Regularity of weak solutions to a certain class of parabolic system

2019/07/15 by Zhong Tan, Tan, Zhong, Jianfeng Zhou +1
Computer Science · Mathematics · #35D30 #35K10 #35K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1907.06307

openalex publication_date 2019/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Hölder continuous with any exponent α∈(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.

Citations

• Parabolic Systems with Polynomial Growth and Regularity

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