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

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 We also construct an example, which shows that H\"older continuity is no longer valid, unless the proposed condition of boundary data is fulfilled.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.