Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity II: close to H1 initial data

Abstract

In this paper, we consider the initial-boundary value problem to the three-dimensional primitive equations for the oceanic and atmospheric dynamics with only horizontal eddy viscosities in the horizontal momentum equations and only vertical diffusivity in the temperature equation in the domain Ω=M×(−h,h), with M=(0,1)×(0,1). Global well-posedness of strong solutions is established, for any initial data (v0,T0)∈H1(Ω)∩L∞(Ω) with (∂zv0,∇HT0)∈Lq(Ω) and v0∈L1z(B1q,2(M)), for some q∈(2,∞), by using delicate energy estimates and maximal regularity estimate in the anisotropic setting.