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Conservative stochastic Cahn--Hilliard equation with reflection

Arnaud Debussche·Lorenzo Zambotti
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Abstract

We consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space--time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution. We propose a method based on infinite dimensional integration by parts formulae, obtaining existence and uniqueness of a strong solution for all continuous nonnegative initial conditions and detailed information on the associated invariant measure and Dirichlet form.

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Arnaud Debussche, Lorenzo Zambotti. 2007-10-12. Conservative stochastic Cahn--Hilliard equation with reflection. https://doi.org/10.1214/009117906000000773

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