On Double Hölder Regularity of the Hydrodynamic Pressure in Bounded Domains
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 $p\in C^{1,2\theta1}(\Omega)$ for $\theta>\frac12$. Moreover, when $\partial \Omega\in C^{3+}$, we prove that an almost double Hölder 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 boundaryfree 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.00929
 arXiv:
 arXiv:2205.00929
 Bibcode:
 2022arXiv220500929D
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 Improved version after referee comments. Version accepted in Calc Var &