Energy Conservation and Vanishing Viscosity Limit for the Primitive Equations

TL;DR

This paper proves energy conservation for weak solutions of the inviscid Primitive Equations with boundaries under Onsager-type conditions, addressing unique challenges posed by the system's structure and domain.

Contribution

It establishes energy conservation for PE with boundary conditions under Onsager assumptions and introduces new methods to handle domain-specific difficulties.

Findings

Energy conservation holds under Onsager-type conditions.

New techniques address boundary and corner challenges.

Provides conditions for no anomalous energy dissipation in vanishing viscosity limit.

Abstract

In this paper, we consider the problem of energy conservation for weak solutions of the inviscid Primitive Equations (PE) in a bounded domain. Based on the work [Bardos et al., Onsager's conjecture with physical boundaries and an application to the vanishing viscosity limit, Comm. Math. Phys., 2019, 291-310], we prove the energy conservation for PE with boundary condition under suitable Onsager-type assumptions. But due to the special structure of PE system and its domain, some new challenging difficulties arise: the lack of information about the vertical velocity, and existing corner points in the domain. We introduce some new ideas to overcome the above obstacles. As a byproduct, we give a sufficient condition for absence of anomalous energy dissipation in the vanishing viscosity limit.