Masayasu Mimura – författare

Interfaces are geometrical objects modelling free or moving boundaries and arise in a wide range of phase change problems in physical and biological sciences, particularly in material technology and in dynamics of patterns. Especially in the end of last century, the study of evolving interfaces in a number of applied fields becomes increasingly important, so that the possibility of describing their dynamics through suitable mathematical models became one of the most challenging and interdisciplinary problems in applied mathematics. The 2000 Madeira school reported on mathematical advances in some theoretical, modelling and numerical issues concerned with dynamics of interfaces and free boundaries. Specifically, the five courses dealt with an assessment of recent results on the optimal transportation problem, the numerical approximation of moving fronts evolving by mean curvature, the dynamics of patterns and interfaces in some reaction-diffusion systems with chemical-biological applications, evolutionary free boundary problems of parabolic type or for Navier-Stokes equations, and a variational approach to evolution problems for the Ginzburg-Landau functional.

This book addresses a new class of billiard problems, focusing on the motion of a self-propelling disk in a nonlinear dissipative system on a rectangular domain. Unlike classical billiards, which have been extensively studied in mathematics, this setting introduces unique dynamics inspired by experiments with camphor disks floating on water—a well-known phenomenon in nonlinear science. Laboratory observations reveal two striking properties: (i) the disk reflects without physical collision at the boundary, and (ii) the reflection angle exceeds the incidence angle, differing from the perfect elastic reflection of classical billiards. These features suggest that the behavior of a self-propelling disk is fundamentally distinct from classical billiard motion.The purpose of this book is to provide a mathematical understanding of such dynamics. We propose three levels of modeling: a moving-boundary (MB) model, a particle model, and a discrete-time model. To demonstrate that the MB model satisfies properties (i) and (ii), we derive the particle model for slow disk motion, describing its position and velocity. Numerical simulations indicate that although classical billiards in a rectangle are simple, the particle model exhibits complex behavior depending on the domain’s shape. To analyze this complexity, we construct discrete-time models that capture the evolution of reflection angles and positions. Using dynamical systems theory, bifurcation analysis, and complementary numerical methods, we show that a self-propelling disk can display intricate and varied billiard motions—even in a rectangular domain—due to angle interactions.This book emphasizes that the trajectory of a billiard disk in a nonlinear dissipative system is determined by inherent dynamics, unlike classical billiards, where outcomes depend heavily on player skill.