American Institute of Mathematical Sciences

March  2010, 28(1): 311-341. doi: 10.3934/dcds.2010.28.311

Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation

 1 Science University of Tokyo, Tokyo 162-8601, Japan 2 Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  August 2009 Revised  January 2010 Published  April 2010

Two theorems concerning strong wellposedness are established for the complex Ginzburg-Landau equation. One of them is concerned with strong $L^{2}$-wellposedness, that is, strong wellposedness for $L^{2}$-initial data. The other deals with $H_{0}^{1}$-initial data as a partial extension. By a technical innovation it becomes possible to prove the convergence of approximate solutions without compactness. This type of convergence is known with accretivity methods when the argument of the complex coefficient is small. The new device yields the generation of a class of non-contraction semigroups even when the argument is large. The results are both obtained as application of abstract theory of semilinear evolution equations with subdifferential operators.
Citation: Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311
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