A conditional regularity result for p-harmonic flows

TL;DR

This paper establishes an epsilon-regularity result for p-harmonic flows, showing that solutions with small mean oscillation are regular, using a sharp nonlinear Gagliardo-Nirenberg inequality.

Contribution

It introduces a new epsilon-regularity criterion for p-harmonic flows based on BMO smallness and employs a novel nonlinear Gagliardo-Nirenberg inequality.

Findings

Proved epsilon-regularity under BMO smallness condition.

Extended regularity results to a broad class of parabolic systems.

Utilized a sharp nonlinear Gagliardo-Nirenberg inequality in the analysis.

Abstract

We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$ u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u) $$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution $u(t,\cdot)$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivi\`ere and the last named author.