Calculus of Variations and Geometric Measure Theory

L. De Rosa - M. Latocca - G. Stefani

On double Hölder regularity of the hydrodynamic pressure in bounded domains

created by stefani on 02 May 2022
modified on 20 Jan 2023


Published Paper

Inserted: 2 may 2022
Last Updated: 20 jan 2023

Journal: Calc. Var. Partial Differ. Equ.
Volume: 62
Year: 2023

ArXiv: 2205.00929 PDF


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.

Keywords: incompressible fluids, boundary regularity, hydrodynamic pressure, Schauder estimates