Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$ 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 $(v_0, T_0)\in H^1$, 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 $\partial_zv_0\in L^m$, 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.

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