Averaging Lemmas and Dispersion Estimates for kinetic equations Pierre-Emmanuel Jabin email: Laboratoire J-A Dieudonne Universite de Nice Parc Valrose, 06108 Nice Cedex 02 Abstract. Averaging lemmas consist in a regularizing effect on the average of the solution to a linear kinetic equation. Some of the main results are reviewed and their proofs presented in as self contained a way as possible. The use of kinetic formulations for the well posedness of scalar conservation laws is eventually explained as an example of application. Key words. Regularizing effects, averaging lemmas, dispersion estimates, conservation laws. Mathematics Subject Classification 35B65, 82C40, 47G10. Introduction Kinetic equations are a particular case of transport equation in the phase space, i.e. on functions f(x, v) of physical and velocity variables like ∂tf + v · ?xf = g, t ≥ 0, x, v ? Rd. As a solution to a hyperbolic equation, the solution cannot be more regular than the initial data or the right hand-side. However a specific feature of kinetic equations is that the averages in velocity, like ?(t, x) = ∫ Rd f(t, x, v)?(v) dv, ? ? C∞c (R d), 1
- direction ? ?
- implies many
- indeed most
- sobolev spaces
- velocity averaging
- unique solution
- many proofs
- variables like