The stochastic MHD equations driven by pure jump noise in $L^p$ spaces

TL;DR

This paper studies the existence and uniqueness of solutions to stochastic magnetohydrodynamic equations driven by jump noise in various domains, including less regular initial data, with global solutions in 2D.

Contribution

It establishes local existence and uniqueness of mild solutions with less regular initial data and proves global solutions in 2D for stochastic MHD equations driven by jump noise.

Findings

Local existence and uniqueness of solutions in $L^q$ spaces.

Global existence of solutions in 2D.

Handling less regular initial data, including marginal cases.

Abstract

We consider the stochastic incompressible magnetohydrodynamic equations driven by additive jump noises on either the whole space $\mathbb{R}^d$, $d=2,3$ or a smooth bounded domain $D$ in $\mathbb{R}^d$. We establish the local existence and uniqueness of a mild solution in the space $L^q(0,T;\mathbb{L}^{p\otimes}_{\sigma}(D))$ allowing for initial data with less regularity, including the marginal case $u_0\in \mathbb{L}^{d\otimes}_{\sigma}(D)$. In the two-dimensional case, we also prove the global existence of mild solutions.