Large BV solutions to the compressible isothermal Euler–Poisson equations with spherical symmetry

Abstract

The Euler–Poisson equations for isothermal flows in charge transport are studied. The global weak entropy solution with spherical symmetry is constructed for arbitrarily large initial data in BV. The global solution is in BV and has large total variations. One difficulty is that the system of Euler–Poisson equations is a hyperbolic–elliptic coupled system, and the property of finite propagation speed does not hold. A modified Glimm scheme is used to construct the approximate solution of the Euler equations while the Poisson equation is approximated by an integral of the solution to the Euler equations. Other difficulties include the geometrical structure, the damping and some additional source terms, which make the analysis more complicated and difficult. To overcome these difficulties, the solution of a discrete ordinary differential equation is used to construct the approximate solution. It is proved that vacuum will not occur in the solution, which is important to obtain the compactness of the approximate solution in BV.