Energy conservation for the Euler equations on mathbbT²\×\n \ℝ₊ for weak solutions defined without reference to the pressure

2018/06/01 by James C. Robinson, Robinson, James C., José L. Rodrigo +3 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1806.00290

openalex publication_date 2018/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study weak solutions of the incompressible Euler equations on\n mathbbT²\× \ℝ₊; we use test functions that are divergence\nfree and have zero normal component, thereby obtaining a definition that does\nnot involve the pressure. We prove energy conservation under the assumptions\nthat u\∈ L³(0,T;L³( mathbbT²\× \ℝ₊)), \n
lim_{|y|
to 0}
frac1|y|
int^T₀
int_
mathbbT²
int^
infty_x3gt;|y|\n|u(x+y)-u(x)|³
mathrmd x
,
mathrmd t=0, and an additional continuity\ncondition near the boundary: for some \δ>0 we require u\∈\nL³(0,T;C⁰( mathbbT²\× [0,\δ]))). We note that all our conditions\nare satisfied whenever u(x,t)\∈ C^\α, for some \α>1/3, with\nH "older constant C(x,t)\∈ L³( mathbbT²\×\ℝ⁺\×(0,T)).\n

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