Boundary layers associated with the 3-D Boussinesq system for Rayleigh–Bénard convection

Abstract

We investigate the boundary layer effects of the 3-D incompressible Boussinesq system for Rayleigh–Bénard convection with vanishing diffusivity limit. By adopting the multi-scale analysis and the asymptotic expansion methods of singular perturbation theory, we construct an exact approximating solution for the viscous and diffusive Boussinesq system with well-prepared initial data. In addition, we obtain the convergence result of the vanishing diffusivity limit.

COMMUNICATED BY:

• Ming Mei

Keywords:

• Boussinesq system
• Rayleigh–Bénard convection
• vanishing diffusivity limit
• boundary layers
• asymptotic expansion

2010 Mathematics Subject Classifications:

• 35Q35
• 76W05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant number 11771031, 11531010] and by Natural Science Fund of Qinghai Province [grant number 2017-ZJ-908].