On convergence of exterior solutions to radial Cauchy solutions for $\square_{1+3}U=0$

TL;DR

This paper investigates how solutions to the 3D linear wave equation with radial initial data can be approximated by exterior solutions with boundary conditions on shrinking balls, establishing convergence in specific Sobolev spaces.

Contribution

It provides explicit formulas and proves convergence of exterior solutions to the Cauchy solution in $H^1$ and $H^2$ spaces, with different boundary conditions, as the domain shrinks to a point.

Findings

Solutions with Neumann conditions converge in $H^2$ as domain shrinks.

Solutions with Dirichlet conditions converge in $H^1$ as domain shrinks.

Explicit solution formulas facilitate the convergence analysis.

Abstract

Consider the Cauchy problem for the 3-d linear wave equation $\square_{1+3}U=0$ with radial initial data $U(0,x)=\Phi(x)=\phi(|x|)$, $U_t(0,x)=\Psi(x)=\psi(|x|)$. A standard result gives that $U$ belongs to $C([0,T];H^s(\mathbb{R}^3))$ whenever $(\Phi,\Psi)\in H^s\times H^{s-1}(\mathbb{R}^3)$. In this note we are interested in the question of how $U$ can be realized as a limit of solutions to initial-boundary value problems on the exterior of vanishing balls $B_\varepsilon$ about the origin. We note that, as the solutions we compare are defined on different domains, the answer is not an immediate consequence of $H^s$ well-posedness for the wave equation. We show how explicit solution formulae yield convergence and optimal regularity for the Cauchy solution via exterior solutions, when the latter are extended continuously as constants on $B_\varepsilon$ at each time. We establish that for $s=2$ the solution $U$ can be realized as an $H^2$-limit (uniformly in time) of exterior solutions on $\mathbb{R}^3\setminus B_\varepsilon$ satisfying vanishing Neumann conditions along $|x|=\varepsilon$, as $\varepsilon\downarrow 0$. Similarly for $s=1$: $U$ is then an $H^1$-limit of exterior solutions satisfying vanishing Dirichlet conditions along $|x|=\varepsilon$.