Nonlinear Stability of Rotating Patterns

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.