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

Abstract

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.