On a Hilbert space framework for evolutionary systems of classical mathematical physics: a survey

Abstract

We survey a space–time Hilbert space approach to familiar dynamical partial differential equation problems from mathematical physics of the operator normal form $$\left( \partial _{0}{\mathcal {M}}+A\right) U=F,$$ F given. Here, $$\partial _{0}$$ denotes the time-derivative, A is a skew-selfadjoint spatial operator and $${\mathcal {M}}$$ describes the material properties. For sake of simplicity, we focus on the case of $${\mathcal {M}}$$ being such that $$\partial _{0}{\mathcal {M}}=\partial _{0}M_{0}+M_{1}$$ , where $$M_{0},M_{1}$$ are continuous spatial operators, the case of materials with no memory. This special case is shown to be closely related to space–time Friedrichs systems. The framework is illustrated with examples such as the classical acoustic, electromagnetic and elastic wave propagation, as well as the Klein–Gordon, Schrodinger and Dirac equation.