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, $L<sup>2$ and $H<sup>1$, 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 $\mathcal{O}\subset \mathbb{R}<sup>n$ is an open convex obstacle with $C<sup>\infty$ boundary, and the governing hyperbolic operator is the wave operator $\Box:=\Delta-\partial_t<sup>2$.