| Global weak solutions of the Navier-Stokes-Poisson equations for self-gravitating viscous gaseous stars are constructed with spherically symmetric initial data and a free boundary connected to a surrounding vacuum state. It indicates that the star keeps the star-shaped and bounded in any finite time, and never collapses under a critical gas adiabatic exponent gamma. We establish the energy estimates by showing that the positive kinetic-internal/dissipation energy can dominate the negative gravitational energy for the critical adiabatic exponent gamma, which is conjectured by astrophysicists. Another feature of this problem is the singularity of solutions near the free boundary and the origin. Our approach is to use an effective difference scheme to construct approximate solutions outside a solid ball and then derive some integrability of rho near the origin and other uniform estimates to pass the limit of the integral form. |
