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The Cauchy Problem for Compressible Hydrodynamic-Maxwell systems: A Local
Theory for Smooth Solutions

#### By: Joseph W. Jerome

The Hydrodynamic-Maxwell equations are studied,
as a compressible model of charge transport
induced by an electromagnetic field in semiconductors.
A local smooth solution theory for the Cauchy problem is established by
the author's modification of the classical semigroup-resolvent
approach of Kato. The
author's theory has three
noteworthy features: (1) stability under vanishing heat flux, which is not
derivable from other theories;
(2)
accommodation to arbitrarily specified terminal time for the regularized
problem; and,
(3) constructive in nature, in that it is based upon time semidiscretization,
and the solution of these semidiscrete problems determines the
localization theory criteria.
The regularization is employed to avoid vacuum states, and
eliminated for the final results which may contract the admissible
time interval.
We also provide a symmetrized formulation in matrix form which is useful for
applications and simulation.
The theory uses the generalized energy estimates of Friedrichs on the
ground function space, and leverages them to the smooth space via
Kato's commutator
estimate.

This paper will appear in (the) Journal of Differential and Integral Equations
16 (2003), 1345--1368,
and can be viewed in the following format: