In the Newtonian mechanics frame work, the motion of a gaseous star is modelled by the PDEs of compressible Euler-Poisson or Navier-Stokes-Poisson equations. The nonlinear stability of two equilibrium configurations, rotating and non-rotating star solutions, will be discussed in this talk.
The rotating star solutions are those for the Euler-Poisson equations with prescribed total mass and angular momentum (or angular velocity), for which the nonlinear orbital stability will be discussed from the energy-minimizing point of view, by using the approach developed by Arnold for Hamiltonian systems, using the conservative laws of total mass, energy and angular momentum.
The key issue here is that the energy functional is not positive definite. This part consists of the nonlinear orbital stability for both the polytropic stars and white dwarfs with the total mass beneath the Chandrasekhar limit, based on the joint work with J. Smoller.
The second part of this talk is on the nonlinear asymptotic stability of non-rotating viscous gaseous stars in the framework of the free boundary problem of compressible Navier-Stokes-Poisson equations with physical vacuum singularity, based on the joint work with Zhouping Xin and Huihui Zeng.
The key issue here is to deal with the strong degeneracy of the system near vacuum states and identify the suitable higher order nonlinear weighted functional to establish the higher order regularity uniform both in time and all the way up to the vacuum boundary.