Highly oscillatory multidimensional shocks

We construct geometric optics expansions of high order for oscillatory multidimensional shocks and then show that the expansions are close to exact shock solutions for small wavelengths. Expansions are constructed both for S_ϵ, the oscillatory function defining the shock surface S_ϵ, and for u, the solutions on each side of S_ϵ. The profile equations yield detailed information on the evolution of (u, ψ_ϵ), showing, for example, how new interior oscillations are produced by a variety of shock—interior and interior—interior interactions. A generic small divisor property, L²‐estimates for linearized shock problems with merely Lipschitz coefficients, and a continuation principle based on an unusual Gagliardo‐Nirenberg inequality all play a role in the proofs. © 1999 John Wiley & Sons, Inc.