Separable Hamiltonian PDEs and Turning Point Principle for Stability of Gaseous Stars

We consider stability of nonrotating gaseous stars modeled by the Euler‐Poisson system. Under general assumptions on the equation of states, we proved a turning point principle (TPP) that the stability of the stars is entirely determined by the mass–radius curve parametrized by the center density. In particular, the stability can only change at extrema (i.e., local maximum or minimum points) of the total mass. For a very general equations of state, TPP implies that for increasing center density the stars are stable up to the first mass maximum and unstable beyond this point until the next mass extremum (a minimum). Moreover, we get a precise counting of unstable modes and exponential trichotomy estimates for the linearized Euler‐Poisson system. To prove these results, we develop a general framework of separable Hamiltonian PDEs. The general approach is flexible and can be used for many other problems, including stability of rotating and magnetic stars, relativistic stars, and galaxies. © 2021 Wiley Periodicals LLC.