Energy conservation for the Euler equations on T2×R+ for weak solutions defined without reference to the pressure

Article type: Research Article

Authors: Robinson, James C.^a | Rodrigo, José L.^a | Skipper, Jack W.D.^{b; *}

Affiliations: [a] Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK | [b] Institute of Applied Mathematics, Leibniz University Hannover, Welfengarten 1, 30167 Hannover, Germany

Correspondence: [*] Corresponding author. E-mail: [email protected].

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 Hölder constant C(x,t)∈L3(T2×R+×(0,T)).

Keywords: Onsager’s conjecture, Euler equations, energy conservation

DOI: 10.3233/ASY-181482

Journal: Asymptotic Analysis, vol. 110, no. 3-4, pp. 185-202, 2018