Expanding solutions near unstable Lane-Emden stars

Abstract

We consider the compressible Euler-Poisson equations for polytropes $P(\rho)=K\rho^{\gamma}$ with $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right]$ and the white dwarf stars. For $\gamma=\frac{4}{3},$ we establish the existence of a global weak solution for the spherically symmetric initial data with mass less than the mass of the Lane-Emden stars (i.e. non-rotating polytropes). For $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right)$, we show the existence of global weak solution for spherical symmetric initial data in an invariant set containing a neighborhood of Lane-Emden stars. Moreover, the support of these solution expands to infinity. As a corollary, this proves the strong instability of the Lane-Emden stars for $\gamma\in \left( \frac{6}{5},\frac{4}{3}\right] $. For $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right),$ our results provide the first example of expanding solutions near the Lane-Emden stars. For white dwarf stars, we prove that the solution cannot collapse if the mass of initial data is less than the Chandrasekhar limit mass, which is the supremum of the mass of the non-rotating white dwarf stars. Our proof strongly uses the variational characterization of the Lane-Emden stars. First, we relate the best constant of a Hardy-Littlewood type inequality with the mass of the Lane-Emden stars with $\gamma=\frac{4}{3}$, which is further shown to equal the Chandrasekhar limit mass. For $\gamma\in\left( \frac{6}{5},\frac{4}{3}\right) $, we show that the Lane-Emden stars are minimizers of an energy-mass functional subject to a Pohozaev type constraint. This is crucial in the construction of the invariant set of expanding solutions.