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Highly oscillatory multidimensional shocks
Author(s) -
Williams Mark
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199902)52:2<129::aid-cpa1>3.0.co;2-w
Subject(s) - mathematics , lipschitz continuity , divisor (algebraic geometry) , continuation , shock (circulatory) , mathematical proof , mathematical analysis , property (philosophy) , variety (cybernetics) , order (exchange) , pure mathematics , geometry , medicine , philosophy , statistics , epistemology , finance , computer science , economics , programming language
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 ± ϵ , the solutions on each side of S ϵ . The profile equations yield detailed information on the evolution of ( u ± ϵ , 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, 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. © 1999 John Wiley & Sons, Inc.