Yoshikazu Giga

Yoshikazu Giga, Hiroyoshi Mitake, Takeshi Ohtsuka et al.

In this paper, we prove the existence of asymptotic speed of solutions to fully nonlinear, possibly degenerate parabolic partial differential equations in a general setting. We then give some explicit examples of equations in this setting and study further properties of the asymptotic speed for each equation. Some numerical results concerning the asymptotic speed are presented.

Takeshi Ohtsuka, Yen-Hsi Richard Tsai, Yoshikazu Giga

We study analytically and numerical the growth rate of a crystal surface growing by several screw dislocations. To describe several spiral steps we use the revised level set method for spirals by the authors (Journal of Scientific Computing 62, 831-874, 2015). We carefully compare our simulation results on the growth rates with predictions in a classical paper by Burton et al. (Philos Trans R Soc Lond Ser A Math Phys Sci 243,299-358, 1951). Then, we propose improved estimates on the growth rate with several different configurations, which are in agreement with our numerical simulations. The influence of distribution of screw dislocations in a group on a line to the growth rate, and the growth rate by a group including different rotational orientations of spirals are also studied in this paper.