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Uniqueness of martingale solutions for three-dimensional stochastic MHD

Establish the uniqueness of the martingale solution for the three-dimensional incompressible resistive stochastic magnetohydrodynamics equations with Gaussian (delta-correlated) forcing, i.e., for the coupled Navier–Stokes and induction equations in a three-dimensional domain under the incompressibility constraints and stochastic forcing as specified in the paper’s MHD model.

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Background

The paper reviews mathematical results for stochastic MHD and notes that existence and uniqueness are established in two dimensions. In three dimensions, the situation is said to be analogous to the stochastic Navier–Stokes equation, where existence of a martingale solution is known but uniqueness remains unresolved.

The authors highlight that this issue persists for the three-dimensional stochastic MHD system due to increased complexity, citing foundational works on stochastic equations and emphasizing that the question of uniqueness is still open.

References

Moreover, for two-dimensional stochastic MHD equations, existence and uniqueness of solutions have been established [Barbu2007, Chueshov2010]. With the three-dimensional case, the situation is, in all likelihood, the same as in the case of the stochastic Navier-Stokes equation (the existence of a martingale solution has been proven, but the question of its uniqueness is open), except that it is more complicated. See, e.g. and references therein.

— Two-Loop Turbulent Helical Magnetohydrodynamics: Large-Scale Dynamo and Energy Spectrum (2506.20578 - Hnatič et al., 25 Jun 2025) in Section 2 (Preliminaries and State of the Art)