Diffusive Mixing of Stable States in the Ginzburg–Landau Equation |
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Authors: | Thierry Gallay Alexander Mielke |
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Affiliation: | Analyse Numérique et EDP, Université de Paris XI, 91405 Orsay, France.?E-mail: Thierry.Gallay@math.u-psud.fr, FR Institut für Angewandte Mathematik, Universit?t Hannover, 30167 Hannover, Germany.?E-mail: mielke@ifam.uni-hannover.de, DE
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Abstract: | The Ginzburg–Landau equation on the real line has spatially periodic steady states of the form , with and . For , , we construct solutions which converge for all t>0 to the limiting pattern as . These solutions are stable with respect to sufficiently small perturbations, and behave asymptotically in time like , where is uniquely determined by the boundary conditions . This extends a previous result of [BrK92] by removing the assumption that should be close to zero. The existence of the limiting profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term. Received: 22 January 1998 / Accepted: 19 April 1998 |
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