Energy conservation for the Euler equations on T 2 × 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 $\mathit{u}\in {\mathit{L}}^3\mathrm{(}0\mathrm{,}\mathit{T};{\mathit{L}}^3\mathrm{(}{\mathbb{T}}^2\times {\mathbb{R}}_{+}\mathrm{)}\mathrm{)}$ , $$\begin{array}{c}\underset{|\mathit{y}|\to 0}{lim}{\displaystyle \frac{1}{|\mathit{y}|}}{{\displaystyle \int }}_0^{\mathit{T}}{{\displaystyle \int }}_{{\mathbb{T}}^2}{{\displaystyle \int }}_{{\mathit{x}}_3\mathrm{>}|\mathit{y}|}^{\infty }{|\mathit{u}\mathrm{(}\mathit{x}+\mathit{y}\mathrm{)}-\mathit{u}\mathrm{(}\mathit{x}\mathrm{)}|}^3\mathrm{d}\mathit{x}\mathrm{d}\mathit{t}=0\mathrm{,}\end{array}$$ and an additional continuity condition near the boundary: for some $\mathit{\delta }\mathrm{>}0$ we require $\mathit{u}\in {\mathit{L}}^3\mathrm{(}0\mathrm{,}\mathit{T};{\mathit{C}}^0\mathrm{(}{\mathbb{T}}^2\times [0\mathrm{,}\mathit{\delta }]\mathrm{)}\mathrm{)}$ . We note that all our conditions are satisfied whenever $\mathit{u}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\in {\mathit{C}}^{\mathit{\alpha }}$ , for some $\mathit{\alpha }\mathrm{>}1\mathrm{/}3$ , with Hölder constant $\mathit{C}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\in {\mathit{L}}^3\mathrm{(}{\mathbb{T}}^2\times {\mathbb{R}}^{+}\times \mathrm{(}0\mathrm{,}\mathit{T}\mathrm{)}\mathrm{)}$ .