Preprint of the project:
08-077 Wolf-Jürgen Beyn, Jens Lorenz.
Nonlinear stability of rotating patterns
We consider 2D localized rotating patterns
which solve a parabolic system of PDEs in 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.
