Local properties of an isothermal charged fluid: Initial-boundary value problem

The Cauchy problem for the one-dimensional isothermal Euler-Poisson system was investigated by Poupaud, Rascle, and Vila in [F. Poupaud, M. Rascle, J.-P. Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data, J. Differential 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 [W. Fang, K. Ito, Weak solutions to a one-dimensional hydrodynamic model of two carrier types for semiconductors, 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 in its vanishing viscosity limit.