Solvability for Stokes system in H\"older spaces in bounded domains and its applications

TL;DR

This paper establishes conditions under which solutions to the Stokes system in bounded domains are H"older continuous, highlighting the regularity requirements for boundary data and demonstrating applications to nonlinear fluid-related systems.

Contribution

It provides new boundary data conditions ensuring H"older continuity of Stokes solutions and applies these results to prove local well-posedness for certain nonlinear coupled systems.

Findings

H"older continuity depends on specific boundary data regularity

Constructed example shows necessity of boundary data conditions

Established local well-posedness for nonlinear fluid systems

Abstract

We consider Stokes system in bounded domains and we present conditions of given data, in particular, boundary data, which ensure H\"older continuity of solutions. For H\"older continuous solutions for the Stokes system the normal component of boundary data requires a bit more regular than boundary data of H\"older continuous solutions for the heat equation. We also construct an example, which shows that H\"older continuity is no longer valid, unless the proposed condition of boundary data is fulfilled. As an application, we consider a certain general types of nonlinear systems coupled to fluid equations and local well-posedness is established in H\"older spaces.