Persistence of regularity for the viscous Boussinesq equations with zero diffusivity

We address the global regularity for the 2D Boussinesq equations with positive viscosity and zero diffusivity. We prove that for data (u₀, ρ₀) in H^s × H^s-1, where 1 < s < 2, the persistence of regularity holds, i.e., the solution (u(t), ρ(t)) exists and belongs to H^s ×H^s-1 for all positive t. Given the existing results, this provides the persistence of regularity for all s ≥ 0. In addition, we address the H^s × H^s persistence and establish it for all s > 1.