Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains

TL;DR

This paper proves that weak solutions to the incompressible Euler equations in bounded domains with certain regularity conditions conserve energy, extending Onsager's conjecture to bounded domains with smooth boundaries.

Contribution

It establishes energy conservation for weak solutions in bounded domains under a Hölder continuity condition with exponent greater than one-third.

Findings

Weak solutions conserve energy if velocity is in L^3 with Hölder exponent > 1/3.

Extension of Onsager's conjecture to bounded domains with smooth boundary.

Energy conservation holds under specified regularity conditions.

Abstract

The goal of this note is to show that, also in a bounded domain $\Omega \subset \mathbb{R}^n$, with $\partial \Omega\in C^2$, any weak solution, $(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in $\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t$, with the impermeability boundary condition: $u\cdot \vec n =0$ on $\partial\Omega\times(0,T)$, is of constant energy on the interval $(0,T)$ provided the velocity field $u \in L^3((0,T); C^{0,\alpha}(\overline{\Omega}))$, with $\alpha>\frac13\,.$