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[Dispersion for the wave equation in the exterior of the torus]{Dispersion for the wave equation in the exterior of the torus in three dimensions}

Ronald Quirchmayr, Alden Waters

TL;DR

This work develops a high-frequency dispersive analysis for the three-dimensional wave equation in the exterior of a torus by introducing a Pöschl-Teller-based operator Δ_P in toroidal coordinates, whose principal symbol matches the Dirichlet Laplacian. By leveraging a Mehler-Fock-type kernel and generalized eigenfunctions, the authors construct a Green operator e^{it√Δ_P} that yields global dispersive estimates away from the torus boundary, circumventing the lack of separability present in non-convex obstacles. The main contribution is a rigorous L^1→L^∞ dispersive bound for high-frequency data, quantified with region-dependent constants and applicable away from the inner boundary, alongside a detailed kernel representation. The results provide a novel parametrix approach for conservative PDEs in non-convex exterior domains and open avenues for addressing boundary effects, inner-ring trapping, and broader frequency ranges in future work.

Abstract

We prove dispersive estimates for the wave equation in the exterior of a torus. Because no separation of variables into a basis of eigenfunctions and eigenvalues exists for the time harmonic problem, we introduce a related approximate operator for the Dirichlet Laplacian in the exterior of a torus. The approximate operator coincides with the Schrödinger operator with a Pöschl-Teller potential and agrees with the Dirichlet Laplacian to leading order. The operator here which we develop is related to the so-called Mehler-Fock kernel. Using the known solution to the eigenvector and eigenvalue problem of Pöschl-Teller, a high-frequency analysis of the approximate operator for the wave equation can be made accurately. The operator for this problem gives a close approximation to the $L^1\rightarrow L^{\infty}$ dispersive estimate at a suitable small distance from the torus for the corresponding exterior wave operator with Dirichlet Laplacian.

[Dispersion for the wave equation in the exterior of the torus]{Dispersion for the wave equation in the exterior of the torus in three dimensions}

TL;DR

This work develops a high-frequency dispersive analysis for the three-dimensional wave equation in the exterior of a torus by introducing a Pöschl-Teller-based operator Δ_P in toroidal coordinates, whose principal symbol matches the Dirichlet Laplacian. By leveraging a Mehler-Fock-type kernel and generalized eigenfunctions, the authors construct a Green operator e^{it√Δ_P} that yields global dispersive estimates away from the torus boundary, circumventing the lack of separability present in non-convex obstacles. The main contribution is a rigorous L^1→L^∞ dispersive bound for high-frequency data, quantified with region-dependent constants and applicable away from the inner boundary, alongside a detailed kernel representation. The results provide a novel parametrix approach for conservative PDEs in non-convex exterior domains and open avenues for addressing boundary effects, inner-ring trapping, and broader frequency ranges in future work.

Abstract

We prove dispersive estimates for the wave equation in the exterior of a torus. Because no separation of variables into a basis of eigenfunctions and eigenvalues exists for the time harmonic problem, we introduce a related approximate operator for the Dirichlet Laplacian in the exterior of a torus. The approximate operator coincides with the Schrödinger operator with a Pöschl-Teller potential and agrees with the Dirichlet Laplacian to leading order. The operator here which we develop is related to the so-called Mehler-Fock kernel. Using the known solution to the eigenvector and eigenvalue problem of Pöschl-Teller, a high-frequency analysis of the approximate operator for the wave equation can be made accurately. The operator for this problem gives a close approximation to the dispersive estimate at a suitable small distance from the torus for the corresponding exterior wave operator with Dirichlet Laplacian.

Paper Structure

This paper contains 14 sections, 17 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction and statement of the main theorem
  2. Toroidal coordinates
  3. Laplacian in toroidal coordinates
  4. Formulation of the obstacle problem
  5. Review of the solutions to the Attractive Pöschl-Teller Equation
  6. The Kernel Representation
  7. Dispersive estimates
  8. Conclusion and Future Directions
  9. Special functions
  10. Gamma function identities
  11. Bessel, modified Bessel & Hankel functions
  12. The Mehler-Fock Kernel
  13. Asymptotics for the kernel
  14. Further error estimates

Key Result

Theorem 1.1

There exists a constant $C$ such that for all $t\geq 0$ and $h\in(0,1)$,

Figures (1)

  • Figure 1: A two-dimensional cross section: the solid torus is indicated by the two dark discs; the outer striped (gray) area shows the set $V_{\epsilon_1}$ from Lemma \ref{['Nbelow']}; solid lines depict $\phi_1$ and $\tau$ isosurfaces.

Theorems & Definitions (37)

  • Theorem 1.1: see e.g. BCDbook
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 27 more