C-0,C-alpha boundary regularity for the pressure in weak solutions of the 2d Euler equations

The purpose of this article is to give a complete proof of a C-0,C-alpha regularity result for the pressure for weak solutions of the two-dimensional 'incompressible Euler equations' when the fluid velocity enjoys the same type of regularity in a compact simply connected domain with C-2 boundary. To accomplish our result, we realize that it is compulsory to introduce a new weak formulation for the boundary condition of the pressure, which is consistent with, and equivalent to, that of classical solutions. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.