Full double H\"older regularity of the pressure in bounded domains

TL;DR

This paper establishes the full double H"older regularity of the pressure in bounded domains for weak solutions of the incompressible Euler equations, extending known results to the boundary case and improving regularity understanding.

Contribution

It proves the double regularity of pressure up to the boundary for weak solutions in bounded domains, combining interior estimates with boundary analysis techniques.

Findings

Pressure is in C^{2γ}_* for γ in (0,1/2], including the borderline case γ=1/2.

Full double regularity of pressure in bounded domains is achieved, extending previous interior results.

Boundary regularity is established using advanced coordinate and pseudodifferential techniques.

Abstract

We consider H\"older continuous weak solutions $u\in C^\gamma(\Omega)$, $u\cdot n|_{\partial \Omega}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\Omega\subset\mathbb{R}^d$. If $\Omega$ is of class $C^{2,1}$ then the corresponding pressure satisfies $p\in C^{2\gamma}_*(\Omega)$ in the case $\gamma\in (0,\frac{1}{2}]$, where $C^{2\gamma}_*$ is the H\"older-Zygmund space, which coincides with the usual H\"older space for $\gamma<\frac12$. This result, together with our previous one in [11] covering the case $\gamma\in(\frac12,1)$, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding $C^{2\gamma}_*$ estimate for the pressure on the whole space $\mathbb{R}^d$, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case $\gamma=\frac{1}{2}$. The boundary regularity features the use of local normal geodesic coordinates, pseudodifferential calculus and a fine Littlewood-Paley analysis of the modified equation in the new coordinate system. We also discuss the relation between different notions of weak solutions, a step which plays a major role in our approach.