A conditional regularity result for p-harmonic flows

Abstract

We prove an $\epsilon$-regularity result for a wide class of parabolic systems $$u_t-\text{div}(|\nabla u{|}^{p-2}\nabla u)=B(u,\nabla u)$$ 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. Rivière and the last named author.

arXiv commentTo appear in NoDEA. Referee suggestions implemented