Strong solutions to the 3D primitive equations with only horizontal dissipation: near H¹ initial data

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 (v₀, T₀) H¹, 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 (v₀, T₀) H¹ L^, such that ᵦv₀ Lᵐ, for some m (2, ), 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 (v₀, T₀) was assumed to have H² regularity.