Type of Publication:  Contribution to a conference 
Publication status:  Published 
Author:  Bianchi, Luigi A.; Bloemker, Dirk 
Year of publication:  2017 
Conference:  Equadiff 2017, Jul 24, 2017  Jul 28, 2017, Bratislava, Slovakia 
Published in:  Proceedings of Equadiff 2017 Conference / Mikula, Karol et al. (ed.).  Bratislava : Slovak University of Technology, Spektrum Stu Publishing, 2017.  pp. 295304.  ISBN 9788022747578 
URL of original publication:  http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/equadiff/article/view/717, Last access on Mar 29, 2018 
Summary: 
We study the impact of small additive spacetime white noise on nonlinear SPDEs on unbounded domains close to a bifurcation, where an infinite band of eigenvalues changes stability due to the unboundedness of the underlying domain. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, and we rely on the approximation via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. One technical problem for establishing error estimates in the stochastic case rises from the spatially translation invariant nature of spacetime white noise on unbounded domains, which implies that at any time the error is always very large somewhere far out in space. Thus we have to work in weighted spaces that allow for growth at infinity. As a first example we study the stochastic onedimensional SwiftHohenberg equation on the whole real line. In this setting, because of the weak regularity of solutions, the standard methods for deterministic modulation equations fail, and we need to develop new tools to treat the approximation. Using energy estimates we are only able to show that solutions of the GinzburgLandau equation are Holder continuous in spaces with a very weak weight, which provides just enough regularity to proceed with the error estimates.

Subject (DDC):  510 Mathematics 
Keywords:  modulation equations, amplitude equations, convolution operator, regularity, RayleighBenard, SwiftHohenberg, GinzburgLandau 
Bibliography of Konstanz:  Yes 
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BIANCHI, Luigi A., Dirk BLOEMKER, 2017. Stochastic Modulation Equations on Unbounded Domains. Equadiff 2017. Bratislava, Slovakia, Jul 24, 2017  Jul 28, 2017. In: MIKULA, Karol, ed. and others. Proceedings of Equadiff 2017 Conference. Bratislava:Slovak University of Technology, Spektrum Stu Publishing, pp. 295304. ISBN 9788022747578
@inproceedings{Bianchi2017Stoch41927, title={Stochastic Modulation Equations on Unbounded Domains}, url={http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/equadiff/article/view/717}, year={2017}, isbn={9788022747578}, address={Bratislava}, publisher={Slovak University of Technology, Spektrum Stu Publishing}, booktitle={Proceedings of Equadiff 2017 Conference}, pages={295304}, editor={Mikula, Karol}, author={Bianchi, Luigi A. and Bloemker, Dirk} }
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