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

We are concerned with the 3D stochastic magnetohydrodynamic (MHD) equations driven by additive noise on torus. For arbitrarily prescribed divergence-free initial data in Lx2, we construct infinitely many probabilistically strong and analytically weak solutions in the class LΩrLtγWxs,p, where r>1 and (s,γ,p) lie in a supercritical regime with respect to 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 [34, 35].