Global Solutions of the Compressible Euler and Euler-Poisson Equations with Large Initial Data of Spherical Symmetry
Time: 2024-05-06
Published By: He Liu
Speaker(s): Yong WANG(CAS)
Time: 15:45-16:45 May 20, 2024
Venue: Siyuan Lecture Hall, Zhihua Building
In this talk, we are concerned with the global existence theory for finite-energy solutions of the multidimensional compressible Euler equations and Euler-Poisson equations (both gaseous stars and plasmas are included) 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. We develop a new approach for the construction of approximate solutions as the solutions of an appropriately formulated 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 equations and 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. The talk is based on joint works with G.Q. Chen, F.M. Huang, T.H. Li, L. He, W.Q. Wang, D.F. Yuan.
