Evolutionary, symmetric p-Laplacian. Interior regularity of time derivatives and its consequences

TL;DR

This paper establishes interior regularity of time derivatives for the evolutionary symmetric p-Laplacian, introducing a novel iteration technique in Nikolskii-Bochner spaces that could extend to other nonlinear PDEs.

Contribution

It develops a new local regularity method in Nikolskii-Bochner spaces for the symmetric p-Laplacian, overcoming growth mismatch issues and potentially applicable to full-gradient cases.

Findings

Proved interior regularity of time derivatives for the symmetric p-Laplacian.

Introduced a novel iteration technique in Nikolskii-Bochner spaces.

Provided auxiliary results on Nikolskii-Bochner spaces that may aid future research.

Abstract

We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, having the regularity result for time derivatives, we obtain respective regularity of the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems on their embeddings and interpolations, may be of independent interest.