Nonlinear Evolution Toward the Linear Diffusive Profile in the Presence of Couette Flow

TL;DR

This paper studies how solutions to 2D Navier-Stokes equations with Couette flow evolve over time, showing they tend toward a linear diffusive profile even with large, low-regularity, or singular initial data.

Contribution

It demonstrates the asymptotic convergence of vorticity to a linear diffusive profile for general perturbations, including singular configurations, in the presence of Couette flow.

Findings

Vorticity approaches a linear diffusive profile over time.

Convergence holds for large, low-regularity, and singular initial data.

Long-time behavior is governed by the linearized vorticity equation.

Abstract

In this paper, we investigate the long-time behavior of solutions to the two-dimensional Navier-Stokes equations with initial data evolving under the influence of the planar Couette flow. We focus on general perturbations, which may be large and of low regularity, including singular configurations such as point vortices, and show that the vorticity asymptotically approaches a constant multiple of the fundamental solution of the corresponding linearized vorticity equation after a long-time evolution determined by the relative Reynolds number.