# Regularity of weak solutions to a certain class of parabolic system

## TL;DR

This paper proves that weak solutions to a certain class of second order parabolic systems are locally Hölder continuous outside a measure-zero singular set, and establishes fractional differentiability and the Hausdorff dimension of the singular set.

## Contribution

It introduces an $A$-caloric approximation method to establish regularity of weak solutions under minimal assumptions of continuous coefficients.

## Key findings

- Weak solutions are locally Hölder continuous outside a zero measure singular set.
- The regularity points form an open set with full measure.
- The singular set has finite Hausdorff dimension.

## Abstract

We study the regularity of weak solutions to a certain class of second order parabolic system under the only assumption of continuous coefficients. By using the $A-$caloric approximation argument, we claim that the weak solution $u$ to such system is locally H\"{o}lder continuous with any exponent $\alpha\in(0,1)$ outside a singular set with zero parabolic measure. In particular, we prove that the regularity point in $Q_T$ is an open set with full measure, and we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. Finally, we deduce the fractional time and fractional space differentiability of $D u$, and at this stage, we obtain the Hausdorff dimension of singular set of $u$.