Stability of 2D partially dissipative Boussinesq equations and 3D rotating Boussinesq equations

Abstract

Examining the stability of nonlinear partial differential systems near a physically relevant equilibrium with suitable perturbation is a fundamental problem in fluid dynamics. This dissertation solves the stability and large-time behavior of solutions to the nonlinear Boussinesq equations.

We study the two-dimensional Boussinesq equations for buoyancy-driven fluids with degenerate dissipations due to their application in a specific physical scenario. Furthermore, these degenerate dissipations help reveal the inner structure of the system when we perform various interactions between the velocity and temperature. We perturb the solutions of two different two-dimensional Boussinesq systems near the hydrostatic equilibrium in a different domain. We prove that the temperature stabilizes the buoyancy-driven fluids for the first system, which has only vertical dissipation and horizontal thermal diffusion. For the second system containing only horizontal dissipation and vertical thermal diffusion, we establish the stability of the solutions and stratifying patterns of the buoyancy-driven fluids as mathematically rigorous facts.

Along with this, we study the stability of the three-dimensional rotating Boussinesq equations with only horizontal dissipation, which have a special two-dimensional solution that is dynamic and independent of depth. On large scales, this unique solution provides the bulk averaged properties of the fluid motion. To achieve the global existence, uniqueness, and stability result, we perturb the three-dimensional rotating Boussinesq equations near this dynamic solution.