The decay and stability of solutions for the 3D density-dependent incompressible Boussinesq system

TL;DR

This paper investigates the long-term decay and stability of solutions to the 3D density-dependent incompressible Boussinesq system, establishing decay rates and demonstrating stability under small initial perturbations.

Contribution

It provides the first decay rate results for solutions of the 3D density-dependent Boussinesq system and proves stability of global smooth solutions under small initial data perturbations.

Findings

Decay rates for solutions of the 3D Boussinesq system are established.

Small perturbations in initial data lead to globally stable solutions.

Solutions remain close to reference solutions over time.

Abstract

This paper deals with stability and the large-time decay to any given global smooth solutions of the 3D density-dependent incompressible Boussinesq system. The decay rate for solutions of the corresponding Cauchy problem is obtained in this work. With the aid of this decay rate, it is shown that a small perturbation of initial data $(\overline{a}_0,\overline{\theta}_0, \overline{u}_0)$ still generates a global smooth solution to the density-dependent Boussinesq system, and this solution keeps close to the reference solution.