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Convex waves grazing convex obstacles to high order

Jian Wang, Mark Williams

TL;DR

The paper develops and analyzes a geometric optics framework for high-frequency oscillations in solutions to hyperbolic boundary problems with convex obstacles, focusing on the wave operator. It formulates and studies two key geometric conditions, the grazing set (GS) and the reflected flow map (RFM), and extends the WW2023 results to incoming plane, spherical, and general convex waves. The authors establish when GS and RFM hold, provide a sharp 3D criterion (Theorem \ref{u1}) ensuring GS for spherical waves, and present explicit examples where GS fails (cusps) or holds only marginally, highlighting the delicate dependence on obstacle geometry and source position. They also prove global RFM results for spherical waves and more general convex incoming waves, and clarify the regularity and potential branching of grazing sets in various dimensions. Overall, the work clarifies when high-frequency asymptotics based on explicit profile equations accurately describe wave transport along grazing rays in convex geometries, with implications for rigorous boundary-layer analyses in hyperbolic PDEs.

Abstract

In a recent paper [WW23] we studied the transport of oscillations in solutions to linear and some semilinear second-order hyperbolic boundary problems along rays that graze a convex obstacle to any order. We showed that high frequency exact solutions are well approximated in $H^1$ by much simpler approximate solutions constructed from explicit solutions to profile equations. That result depends on two geometric assumptions, referred to here as the grazing set (GS) and reflected flow map (RFM) assumptions, that are both difficult to verify in general. The GS assumption states that the grazing set, that is, the set of points on the spacetime boundary at which incoming characteristics meet the boundary tangentially, is a codimension two, $C^1$ submanifold of spacetime. The second is that the reflected flow map, which sends points on the spacetime boundary forward in time to points on reflected and grazing rays, is injective and has appropriate regularity properties near the grazing set. In this paper we analyze these assumptions for incoming plane, spherical, and more general ``convex waves" when the governing hyperbolic operator is the wave operator $\Box:=Δ-\partial_t^2$. We prove general results describing when the assumptions hold, and provide explicit examples where the GS assumption fails.

Convex waves grazing convex obstacles to high order

TL;DR

The paper develops and analyzes a geometric optics framework for high-frequency oscillations in solutions to hyperbolic boundary problems with convex obstacles, focusing on the wave operator. It formulates and studies two key geometric conditions, the grazing set (GS) and the reflected flow map (RFM), and extends the WW2023 results to incoming plane, spherical, and general convex waves. The authors establish when GS and RFM hold, provide a sharp 3D criterion (Theorem \ref{u1}) ensuring GS for spherical waves, and present explicit examples where GS fails (cusps) or holds only marginally, highlighting the delicate dependence on obstacle geometry and source position. They also prove global RFM results for spherical waves and more general convex incoming waves, and clarify the regularity and potential branching of grazing sets in various dimensions. Overall, the work clarifies when high-frequency asymptotics based on explicit profile equations accurately describe wave transport along grazing rays in convex geometries, with implications for rigorous boundary-layer analyses in hyperbolic PDEs.

Abstract

In a recent paper [WW23] we studied the transport of oscillations in solutions to linear and some semilinear second-order hyperbolic boundary problems along rays that graze a convex obstacle to any order. We showed that high frequency exact solutions are well approximated in by much simpler approximate solutions constructed from explicit solutions to profile equations. That result depends on two geometric assumptions, referred to here as the grazing set (GS) and reflected flow map (RFM) assumptions, that are both difficult to verify in general. The GS assumption states that the grazing set, that is, the set of points on the spacetime boundary at which incoming characteristics meet the boundary tangentially, is a codimension two, submanifold of spacetime. The second is that the reflected flow map, which sends points on the spacetime boundary forward in time to points on reflected and grazing rays, is injective and has appropriate regularity properties near the grazing set. In this paper we analyze these assumptions for incoming plane, spherical, and more general ``convex waves" when the governing hyperbolic operator is the wave operator . We prove general results describing when the assumptions hold, and provide explicit examples where the GS assumption fails.

Paper Structure

This paper contains 19 sections, 15 theorems, 195 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions, results, and applications
  3. Incoming and reflected phases
  4. Orders of grazing
  5. Main results
  6. Applications
  7. The reflected flow map for spherical incoming waves
  8. The grazing set (GS) assumption
  9. Spherical waves grazing 2D obstacles
  10. Spherical waves grazing nD obstacles with an extra symmetry
  11. A class of 3D obstacles for which the GS assumption holds
  12. Branching does not occur in 3D near an isolated zero of Gauss curvature
  13. No branching
  14. Examples where the grazing set assumption fails or just barely holds
  15. Cusped grazing sets: examples where the grazing curve is continuous but not differentiable at $v=0$; spherical waves.
  16. ...and 4 more sections

Key Result

Proposition 2.10

Let $\beta(x,t)=x_1-F(\overline{x})$ and let Under Assumptions n1z (strict concavity of $F$) and grill5 ($(1,0,t)\in G_{\phi_i}$), we have $\underline{\sigma}\in\mathcal{G}_d$.

Figures (2)

  • Figure 1: Light Source at $(1,-1,0)$. First row: Cusped grazing set for the obstacle given by $x_1=1-(x_2^4+x_3^2)$. Second row: Cusped grazing set for the obstacle given by $x_1=1-(x_2^4+x_2^2 x_3^2+x_3^2)$. Colored according to level sets $x_1=\mathrm{const.}$.
  • Figure 2: Light source at $(1,-\infty,0)$. First row: Cusped grazing set for the obstacle given by $x_1=1-(x_2^4+2x_2^2 x_3^2 + x_2 x_3^2+x_3^2)$. Second row: $C^1$ grazing set for the obstacle given by $x_1=1-(x_2^4+x_2 x_3^4 + x_2^2 x_3^4+x_3^2)$. Colored according to level sets $x_1=\mathrm{const.}$.

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.3
  • Remark 2.5
  • Definition 2.7: melrosesjostrand1978cpam
  • Definition 2.8: Grazing points of order $\infty$
  • Definition 2.9: Glancing points of diffractive type
  • Proposition 2.10
  • proof
  • Remark 2.11
  • Remark 2.13
  • ...and 31 more