Quantum mechanics

Taylor–Couette flow

Rayleigh number

Stability (learning theory)

Perturbation (astronomy)

Computer science

Natural convection

Classical mechanics

Dissipation

Thermodynamics

Physics

Nonlinear system

Mathematics

Flow (mathematics)

Boussinesq approximation (buoyancy)

Mathematical analysis

Geometry

Couette flow

Buoyancy

Mechanics

Convection

Machine learning

Journal of Functional Analysis

This paper establishes the nonlinear stability of the Couette flow for the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations concerned here model buoyancy-driven fluids such as atmospheric and oceanographic flows. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non-self-adjoint operator y∂x−ν∂yy in the perturbation equation that makes the nonlinear stability possible. When the initial perturbation from the Couette flow (y,0) is no more than the viscosity to a suitable power (in the Sobolev space Hb with b>43), we prove that the solution of the 2D Boussinesq system with only vertical dissipation on T×R remains close to the Couette at the same order. A special consequence of this result is the stability of the Couette for the 2D Navier-Stokes equations with only vertical dissipation.

Stability of Couette flow for 2D Boussinesq system with vertical dissipation