Quantum mechanics

Heat transfer

Metallurgy

Materials science

Rayleigh number

Applied mathematics

Coupling (piping)

Degenerate energy levels

Stability (learning theory)

Computer science

Natural convection

Classical mechanics

Dissipation

Thermodynamics

Physics

Mathematics

Boussinesq approximation (buoyancy)

Mathematical analysis

Hydrostatic equilibrium

Partial differential equation

Buoyancy

Mechanics

Machine learning

Journal of Differential Equations

This paper focuses on a special two-dimensional (2D) Boussinesq system modeling buoyancy driven fluids. It governs the motion of the velocity and temperature perturbations near the hydrostatic balance. This is a partially dissipated system with the velocity involving only the vertical dissipation. We are able to establish the global stability and the large-time behavior of the solutions. In particular, our results reveal that the buoyancy force actually stabilizes the fluids through the coupling and interaction. Without the coupling, the 2D Navier-Stokes equation with only vertical dissipation is not known to be stable. Mathematically the coupling allows us to deduce that both the velocity and the temperature obey degenerate damped wave equations, which generates the stabilization effect.

Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation