$\mathcal{C}^{\alpha}$-regularity for nonlinear non-diagonal parabolic systems

TL;DR

This paper extends the regularity theory for nonlinear parabolic systems, proving Hölder continuity of solutions under conditions similar to elliptic cases but adapted for time-dependent problems, for growth parameters p>d/2.

Contribution

It generalizes the known elliptic regularity results to the parabolic setting for systems with non-diagonal structure and p-growth, covering cases where p>d/2.

Findings

Established $^{lpha}$-regularity for weak solutions in space and time.

Extended classical regularity results to a broader class of parabolic systems.

Provided the first regularity result for systems far from the radial (Uhlenbeck) structure.

Abstract

In the elliptic theory for $p$-Laplacian-like problems, the H\"{o}lder continuity of solutions has been proven for problems arising as Euler--Lagrange equations of a convex potential with $p$-growth that additionally satisfies the splitting condition. In this article, we extend these results to the parabolic setting. We investigate nonlinear parabolic systems whose structure parallels the elliptic case but incorporates time dependence. Assuming suitable space-time regularity of $F$ and natural structural conditions analogous to the stationary theory, we establish $\mathcal{C}^{\alpha}$-regularity of weak solutions in space and time whenever the growth parameter $p>d/2$. This extends the classical result for parabolic systems, which is valid only for $p>d-2$. This is the only regularity result for systems that are far from the radial (Uhlenbeck) structure.