H\"older regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains

TL;DR

This paper proves that for weak solutions of the 3D Euler equations in bounded domains, the pressure inherits Hölder regularity from the velocity, which is crucial for advancing the Onsager Conjecture proof.

Contribution

It introduces a new very weak boundary condition formulation for pressure and proves its necessity and derivation, aiding the Onsager Conjecture analysis.

Findings

Pressure inherits Hölder regularity from velocity in bounded domains.

Away from the boundary, pressure has twice the Hölder regularity.

New weak boundary condition formulation for pressure is necessary and rigorously derived.

Abstract

We consider the three-dimensional incompressible Euler equations on a bounded domain $\Omega$ with $C^4$ boundary. We prove that if the velocity field $u \in C^{0,\alpha} (\Omega)$ with $\alpha > 0$ (where we are omitting the time dependence), it follows that the corresponding pressure $p$ of a weak solution to the Euler equations belongs to the H\"older space $C^{0, \alpha} (\Omega)$. We also prove that away from the boundary $p$ has $C^{0,2\alpha}$ regularity. In order to prove these results we use a local parametrisation of the boundary and a very weak formulation of the boundary condition for the pressure of the weak solution, as was introduced in [C. Bardos and E.S. Titi, Philos. Trans. Royal Soc. A, 380 (2022), 20210073], which is different than the commonly used boundary condition for classical solutions of the Euler equations. Moreover, we provide an explicit example illustrating the necessity of this new very weak formulation of the boundary condition for the pressure. Furthermore, we also provide a rigorous derivation of this new formulation of the boundary condition for weak solutions of the Euler equations. This result is of importance for the proof of the first half of the Onsager Conjecture, the sufficient conditions for energy conservation of weak solutions to the three-dimensional incompressible Euler equations in bounded domains. In particular, the results in this paper remove the need for separate regularity assumptions on the pressure in the proof of the Onsager conjecture.