Nonresonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system. (English) Zbl 1145.82338

Ben Abdallah, Naoufel (ed.) et al., Dispersive transport equations and multiscale models. Papers presented at the IMA workshops “Dispersive corrections to transport equations”, May 1–5, 2000, “Simulation of transport in transition systems”, May 22–26, 2000, and “Multiscale models for surface evolution and reacting flows”, June 5–9, 2000, Minneapolis, MN, USA. New York, NY: Springer (ISBN 0-387-40496-1/hbk). IMA Vol. Math. Appl. 136, 37-50 (2004).
Summary: Consider a system (see (1.1) below) consisting of a linear wave equation coupled to a transport equation. Such a system is called nonresonant when the maximum speed for particles governed by the transport equation is less than the propagation speed in the wave equation. Velocity averages of solutions to such nonresonant coupled systems are shown to be more regular than those of either the wave or the transport equation alone. This smoothing mechanism is reminiscent of the proof of existence and uniqueness of \(C^1\) solutions of the Vlasov-Maxwell system by Glassey-Strauss for time intervals on which particle momenta remain uniformly bounded. Applications of our smoothing results to solutions of the Vlasov-Maxwell system are discussed.
For the entire collection see [Zbl 1024.00072].


82C40 Kinetic theory of gases in time-dependent statistical mechanics
35B65 Smoothness and regularity of solutions to PDEs
35B34 Resonance in context of PDEs
35L05 Wave equation
35Q75 PDEs in connection with relativity and gravitational theory
82D10 Statistical mechanics of plasmas