Full double Hölder regularity of the pressure in bounded domains

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 $p\in C_{\ast }^{2\gamma }(\Omega )$ 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.

Extended version published on IMRN. Section 2.1 and Section 6.1 have been added, some minor mistakes have been corrected after the referee reports

References in corpus (4)

• Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains
• Boundary regularity estimates in Hölder spaces with variable exponent
• $C^{0,\alpha }$ boundary regularity for the pressure in weak solutions of the $2d$ Euler equations
• On Double Hölder Regularity of the Hydrodynamic Pressure in Bounded Domains

Cited by in corpus (1)

• Hölder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains