Interior Second Order Hölder Regularity for Stokes systems

2024/01/18 by Rong Dong, Dongsheng Li, Dong, Rong +3
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2401.09841

openalex publication_date 2024/01/18 · openalex created_date 2024/01/20 · openalex updated_date 2026/07/28

Abstract

Global second order Hölder regularity for Stokes systems can be obtained by global Schauder estimates, which are actually a priori estimates and were established by Solonnikov [20] and [23] with appropriate compatible conditions. This paper will investigate the corresponding interior regularity which unfortunately may fail in general from Serrin's counterexample (cf. [19]). However, we discover interior C^2,α regularity for velocity and interior C^1,α regularity for pressure in spatial variables, and furthermore, for curl of velocity, we find its gradient belongs to C^{α, \fracα2}, that is, possesses Hölder continuity in both space and time directions. The interesting phenomenon here is that no continuity in time variable is assumed for both the coefficients and the righthand side terms. The estimates for velocity and its curl are achieved pointwisely and the results are sharp indicated by a counterexample.

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