Convex waves grazing convex obstacles to high order

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 $\Box:=\Delta-\partial_t^2$. We prove general results describing when the assumptions hold, and provide explicit examples where the GS assumption fails.