Regularity of weak solutions to a certain class of parabolic system

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 $u$.