$C^{0,\alpha }$ boundary regularity for the pressure in weak solutions of the $2d$ Euler equations

The purpose of this note is to give a complete proof of a $C^{0,\alpha }$ 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.