ABSTRACT

In this talk we will highlight recent discoveries in kinetic theory from two recent papers. We will develop an uncertainty principle, a new a-priori bound and the concept of a blind cone with respect to an observer for the evolution particles in the mesoscopic scale solely based on conservation laws. We will show that the energy within any bounded set of the spatial variable is integrable over time and that the total mass of the particles concentrates within a specific collection of arbitrarily acute blind cones with respect to any observer. This shows that, as uncertainty inevitably increases, particles will move away in a radial manner from any fixed observer thereby erasing the angular component of momentum. We will also discuss a generalization of these results for the interactions instead of the particles and establish analogies to Morawetz and interaction Morawetz estimates for the nonlinear Schrodinger equation. These results are independent of the specific structure of interactions, therefore they are also true for the Boltzmann equation. We will end the talk with results about the case of the Boltzmann equation. We will introduce the concept of a scattering frame of reference and show the existence and uniqueness of a specific class of classical solutions to the Boltzmann equation. These solutions scatter to linear states in the L^{\infty} norm. Furthermore, we will discuss the asymptotic completeness of this class of solutions and establish another connection with the case of the nonlinear Schrodinger equation. Notably, this shows that solutions of the Boltzmann equation do not necessarily converge to a Maxwellian but can scatter to linear states arbitrarily close to any prescribed linear state.

Meeting ID: 974 8308 6174

Password: 207528