Abstract
The so-called Ginzburg-Landau formalism applies for parabolic systems which are defined on cylindrical domains, which are, close to the threshold of instability, and for which the unstable Fourier modes belong to non-zero wave numbers. This formalism allows to describe an attracting set of solutions by a modulation equation, here the Ginzburg-Landau equation. If the coefficient in front of the cubic term of the formally derived Ginzburg-Landau equation has negative real part the method allows to show global existence in time in the original system of all solutions belonging to small initial conditions, inL ∞. Another aim of this paper is to construct a pseudo-orbit of Ginzburg-Landau approximations which is close to a solution of the original system up tot=∞. We consider here as an example the socalled Kuramoto-Shivashinsky equation to explain the methods, but it applies also to a wide class of other problems, like e.g. hydrodynamical problems or reaction-diffusion equations, too.
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Communicated by J.-P. Eckmann
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Schneider, G. Global existence via Ginzburg-Landau formalism and pseudo-orbits of Ginzburg-Landau approximations. Commun.Math. Phys. 164, 157–179 (1994). https://doi.org/10.1007/BF02108810
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DOI: https://doi.org/10.1007/BF02108810