Energy Conservation and Vanishing Viscosity Limit for the Primitive Equations

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 diffculties 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.