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 T2×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∈L3(0,T;L3(T2×R+)), lim|y|→01|y|∫0T∫T2∫x3>|y|∞|u(x+y)−u(x)|3dxdt=0, and an additional continuity condition near the boundary: for some δ>0 we require u∈L3(0,T;C0(T2×[0,δ])). We note that all our conditions are satisfied whenever u(x,t)∈Cα, for some α>1/3, with Holder constant C(x,t)∈L3(T2×R+×(0,T)).