Energy conservation in the 3D Euler equations on ${\mathbb{T}}^2\times {\mathbb{R}}_{+}$

The aim of this paper is to prove energy conservation for the incompressible Euler equations in a domain with boundary. We work in the domain ${\mathbb{T}}^2\times {\mathbb{R}}_{+}$, where the boundary is both flat and has finite measure. However, first we study the equations on domains without boundary (the whole space ${\mathbb{R}}^3$, the torus ${\mathbb{T}}^3$, and the hybrid space ${\mathbb{T}}^2\times \mathbb{R}$). We make use of some of the arguments of Duchon & Robert ({\it Nonlinearity} {\bf 13} (2000) 249–255) to prove energy conservation under the assumption that $u\in L^3(0,T;L^3({\mathbb{R}}^3))$ and one of the two integral conditions \begin{equation*} \lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{$\mathbb{R}$^3} |u(x+y)-u(x)|^3,d x,d t=0 \end{equation*} or \begin{equation*} \int_0^T\int_{$\mathbb{R}$^3}\int_{$\mathbb{R}$^3}\frac{|u(x)-u(y)|^3}{|x-y|^{4+\delta}},d x,d y<\infty,\qquad\delta>0, \end{equation*} the second of which is equivalent to requiring $u\in L^3(0,T;W^{\alpha ,3}({\mathbb{R}}^3))$ for some $\alpha >1\mathrm{/}3$. We then use the first of these two conditions to prove energy conservation for a weak solution $u$ on $D_{+}:={\mathbb{T}}^2\times {\mathbb{R}}_{+}$: we extend $u$ a solution defined on the whole of ${\mathbb{T}}^2\times \mathbb{R}$ and then use the condition on this domain to prove energy conservation for a weak solution $u\in L^3(0,T;L^3(D_{+}))$ that satisfies \begin{equation*} \lim_{|y|\to 0} \frac{1}{|y|}\int^{T}{0}\iint{$\mathbb{T}$^2}\int^\infty_{|y|}|u(t,x+y)-u(t,x)|^3 ,d x_3 ,d x_1 ,d x_2 ,d t=0, \end{equation*} and certain continuity conditions near the boundary $\mathrm{\partial }{D}_{+}=\{x_3=0\}$.