Non-Stationary Abstract Friedrichs Systems

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.