12/11/2006

Enno Lenzman
MIT

Pseudo-relativistic nonlinear Schrodinger equations

This talk deals with a novel class of dispersive PDEs called pseudo-relativisitic nonlinear Schrodinger equations. These equations have recently found a significant application as effective descriptions for the dynamical evolution of self-gravitating, relativistic matter. Based on this physical motivation, I will present results that aim at understanding the qualitative behavior of solutions for these model equations. Here great emphasis is put on the pseudo-relativistic Hartree equation whose focusing nonlinearity is of critical strength in the sense that large initial data can lead to singularity formation resulting in finite-time blow-up of the solution. Such a breakdown indicates a ``gravitational collapse'' of the physical system (a boson star) modeled by this equation, and it substantiates the intuitive picture of collapsing stellar matter that exceeds a critical total mass. If time permits, I will also discuss very recent work on pseudo-relativistic Hartree-Fock equations} which provide an effective model for the dynamical evolution of fermion stars (such as white dwarfs). Part of the material covered in my talk is joint work with J. Frohlich (ETH Zurich) and L. Jonsson (KTH Stockholm).