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Item Open Access A new estimate for the Ginzburg-Landau approximation on the real axis(1994) Schneider, GuidoModulation equations play an essential role in the understanding of complicated systems near the threshold of instability. For scalar parabolic equations for which instability occurs at nonzero wavelength, we show that the associated Ginzburg-Landau equation dominates the dynamics of the nonlinear problem locally, at least over a long timescale. We develop a method which is simpler than previous ones and allows initial conditions of lower regularity. It involves a careful handling of the critical modes in the Fourier-transformed problem and an estimate of Gronwall's type. As an example, we treat the Kuramoto-Shivashinsky equation. Moreover, the method enables us to handle vector-valued problems.Item Open Access Global existence via Ginzburg-Landau formalism and pseudo- orbits of Ginzburg-Landau approximations(1994) Schneider, GuidoThe 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 in L∞. 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 to t = ∞. 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.Item Open Access Error estimates for the Ginzburg-Landau approximation(1994) Schneider, GuidoModulation equations play an essential role in the understanding of complicated dynamical systems near the threshold of instability. Here we look at systems defined over domains with one unbounded direction and show that the Ginzburg-Landau equation dominates the dynamics of the full problem, locally, at least over a long time-scale. As an application of our approximation theorem we look here at Bénard's problem. The method we use involves a careful handling of critical modes in the Fourier-transformed problem and an estimate of Gronwall's type.