This seminar will be presented in hybrid mode. The speaker will deliver his talk remotely.
Title: Diffusive and superdiffusive limits for a kinetic equation with a boundary condition
Abstract: We consider the limit of a linear kinetic equation in one spatial dimension, with reflection-transmission-absorption at a point interface. The scattering kernel is degenerate. The equation with this type of boundary condition arises from considering the kinetic limit for a microscopic one-dimensional chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit either diffusive, or superdiffusive behaviour in the long time limit, depending on whether the dispersion relation for the chain is optical, or acoustic. The latter corresponds to either the presence, or absence of an external potential acting on the chain. With the interface, the long time limit is either the unique solution of a heat equation with a Dirichlet boundary condition (in the optical case), or a version of the fractional in space heat equation, with reflection-transmission-absorption at the interface (in the acoustic case). In the latter case the limit problem corresponds to a certain stable process that is either absorbed, reflected, or transmitted upon crossing the interface.
The results have been obtained in collaboration with G. Basile (Univ. Roma I), S. Olla (Univ. Paris-Dauphine), L. Ryzhik (Stanford Univ.) and H. Spohn (TU München).
The talk will be followed by refreshments in the Huxley Common Room at 4pm.