*Submitted Paper*

**Inserted:** 16 jan 2023

**Last Updated:** 28 jan 2023

**Year:** 2023

**Abstract:**

We consider Hölder 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 $C^{2,\delta}$, for some $\delta>0$, 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 Hölder-Zygmund space, which coincides with the usual Hölder space for $\gamma<\frac12$. This result, together with our previous one 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.

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