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

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.