Existence and regularity results for the primitive equations in two space dimensions

Abstract

Abstract. Our aim in this article is to present some existence, uniqueness and regularity results for the Primitive Equations of the ocean in space dimension two with periodic boundary conditions. We prove the existence of weak solutions for the PEs, the existence and uniqueness of strong solutions and the existence of more regular solutions, up to C ∞ regularity.1. Introduction. The objective of this article is to derive various results of existence and regularity of solutions for the Primitive Equations of the ocean (PEs) in two space dimensions. These results, besides their intrinsic interest, are needed in [9] which is another motivation of this work.We consider the PEs in their nondimensional form (see Section 5) :

Pathwise Solutions of the 2-D Stochastic Primitive Equations

“…Subsequent work of [27] developed alternative proofs, which allow for the treatment of physically relevant boundary conditions. For the two dimensional deterministic setting we mention [38], [9] where both the cases of weak and strong solutions are considered. Despite these breakthroughs in the 3-D system, the 2-D primitive equations seem to be significantly more difficult mathematically than the 2-D Navier-Stokes equations.…”

Global existence of weak solutions to 3D compressible primitive equations with degenerate viscosity

“…The PEs have attracted many attentions in fluid mechanics and applied mathematics such as [14,23,15,25,8,28,11], due to its physical importance,complexity, rich phenomena and mathematical challenges. Mathematical arguments of incompressible primitive equations were studied from 1990s.…”

“…In [28] Temam and Ziane considered the local existence of strong solutions for the PEs of the atmosphere, the ocean and the coupled atmosphere-ocean. Petcu et al [23] considered some regularity results for the two dimensional PEs with periodical boundary conditions. Cao and Titi [8] proved the global existence and uniqueness of strong solutions to the three-dimensional viscous incompressible primitive equations of large scale ocean and atmosphere dynamics.…”

Pullback Attractor for Nonautonomous Primitive Equations of Large-Scale Ocean and Atmosphere Dynamics

“…In recent years, the primitive equations of the atmosphere, the ocean, and the coupled atmosphere-ocean have been extensively studied from the mathematical point of view (see [14][15][16][19][20][21][22][23][24][25][26], etc.). The mathematical framework of the primitive equations of the ocean was formulated, and the existence of weak solutions was proved by Lions et al in [15].…”