Global Finite-Energy Solutions of the Compressible Euler-Poisson Equations for General Pressure Laws with Spherical Symmetry

Abstract

We are concerned with global finite-energy solutions of the three-dimensional compressible Euler-Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. We construct global finite-energy solutions of the Cauchy problem for the Euler-Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the Navier-Stokes-Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in $L^p$, and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the viscosity coefficient. Then a special entropy pair is carefully designed by solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum ($ρ=0$) and far-field ($ρ=\infty$) are carefully analyzed. Owing to the generality of the pressure law, only the $W^{-1,p}_{\rm loc}$-compactness of weak entropy dissipation measures with $p\in [1,2)$ can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the $L^p$ compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.