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

TL;DR

This paper establishes energy conservation for weak solutions of the incompressible Euler equations on a half-torus cross the positive real line, without involving pressure, under specific regularity and boundary conditions.

Contribution

It introduces a pressure-free weak formulation for Euler equations on a half-torus and proves energy conservation under new regularity and boundary assumptions.

Findings

Energy conservation holds under specified regularity conditions.

Conditions are satisfied by solutions with Hölder continuity of order greater than 1/3.

The approach avoids explicit pressure dependence in the weak formulation.

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 $u\in L^3(0,T;L^3(\mathbb{T}^2\times \mathbb{R}_+))$, $$ \lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\mathbb{T}^2}\int^\infty_{x_3>|y|} |u(x+y)-u(x)|^3\mathrm{d} x\, \mathrm{d} t=0, $$ and an additional continuity condition near the boundary: for some $\delta>0$ we require $u\in L^3(0,T;C^0(\mathbb{T}^2\times [0,\delta])))$. We note that all our conditions are satisfied whenever $u(x,t)\in C^\alpha$, for some $\alpha>1/3$, with H\"older constant $C(x,t)\in L^3(\mathbb{T}^2\times\mathbb{R}^+\times(0,T))$.