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Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain

Published online by Cambridge University Press:  03 June 2015

Jing Wang  and
Feng Xie
Jing Wang
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai 200234, People’s Republic of China, (matjwang@shnu.edu.cn)
Feng Xie
Affiliation:
Department of Mathematics and Ministry of Education–Lab of Scientific Computation, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China, (tzxief@sjtu.edu.cn)
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Abstract

In this paper, we study the zero dissipation limit of the initial boundary-value problem of the multi-dimensional Boussinesq equations with viscosity and heat conductivity. Such equations are used as models for the motion of multi-dimensional incompressible fluids in atmospheric and oceanographic turbulence. In particular, they describe the thermal convection of an incompressible flow, and constitute the relations between the velocity field, the pressure and the local temperature. Under the Navier slip boundary condition in the velocity field and the thermal isolation boundary condition for the temperature, we prove the existence of weak amplitude characteristic boundary layers. Then, by a standard energy method, we prove the L2 convergence of the solutions when both the viscosity and the heat conductivity coefficients tend to 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 611 - 637
Copyright
Copyright © Royal Society of Edinburgh 2015 

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