Resonant solutions and (in)stability of the linear wave equation

TL;DR

This paper investigates the stability and solution-data relationships of the linear wave equation, revealing resonance effects that challenge classical assumptions and proposing resonance-aware norms for improved isomorphism.

Contribution

It demonstrates that classical solution and data spaces may be incompatible due to resonant waves and introduces resonance-aware norms to establish isomorphisms.

Findings

Classical Bochner spaces may not support an isomorphism due to resonant waves.

Resonance-aware norms can restore the isomorphism between solutions and data.

Results extend to other linear time-dependent PDEs.

Abstract

We revise the analysis of the acoustic wave equation, addressing the question whether the classical well-posedness implies the existence of an isomorphism between prescribed solution and data spaces. This question is of interest for the design and the analysis of discretization methods. Expanding on existing results, we point out that established choices of solution and data space in terms of classical Bochner spaces must be expected to be incompatible with the existence of such an isomorphism, because of resonant waves. We formulate this observation in the language of the so-called inf-sup theory, with the help of an eigenfunction expansion, which reduces the original partial differential equation to a system of ordinary differential equations. We further verify that an isomorphism can be established, for each equation in the system, upon equipping the data space with a suitable resonance-aware norm. In the appendix, we extend our results to other time-dependent linear PDEs.