Full double Hölder regularity of the pressure in bounded domains

We consider Hölder continuous weak solutions $u\in C^{\gamma }(\mathrm{\Omega })$, $u\cdot n{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$, of the incompressible Euler equations on a bounded and simply connected domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$. If $\mathrm{\Omega }$ is of class $C^{2,1}$ then the corresponding pressure satisfies $p\in C_{\ast }^{2\gamma }(\mathrm{\Omega })$ in the case $\gamma \in (0,\frac{1}{2}]$, where $C_{\ast }^{2\gamma }$ is the Hölder-Zygmund space, which coincides with the usual Hölder space for $\gamma <\frac{1}{2}$. This result, together with our previous one in [11] covering the case $\gamma \in (\frac{1}{2},1)$, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding $C_{\ast }^{2\gamma }$ 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.