Global Regularity for 2D Boussinesq Temperature Patches with No Diffusion

Abstract

This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space R 2 . We prove that for initial patches with W 2,∞ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound ∇ 2 u, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the C 1+γ regularity result in [23], 0 < γ < 1, using the scale of Sobolev spaces for the velocity. Furthermore, taking advantage of the new cancellations, we go beyond to show the persistence of regularity for C 2+γ patches.

Global Regularity of 2D Density Patches for Inhomogeneous Navier–Stokes

“…Since (19) and (28) give u L 8 ≤ c ( u 0 L 2 ) u 0 H 1 , using (19), (32) and Young's inequality the terms I 32 and I 33 are bounded as I 31 . Therefore,…”

“…Without considering regularity in the tangential direction to the density patch for the initial velocity, the initial conditions in [30], [16] are at the level of u 0 ∈ B 1+ǫ 2,1 (ǫ > 0), u 0 ∈ B γ 2,1 , respectively. Indeed, as in [19], from the results of maximum regularity of the linear heat equation, we deem u 0 ∈ H γ+s is sharp at the scale of Sobolev spaces from this approach.…”

“…The quasilinear character of the coupling between density and velocity makes it harder to propagate the regularity of the velocity and hence that of the patch. This extra difficulty, compared to the same problem for Boussinesq system [19], is overcome by noticing that the characteristic function of a Lipschitz patch belongs to the multiplier space M(Ḃ −1+γ ∞,∞ ) (see section below). Hence we will prove that u ∈ L 1 (0, T ; C 1+γ ) and thus the propagation follows by the particle trajectories system…”

On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term

“…In the class of temperature-patch type solutions with no diffusion and viscosity in the whole space, there is a vast literature, see for example [10], [18] and references therein.…”

On the Yudovich's type solutions for the 2D Boussinesq system with thermal diffusivity

“…As for Boussinesq system, the vortex patch problem for inviscid Boussinesq equations has been discussed by Hassainia and Hmidi in [21]. Then Danchin and Zhang in [15], Gancedo and García-Juárez in [19] considered the temperature patch problem associate to the Boussinesq system with full Laplacian dissipation in velocity and no diffusion in temperature. Then for the stratified Euler system, which is system (2) with constant temperature diffusion, Hmidi and Zerguine studied the vortex patch problem in [25].…”