Long-time behaviour of two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane

TL;DR

This paper analyzes the long-term behavior of solutions to 2D Navier-Stokes equations with Couette flow on a half plane, revealing the asymptotic vorticity profile and introducing a novel spectral analysis method for boundary operators.

Contribution

It provides a rigorous description of the asymptotic vorticity distribution and introduces a new spectral analysis approach for Fokker-Planck type operators with boundary conditions.

Findings

Vorticity approaches a specific profile involving a kernel of a Fokker-Planck operator.

Derived explicit asymptotic formula for vorticity in the presence of Couette flow.

Introduced a new method for spectral analysis of boundary operators.

Abstract

In this paper, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane with Navier-slip boundary conditions. We prove that the total vorticity will approach \begin{align*} -1+\frac{M_2(\omega_{0})}{\nu^{3/2}(1+t)^{5/2}} \bar{\Omega}\left( \frac{x}{\sqrt{\nu(1+t)^3}}, \frac{y}{\sqrt{\nu(1+t)}} \right), \end{align*} where $-1$ is the vorticity of the Couette flow and $\bar{\Omega}$ is the kernel of a Fokker-Planck type operator $\mathcal{L}=\partial_Y^2+\frac32 X\partial_X+\frac12 Y\partial_Y+\frac52-Y\partial_X$. In the proof, we introduce a new idea of studying the spectrum of such type operators with boundary.