Regularity theory for the dissipative solutions of the MHD equations

TL;DR

This paper extends partial regularity theory for weak solutions of the MHD equations by introducing dissipative solutions, which relax pressure hypotheses and achieve local Hölder regularity in space.

Contribution

It introduces dissipative solutions for MHD equations, weakening pressure assumptions and establishing spatial Hölder regularity for weak solutions.

Findings

Dissipative solutions generalize classical weak solutions.

Hölder regularity in space is achieved under weaker pressure conditions.

The theory broadens understanding of solution regularity in MHD equations.

Abstract

We study here a new generalization of Caffarelli, Kohn and Nirenberg's partial regularity theory for weak solutions of the MHD equations. Indeed, in this framework some hypotheses on the pressure P are usually asked (for example P $\in$ L q t L 1 x with q > 1) and then local H{\"o}lder regularity, in time and space variables, for weak solutions can be obtained over small neighborhoods. By introducing the notion of dissipative solutions, we weaken the hypothesis on the pressure (we will only assume that P $\in$ D) and we will obtain H{\"o}lder regularity in the space variable for weak solutions.