Existence and Uniqueness of Weak Solutions to Viscous Primitive Equations for a Certain Class of Discontinuous Initial Data

The primitive equations are derived from the Boussinesq system of incompressible flow under the hydrostatic balance assumption, by taking the zero singular perturbation limit of the small aspect ratio. This singular perturbation limit of small aspect ratio can be rigorously justified, see, Azérad–Guillén [1] and Li–Titi [26]. The primitive equations play a fundamental role for weather prediction models, see, e.g., the books by Lewandowski [24], Majda [30], Pedlosky [32], Vallis [38], and Washington– Parkinson [39]. In this paper, we consider the following primitive equations (without considering the coupling to the temperature equation): ∂tv + (v · ∇H)v + w∂zv +∇Hp(x , t)−∆v + f0k × v = 0, (1.1) ∇H · v + ∂zw = 0, (1.2)