Well-posedness of a needle crystal growth problem with anisotropic surface tension

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 in Sobolev spaces for the needle crystal problem with arbitrary initial interface. Furthermore, we obtain that, on average in time, the solution gains 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.

Keywords:

• unsteady needle crystal
• initial value problem
• well posedness
• Sobolev space
• smothing effect

AMS Subject Classifications:

• 35Q72
• 74N05
• 80A22

Acknowledgments

This work is dedicated to professor Alexander Pankov on the occasion of his 65th birthday, the author thanks him for his help over the years. Part of this work was done when the author was in residence at MSRI in Berkeley in the Spring of 2011, and the author was grateful for its support and hospitality. The author also thanks professor Sijue Wu of University of Michigan for very helpful discussions.