Nonlinear instability theory of Lane-Emden stars (Q2875806)

In this paper the author considers the compressible Euler-Poisson system. It is the simplest hydrodynamical model describing the motion of self-gravitating Newtonian inviscid gaseous stars. This system contains PDEs having the form: NEWLINE\[NEWLINE \partial_t\rho + \nabla\cdot (\rho \mathbf{u}) = 0, \;\;\rho (\partial_t\mathbf{u}+\mathbf{u}\cdot\nabla\mathbf{u})+ \nabla p = - \rho\nabla\Phi , \;\;\triangle\Phi = 4\pi\rho , NEWLINE\]NEWLINE where \((t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{3}\) and \(\rho \), \(\mathbf{u}\), \(p\) are the density, velocity, and pressure of the gas, respectively; \(\Phi \) is the gravitational potential. Here \(p=K\rho^{\gamma }\), where \(K\) is the entropy constant, and \(\gamma > 1\) is the adiabatic gas exponent. It turns out that the above stated system under the spherically symmetric transformation \(\rho (t,x)=\rho (t,x)\), \(\mathbf{u}(t,x)=u(t,r)x/r\) (\(r=|x|\)) takes the form: NEWLINE\[NEWLINE \rho_t + r^{-2}(r^2\rho u)_{r} = 0, \;\;\rho u_t+\rho u u_r+p_r+4\pi\rho r^{-2}\int_{0}^{r} \rho s^2ds=0 . NEWLINE\]NEWLINE An interesting fact is that the stationary solutions (\(\rho_0(r), \;u_0=0\)) satisfy the ODE NEWLINE\[NEWLINE dp/dr+4\pi\rho r^{-2}\int_{0}^{r} \rho s^2ds = 0, NEWLINE\]NEWLINE which may transform in the known Lane-Emden equation. Note that the total energy \(E(\rho ,u)\) of the Euler-Poisson system is not positive definite, which is an obstacle to solving the stability problem. Therefore it is meaningfully to consider the corresponding Lane-Emden energy \(E(\rho )={4-3\gamma \over \gamma -1} \int pdx\). Thus, \(\gamma \) is a parameter of interest for the stability discussion in this paper. The author considers the case \(6/5<\gamma <4/3\) that seems of special physical interest. The compactly supported Lane-Emden solutions are estimated by \(\rho_0(r)\sim (R-r)^{1/(\gamma -1)}\) for \(r\sim R\) near the vacuum boundary, thus the stability gets an open problem being of interest. Furthermore, the author establishes a nonlinear instability theory of ``compactly supported Lane-Emden steady stars provided that \(6/5<\gamma <4/3\) in the presence of the vacuum boundary''.NEWLINENEWLINEThe main result stated here is that there exists a family of solutions to the Euler-Poisson system which belong to the functional space \(Z_{1}^{\alpha }\) (\(3<\alpha <5\)) with suitable norm, and sufficiently small initial data in the functional space \(Z_{0}^{\alpha }\) for \(t\in[0,T)\) (\(T>T^{\delta }\), \(\delta >0\)).