Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$ initial data

Abstract

In this paper, we consider the initial-boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity. We establish the local, in time, well-posedness of strong solutions, for any initial data $(v_0,T_0)\in H^1$, by using the local, in space, type energy estimate. We also establish the global well-posedness of strong solutions for this system, with any initial data $(v_0,T_0)\in H^1\cap L^{\infty }$, such that ${\partial }_z{v}_0\in L^m$, for some $m\in (2,\infty )$, by using the logarithmic type anisotropic Sobolev inequality and a logarithmic type Gronwall inequality. This paper improves the previous results obtained in [Cao, C.; Li, J.; Titi, E.S.: Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl.Math., Vol. 69 (2016), 1492-1531.], where the initial data $(v_0,T_0)$ was assumed to have $H^2$ regularity.

References in corpus (5)

• Local and Global Well-posedness of Strong Solutions to the 3D Primitive Equations with Vertical Eddy Diffusivity
• Global well-posedness of the 2D Boussinesq equations with vertical dissipation
• 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
• Existence and uniqueness of weak solutions to viscous primitive equations for certain class of discontinuous initial data

Cited by in corpus (2)

• Global well-posedness of the 2D Boussinesq equations with vertical dissipation
• A Tropical Atmosphere Model with Moisture: Global Well-posedness and Relaxation Limit