On the Attractor for the Semi-Dissipative Boussinesq Equations

TL;DR

This paper investigates the long-term dynamics of a semi-dissipative Boussinesq system, demonstrating the existence of a global attractor with properties akin to classical fluid models and containing invariant manifolds related to turbulence theories.

Contribution

It establishes the existence of a global attractor for a semi-dissipative Boussinesq system, linking it to turbulence theories and extending understanding of long-term behavior in such models.

Findings

Existence of a global attractor for the system

Attractor retains properties of Navier-Stokes attractors

Contains invariant manifolds related to turbulence theories

Abstract

In this article, we study the long time behavior of solutions of a variant of the Boussinesq system in which the equation for the velocity is parabolic while the equation for the temperature is hyperbolic. We prove that the system has a global attractor which retains some of the properties of the global attractors for the 2D and 3D Navier-Stokes equations. Moreover, this attractor contains infinitely many invariant manifolds in which several universal properties of the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.