Communications in Mathematical Sciences

Volume 17 (2019)

Number 8

Stability and decay rates for a variant of the 2D Boussinesq–Bénard system

Pages: 2325 – 2352

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n8.a11

Authors

Jiahong Wu (Department of Mathematics, Oklahoma State University, Stillwater, Ok., U.S.A.)

Xiaojing Xu (School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, China)

Ning Zhu (School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, China)

Abstract

This paper investigates the stability and large-time behavior of perturbations near the trivial solution to a variant of the 2D Boussinesq–Bénard system. This system does not involve thermal diffusion. Our research is partially motivated by a recent work [C.R. Doering, J. Wu, K. Zhao, and X. Zheng, Phys. D, 376/377:144–159, 2018] on the stability and large-time behavior of solutions near the hydrostatic balance concerning the 2D Boussinesq system. Due to the lack of thermal diffusion, these stability problems are difficult. The energy method and classical approaches are no longer effective in dealing with these partially dissipated systems. This paper presents a new approach that takes into account the special structure of the linearized system. The linearized parts of the vorticity equation and the temperature equation both obey a degenerate damped wave-type equation. By representing the nonlinear system in an integral form and carefully crafting the functional setting for the initial data and solution spaces, we are able to establish the long-term stability and global (in time) existence and uniqueness of smooth solutions. Simultaneously, we also obtain exact decay rates for various derivatives of the perturbations.

Keywords

Boussinesq–Bénard equations, global solution, large-time behavior, stability, velocity damping

2010 Mathematics Subject Classification

35Q35, 76D03, 76D05

Full Text (PDF format)

J. Wu was partially supported by NSF grant DMS 1624146, by the AT&T Foundation at Oklahoma State University (OSU) and by NSFC (No. 11471103, a grant awarded to Professor Baoquan Yuan). X. Xu was partially supported by NSFC (No. 11771045, No. 11871087), BNSF (No. 2112023) and by the Fundamental Research Funds for the Central Universities of China. N. Zhu was partially supported by NSFC (No. 11771045, No. 11771043).

Received 29 October 2018

Accepted 3 September 2019

Published 3 February 2020