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Malliavin Calculus for the stochastic Cahn- Hilliard/Allen-Cahn equation with unbounded noise diffusion
Antonopoulou, Dimitra Farazakis, Dimitris Karali, Georgia D.
Antonopoulou, Dimitra
Farazakis, Dimitris
Karali, Georgia D.
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Publication Date
2018-05-08
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Abstract
The stochastic partial di erential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic
driving force. This equation is a combination of Cahn-Hilliard and Allen-Cahn type operators with
a multiplicative, white, space-time noise of unbounded di usion. We apply Malliavin calculus, in
order to investigate the existence of a density for the stochastic solution u. In dimension one,
according to the regularity result in [5], u admits continuous paths a.s. Using this property, and
inspired by a method proposed in [8], we construct a modi ed approximating sequence for u, which
properly treats the new second order Allen-Cahn operator. Under a localization argument, we prove
that the Malliavin derivative of u exists locally, and that the law of u is absolutely continuous,
establishing thus that a density exists.
Citation
Antonopoulou, D., Farazakis, D., & Karali, G. D. (2018). Malliavin Calculus for the stochastic Cahn- Hilliard/Allen-Cahn equation with unbounded noise diffusion. Journal of Differential Equations, 265(7), 3168-3211.
Publisher
Elsevier
Journal
Journal of Differential Equations
Research Unit
DOI
10.1016/j.jde.2018.05.004
PubMed ID
PubMed Central ID
Type
Article
Language
en
Description
Series/Report no.
ISSN
0022-0396