Local Properties of an Isothermal Charged Fluid:
Initial-Boundary Value Problem
By: Joseph W. Jerome
The Cauchy problem for the one-dimensional isothermal Euler-Poisson system was
investigated by F. Poupaud, M. Rascle, and J.-P. Vila in [J. Diff.
Equations 123 (1995), 93--121]. Glimm's scheme was employed to obtain a
global entropic solution. It appears that the initial-boundary value
problem has not
been investigated previously for this system, except in the isentropic
case,
via an approach based on
compensated compactness.
While the isothermal case, employing the ideal gas law for the pressure,
suggests artificial diffusion/viscosity,
the underlying infrastructure
(the analog of Glimm's scheme) has not yet been established to
analyze the initial-boundary value problem.
We begin a program here,
utilizing diffusion and viscosity. By employing Kato's theory of
evolution operators,
we provide a local smooth existence/uniqueness theory.
A theorem of Smoller, as
generalized by Fang and Ito [Nonlinear Anal. 28 (1997), 947--963],
is used to obtain
invariant region bounds for the evolution. Because the theory is local
shocks do not appear, either in the parabolic system, or its vanishing
viscosity limit.
This paper will appear in Nonlinear Analysis, vol. 69 (3) (2008),
866--873. DOI: 10.1016/j.na.2008.02.045.
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