We are concerned with a global existence theory for finite-energy solutions of the multidimensional Euler-Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main challenges is the strengthening of waves as they move radially inward towards the origin, especially under the self-consistent gravitational field for gaseous stars. A fundamental unsolved problem is whether the density of the global solution forms a delta measure (i.e., concentration) at the origin. To solve this problem, we develop a new approach for the construction of approximate solutions as the solutions of an appropriately formulated free boundary problem for the compressible Navier-Stokes-Poisson equations with a carefully adapted class of degenerate density-dependent viscosity terms, so that a rigorous convergence proof of the approximate solutions to the corresponding global solution of the compressible Euler-Poisson equations with large initial data of spherical symmetry can be obtained. Even though the density may blow up near the origin at a certain time, it is proved that no delta measure (i.e., concentration) in space-time is formed in the vanishing viscosity limit for the finite-energy solutions of the compressible Euler-Poisson equations for both gaseous stars and plasmas in the physical regimes under consideration.

中文翻译：

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球对称性大初始数据可压缩欧拉-泊松方程的全局解

我们关注的是具有大量球对称初始数据的可压缩气态恒星和等离子体的多维欧拉-泊松方程的有限能量解的全局存在理论。主要挑战之一是当波径向向内移向原点时波的增强，特别是在气态恒星的自洽引力场下。一个尚未解决的基本问题是全局解的密度是否在原点形成增量测度（即浓度）。为了解决这个问题，我们开发了一种构造近似解的新方法，作为可压缩纳维-斯托克斯-泊松方程的适当表述的自由边界问题的解，该方程具有精心调整的简并密度相关粘度项类，因此可以得到具有大量球对称初始数据的可压缩Euler-Poisson方程组相应全局解的近似解的严格收敛性证明。尽管密度可能在某个时间在原点附近爆炸，但证明了可压缩欧拉-泊松有限能量解的消失粘度极限中不会形成时空δ测度（即浓度）所考虑的物理状态中气态恒星和等离子体的方程。