Energy conservation for the Euler equations on ${\mathbb{T}}^2\times {\mathbb{R}}_{+}$ for weak solutions defined without reference to the pressure

Abstract

We study weak solutions of the incompressible Euler equations on ${\mathbb{T}}^2\times {\mathbb{R}}_{+}$; we use test functions that are divergence free and have zero normal component, thereby obtaining a definition that does not involve the pressure. We prove energy conservation under the assumptions that $u\in L^3(0,T;L^3({\mathbb{T}}^2\times {\mathbb{R}}_{+}))$, $$\underset{|y|\to 0}{\lim }\frac{1}{|y|}{\int }_0^T{\int }_{{\mathbb{T}}^2}{\int }_{x_3>|y|}^{\infty }|u(x+y)-u(x){|}^3\mathrm{d}x\mathrm{d}t=0,$$ and an additional continuity condition near the boundary: for some $\delta >0$ we require $u\in L^3(0,T;C^0({\mathbb{T}}^2\times [0,\delta ])))$. We note that all our conditions are satisfied whenever $u(x,t)\in C^{\alpha }$, for some $\alpha >1/3$, with Hölder constant $C(x,t)\in L^3({\mathbb{T}}^2\times {\mathbb{R}}^{+}\times (0,T))$.

21 pages