Transport of Nonlinear Oscillations Along Rays that Graze a Convex Obstacle to any Order

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 \(\Box :=\Delta -\partial _t^2\). The main theorem says that high frequency exact solutions are well approximated in spaces of low regularity by approximate solutions constructed from fairly explicit solutions to relatively simple profile equations. The theorem has two main assumptions. The first is 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. Both assumptions are in general hard to verify, but we show that they are satisfied for the diffraction of incoming plane waves by a large class of strictly convex obstacles in all dimensions, involving grazing points of arbitrarily high finite or infinite order.