Calculus of Variations and Geometric Measure Theory
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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 03 May 2022

[BibTeX]

Submitted Paper

Inserted: 2 may 2022
Last Updated: 3 may 2022

Year: 2022

ArXiv: 2205.00929 PDF

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 $\theta>\frac12$. This extends the recent result of Bardos-Titi obtained in the planar case to every dimension $d\ge2$ and it also doubles the pressure regularity for $\theta>\frac12$. Differently from the work of Bardos-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


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