Some non-linear geometric and kinetic evolutions and their approximations

Doctoral thesis, 2004

This thesis deals with three non-linear evolution problems: mean curvature flow, Willmore flow, and the evolution of solutions to the space homogeneous Boltzmann equation. Generalized mean curvature flows are also considered. A major part of the study focuses on the construction of approximations to these evolutions. The thesis consists of four papers. A unified convolution-thresholding approach to the generalized mean curvature flow and to the Willmore flow is suggested in the first and the second paper. The convergence of the approximations to viscosity solutions of the corresponding PDE is shown for the generalized mean curvature flow. For the Willmore flow the convergence of the convolution-thresholding scheme is shown in the case when the evolution is smooth and embedded. The third paper concentrates on an analytical and numerical study of self-similar (homothetic) mean curvature and Willmore flows. Two qualitatively different families of such evolutions for each type of the flows are found. Representatives from the first family become singular in finite time, while representatives from the second family come from singular initial data. In particular considerably new examples of smooth surfaces that develop a singularity in finite time during the Willmore evolution are obtained numerically. Another new result of this study is the construction of the mean curvature and Willmore evolutions starting from certain surfaces with singularities. In the fourth paper a new deterministic numerical method for solution of the Boltzmann equation is constructed. The suggested method is the only known deterministic scheme that effectively handles discontinuous solutions. It is based on a combination of two qualitatively different approaches: the approximation of discontinuous solutions on a non-uniform adaptive grid, and the approximation of smooth terms in the Boltzmann equation by a Fourier based spectral method.