Diffusive expansion in kinetic theory and dynamics of gaseous stars

This dissertation investigates the mathematical properties of nonlinear partial differential equations arising in kinetic theory, plasma physics, astrophysics, and fluid and gas dynamics. The first part of the thesis concerns the diffusive expansion to the Vlasov-Maxwell-Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDEs, the incompressible Vlasov-Navier-Stokes-Fourier system and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a unified nonlinear energy method and it guarantees the global in time validity of such an expansion up to any order. The Euler-Poisson system for inviscid gases and the Navier-Stokes-Poisson system for viscous gases are the fundamental models for the dynamics of self-gravitating gaseous stars. The second part of the thesis contributes to the rigorous study of the nonlinear instability of the Euler-Poisson system with the adiabatic exponent g=65 . The instability is established by developing a bootstrap argument from the linear instability to the nonlinear model. A number of weighted energy norms are constructed to close the nonlinear energy estimates. The third part of the thesis is devoted to investigating the local in time well-posedness of strong solutions to the full Navier-Stokes-Poisson system with spherical symmetry as a vacuum free boundary problem. In particular, the result captures the behavior of the Lane-Emden steady star configurations for all ranges of g∈65,2 . A weak solution is constructed via an iteration scheme defined in Lagrangian coordinates. A key idea in proving the uniform estimates as well as the regularity is to utilize both the Eulerian and Lagrangian formulations.