Global well-posedness for the 2D Boussinesq equations with a velocity damping term

Abstract

In this paper, we prove global well-posedness of smooth solutions to the twodimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state (0, x2). As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [1] (see P.3597).2010 Mathematics Subject Classification. 35Q35, 76B03.

Asymptotic behavior of 2D stably stratified fluids with a damping term in the velocity equation

“…The global in time well-posedness of system (4.1) has been established in [19] in the framework of homogeneous Sobolev spaces Ḣs ∩ Ḣ−1 with s ≥ 6 in the whole R 2 domain. The same type of results for non-homogeneous Sobolev spaces (in terms of the velocity variable) in a bounded domain (the periodic strip T × [−1, 1] with no-slip conditions) can be found in [6] and are based on a different strategy.…”

“…Starting from the results of [19], where the global in time well-posedness of solutions to system (4.1) in homogeneous Sobolev spaces is obtained, here we provide explicit decay rates of the smooth solutions. We first state the global in time existence result due to Wan [19].…”

“…In [6], global existence in H s , s > 14 and decay rates in H 4 of the solutions (b, u) to system (1.4) in the periodic strip with no-flux conditions on the horizontal boundaries are obtained. However, our true starting point is [19], where global existence of solutions (b, ω)…”

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

“…While this article was being written, a preprint by Wan [38] appeared, proving global existence for 2D damping Boussinesq with stratification in R 2 . However, there are some important differences between [38] and our paper. Indeed, we consider horizontal boundaries and periodicity in the x−varibles and we get certain asymptotic stability of solutions.…”

“…Indeed, we consider horizontal boundaries and periodicity in the x−varibles and we get certain asymptotic stability of solutions. In addition, we work at the velocity level instead of working with the vorticity, as it is done in [38], and the functional we use for our energy estimates is also different. These points make both proofs different.…”