Caustics for dissipative semilinear oscillations

Memoirs of the American Mathematical Society

Volumn , Issue 685, 2000, Pages 1-72

  Joly, Jean Luc ^a   Metivier, Guy ^b   Rauch, Jeffrey ^c  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b UNIVERSITÉ DE RENNES 1   (France)

^c UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034147311     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (28)

1. [A] G. Allaire, Homogeneization and two-scale analysis, S.I.A.M. J. Math. Anal., 23, 1992, 1482-1518.
2. [CDMM] M. Cowling, S. Disney, G. Mauceri and D. Müller, Damping oscillatory integrals, Invent. Math., 101, 1990, 237-260.
3. [Du] J.J. Duistermaat, Oscillatory integrals, Lagrangian immersions and unfolding of singularities, Comm. Pure Appl. Math. 27,1974, 207-281.
4. [DH] J.J. Duistermaat and L. Hörmander, Fourier integral operators II, Acta. Math., 128, 1972, 183-269.
5. [E] W. E, Homogeneization of linear and nonlinear transport equations, Comm. Pure Appl. Math, 45, 1992, 301-326.
6. [EG] L. Evans, R. Gariepy , Measure theory and fine properties of functions, Studies in Advanced Mathematic. Boca Raton, CRC Press, 1992.
7. [ES] W. E, D. Serre, Correctors for the homogeneization of conservation laws with oscillatory forcing terms, Asymptotic Analysis, 5, 1992, 311-316.
8. [Ha] A. Haraux, Semilinear Hyperbolic Problems in Bounded Domains, Mathematical Reports, vol. 3, Part 1, Harwood Academic Publishers, 1987.
9. [Hö 1] L. Hörmander, Fourier integral operators I, Acta. Math., 127, 1971, 79-183.
10. [Hö 2] L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, 1991.
11. [HMR] J. Hunter, A. Majda and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves II: several space variables, Stud. Appl. Math., 75,1986, 187-226.
12. [JMR 1] J.-L. Joly, G. Métivier and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics, Annales Scientifique de l'École Normale Supérieure, 28, 1995, 51-113.
13. [JMR 2] J.-L. Joly, G. Métivier and J. Rauch, Focusing at a point and absorption of nonlinear oscillations, Trans. Amer. Math. Soc., 347, 10, 1995, 3921-3969.
14. [JMR 3] J.-L. Joly, G. Métivier and J. Rauch, Nonlinear oscillations beyond caustics, Comm. Pure Appl. Math., Vol XLIX, 5, 1996, 443-527.
15. [JMR 4] J.-L. Joly, G. Métivier and J. Rauch, Several Recent Results in Nonlinear Geometric Optics, Partial Differential Equations and Mathematical Physics, The Danish-Swedish Analysis Seminar, Birkhauser, 1995, 181-207.
16. [JR] J.L. Joly and J. Rauch, Justification of multidimensional single phase semilinear geometric optics, Trans. Amer. Math. Soc., 330, 1992, 599-625.
17. [La] P. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. Journal, 24, 1957, 627-645.
18. [L] J.-L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
19. [LS] J.-L. Lions and W. Strauss, Some nonlinear evolution equations, Bull.Soc. Math. France, 93, 1965, 43-96.
20. [Lu] D. Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure Appl. Math., 13, 1966, 85-114.
21. [McLPT] D.W. McLaughlin, G. Papanicolaou and L. Tartar, Weak limits of semi-linear hyperbolic systems with oscillating data, in Macroscopic Modelling of Turbulent Flow, Lecture Notes in Physics, 230, 1985, 277-298.
22. [N] G. Nguetseng, A general convergence result for a functional related to the theory of homogeneization, Siam J. Math. Anal., 20, 1989, 608-623.
23. [R 1] B. Randol, On the Fourier transform of the indicator function of a planar set, Trans. Amer. Math. Soc., 139, 1969, 271-278.
24. [R 2] B. Randol, On the asymptotic behavior of the Fourier transform of the indicator function of a convex set, Trans. Amer. Math. Soc., 139, 1969, 279-285.
25. [Ste] E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, 1993.
26. [Str] W. Strauss, The Energy Method in Nonlinear Partial Differential Equations, Notas de matematica 47, Instituto de Matematica Pura e Applicata, Rio de Janeiro 1969.
27. [So] C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105, Cambridge University Press, 1993
28. [Sv] I. Svensson, Estimates for the Fourier transform of the characteristic function of a convex set, Afk. för Mat., 9, 1971, 11-22.