We consider Coulomb particles of positive and negative charge moving on the real line. The velocity of each particle is given by the electric field induced by the other particles. Particles typically collide in finite time. Upon collision, we remove them from the system. The figure illustrates the trajectories of the particles.
Our main aim is to pass to the limit as the number of particles tends to infinity. The difficulty is due to both the singularity of the electric field and the particle collisions. We overcome this difficulty by using nifty ODE tricks and by fine-tuning the notion of viscosity solutions.
This is joint work with M.A. Peletier (Eindhoven U, Netherlands) and N. Pozar (Kanazawa U).