Resonant solutions and (in)stability of the linear wave equation
This addresses a foundational issue in the analysis of discretization methods for linear PDEs, offering incremental improvements to existing theoretical frameworks.
The paper investigates whether classical well-posedness of the acoustic wave equation implies an isomorphism between solution and data spaces, finding that standard Bochner spaces are incompatible due to resonant waves, and proposes a resonance-aware norm to establish such an isomorphism.
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.