Strong solutions for a class of stochastic thermo-magneto-hydrodynamic-type systems with multiplicative noise

We establish the existence and uniqueness of strong solutions, in both the PDE and probabilistic sense, for a broad class of nonlinear stochastic partial differential equations (SPDEs) on a bounded domain $\mathscr{O}\subset \mathbb{R}^d$ ($d=2,3$), driven by multiplicative Gaussian noise. The solutions are global in time for $d=2$. This theory simultaneously covers several physically relevant systems, including stochastic convective Brinkman–Forchheimer equations, stochastic magnetohydrodynamics (MHD), stochastic Bénard convection in porous media, stochastic convective dynamo system, stochastic thermo-magneto-micropolar fluids, and stochastic diffusive tropical climate model, for which previous results only provide analytically weak martingale or pathwise solutions. The proof relies on Galerkin approximation and compactness argument. Up to a suitable stopping time, we derive strong moment bounds and verify a Cauchy property for the approximate solutions, in the absence of any inherent cancellation structure. By applying a Gronwall-type lemma for stochastic processes, we establish the existence and uniqueness of maximal strong pathwise solutions, which are global in two spatial dimensions.

💡 Research Summary

This paper addresses the well‑posedness of a broad class of nonlinear stochastic partial differential equations (SPDEs) that model thermally coupled fluid–magnetic phenomena. The abstract formulation is

  dΦ/dt + A Φ + B(Φ,Φ) + R(Φ) = g(Φ)·Ẇ, Φ(0)=Φ₀,

where the state vector Φ may contain velocity, temperature, magnetic field, and possibly micro‑rotation variables. The domain 𝒪⊂ℝᵈ is bounded with d = 2 or 3, and the driving noise is a multiplicative Gaussian Wiener process.

Functional setting and assumptions
The linear operator A is self‑adjoint, positive, and has a compact inverse, giving rise to a Hilbert scale D(A^α). The energy space V = D(A^{1/2}) is continuously embedded in H = L²(𝒪) and its dual V′, forming the Gelfand triple V⊂H⊂V′. The bilinear form B: V×V→V′ is continuous, antisymmetric (⟨B(v₁,v₂),v₃⟩ = −⟨B(v₁,v₃),v₂⟩), and satisfies a family of Sobolev‑type estimates (2.2)–(2.5) that guarantee ‖B(v,v)‖_{D(A^{1/8})} ≤ C‖v‖²|Av|. The nonlinear operator R: V→H is decomposed as R = S+F with ⟨S(v),v⟩ ≥ 0 and polynomial growth; it fulfills (2.6)–(2.10), including a Lipschitz‑type bound in H. The stochastic coefficient g is progressively measurable and Lipschitz simultaneously in H, V and D(A) (condition (G)).

Physical models covered
By specifying A, B, R, and g appropriately, the abstract framework captures six important systems:

1. Stochastic convective Brinkman–Forchheimer equations (viscous flow in porous media with a power‑law drag).
2. Stochastic magnetohydrodynamics (MHD).
3. Stochastic Bénard convection (coupled velocity–temperature).
4. Stochastic convective dynamo equations.
5. Stochastic thermo‑magneto‑micropolar fluids (including micro‑rotation).
6. Stochastic diffusive tropical climate models.

All these models are considered on bounded domains with Dirichlet (no‑slip) boundary conditions; the paper also discusses Navier‑type and perfect‑slip conditions. The authors point out that previous literature only provided analytically weak martingale or pathwise weak solutions, and in some cases (e.g., Dirichlet Brinkman–Forchheimer) contained gaps due to incorrect commutation of the Leray projector with the Laplacian. This work rectifies those issues.

Main results

• Theorem 5.2: For d = 2 or 3, given Φ₀∈L²(Ω;V) and under the structural assumptions, there exists a unique maximal strong pathwise solution (Φ,τ*) such that Φ∈L²(Ω;C(