✦ LIBER ✦

Highly oscillatory multidimensional shocks

✍ Scribed by Mark Williams

Publisher: John Wiley and Sons
Year: 1999
Tongue: English
Weight: 811 KB
Volume: 52
Category: Article
ISSN: 0010-3640
DOI: 10.1002/(sici)1097-0312(199902)52:2<129::aid-cpa1>3.0.co;2-w

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✦ Synopsis

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 ψε, the oscillatory function defining the shock surface Sε, and for u ± ε , the solutions on each side of Sε. The profile equations yield detailed information on the evolution of (u ± ε , ψε), showing, for example, how new interior oscillations are produced by a variety of shock-interior and interior-interior interactions. A generic small divisor property, L 2 -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.

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