Stability and instability of generalized Kolmogorov flows on the two-dimensional sphere. (English) Zbl 1099.37059

Ilyin, A. A.

Adv. Differ. Equ. 9, No. 9-10, 979-1008 (2004).

Summary: The Navier-Stokes equations on the two-dimensional rotating sphere with a family of forcing terms, whose stream functions are the Legendre polynomials \(P_s\), are considered. The stability and instability properties of the corresponding generalized Kolmogorov flows are studied both analytically and numerically. Logarithmically sharp lower bounds for the dimension of the global attractor are obtained. The effect of rotation on the stability properties of the Kolmogorov flows is discussed.

MSC:

37L10	Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
35Q30	Navier-Stokes equations
76D05	Navier-Stokes equations for incompressible viscous fluids
76U05	General theory of rotating fluids
76E30	Nonlinear effects in hydrodynamic stability
37L30	Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems

Keywords:

Navier-Stokes equations; Legendre polynomials; global attractor; effect of rotation; stability properties