TY - JOUR

T1 - Radial solutions to the Cauchy problem for 1+3U = 0 as limits of exterior solutions

AU - Jenssen, Helge Kristian

AU - Tsikkou, Charis

N1 - Funding Information:
The work of Jenssen was supported in part by the National Science Foundation under Grant DMS-1311353, and the work of Tsikkou was supported in part by the WVU ADVANCE Sponsorship Program.
Publisher Copyright:
© 2016 World Scientific Publishing Company.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - We consider the strategy of realizing the solution of a Cauchy problem (CP) 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 three-dimensional (3d), linear wave equation 1+3U = 0 with radial Cauchy data U(0,x) = Φ(x) = φ(/x/), Ut(0,x) = ψ(x) = φ(/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 CP. In treating the 3d 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 3d 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); Hs(R3)) whenever (Φ, Ψ) Hs × Hs-1(R3). 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 H2 × H1(R3), we obtain H1-solutions via exterior Neumann solutions, and only L2-solutions via exterior Dirichlet solutions.

AB - We consider the strategy of realizing the solution of a Cauchy problem (CP) 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 three-dimensional (3d), linear wave equation 1+3U = 0 with radial Cauchy data U(0,x) = Φ(x) = φ(/x/), Ut(0,x) = ψ(x) = φ(/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 CP. In treating the 3d 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 3d 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); Hs(R3)) whenever (Φ, Ψ) Hs × Hs-1(R3). 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 H2 × H1(R3), we obtain H1-solutions via exterior Neumann solutions, and only L2-solutions via exterior Dirichlet solutions.

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U2 - 10.1142/S0219891616500235

DO - 10.1142/S0219891616500235

M3 - Article

AN - SCOPUS:85006983985

VL - 13

SP - 833

EP - 860

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

SN - 0219-8916

IS - 4

ER -