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 $\theta \le \frac{1}{2}$ and $p\in C^{1,2\theta -1}(\Omega )$ for $\theta >\frac{1}{2}$. Moreover, when $\partial \Omega \in C^{3+}$, we prove that an almost double Hölder regularity $p\in C^{2\theta -}(\Omega )$ holds even for $\theta <\frac{1}{2}$. This extends and improves the recent result of Bardos and Titi obtained in the planar case to every dimension $d\ge 2$ 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.

Improved version after referee comments. Version accepted in Calc Var & PDEs

References in corpus (3)

• 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

Cited by in corpus (2)

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