A note on the space-time variational formulation for the wave equation with source term in $L^2(Q)$

TL;DR

This paper develops a new variational formulation for the wave equation with source terms in L^2, providing a foundation for advanced space-time numerical methods and boundary integral equation analysis.

Contribution

It introduces a novel solution space and test space framework for the wave equation, ensuring well-posedness and applicability to space-time discretizations.

Findings

Proves existence and uniqueness in the new variational setting.

Shows the solution space is not a subspace of H^2(Q).

Establishes the framework's suitability for space-time boundary element methods.

Abstract

We derive a variational formulation for the scalar wave equation in the second-order formulation on bounded Lipschitz domains and homogeneous initial conditions. We investigate a variational framework in a bounded space-time cylinder $Q$ with a new solution space and the test space $L^2(Q)$ for source terms in $L^2(Q)$. Using existence and uniqueness results in $H^1(Q)$, we prove that this variational setting fits the inf-sup theory, including an isomorphism as solution operator. Moreover, we show that the new solution space is not a subspace of $H^2(Q)$. This new uniqueness and solvability result is not only crucial for discretizations using space-time methods, including least-squares approaches, but also important for regularity results and the analysis of related space-time boundary integral equations, which form the basis for space-time boundary element methods.