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

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 $\alpha >\frac{1}{3}.$

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