Functional Analytic Methods for Evolution Systems

By: Joseph W. Jerome

By using two prototypical applications, the hydrodynamic-Maxwell system and the Navier-Stokes/charge transport system, we discuss the current relevance of local smooth theories for the Cauchy problem based on semigroup methods, and inspired by the Friedrichs and Kato inequalities. There appear to be three major advantages to the use of this theory: stability under the vanishing of diffusion or viscosity terms; flexibility in handling block systems, which are only partially symmetrizable; the use of implicit semidiscretization to determine estimates relating the size of the initial datum and the admissible terminal time.
This review paper has appeared in Contemporary Mathematics, volume 371, pp. 193--204, and can be viewed in the following format: