Non-Stationary Abstract Friedrichs Systems

Abstract

Inspired by the results of Ern et al. (Commun Partial Differ Equ 32:317–341, 2007) on the abstract theory for Friedrichs symmetric positive systems, we give the existence and uniqueness result for the initial- (boundary) value problem for the non-stationary abstract Friedrichs system. Despite the absence of the well-posedness result for such systems, there were already attempts for their numerical treatment by Burman et al. (SIAM J Numer Anal 48:2019–2042, 2010) and Bui-Thanh et al. (SIAM J Numer Anal 51:1933–1958, 2013). We use the semigroup theory approach and prove that the operator involved satisfies the conditions of the Hille–Yosida generation theorem. We also address the semilinear problem and apply the new results to a number of examples, such as the symmetric hyperbolic system, the unsteady div–grad problem, and the wave equation. Special attention was paid to the (generalised) unsteady Maxwell system.