On the Continuity of Bounded Weak Solutions to Parabolic Equations and Systems with Quadratic Growth in Gradients

TL;DR

This paper proves the pointwise continuity of bounded weak solutions for certain scalar and coupled parabolic equations with quadratic gradient growth, using an elementary approach that avoids higher integrability assumptions.

Contribution

It introduces a simple method to establish regularity for scalar and coupled parabolic systems with quadratic gradient growth, without requiring higher $L^p$ integrability.

Findings

Bounded weak solutions are pointwise continuous.

Elementary approach simplifies regularity proofs.

Applicable to strongly coupled systems without higher integrability.

Abstract

We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and its applications to strongly coupled systems does not require higher $L^p$ integrability of derivatives.