Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion

Abstract

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation. Global well-posedness of strong solution is established for any $H^2$ initial data. An $N$-dimensional logarithmic Sobolev embedding inequality, which bounds the $L^{\infty }$ norm in terms of the $L^q$ norms up to a logarithm of the $L^p$-norm, for $p>N$, of the first order derivatives, and a system version of the classic Gronwall inequality are exploited to establish the required a priori $H^2$ estimates for the global regularity.

References in corpus (1)

• Local and Global Well-posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity

Cited by in corpus (7)

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• A Tropical Atmosphere Model with Moisture: Global Well-posedness and Relaxation Limit
• Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations with Heat and Salinity Diffusion
• Primitive Equations with Horizontal Viscosity: The Initial Value and the Time-Periodic Problem for Physical Boundary Conditions
• Existence and uniqueness of weak solutions to viscous primitive equations for certain class of discontinuous initial data
• Global well-posedness of strong solutions to a tropical climate model