Velocity averaging lemmas: classical, quantum and semi-classical

Averaging lemmas were introduced as a tool of the mathematical analysis of kinetic equations, i.e. PDEs for functions in phase space ( x , v ) containing a transport (“advection”) term. By integrating over v in velocity space ℝ v d (velocity averaging), one gains regularity for the density in position space ℝ x d . The concept was invented independently by V. I. Agoshkov and by F. Golse, B. Perthame, R. Sentis and P.-L. Lions, and successfully applied to the analysis of Vlasov or Boltzmann equations in classical kinetic theory. In quantum kinetic theory, the Schrödinger equation for the complex-valued wave function in the physical space is converted into the Wigner equation for the real-valued Wigner function in phase space (which can take negative values). The Wigner ( quantum Vlasov ) equation contains the transport term of classical kinetic equations plus a pseudo-differential operator containing the potential. We give answers to the long-standing question of whether and to which extent averaging lemmas apply to the quantum case of the Wigner equation. The hard part are the semi-classical averaging lemmas, where one considers the asymptotics of vanishing Planck constant towards the nonnegative Wigner measure. In that context pure vs. mixed states play a crucial role, as well as the connection between the Schrödinger equation and Quantum Hydrodynamics (QHD). We present the results for the classical and quantum cases and sketch the semi-classical case which is worked out in full detail in a follow-up article.