A note for double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of Euler equations

TL;DR

This paper provides an elementary proof for the interior double Hölder regularity of the hydrodynamic pressure in weak solutions of the Euler equations within bounded domains, using standard elliptic PDE techniques.

Contribution

It introduces a new elementary approach to establish pressure regularity for Euler solutions, avoiding pseudodifferential calculus and relying on standard PDE methods.

Findings

Proves pressure regularity in $C^{0,2eta}_{ m int}$ for velocity in $C^{0,eta}$ with $0<eta<1/2$.

Develops a novel cutoff function technique based on boundary distance.

Simplifies previous complex proofs using standard elliptic PDE tools.

Abstract

We give an elementary proof for the interior double H\"{o}lder regularity of the hydrodynamic pressure for weak solutions of the Euler Equations in a bounded $C^2$-domain $\Omega \subset \mathbb{R}^d$; $d\geq 3$. That is, for velocity $u \in C^{0,\gamma}(\Omega;\mathbb{R}^d)$ with some $0<\gamma<1/2$, we show that the pressure $p \in C^{0,2\gamma}_{\rm int}(\Omega)$. This is motivated by the studies of turbulence and anomalous dissipation in mathematical hydrodynamics and, recently, has been established in [L. De Rosa, M. Latocca, and G. Stefani, Int. Math. Res. Not. 2024.3 (2024), 2511--2560] over $C^{2,1}$-domains by means of pseudodifferential calculus. Our approach involves only standard elliptic PDE techniques, and relies on a variant of the modified pressure introduced in [C. W. Bardos, D. W. Boutros, and E. S. Titi, H\"{o}lder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains, Arch. Rational Mech. Anal. 249 (2025), 28] and the potential estimates in [L. Silvestre, unpublished notes]. The key novel ingredient of our proof is the introduction of two cutoff functions whose localisation parameters are carefully chosen as a power of the distance to $\partial\Omega$.