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.
Categories: math.AP