Diffusive stability of rolls in the two–dimensional real and complex swift–hohenberg equation

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Abstract
We show the nonlinear stability of small bifurcating stationary rolls u , , for the Swift–Hohenberg–equation on the domain R2. In Bloch wave representation the linearization around a marginal stable roll u∊,x, has continuous spectrum up to 0 with a locally parabolic shape at the critical Bloch vector 0. Using an abstract renormalization theorem we show that small spatially localized integrable perturbations decay diffusively to zero. Moreover we estimate the size of the domain of attraction of a roll u∊,x, in terms of its modulus and Fourier wavenumber. To explain the method we also treat the nonlinear stability of stationary rolls for the complex Swift–Hohenberg equation on R2

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