Dynamics of Partial Differential Equations

Volume 5 (2008)

Number 4

Nonlinear stability of rotating patterns

Pages: 349 – 400

DOI: http://dx.doi.org/10.4310/DPDE.2008.v5.n4.a4

Authors

Wolf-Jürgen Beyn (Fakultät für Mathematik, Universität Bielefeld, Germany)

Jens Lorenz (Department of Mathematics and Statistics, University of New Mexico, Albuquerque)

Abstract

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.

Keywords

rotating patterns, asymptotic stability, nonlinear stability, relative equilibria, group action, Ginzburg–Landau equation

2010 Mathematics Subject Classification

35B35, 35B40, 35K57

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