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

2023/09/12 by Jian Wang, Mark Williams, Wang, Jian +1
Mathematics · #35L20 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2309.05910

openalex publication_date 2023/09/12 · openalex created_date 2023/09/14 · openalex updated_date 2026/07/28

Abstract

We provide a geometric optics description in spaces of low regularity, L² and H¹, 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=(ℝⁿ∖ O)× ℝ_t, where O⊂ ℝⁿ is an open convex obstacle with C^∞ boundary, and the governing hyperbolic operator is the wave operator \Box:=Δ-∂_t².

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