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

TL;DR

This paper proves that the pressure in weak solutions of 2D incompressible Euler equations is $C^{0,eta}$ regular up to the boundary, using a new boundary condition formulation for pressure.

Contribution

It provides a complete proof of boundary regularity for pressure in weak solutions, introducing a novel weak boundary condition formulation.

Findings

Pressure is $C^{0,eta}$ regular up to the boundary.

A new weak boundary condition formulation is established.

The result applies to solutions with velocity regularity in a $C^2$ domain.

Abstract

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.