Global regularity and infinite Prandtl-number limit for 2D Boussinesq temperature patches in $W^{k,\infty}$ and $C^{k,γ}$: lower-order case

We study global regularity and the infinite Prandtl number limit for temperature patches of the two-dimensional nondiffusive Boussinesq system on $\mathbb R^2$, with velocity dissipation $Λ^{2α}$, $\frac12<α\leq1$. We establish global well-posedness for $θ_0\in L^1\cap L^\infty$ and divergence-free