Attractivity of the Ginzburg–Landau mode distribution for a pattern forming system with marginally stable long modes

The mathematical theory of the description of pattern forming systems close to the first instability via the Ginzburg–Landau equation is based on approximation and attractivity results. This theory allowed to prove global existence results and upper semicontinuity of attractors for classical hydrodynamical stability problems such as the Couette–Taylor problem. Recently, approximation results for the Ginzburg–Landau approximation for pattern forming systems with marginally stable long modes, such as the Bénard–Marangoni system, have been shown. It is the purpose of this paper to prove the second fundamental property, namely the attractivity of the Ginzburg–Landau mode distribution, for such systems.