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Higher moments for the Stochastic Cahn - Hilliard Equation with multiplicative Fourier noise
Antonopoulou, Dimitra
Antonopoulou, Dimitra
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2023-01-03
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We consider in dimensions $d=1,2,3$ the $\eps$-dependent stochastic Cahn-Hilliard equation with a multiplicative and
sufficiently regular in space infinite dimensional Fourier noise with strength of order $\mathcal{O}(\eps^\gamma)$, $\gamma>0$. The initial condition is non-layered and independent from $\eps$. Under general assumptions on the noise diffusion $\sigma$, we prove moment estimates in $H^1$ (and in $L^\infty$ when $d=1$). Higher $H^2$ regularity $p$-moment estimates are derived when $\sigma$ is bounded, yielding as well space H\"older and $L^\infty$ bounds for $d=2,3$, and path a.s. continuity in space. All appearing constants are expressed in terms of the small positive parameter $\eps$. As in the deterministic case, in $H^1$, $H^2$, the bounds admit a negative polynomial order in $\eps$. Finally, assuming layered initial data of initial energy uniformly bounded in $\eps$, as proposed by X.F. Chen in \cite{chenjdg}, we use our $H^1$ $2$d-moment estimate and prove the stochastic solution's convergence to $\pm 1$ as $\eps\rightarrow 0$ a.s., when the noise diffusion has a linear growth.
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Antonopoulou, D. (2023). Higher moments for the Stochastic Cahn - Hilliard Equation with multiplicative Fourier noise. Nonlinearity, 36(2), 1053. https://doi.org/10.1088/1361-6544/acadc9
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IOP Publishing
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Nonlinearity
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‘This is the Accepted Manuscript version of an article accepted for publication in [Nonlinearity]. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at [https://doi.org/10.1088/1361-6544/acadc9].’
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0951-7715
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1361-6544