English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study weak solutions of the incompressible Euler equations on$\mathbb{T}^2\times \mathbb{R}_+$; we use test functions that are divergencefree and have zero normal component, thereby obtaining a definition that doesnot involve the pressure. We prove energy conservation under the assumptionsthat $u\in L^3(0,T;L^3(\mathbb{T}^2\times \mathbb{R}_+))$, $$  \lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\mathbb{T}^2}\int^\infty_{x_3>|y|}|u(x+y)-u(x)|^3\mathrm{d} x\, \mathrm{d} t=0, $$ and an additional continuitycondition near the boundary: for some $\delta>0$ we require $u\inL^3(0,T;C^0(\mathbb{T}^2\times [0,\delta])))$. We note that all our conditionsare satisfied whenever $u(x,t)\in C^\alpha$, for some $\alpha>1/3$, withH\"older constant $C(x,t)\in L^3(\mathbb{T}^2\times\mathbb{R}^+\times(0,T))$.

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