On Energy Conservation for the Hydrostatic Euler Equations: An Onsager Conjecture

TL;DR

This paper investigates an Onsager-type conjecture for hydrostatic Euler equations, establishing the regularity threshold for energy conservation and introducing new weak solution notions to handle anisotropic regularity issues.

Contribution

It extends Onsager's conjecture to hydrostatic Euler equations, identifying a higher regularity threshold for energy conservation and proposing new weak solution frameworks.

Findings

Energy is conserved if horizontal velocity is sufficiently regular with Hölder exponent ≥ 1/2.

The vertical velocity is less regular, affecting the anisotropic regularity considerations.

New notions of weak solutions are introduced to properly interpret the nonlinear terms.

Abstract

Onsager's conjecture, which relates the conservation of energy to the regularity of weak solutions of the Euler equations, was completely resolved in recent years. In this work, we pursue an analogue of Onsager's conjecture in the context of the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). In this case the relevant conserved quantity is the horizontal kinetic energy. We first consider the standard notion of weak solution which is commonly used in the literature. We show that if the horizontal velocity $(u,v)$ is sufficiently regular then the horizontal kinetic energy is conserved. Interestingly, the spatial H\"older regularity exponent which is sufficient for energy conservation in the context of the hydrostatic Euler equations is $\frac{1}{2}$ and hence larger than the corresponding regularity exponent for the Euler equations (which is $\frac{1}{3}$). This is due to the anisotropic regularity of the velocity field: Unlike the Euler equations, in the case of the hydrostatic Euler equations the vertical velocity $w$ is one degree spatially less regular with respect to the horizontal variables, compared to the horizontal velocity $(u,v)$. Since the standard notion of weak solution is not able to deal with this anisotropy properly, we introduce two new notions of weak solutions for which the vertical part of the nonlinearity is interpreted as a paraproduct. We finally prove several sufficient conditions for such weak solutions to conserve energy.