Incompletely Parabolic Systems from Friedrichs Theory Point of View

Abstract

An incompletely parabolic system of partial differential equations consist of one parabolic subsystem coupled to a hyperbolic subsystem. For the initial-boundary value problem, it has been shown, by requiring that a solution remains bounded at any time by the data, that boundary conditions which make both subsystems well-posed render the global system well-posed too. In this paper, we establish the same type of result with the help of the notion of semi-admissible boundary conditions in the theory of Friedrichs positive systems of differential equations. Our aim is not to obtain the same results of existence and uniqueness as those for the Cauchy problem, but rather to find a way to establish boundary conditions on the subsystems which can be used for the global system too. The theory is illustrated by the two examples of the compressible Navier–Stokes equations and of the hydrodynamic model for semiconductor devices, both in two space dimensions.