$C^{0,α}$ 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 $C^{0,α}$ 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.

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