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

Abstract

The goal of this note is to show that, 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 ${\alpha > \frac13.}$