Stochastic magnetohydrodynamics system: cross and magnetic helicity in ideal case; non-uniqueness up to Lions' exponents from prescribed initial data

TL;DR

This paper investigates the behavior of 3D magnetohydrodynamics systems under stochastic forcing, demonstrating invariance properties in the ideal case and non-uniqueness of solutions with prescribed initial data using convex integration techniques.

Contribution

It introduces new convex integration constructions for stochastic MHD, showing invariance of helicities in ideal cases and non-uniqueness up to Lions' exponents.

Findings

Invariance of cross helicity in ideal case

Construction of solutions with energy and helicities more than doubled

Non-uniqueness of solutions starting from prescribed initial data

Abstract

We consider the three-dimensional magnetohydrodynamics system forced by random noise. First, for smooth solutions in the ideal case, the cross helicity remains invariant while the magnetic helicity precisely equals the initial magnetic helicity added by a linear temporal growth and multiplied by an exponential temporal growth respectively in the additive and the linear multiplicative case. We employ the technique of convex integration to construct an analytically weak and probabilistically strong solution such that, with positive probability, all of the total energy, cross helicity, and magnetic helicity more than double from initial time. Second, we consider the three-dimensional magnetohydrodynamics system forced by additive noise and diffused up to the Lions' exponent and employ convex integration with temporal intermittency to prove non-uniqueness of solutions starting from prescribed initial data.