Existence and non-uniqueness of probabilistically strong solutions to 3D stochastic magnetohydrodynamic equations

TL;DR

This paper demonstrates the existence of infinitely many probabilistically strong solutions to 3D stochastic MHD equations, showing non-uniqueness in certain regimes and analyzing the behavior as noise diminishes.

Contribution

It constructs multiple solutions in supercritical regimes, revealing non-uniqueness and extending understanding of stochastic MHD equations beyond previous limits.

Findings

Infinitely many solutions exist in supercritical regimes.

Non-uniqueness is sharp at one Ladyzhenskaya-Prodi-Serrin endpoint.

Solutions converge to deterministic solutions as noise tends to zero.

Abstract

We are concerned with the 3D stochastic magnetohydrodynamic (MHD) equations driven by additive noise on torus. For arbitrarily prescribed divergence-free initial data in $L^{2}_x$, we construct infinitely many probabilistically strong and analitically weak solutions in the class $L^{r}_{\Omega}L_{t}^{\gamma}W_{x}^{s,p}$, where $r>1$ and $(s, \gamma, p)$ lie in a supercritical regime with respect to the the Lady\v{z}henskaya-Prodi-Serrin (LPS) criteria. In particular, we get the non-uniqueness of probabilistically strong solutions, which is sharp at one LPS endpoint space. Our proof utilizes intermittent flows which are different from those of Navier-Stokes equations and derives the non-uniqueness even in the high viscous and resistive regime beyond the Lions exponent 5/4. Furthermore, we prove that as the noise intensity tends to zero, the accumulation points of stochastic MHD solutions contain all deterministic solutions to MHD solutions, which include the recently constructed solutions in [28, 29] to deterministic MHD systems.