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Nonresonant smoothing for coupled wave and transport equations and the Vlasov-Maxwell system. (English) Zbl 1064.35097

Summary: Consider a system consisting of a linear wave equation coupled to a transport equation: \[ \square_{t,x}u=f,\quad (\partial_t+ v(\xi)\cdot \nabla_x)f =P(t,x,\xi,D_\xi)g, \] 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 R. Glassey and W. Strauss for time intervals on which particle momenta remain uniformly bounded, in [Arch. Ration. Mech. Anal. 92, 59–90 (1986; Zbl 0595.37072)]. Applications of our smoothing results to solutions of the Vlasov-Maxwell system are discussed.

MSC:

35L05 Wave equation
35B65 Smoothness and regularity of solutions to PDEs
35F20 Nonlinear first-order PDEs
35Q75 PDEs in connection with relativity and gravitational theory
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D10 Statistical mechanics of plasmas

Citations:

Zbl 0595.37072
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References:

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