Galilean Theory of Dispersion and Scattering: Conservation laws, Blind Cones and the Increase of Uncertainty

The HADES seminar on Tuesday, April 5 will be at 3:30 pm in Room 740.

Speaker: Nima Moini

Abstract: In this talk, I will sketch a new approach to the study of kinetic equations solely under the assumption of conservation laws. The new idea is based on an uncertainty principle, the introduction of blind cones with respect to an observer and the Galilean invariance of different inertial frames of reference. In fact, as the uncertainty inevitably increases with time, particles will move away in an asymptotically radial manner from any fixed observer thereby establishing a new notion of dispersion. The generality of this approach reveals a mathematical relationship between the Landau and Boltzmann equations in the context of “the grazing collisions”, which until now was solely phenomenological. Moreover, I will discuss a new scattering theory for the kinetic equations and demonstrate its utility in the case of the Boltzmann equation for hard spheres. The new framework improves upon the existing results by proving the asymptotic completeness of the solutions of the Boltzmann equation near an equilibrium in the  $L^\infty$ setting. In particular, for any solution to the transport equation, there are arbitrarily close in $L^\infty$  norm, scattered solutions of the Boltzmann equation, this implies that solutions of the Boltzmann equation defined over the whole space will not converge to the state of thermodynamic equilibrium.

Leave a Reply

Your email address will not be published. Required fields are marked *