Local Well-Posedness of Dynamics of Viscous Gaseous Stars

We establish the local in time well-posedness of strong solutions to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system in the spherically symmetric and isentropic motion. Our result captures the physical vacuum boundary behavior of the Lane-Emden star configurations for all adiabatic exponents γ > 6 5 . 1 Formulation and Notation The motion of self-gravitating viscous gaseous stars can be described by the compressible NavierStokes-Poisson system: ∂ρ ∂t +∇ · (ρu) = 0, ∂(ρu) ∂t +∇ · (ρu⊗ u) +∇p = −ρ∇Φ+ μ△u, △Φ = 4πρ, (1.1) where t ≥ 0, x ∈ R, ρ ≥ 0 is the density, u ∈ R the velocity, p the pressure of the gas, Φ the potential function of the self-gravitational force, and μ > 0 the constant viscosity coefficient. We consider polytropic gases and the equation of state is given by p = Aρ where A is an entropy constant and γ > 1 is an adiabatic exponent; in this case, the motion is called barotropic, which means the pressure does not depend on the temperature or specific entropy. Values of γ have their own physical significance [3]; for example, γ = 5 3 stands for monatomic gas, 7 5 for diatomic gas, γ ց 1 for heavier molecules. These γ’s also take important part in the existence, uniqueness, and stability of stationary solutions, for instance, see [1, 5, 6] for inviscid gaseous stars modeled by the Euler-Poisson system. For the spherically symmetric motion, i.e. ρ(t,x) = ρ(t, r) and u(t,x) = u(t, r)x r , where u is a scalar function and r = |x|, (1.1) can be written as follows: ρt + 1 r2 (rρu)r = 0, ρut + ρuur + pr + 4πρ r2 ∫ r 0 ρsds = μ(urr + 2ur r − 2u r2 ). (1.2) Stationary solutions ρ = ρ0(r) and u = 0, non-moving gaseous spheres, satisfy the following: (p0)r + 4πρ0 r2 ∫ r