Transport of nonlinear oscillations along rays that graze a convex obstacle to any order

TL;DR

This paper develops a geometric optics framework to analyze how oscillations propagate along grazing rays near convex obstacles in low-regularity spaces, extending understanding of wave behavior in complex boundary conditions.

Contribution

It introduces a novel geometric optics approach for low-regularity spaces to describe oscillation transport along grazing rays near convex obstacles, including high-order grazing effects.

Findings

Describes oscillation transport along grazing rays in $L^2$ and $H^1$ spaces.

Extends analysis to high-order grazing and infinite order cases.

Provides a framework applicable to wave equations with boundary interactions.

Abstract

We provide a geometric optics description in spaces of low regularity, $L^2$ and $H^1$, of the transport of oscillations in solutions to linear and some semilinear second-order hyperbolic boundary problems along rays that graze the boundary of a convex obstacle to arbitrarily high finite or infinite order. The fundamental motivating example is the case where the spacetime manifold is $M=(\mathbb{R}^n\setminus \mathcal{O})\times \mathbb{R}_t$, where $\mathcal{O}\subset \mathbb{R}^n$ is an open convex obstacle with $C^\infty$ boundary, and the governing hyperbolic operator is the wave operator $\Box:=\Delta-\partial_t^2$.