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Global solutions to the isothermal Euler-Poisson system with arbitrarily large data. (English) Zbl 0845.35123
Summary: We prove the global existence of a solution to the Euler-Poisson system, with arbitrarily large data, in a one-dimensional geometry. The pressure law we consider, is deduced from an isothermal assumption for the electrons gas.
In this case, Nishida has already pointed out that the linear part of the Glimm functional is decreasing with respect to time. Using a Glimm scheme, he used this property to construct globally defined weak solutions for the Euler system with arbitrary large data. We follow his outline of proof. Here, a new difficulty arises with the source term, due to the electric field. A key point is that the Glimm scheme is almost conservative. This quasi-conservation of charge leads to a uniform estimate of the total variation of the electric field. This estimate allows to prove the convergence of the scheme.

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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