On the Stochastic Cahn-Hilliard/Allen-Cahn equation - Isaac Newton ...
On the Stochastic Cahn-Hilliard/Allen-Cahn equation - Isaac Newton ...
On the Stochastic Cahn-Hilliard/Allen-Cahn equation - Isaac Newton ...
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Previous results with stochastic forcing<br />
<strong>Stochastic</strong> <strong>Cahn</strong>-<strong>Hilliard</strong> <strong>equation</strong> on a rectangle (only <strong>the</strong> bi-Laplacian):<br />
f = F ′ where F is some ”double well potential” polynomial<br />
∂tu(x, t) = −∆[∆u(x, t) − f (u(x, t))] + σ(u(x, t)) ˙W (x, t)<br />
dimension 1 with additive space-time white noise distribution<br />
solutions (dimension d ≥ 2, L 2 (O) solution for addive colored noise)<br />
Da Prato, Debussche, 1996)<br />
dimension 1 to 3: Cardon-Weber with multiplicative space time white<br />
noise on a rectangle 2001 (dimension 4 and 5, Cardon-Weber & M.<br />
with colored noise, 2004); <strong>the</strong> diffusion coefficient σ is Lipschitz and<br />
bounded.<br />
O = (0, 1) and f (u) = θ/2 ln( 1+u<br />
1−u ) − θcu with θ < θc with additive<br />
noise and reflection measures Debussche & Goudenège (2011)<br />
A. Millet (SAMM, Paris 1 and PMA) <strong>Cahn</strong>-<strong>Hilliard</strong>/<strong>Allen</strong>-<strong>Cahn</strong> <strong>equation</strong> Workshop <strong>Stochastic</strong> PDEs 5 / 18
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