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.
Categories: math.AP