Global regularity for the 2D Boussinesq equations with partial viscosity terms

Abstract

In this paper, we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the

Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

“…A similar result has been stated for (B κ,0 ) in the case s ≥ 3 by D. Chae in [6], then extended to rough data by T. Hmidi and S. Keraani in [13]. There, global well-posedness is shown whenever the initial velocity u 0 belongs to B 1+ 2 p p,1 and the initial temperature θ 0 is in L r for some (p, r) satisfying 2 < r ≤ p ≤ ∞ (plus a technical condition if p = r = ∞).…”

“…In addition, those smooth data belong to all the Sobolev spaces H s . Hence, applying Chae's result [6] provides us with a sequence of smooth global solutions (θ n , u n ) n∈N which belong to all the spaces C(R + ; H s ). From system (B κ,0 ) and standard product laws in Sobolev spaces, we deduce that (θ n , u n ) belongs to C 1 (R + ; H s ) for all s ∈ R, and thus also to…”

Dynamic stability of the three‐dimensional axisymmetric Navier‐Stokes equations with swirl

“…The two-dimensional Boussinesq equations are closely related to the three-dimensional axisymmetric Navier-Stokes equations with swirl (away from the symmetry axis). Recently Chae [5] and Hou and Li [14] proved independently the global existence of the two-dimensional viscous Boussinesq equations with viscosity entering only in the fluid equation, while the density equation remains inviscid. Recent studies by Constantin, Fefferman, and Majda [7] and Deng, Hou, and Yu [10,11] show that the local geometric regularity of the unit vorticity vector can play an important role in depleting vortex stretching dynamically.…”

Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity

“…Note that both systems (1.5)-(1.6) are much more degenerate than the partially viscous Boussinesq system with full Laplacian dissipation considered by Chae [4] and Hou-Li [8]. The deep observation of Hmidi-Keraani-Rousset is to utilize a maximumprinciple type structure hidden in (1.5) (resp.…”

“…For the partially viscous Boussinesq system (1.2) (with constant viscosity), that is, ν > 0 is a positive constant, κ = 0; or κ > 0 is a positive constant, ν = 0, Chae [4] and Hou-Li [8] independently settled global regularity for large initial data. In both works, a key observation is the use of the following Brezis-Wainger inequality [2] (to control ∇θ…”