Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains

Abstract

We address the long-time behavior of the 2D Boussinesq system, which consists of the incompressible Navier-Stokes equations driven by a non-diffusive density. We construct globally persistent solutions on a smooth bounded domain, when the initial data belongs to $(H^k\cap V)\times H^k$ for $k\in\mathbb{N}$ and $H^s\times H^s$ for $0<s<2$. The proofs use parabolic maximal regularity and specific compatibility conditions at the initial time. Additionally, we also deduce various asymptotic properties of the velocity and density in the long-time limit and present a necessary and sufficient condition for the convergence to a steady state.