Stability of stationary solutions of the forced Navier-Stokes equations on the two-torus

,Chuong V Tran,Theodore G Shepherd,Han-Ru Cho

doi:10.3934/dcdsb.2002.2.483

Abstract

We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain $[0,2\pi]\times[0,2\pi/\alpha]$, where $\alpha\in(0,1]$, with doubly periodic boundary conditions. For the linear problem we employ the classical energy--enstrophy argument to derive some fundamental properties of unstable eigenmodes. From this it is shown that forces of pure $x_2$-modes having wavelengths greater than $2\pi$ do not give rise to linear instability of the corresponding primary stationary solutions. For the nonlinear problem, we prove the equivalence of nonlinear stability with respect to the energy and enstrophy norms. This equivalence is then applied to derive optimal conditions for nonlinear stability, including both the high- and low-Reynolds-number limits.

Highlights

We consider 2D incompressible fluid flow in a doubly periodic rectangular domain T 2 = [0, 2π] × [0, 2π/α], where α ∈
We present the main result in the nonlinear stability analysis
The problem of determining the nonlinear stability condition for unow reduces to determining the greatest value of the left-hand side of (42) for all integers k2 and setting it ≤ 1

Summary

Introduction

Where a positive (negative) real part of σ corresponds to an unstable (stable) eigenmode. For the inviscid case, the Fourier series of an unstable eigenmode of the basic flow uis non-terminating. Since the eigenmode cannot be entirely in H(|s|), for which (30) would hold trivially, this implies that an unstable disturbance must have a component with a wavenumber smaller than |s|. This recovers a well-known result in the inviscid linear problem (see, for example, [2, 3]).

Upon substitution of the above identity the trilinear form then reads

With this result uis nonlinearly stable when

This condition is satisfied when