Global regularity of temperature patches for the 3D non-diffusive Boussinesq system with large Prandtl number

Abstract

So far the global well-posedness of strong solutions for the 3D non-diffusive Boussinesq system with large initial data remains a remarkable open problem. In this paper, we solve this problem in the regime of large Prandtl number. More precisely, we prove the global existence and uniqueness of strong solution for this 3D Boussinesq system associated with initial data $(u_0,\theta_0)\in H^{\frac{1}{2}}(\mathbb{R}^3) \times (L^1\cap L^s(\mathbb{R}^3))$ with $s>3$, provided that the Prandtl number is sufficiently large (the threshold depends only on a scale-invariant norm of $(u_0,\theta_0)$); moreover, for the non-constant temperature patch initial data, we establish the global persistence of $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$ ($0<\gamma<1$) boundary regularity of the evolved temperature patch, with corresponding estimates uniform in the large Prandtl number regime. Furthermore, we rigorously justify the limit as the Prandtl number tends to infinity and show that the patch solution of the 3D Boussinesq system converges to the unique patch solution of the 3D Stokes-transport system, and that the patch boundary regularity in $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$ is preserved globally in time. In particular, our result for the 3D Stokes-transport system can be viewed as the 3D analogue of the main result in Grayer II [ARMA 2023] concerning 2D Stokes-transport system.