Energy conservation of weak solutions for the incompressible Euler equations via vorticity

TL;DR

This paper investigates how controlling vorticity in specific L^p spaces ensures energy conservation in incompressible Euler equations, extending previous results to nonhomogeneous flows and generalizing Onsager's critical spaces.

Contribution

It provides a new criterion for energy conservation based on vorticity control in L^{3} spaces for both homogeneous and nonhomogeneous flows, addressing open problems in the field.

Findings

Vorticity in L^{3}(0,T;L^{3n/(n+2)}(Ω)) ensures energy conservation for homogeneous flows.

Energy is conserved for nonhomogeneous flows if vorticity and density gradient conditions are met.

The results generalize previous criteria and affirmatively answer a problem posed by Chen and Yu.

Abstract

Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the $L^p$ control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion $\omega=\text{curl}u\in L^{3}(0,T;L^{\frac{3n}{n+2}}(\Omega))\,(n=2,3)$, which generalizes the corresponding result in [8] and can be viewed as in Onsager critical spatio-temporal spaces. Regarding the nonhomogeneous flow, it is shown that the energy is conserved as long as the vorticity lies in the same space as before and $\nabla\sqrt{\rho}$ belongs to $L^{\infty}(0,T;L^{n}(\mathbb{T}^{n}))\,(n=2,3)$, which gives an affirmative answer to a problem proposed by Chen and Yu in [5].