TY - JOUR

T1 - Stability Near Hydrostatic Equilibrium to the 2D Boussinesq Equations Without Thermal Diffusion

AU - Tao, Lizheng

AU - Wu, Jiahong

AU - Zhao, Kun

AU - Zheng, Xiaoming

N1 - Funding Information:
The authors thank the referee for the insightful comments that have helped significantly improve the manuscript. Support for this work came in part from NSF Award DMS-1614246 (JW), a Simons Foundation Collaboration Grant for Mathematicians 413028 (KZ), and from Louisiana Board of Regents Research Competitiveness Subprogram Award LEQSF(201518)-RD-A-24 (KZ). KZ also gratefully acknowledges start up funding from the Department of Mathematics at Tulane University.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - This paper furthers our studies on the stability problem for perturbations near hydrostatic equilibrium of the 2D Boussinesq equations without thermal diffusion and solves some of the problems left open in Doering et al. (Physica D 376(377):144–159, 2018). We focus on the periodic domain to avoid the complications due to the boundary. We present several results at two levels: the linear stability and the nonlinear stability levels. Our linear stability results state that the velocity field u associated with any initial perturbation converges uniformly to 0 and the temperature θ converges to an explicit function depending only on y as t tends to infinity. In addition, we obtain an explicit algebraic convergence rate for the velocity field in the L2-sense. Our nonlinear stability results state that any initial velocity small in L2 and any initial temperature small in L2 lead to a stable solution of the full nonlinear perturbation equations in large time. Furthermore, we show that the temperature is eventually stratified and converges to a function depending only on y if we know it admits a certain uniform-in-time bound. An explicit decay rate for the velocity in L2 is also ensured if we make assumption on the high-order norms of u and θ.

AB - This paper furthers our studies on the stability problem for perturbations near hydrostatic equilibrium of the 2D Boussinesq equations without thermal diffusion and solves some of the problems left open in Doering et al. (Physica D 376(377):144–159, 2018). We focus on the periodic domain to avoid the complications due to the boundary. We present several results at two levels: the linear stability and the nonlinear stability levels. Our linear stability results state that the velocity field u associated with any initial perturbation converges uniformly to 0 and the temperature θ converges to an explicit function depending only on y as t tends to infinity. In addition, we obtain an explicit algebraic convergence rate for the velocity field in the L2-sense. Our nonlinear stability results state that any initial velocity small in L2 and any initial temperature small in L2 lead to a stable solution of the full nonlinear perturbation equations in large time. Furthermore, we show that the temperature is eventually stratified and converges to a function depending only on y if we know it admits a certain uniform-in-time bound. An explicit decay rate for the velocity in L2 is also ensured if we make assumption on the high-order norms of u and θ.

UR - http://www.scopus.com/inward/record.url?scp=85084465686&partnerID=8YFLogxK

U2 - 10.1007/s00205-020-01515-5

DO - 10.1007/s00205-020-01515-5

M3 - Article

AN - SCOPUS:85084465686

VL - 237

SP - 585

EP - 630

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -