Radial solutions to the Cauchy problem for $\square_{1+3}U=0$ as limits of exterior solutions

Abstract

We consider the strategy of realizing the solution of a Cauchy problem with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the 3-dimensional, linear wave equation $\square_{1+3}U=0$ with radial Cauchy data $U(0,x)=\Phi(x)=\phi(|x|)$, $U_t(0,x)=\Psi(x)=\psi(|x|)$. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric Cauchy problem. In treating the 3-d wave equation we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3-d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in $C([0,T);H^s(\mathbb{R}^3))$ whenever $(\Phi,\Psi)\in H^s\times H^{s-1}(\mathbb{R}^3)$. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in $H^2\times H^1(\mathbb{R}^3)$ we obtain $H^1$-solutions via exterior Neumann solutions, and only $L^2$-solutions via exterior Dirichlet solutions.