  In this paper we study a self-consistent Vlasov-Fokker-Planck equations which
describes the longitudinal dynamics of an electron bunch in the storage ring of
a synchrotron particle accelerator. We show existence and uniqueness of global
classical solutions under physical hypotheses on the initial data. The proof
relies on a mild formulation of the equation and hypoelliptic regularization
estimates. We also address the problem of the long-time behavior of solutions.
We prove the existence of steady states, called Haissinski solutions, given
implicitly by a nonlinear integral equation. When the beam current (i.e. the
nonlinearity) is small enough, we show uniqueness of steady state and local
asymptotic nonlinear stability of solutions in appropriate weighted Lebesgue
spaces. The proof is based on hypocoercivity estimates. Finally, we discuss the
physical derivation of the equation and its particular asymmetric interaction
potential.
