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Existence of dendritic crystal growth with stochastic perturbations. (English) Zbl 0914.60079

Summary: We prove the first mathematical existence result for a model of dendritic crystal growth with thermal fluctuations. The incorporation of noise is widely believed to be important in solidification processes. Our result produces an evolving crystal shape and a temperature field satisfying the Gibbs-Thomson condition at the crystal interface and a heat equation with a driving force in the form of a spatially correlated white noise. We work in the regime of infinite mobility, using a sharp interface model with a smooth and elliptic anisotropic surface energy. Our approach permits the crystal to undergo topological changes. A time discretization scheme is used to approximate the evolution. We combine techniques from geometric measure theory and stochastic calculus to handle the singular geometries and take advantage of the cancellation properties of the white noise.

MSC:

60K40 Other physical applications of random processes
80A22 Stefan problems, phase changes, etc.
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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