic non-stationary case.

Stability and structure preservation

Two properties are established. First, since both $\boldsymbol{B}$2 and the projected noise coefficients lie in $\boldsymbol{B}$3, the magnetic update is an identity in $\boldsymbol{B}$4, whence

$\boldsymbol{B}$5

elementwise, so $\boldsymbol{B}$6 holds exactly for all $\boldsymbol{B}$7 given divergence-free initial data — up to solver tolerance, confirmed numerically.

Second, for every integer $\boldsymbol{B}$8 there exists a threshold $\boldsymbol{B}$9 such that if the Hall-noise smallness parameter

$$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$0

satisfies $$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$1, then uniform-in-$$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$2 moment bounds hold for all moments of the discrete energy, the jump sums, and the dissipation ($$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$3 accumulated over time). The proof combines a pathwise energy identity, conditional Itô isometries, higher-moment estimates for conditionally Gaussian increments, a discrete Burkholder–Davis–Gundy inequality applied to the martingale increment, and absorption of the shifted current-dissipation term enabled precisely by the smallness of $$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$4. This smallness assumption is a genuine restriction: without it, the fluctuating Hall contribution would feed energy into the induction equation faster than resistive dissipation removes it, and the estimates would close only under stronger hypotheses not explored here.

Convergence to a martingale solution

The convergence argument proceeds through the standard stochastic compactness pipeline, executed with care for the structure-preserving discretisation.

Time-translate and tightness estimates. Using Stokes projections of smooth solenoidal test functions and the commuting projection identities, the author derives expected time-translate bounds of order $$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$5 for the velocity in $$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$6 and for the magnetic field in $$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$7, $$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$8. Maximal block increment estimates of fourth order, combined with a covering argument, yield stochastic equicontinuity in these negative norms; compact containment follows from the $$\boldsymbol{E} = \sigma \boldsymbol{J} + \eta\, \boldsymbol{J}\times\boldsymbol{B} - \boldsymbol{u}\times\boldsymbol{B},$$9 bounds. Joint tightness of the full vector of interpolants then holds on a product path space combining strong $\sigma>0$0 compactness (via Simon's criterion, applied to reconstructed magnetic fields for the $\sigma>0$1 component) with weak and weak-star compactness of the gradient and current variables.

Skorokhod representation and identification. The Jakubowski–Skorokhod theorem transfers a subsequence to a new probability space with almost-sure convergence: strong in $\sigma>0$2 for all interpolated variables, continuous in the negative path spaces, weak in $\sigma>0$3 for the velocity, and weak-star in $\sigma>0$4 for $\sigma>0$5. Divergence constraints and the normal boundary condition pass to the limit directly; the current is identified distributionally as $\sigma>0$6, which via the Gaffney estimate upgrades the limit to $\sigma>0$7. All nonlinear deterministic terms — convection in skew-symmetrised form, Lorentz force, Hall term $\sigma>0$8, and transport $\sigma>0$9 — are identified using strong/weak pairing arguments.

Martingale identification. The discrete residuals are shown to be square-integrable martingales with respect to the discrete filtration; equality of joint laws transfers the martingale-test identities to the new space. Uniform fourth moments of the residuals give uniform integrability, and the covariance densities converge by the Lipschitz/growth assumptions on $\eta\ge 0$0 and the summability of $\eta\ge 0$1. The limits $\eta\ge 0$2, $\eta\ge 0$3 are therefore continuous square-integrable martingales with respect to the natural filtration of $\eta\ge 0$4, with predictable covariations

$\eta\ge 0$5

and analogously for the magnetic martingales with coefficients $\eta\ge 0$6; the cross-covariation vanishes by independence of the two Brownian families. The Brownian representation theorem for cylindrical martingales then yields the driving Brownian motions, completing the main theorem: every sequence $\eta\ge 0$7 admits a subsequence whose discrete laws converge to a weak martingale solution in the sense of the paper's Definition 2.4.

Because uniqueness of finite-energy martingale solutions is unknown in three dimensions, the convergence is necessarily subsequential, and no convergence rate is asserted — a point the author states explicitly rather than leaving implicit.

Numerical experiments

Experiments on the unit cube illustrate a magnetic island reconnection scenario with $\eta\ge 0$8, $\eta\ge 0$9, $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$0, a single momentum noise mode $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$1, and a single current-fluctuation mode of amplitude $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$2. Across 20 sample paths driven by common Brownian paths, the cellwise magnetic-divergence defect remains at machine precision throughout, empirically confirming the exact Gauss-law preservation. The discrete total energy exhibits a dominant dissipative trend under the chosen noise amplitudes.

Self-convergence studies against finer reference solutions (with matched Brownian paths across levels) show monotone error reduction. Temporal rates approach first order, e.g. observed rates of 1.52 for $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$3, 1.42 for $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$4, and 1.33 for $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$5 at the finest step tested. Spatial rates on very coarse meshes ($$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$6 to $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$7 per direction) are less uniform, ranging roughly between 0.3 and 1.3 depending on the variable and level. These figures are presented strictly as self-convergence indicators, not as proved rates.

Limitations and open questions

Several restrictions qualify the results. The smallness condition $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$8 on the Hall-noise intensity is essential to closing the moment estimates; whether stability and convergence hold for arbitrary Hall-noise amplitude is open. The convergence is subsequential only, reflecting the absence of any uniqueness theory for finite-energy martingale solutions of the three-dimensional stochastic Hall–MHD system; consequently no error rates are available, and the numerical rates reported are empirical only. The analysis is confined to bounded convex polyhedral domains, where the Gaffney inequality and Maxwell regularity estimates hold; extensions to nonconvex or unbounded domains are not addressed. Finally, the definition of weak solution omits the electric field as a primary unknown, recovering it only as an $$\sum_i \boldsymbol{g}_i(\boldsymbol{u},\boldsymbol{B})\,\mathrm{d}\beta_i^u$$9 object; whether stronger solution concepts (e.g., pathwise or analytically strong solutions) are attainable under this noise structure remains unanswered.

Conclusion

The paper delivers two coupled contributions: a constructive existence proof of weak martingale solutions for stochastic resistive Hall–MHD with $curl$0-type Gaussian current noise and perfectly conducting boundary conditions, and the first convergent fully discrete structure-preserving finite element method for this class. The compatible de Rham discretisation preserves the magnetic Gauss law exactly at the discrete level, and the Maxwell reconstruction operator supplies the regularity needed to control the state-dependent Hall noise. The price paid is a smallness assumption on the fluctuating Hall coefficient relative to the resistivity and a subsequential, rate-free convergence statement inherited from the lack of uniqueness. Within these constraints, the work establishes a rigorous bridge between structure-preserving numerical analysis and stochastic fluid-plasma modelling that was previously absent.