Motion of Winterbottom-like droplets in SPDEs with smooth noise

Tadahisa Funaki (Univ. Tokyo)

We shall discuss the singular limit as ϵ ↓ 0 for a parabolic stochastic partial differential equation (SPDE) of fourth order:

(1)  \frac{∂u}{∂t} = −\frac{1}{ϵ}Δ\{Δu + f(u)\} + div ẇ(t, x), x ∈ 𝕋^{d},

on a torus 𝕋^{d} with a smooth Gaussian noise w(t, x) = (w^{i}(t, x))_{i=1}^{d}, and derive a diffusive motion of Winterbottom-like droplets in the limit.

In general, for an interacting system, under the situation that the corresponding hydrodynamic (macroscopic) equation admits a continuous family of stationary solutions, one can expect to observe their fluctuations under a properly chosen time scale beyond the hydrodynamic one. One possible choice for such microscopic system might be the Ginzburg-Landau ∇φinterface model of conservative type with pinning and repulsion, cf. [2]. In this system, although the hydrodynamic limit is not established at present, the stationary solutions of the corresponding macroscopic equation ought to be the so-called Winterbottom shapes [1] which have the degree of freedom of translation.

Since this model itself has several technical difficulties, we modify it in the following aspects and introduce the SPDE (1):

(a) We start with the macroscopic continuous system on a torus instead of the microscopic lattice system replacing the noise with smooth one.

(b) At the same time, the microscopic self potential describing pinning and repulsion is replaced by the macroscopic one, which is smooth.

In fact, for the ∇φ-interface dynamics of conservative type with a quadratic interaction potential made the modification (b) being defined on a lattice torus with side length N, the hydrodynamic time scale is of O(N^{4}), which leads to the macroscopic equation ∂u/∂t = −Δ{Δu + f(u)} in the limit. Then, choosing the longer time scale of O(N^{4+d}), which keeps the noise term of O(1), the SPDE (1) naturally arises with ϵ = N^{−d}.

References

[1] E. Bolthausen and D. Ioffe, Harmonic crystal on the wall: a microscopic approach, Commun. Math. Phys., 187 (1997), pp. 523–566.

[2] T. Funaki, Stochastic Interface Models, In: Lectures on Probability Theory and Statistics, Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003 (ed. J. Picard), Lect. Notes Math., 1869 (2005), Springer.