On Double H\"older Regularity of the Hydrodynamic Pressure in Bounded Domains

TL;DR

This paper establishes new regularity results for the hydrodynamic pressure in bounded domains for incompressible fluids, extending previous planar results to higher dimensions and improving the regularity estimates.

Contribution

It proves double H"older regularity of the pressure in bounded domains for velocities with fractional regularity, extending prior planar results to all dimensions and relaxing boundary conditions.

Findings

Pressure regularity depends on velocity H"older exponent

Double H"older regularity achieved under certain boundary smoothness

Results extend to higher dimensions and non-zero divergence cases

Abstract

We prove that the hydrodynamic pressure $p$ associated to the velocity $u\in C^\theta(\Omega)$, $\theta\in(0,1)$, of an inviscid incompressible fluid in a bounded and simply connected domain $\Omega\subset \mathbb R^d$ with $C^{2+}$ boundary satisfies $p\in C^{\theta}(\Omega)$ for $\theta \leq \frac12$ and $p\in C^{1,2\theta-1}(\Omega)$ for $\theta>\frac12$. Moreover, when $\partial \Omega\in C^{3+}$, we prove that an almost double H\"older regularity $p\in C^{2\theta-}(\Omega)$ holds even for $\theta<\frac12$. This extends and improves the recent result of Bardos and Titi obtained in the planar case to every dimension $d\ge2$ and it also doubles the pressure regularity. Differently from Bardos and Titi, we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the $d$-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.