Nonresonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system

Abstract

Consider a system consisting of a linear wave equation coupled to a transport equation: \begin{equation*} \Box_{t,x}u =f , \end{equation*} \begin{equation*} (\partial_t + v(\xi) \cdot \nabla_x)f =P(t,x,\xi, D_\xi)g , \end{equation*} Such a system is called \textit{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 R. Glassey and W. Strauss for time intervals on which particle momenta remain uniformly bounded, see ``Singularity formation in a collisionless plasma could occur only at high velocities'', \textit{Arch. Rational Mech. Anal.} \textbf{92} (1986), no. 1, 59-90. Applications of our smoothing results to solutions of the Vlasov-Maxwell system are discussed.