Higher regularity of Holder continuous solutions of parabolic equations with singular drift velocities

TL;DR

This paper proves that Holder continuous weak solutions to a certain nonlinear parabolic equation with singular drift velocities are actually classical solutions, using advanced functional analysis techniques without minimality constraints.

Contribution

It establishes higher regularity of solutions to a magnetohydrodynamics-inspired equation using Besov spaces and energy estimates, without minimal Holder exponent assumptions.

Findings

Holder solutions are classical solutions

No minimal Holder exponent required

Uses space-time Besov spaces and energy estimates

Abstract

Motivated by an equation arising in magnetohydrodynamics, we prove that Holder continuous weak solutions of a nonlinear parabolic equation with singular drift velocity are classical solutions. The result is proved using the space-time Besov spaces introduced by Chemin and Lerner, combined with energy estimates, without any minimality assumption on the Holder exponent of the weak solutions.