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

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.