Local smoothing effect and existence for a needle crystal growth problem with anisotropic surface tension

TL;DR

This paper investigates a two-dimensional needle crystal growth model with anisotropic surface tension, proving local existence, uniqueness, and a smoothing effect over time, along with continuous dependence on initial data.

Contribution

It introduces a new initial value problem derived from a one-sided model and establishes local well-posedness and smoothing properties for the crystal growth problem.

Findings

Existence and uniqueness of local solutions for any initial interface.

Solutions gain 3/2 derivatives of smoothness over time.

Continuous dependence of solutions on initial data is proven.

Abstract

We study an initial value problem for two-dimensional needle crystal growth with anisotropic surface tension. The initial value problem is derived from the so called one-sided model based on complex variables method. We then obtain the existence and uniqueness of local solution of the needle crystal problem for any initial interface. Furthermore, we obtain that, on average in time, the solution gains 3/2 derivative of smoothness in spatial variable compared to the initial data. The continuous dependence on the initial data of the solution map is also established.