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

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 (∂z v₀, ∇H T₀) ∈ Lq(Ω) and v₀ ∈ Lz¹(B¹q,2(M)), for some q ∈ (2,∞), by using delicate energy estimates and maximal regularity estimate in the anisotropic setting.