On Some Nonlocal Evolution Equations Arising in Materials Science

摘要

Equations for a material that can exist stably in one of two homogeneous states are derived from a microscopic or lattice viewpoint with the assumption that the evolution follows a gradient flow of the free energy with respect to some metric. Alternatively, Newtonian dynamics can be considered. The resulting lattice dynamical systems are analyzed, as are equations on the continuum where the lattice interaction energy is viewed as an approximation to a Riemann integral. These equations are lattice or nonlocal versions of the Allen-Calm, Cahn-Hilliard, Phase-Field, or Klein-Gordon equations. Some results presented here provide for the well-posedness of the equations, while others give asymptotics or qualitative behavior of special solutions, such as traveling waves or pulses. This summarizes results previously reported in papers with coauthors Xinfu Chen, Adam Chmaj, Jianlong Han, Chunlei Zhang, and Guangyu Zhao.