Dynamics of Partial Differential Equations
Volume 5 (2008)
Nonlinear stability of rotating patterns
Pages: 349 – 400
We consider 2D localized rotating patterns which solve a parabolic system of PDEs on the spatial domain R². Under suitable assumptions, we prove nonlinear stability with asymptotic phase with respect to the norm in the Sobolev space H². The stability result is obtained by a combination of energy and resolvent estimates, after the dynamics is decomposed into an evolution within a three–dimensional group orbit and a transversal evolution towards the group orbit.
The stability theorem is applied to the quintic–cubic Ginzburg–Landau equation and illustrated by numerical computations.
rotating patterns, asymptotic stability, nonlinear stability, relative equilibria, group action, Ginzburg–Landau equation
2010 Mathematics Subject Classification
35B35, 35B40, 35K57