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

2024/08/10 by Cao, Wenping, Li, Yachun, Zhang, Deng
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2408.05450

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 L²_x, we construct infinitely many probabilistically strong and analitically weak solutions in the class L^r_ΩL_t^γW_x^s,p, where r>1 and (s, γ, 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.

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