REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM

Jianfeng ZHOU; Zhong TAN

doi:10.1007/s10473-021-0419-3

Acta mathematica scientia, Series B >

2021 , Vol. 41 >Issue 4: 1333 - 1365

DOI: https://doi.org/10.1007/s10473-021-0419-3

Articles

REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM

Jianfeng ZHOU ,
Zhong TAN

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1. School of Mathematical Sciences, Peking University, Beijing 100871, China;
2. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received date: 2020-04-15

Online published: 2021-09-01

Supported by

The first author is partially supported by the Postdoctoral Science Foundation of China (2019TQ0006); the second author is partially supported by the National Natural Science Foundation of China (11726023, 11531010).

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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

Cite this article

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 . DOI: 10.1007/s10473-021-0419-3

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