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

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 $L_x^2$, we construct infinitely many probabilistically strong and analitically weak solutions in the class $L_{\Omega }^r{L}_t^{\gamma }{W}_x^{s,p}$, where $r>1$ and $(s,\gamma ,p)$ lie in a supercritical regime with respect to the the Ladyžhenskaya-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.