TY - JOUR

T1 - Sharp interface limit for stochastically perturbed mass conserving Allen-Cahn equation

AU - Funaki, Tadahisa

AU - Yokoyama, Satoshi

N1 - Publisher Copyright:
© Institute of Mathematical Statistics, 2019.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - This paper studies the sharp interface limit for a mass conserving Allen- Cahn equation, added an external noise and derives a stochastically perturbed mass conserving mean curvature flow in the limit. The stochastic term destroys the precise conservation law, instead the total mass changes like a Brownian motion in time. For our equation, the comparison argument does not work, so that to study the limit we adopt the asymptotic expansion method, which extends that for deterministic equations used originally in de Mottoni and Schatzman [Interfaces Free Bound. 12 (2010) 527-549] for the nonconservative case and then in Chen et al. [Trans. Amer. Math. Soc. 347 (1995) 1533-1589] for the conservative case. Differently from the deterministic case, each term except the leading term appearing in the expansion of the solution in a small parameter ε diverges as ε tends to 0, since our equation contains the noise which converges to a white noise and the products or the powers of the white noise diverge. To derive the error estimate for our asymptotic expansion, we need to establish the Schauder estimate for a diffusion operator with coefficients determined from higher order derivatives of the noise and their powers.We show that one can choose the noise sufficiently mild in such a manner that it converges to the white noise and at the same time its diverging speed is slow enough for establishing a necessary error estimate.

AB - This paper studies the sharp interface limit for a mass conserving Allen- Cahn equation, added an external noise and derives a stochastically perturbed mass conserving mean curvature flow in the limit. The stochastic term destroys the precise conservation law, instead the total mass changes like a Brownian motion in time. For our equation, the comparison argument does not work, so that to study the limit we adopt the asymptotic expansion method, which extends that for deterministic equations used originally in de Mottoni and Schatzman [Interfaces Free Bound. 12 (2010) 527-549] for the nonconservative case and then in Chen et al. [Trans. Amer. Math. Soc. 347 (1995) 1533-1589] for the conservative case. Differently from the deterministic case, each term except the leading term appearing in the expansion of the solution in a small parameter ε diverges as ε tends to 0, since our equation contains the noise which converges to a white noise and the products or the powers of the white noise diverge. To derive the error estimate for our asymptotic expansion, we need to establish the Schauder estimate for a diffusion operator with coefficients determined from higher order derivatives of the noise and their powers.We show that one can choose the noise sufficiently mild in such a manner that it converges to the white noise and at the same time its diverging speed is slow enough for establishing a necessary error estimate.

KW - Allen-Cahn equation

KW - Asymptotic expansion

KW - Mass conservation law

KW - Mean curvature flow

KW - Sharp interface limit

KW - Stochastic perturbation

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U2 - 10.1214/18-AOP1268

DO - 10.1214/18-AOP1268

M3 - Article

AN - SCOPUS:85061793840

SN - 0091-1798

VL - 47

SP - 560

EP - 612

JO - Annals of Probability

JF - Annals of Probability

IS - 1

ER -