Various blow-up regimes for the Complex Ginzburg-Landau equation

Abstract:
The cubic Complex Ginzburg-Landau (CGL) equation is one of the most studied nonlinear equations in the physics community, where it attracted a lot of attention in the 1970’. It was first derived by Newell and Whitehead in 1969 as a model for instability in fluid convection problems, and used by Stewartson and Stuart in 1971, for the description of unstable plane Poiseuille flows.
As for the math community, the first results were given around the year 2000. This delay is mainly due to the lack of structure in this parabolic system, where no variational structure exists and no maximum principle is available.
Introducing new tools, we were able to rigorously construct some blow-up solutions for CGL, in different parameters’ regimes. This talk is dedicated to the proof of those results, in relation with the earlier physicists’ predictions.
This is a joint work with G.K. Duong and H.Zaag.