Concerning the Existence of Classical Solutions to the Stokes System. On the Minimal Assumptions Problem

In this notes we consider the stationary Stokes system in a bounded, connected, three-dimensional smooth domain, with homogeneous Dirichlet boundary condition. Proofs also apply to the n-dimensional case, and to other boundary conditions, like Navier-slip ones. We say here that a solution is classical if all derivatives appearing in the equations are continuous up to the boundary. It is well known, for long time, that solutions of the Stokes system are classical if the external forces belong to the H?lder space \({C^{0, \lambda}(\bar{\Omega})}\) . It is also well known that, in general, solutions are not classical in the presence of continuous external forces. Hence, a very challenging problem is to find Banach spaces, strictly containing the H?lder spaces \({C^{0, \lambda}(\bar{\Omega})}\) such that solutions to the Stokes problem corresponding to forces in the above space are classical. We prove this result for external forces in a suitable functional space, denoted \({{\rm C}_*(\bar{\Omega})}\) , introduced in references Beirão da Veiga (On the solutions in the large of the two-dimensional flow of a non-viscous incompressible fluid, 1982) and Beirão da Veiga (J Differ Equ 54(3):373–389, 1984) in connection with the Euler equations.