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

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.