On some regularity results for the stationary Stokes system and the 2 – $D$ Euler equations

Abstract

We revisit "minimal assumptions" on the data which guarantee that solutions to the 2 –
$D$ evolution Euler equations in a bounded domain are classical. Classical means here that all the derivatives appearing in the equations and boundary conditions are continuous up to the boundary. Following a well known device, the above problem led us to consider this same regularity problem for the Poisson equation under homogeneous Dirichlet boundary conditions. At this point, one was naturally led to consider the extension of this last problem to more general linear elliptic boundary value problems, and also to try to extend the results to more general data spaces. Pursuing and developing results that remained unpublished about thirty years, we survey the route followed in the study of these problems and we consider new results and open problems. In particular, we extend some minimal assumption results for the stationary Stokes system, and for and for the planar, evolution, Euler equations, to larger data spaces.