We investigate the regularity of weak solutions of the relativistic Vlasov–Maxwell system by using Fourier analysis and the smoothing effect of low velocity particles. This smoothing effect has been used by several authors (see Glassey and Strauss 1986; Klainerman and Staffilani, 2002) for proving existence and uniqueness of [Formula: see text]-regular solutions of the Vlasov–Maxwell system. This smoothing mechanism has also been used to study the regularity of solutions for a kinetic transport equation coupled with a wave equation (see Bouchut, Golse and Pallard 2004). Under the same assumptions as in the paper “Nonresonant smoothing for coupled wave[Formula: see text]+[Formula: see text]transport equations and the Vlasov–Maxwell system”, Rev. Mat. Iberoamericana 20 (2004) 865–892, by Bouchut, Golse and Pallard, we prove a slightly better regularity for the electromagnetic field than the one showed in the latter paper. Namely, we prove that the electromagnetic field belongs to [Formula: see text], with [Formula: see text].
Partial Regularity of Weak Solutions of the Viscoelastic Navier--Stokes Equations with Damping
