Nonresonant smoothing in Sobolev spaces for coupled wave+transport equations

Abstract

Consider a system consisting of a linear wave equation coupled to a transport equation: □ t,xu=f, (∂ t+v(ξ)·∇ x)f=P(t,x,ξ,D ξ)g, where P( t, x, ξ, D ξ ) is a linear differential operator of order m in ξ. Such a system is called nonresonant when the maximum speed in 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 question was investigated first in terms of Sobolev spaces H s in the paper of F. Bouchut, F. Golse and C. Pallard, Non-resonant smoothing for coupled wave+transport equations and the Vlasov–Maxwell system, (Rev. Mat. Iberoamericana, 2003, in press.) The same authors also studied a related question in On classical solutions to the 3D relativistic Vlasov–Maxwell system: Glassey–Strauss' theorem revisited (Arch. Rational Mech. Anal., in press). Here we state a result in Sobolev spaces W s, p . More precisely, if f, g belong to L p loc ( R ∗ +× R N× R M) and with initial data for u regular enough, then for any test function χ∈ C m c( R M ξ) we show that ∫u(·,·,ξ)χ(ξ) dξ∈W 1+γ,p loc R ∗ +× R N , when γ=1−(N−1)| 1 2 − 1 p |⩾0 and 1< p<+∞. We also study the limit cases p=1 and p=+∞ when N=3.