Interior Second Order H\"{o}lder Regularity for Stokes systems

Global second order H\"{o}lder 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 $C^{2,\alpha}$ regularity for velocity and interior $C^{1,\alpha}$ regularity for pressure in spatial variables, and furthermore, for curl of velocity, we find its gradient belongs to $C^{\alpha, \frac{\alpha}2}$, that is, possesses H\"{o}lder 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.