| Abstract: |
| The phase separation process in a binary alloy is
parameterized by a small positive parameter $\varepsilon$
representing the order of the width of the evolving layers.
Stochastic versions for the equations modelling the evolution of
the concentration of the phases arise due to thermal
fluctuations, external mass supply or impurities in the alloy. We
consider the $\varepsilon$-dependent stochastic Cahn-Hilliard
equation with multiplicative and sufficiently regular in space
noise with strength of order $O(\varepsilon^\gamma)$, $\gamma > 0$
in dimensions $d = 1,2,3$. We prove $p$-moments estimates in
$H^1$, $H^2$ and $L^\infty$ norms. When the initial data are
layered by using the energy ($H^1$) estimate we prove that, as
$\varepsilon$ tends to $0$, the solution $u\rightarrow \pm 1$ in
the $L^2$ norm with probability tending to $1$. This implies the
complete separation of the two phases on the sharp interface
limit even in the presence of noise. |
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