REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM

doi:10.1007/s10473-021-0419-3

摘要/Abstract

摘要： We study the regularity of weak solutions to a class of second order parabolic system under only the assumption of continuous coefficients. We prove that the weak solution $u$ to such system is locally Hölder 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 the 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 a singular set of $u$.

关键词: Parabolic system, regularity, weak solution, Hausdorff dimension

Abstract: We study the regularity of weak solutions to a class of second order parabolic system under only the assumption of continuous coefficients. We prove that the weak solution $u$ to such system is locally Hölder 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 the 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 a singular set of $u$.

Key words: Parabolic system, regularity, weak solution, Hausdorff dimension

中图分类号:

35D30

周建丰, 谭忠. REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM[J]. 数学物理学报(英文版), 2021, 41(4): 1333-1365.

Jianfeng ZHOU, Zhong TAN. REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM[J]. Acta mathematica scientia,Series B, 2021, 41(4): 1333-1365.

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