 9826 Th. Gallay (Paris XI) and A. Mielke (Hannover)
 Diffusive Mixing of Stable States in the GinzburgLandau Equation
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Jan 23, 98

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Abstract. The GinzburgLandau equation $\partial_t u = \partial_x^2u+uu^2u$
on the real line has spatially periodic steady states of the the form
$U_{\eta,\beta}(x)=(1{}\eta^2)^{1/2}\,{\mathrm e}^{{\mathrm i}
(\eta x+\beta)}$, with $\eta \leq 1$ and $\beta \in {\mathbb R}$.
For $\eta_+,\eta_{\in} (1/\sqrt{3},1/\sqrt{3})$, $\beta_+,\beta_{\in}
{\mathbb R}$, we construct solutions which converge for all $t>0$ to
the limiting pattern $U_{\eta_\pm,\beta_\pm}$ as $x\to \pm \infty$.
These solutions are stable with respect to sufficiently small
${\mathrm H}^2$ perturbations, and behave asymptotically in time
like $(1\widetilde\eta(x/\sqrt t)^2)^{1/2}\,\exp({\mathrm i}\sqrt t
\,\widetilde N(x/ \sqrt t\,))$, where $\widetilde N'=\widetilde\eta
\in {\mathcal C}^\infty({\mathbb R})$ is uniquely determined by the
boundary conditions $\widetilde\eta(\pm\infty) = \eta_\pm$. This
extends a previous result of Bricmont and Kupiainen by removing
the assumption that $\eta_\pm$ should be close to zero. The existence
of the limiting profile $\widetilde\eta$ is obtained as an application
of the theory of monotone operators, and the longtime behavior of
our solutions is controlled by rewriting the system in scaling variables
and using energy estimates involving an exponentially growing damping
term.
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