Yet another criterion for global existence in the 3D relativistic Vlasov-Maxwell system

We prove that solutions of the 3D relativistic Vlasov-Maxwell system can be extended, as long as the quantity $\sigma_{-1}(t, x) = \max_{|\omega|=1} \,\int_{R^3} \frac{dp}{\sqrt{1+p^2}}\, \frac{1}{(1+v\cdot\omega)}\, f(t, x, p)$ is bounded in $L^2_x$.Comment: 24 page

Slow Motion of Charges Interacting Through the Maxwell Field

We study the Abraham model for $N$ charges interacting with the Maxwell field. On the scale of the charge diameter, $R_{\phi}$, the charges are a distance \eps^{-1}R_{\phi} apart and have a velocity \sqrt{\eps} c with \eps a small dimensionless parameter. We follow the motion of the charges over times of the order \eps^{-3/2}R_{\phi}/c and prove that on this time scale their motion is well approximated by the Darwin Lagrangian. The mass is renormalized. The interaction is dominated by the instantaneous Coulomb forces, which are of the order \eps^{2}. The magnetic fields and first order retardation generate the Darwin correction of the order \eps^{3}. Radiation damping would be of the order \eps^{7/2}

The Vlasov-Poisson system with radiation damping

We set up and analyze a model of radiation damping within the framework of continuum mechanics, inspired by a model of post-Newtonian hydrodynamics due to Blanchet, Damour and Schaefer. In order to simplify the problem as much as possible we replace the gravitational field by the electromagnetic field and the fluid by kinetic theory. We prove that the resulting system has a well-posed Cauchy problem globally in time for general initial data and in all solutions the fields decay to zero at late times. In particular, this means that the model is free from the runaway solutions which frequently occur in descriptions of radiation reaction