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

TL;DR

This work develops a space-time variational framework for the scalar wave equation in second-order form on a bounded Lipschitz domain, targeting source terms $f\in L^2(Q)$. By introducing the new solution space $H^1_{0;0,}(Q;\square)$, which enforces $\square v\in L^2(Q)$ and $\partial_t v(\cdot,0)=0$, and defining $a(v,w)=\langle \square v, w\rangle_{L^2(Q)}$, the authors establish an inf-sup compliant formulation that yields a unique solution operator with $\|\square u\|_{L^2(Q)} = \|f\|_{L^2(Q)}$. They also prove that the new space is not contained in $H^2(0,T;L^2(\Omega))$, clarifying regularity limitations. The results enable robust space-time discretizations, least-squares approaches, and space-time boundary integral methods, providing a solid theoretical foundation for space-time analysis and numerical schemes for hyperbolic problems.

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.

Theorems & Definitions (7)

• Theorem 1
• Lemma 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof