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

2025/01/01 by Arbon, Ryan
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.00690

Abstract

We consider the quantitative asymptotic stability of the stably stratified Couette flow solution to the 2D fully dissipative nonlinear Boussinesq system on ℝ² with large Richardson number R > 1/4, viscosity ν and density dissipation κ. For an initial perturbation (ω_in, θ_in) of size μ^{1/2 + ε} in a low-order anisotropic Sobolev space, for μ roughly min(ν, κ)(1 - O(1/√(R))) and ν, κ 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 θ. This is the first result of its type for the Boussinesq system on the fully unbounded domain ℝ². We also translate some known linear results from \mathbbT × ℝ to ℝ², and we give an alternative theorem for the nonlinear result.

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