Singular stochastic Allen–Cahn equations with dynamic boundary conditions

We prove a well-posedness result for stochastic Allen–Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular cha...

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Bibliographic Details
Published in:Journal of Differential Equations 2019-04, Vol.266 (8), p.4624-4667
Main Authors: Orrieri, Carlo, Scarpa, Luca
Format: Article
Language:English
Subjects:
Allen–Cahn equation
Asymptotic estimates
Dynamic boundary conditions
Singular potentials
Well-posedness
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Summary:We prove a well-posedness result for stochastic Allen–Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2018.10.007