Nonlinear stability of the two-jet Kolmogorov type flow on the unit sphere under a perturbation with nondissipative part

Abstract

We consider the vorticity form of the Navier–Stokes equations on the two-dimensional unit sphere and study the nonlinear stability of the two-jet Kolmogorov type flow which is a stationary solution given by the zonal spherical harmonic function of degree two. In particular, we assume that a perturbation contains a nondissipative part given by a linear combination of the spherical harmonics of degree one and investigate the effect of the nondissipative part on the long-time behaviour of the perturbation through the convection term. We show that the nondissipative part of a weak solution to the nonlinear stability problem is preserved in time for all initial data. Moreover, we prove that the dissipative part of the weak solution converges exponentially in time towards an equilibrium which is expressed explicitly in terms of the nondissipative part of the initial data and does not vanish in general. In particular, it turns out that the asymptotic behaviour of the weak solution is finally determined by a system of linear ordinary differential equations. To prove these results, we make use of properties of Killing vector fields on a manifold. We also consider the case of a rotating sphere.