TY - JOUR

T1 - Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion

AU - Doering, Charles R.

AU - Wu, Jiahong

AU - Zhao, Kun

AU - Zheng, Xiaoming

N1 - Funding Information:
Support for this work came in part from NSF Awards DMS-1515161 (CRD) and DMS-1614246 (JW), a John Simon Guggenheim Foundation Fellowship (CRD), a Simons Foundation Collaboration Grant for Mathematicians 413028 (KZ), and from Louisiana Board of Regents Research Competitiveness Subprogram Award LEQSF(2015–18)-RD-A-24 (KZ). KZ also gratefully acknowledges start up funding from the Department of Mathematics at Tulane University . We thank Professor Chi-Wang Shu for helpful discussions regarding numerical methods.
Funding Information:
Support for this work came in part from NSF Awards DMS-1515161 (CRD) and DMS-1614246 (JW), a John Simon Guggenheim Foundation Fellowship (CRD), a Simons Foundation Collaboration Grant for Mathematicians413028 (KZ), and from Louisiana Board of Regents Research Competitiveness Subprogram AwardLEQSF(2015?18)-RD-A-24 (KZ). KZ also gratefully acknowledges start up funding from the Department of Mathematics at Tulane University. We thank Professor Chi-Wang Shu for helpful discussions regarding numerical methods.
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - We study the global well-posedness and stability/instability of perturbations near a special type of hydrostatic equilibrium associated with the 2D Boussinesq equations without buoyancy (a.k.a. thermal) diffusion on a bounded domain subject to stress-free boundary conditions. The boundary of the domain is not necessarily smooth and may have corners such as in the case of rectangles. We achieve three goals. First, we establish the global-in-time existence and uniqueness of large-amplitude classical solutions. Efforts are made to reduce the regularity assumptions on the initial data. Second, we obtain the large-time asymptotics of the full nonlinear perturbation. In particular, we show that the kinetic energy and the first order derivatives of the velocity field converge to zero as time goes to infinity, regardless of the magnitude of the initial data, and the flow stratifies in the vertical direction in a weak topology. Third, we prove the linear stability of the hydrostatic equilibrium T(y) satisfying T′(y)=α>0, and the linear instability of periodic perturbations when T′(y)=α<0. Numerical simulations are supplemented to corroborate the analytical results and predict some phenomena that are not proved. The authors are pleased to dedicate this paper to Professor Edriss Saleh Titi on the occasion of his 60th birthday. Professor Titi's myriad research contributions – including contributions to the problem constituting the focus of this work – and leadership in the mathematical fluid dynamics community serve as an inspiration to his students, collaborators and colleagues.

AB - We study the global well-posedness and stability/instability of perturbations near a special type of hydrostatic equilibrium associated with the 2D Boussinesq equations without buoyancy (a.k.a. thermal) diffusion on a bounded domain subject to stress-free boundary conditions. The boundary of the domain is not necessarily smooth and may have corners such as in the case of rectangles. We achieve three goals. First, we establish the global-in-time existence and uniqueness of large-amplitude classical solutions. Efforts are made to reduce the regularity assumptions on the initial data. Second, we obtain the large-time asymptotics of the full nonlinear perturbation. In particular, we show that the kinetic energy and the first order derivatives of the velocity field converge to zero as time goes to infinity, regardless of the magnitude of the initial data, and the flow stratifies in the vertical direction in a weak topology. Third, we prove the linear stability of the hydrostatic equilibrium T(y) satisfying T′(y)=α>0, and the linear instability of periodic perturbations when T′(y)=α<0. Numerical simulations are supplemented to corroborate the analytical results and predict some phenomena that are not proved. The authors are pleased to dedicate this paper to Professor Edriss Saleh Titi on the occasion of his 60th birthday. Professor Titi's myriad research contributions – including contributions to the problem constituting the focus of this work – and leadership in the mathematical fluid dynamics community serve as an inspiration to his students, collaborators and colleagues.

KW - 2D Boussinesq equations

KW - Classical solution

KW - Initial–boundary value problem

KW - Long-time behavior

UR - http://www.scopus.com/inward/record.url?scp=85041069528&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2017.12.013

DO - 10.1016/j.physd.2017.12.013

M3 - Article

AN - SCOPUS:85041069528

SN - 0167-2789

VL - 376-377

SP - 144

EP - 159

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -