Long-time behavior of subcritical 2D vorticity Boussinesq equations

Abstract

In this paper the subcritical Boussinesq system with fractional dissipation in T2 is considered. Our aim is to study the long-time behavior of solutions of Boussinesq system in its natural scale-invariant Sobolev space and prove the existence of a global attractor of optimal regularity. To this end we investigate the global well-posedness and global attractor for Boussinesq system in H2−α(T2)×H2−α(T2) via commutator estimates for nonlinear terms and a new energy estimate in Sobolev spaces to bootstrap the regularity, derived by means of nonlinear lower bounds on the fractional Laplacian. This estimate allows a sharp use of the subcritical nature of the L∞ bounds for this problem. Besides, we study the critical limit of the attractors and prove their stability and upper semicontinuity when α→1+.