Full Double Hölder Regularity of the Pressure in Bounded Domains

Abstract

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 $C^{2,1}$ 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 Hölder–Zygmund space, which coincides with the usual Hölder space for $\gamma <\frac 12$. This result, together with our previous one in [ 11] covering the case $\gamma \in (\frac 12,1)$, 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 that plays a major role in our approach.