Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussinesq system on the Plane

Abstract: We consider the quantitative asymptotic stability of the stably stratified Couette flow solution to the 2D fully dissipative nonlinear Boussinesq system on $\mathbb{R}^2$ with large Richardson number $R > 1/4$, viscosity $\nu$ and density dissipation $\kappa$. For an initial perturbation $(\omega_{in}, \theta_{in})$ of size $\mu^{1/2 + \epsilon}$ in a low-order anisotropic Sobolev space, for $\mu$ roughly $\min(\nu, \kappa)\left(1 - O(1/\sqrt{R})\right)$ and $\nu$, $\kappa$ comparable, we demonstrate asymptotic stability with explicit enhanced dissipation and Taylor dispersion rates of decay. We also give inviscid damping estimates on the velocity $u$ and the density $\theta$. This is the first result of its type for the Boussinesq system on the fully unbounded domain $\mathbb{R}^2$. We also translate some known linear results from $\mathbb{T} \times \mathbb{R}$ to $\mathbb{R}^2$, and we give an alternative theorem for the nonlinear result.