Velocity Averaging (or not…) and Quantum Kinetic Theory

Abstract:
Velocity averaging is a class of regularity results for the transport equation which says that macroscopic observables attached to the distribution function solution of a kinetic equation are more regular than the distribution function itself. It has important applications to the existence theory for the Boltzmann and Vlasov equations, and to hydrodynamic limits of the Boltzmann equation. This talk will discuss the possibility of applying velocity averaging in the context of quantum dynamics. Our analysis of this problem will lead us to discuss the so-called Madelung quantum hydrodynamic equations and the magnetic or gauge-invariant Wigner transform due to Stratonovich.