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 $\alpha>1/3$. 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 $\partial D_+=\{ x_3=0\}$.
Categories: math.AP