Trilinear resonant interactions of semilinear hyperbolic waves
G. Métivier and S. Schochet
Source: Duke Math. J. Volume 95, Number 2
(1998), 241-304.
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077229698
Mathematical Reviews number (MathSciNet): MR1652009
Zentralblatt MATH identifier: 0955.35007
Digital Object Identifier: doi:10.1215/S0012-7094-98-09508-4
References
[A] G. Allaire, Homogenization and two scale convergence, SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518.
Mathematical Reviews (MathSciNet): MR93k:35022
Zentralblatt MATH: 0770.35005
Digital Object Identifier: doi:10.1137/0523084
[BB] W. Blaschke and G. Bol, Geometrie der Gewebe, Springer-Verlag, New York, 1938.
Zentralblatt MATH: 0020.06701
[Br] L. Breiman, Probability, Addison-Wesley, Reading, Mass., 1968.
Mathematical Reviews (MathSciNet): MR37:4841
Zentralblatt MATH: 0174.48801
[D] R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70.
Mathematical Reviews (MathSciNet): MR84k:35091
Zentralblatt MATH: 0519.35054
Digital Object Identifier: doi:10.1007/BF00251724
[E] W. E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math. 45 (1992), no. 3, 301–326.
Mathematical Reviews (MathSciNet): MR92k:35026
Zentralblatt MATH: 0794.35014
Digital Object Identifier: doi:10.1002/cpa.3160450304
[ES] W. E and D. Serre, Correctors for the homogenization of conservation laws with oscillatory forcing terms, Asymptotic Anal. 5 (1992), no. 4, 311–316.
Mathematical Reviews (MathSciNet): MR92m:35170
Zentralblatt MATH: 0766.35026
[Ev] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conf. Ser. in Math., vol. 74, Amer. Math. Soc., Providence, 1990.
Mathematical Reviews (MathSciNet): MR91a:35009
Zentralblatt MATH: 0698.35004
[G] P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations 16 (1991), no. 11, 1761–1794.
Mathematical Reviews (MathSciNet): MR92k:35027
Zentralblatt MATH: 0770.35001
Digital Object Identifier: doi:10.1080/03605309108820822
[Her] I. N. Herstein, Topics in Algebra, 2nd ed., Xerox College Publishing, Lexington, Mass., 1975.
Mathematical Reviews (MathSciNet): MR50:9456
Zentralblatt MATH: 0122.01301
[HR] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press Inc., Publishers, New York, 1963.
Mathematical Reviews (MathSciNet): MR28:158
Zentralblatt MATH: 0115.10603
[H] J. K. Hunter, Interacting weakly nonlinear hyperbolic and dispersive waves, Microlocal Analysis and Nonlinear Waves (Minneapolis, Minn., 1988–1989) eds. M. Beals, R. B. Melrose, and J. Rauch, IMA Vol. Math. Appl., vol. 30, Springer-Verlag, New York, 1991, pp. 83–111.
Mathematical Reviews (MathSciNet): MR92e:35103
Zentralblatt MATH: 0794.35024
[HK] J. Hunter and J. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math. 36 (1983), no. 5, 547–569.
Mathematical Reviews (MathSciNet): MR85h:35143
Zentralblatt MATH: 0547.35070
Digital Object Identifier: doi:10.1002/cpa.3160360502
[HMR] J. Hunter, A. Majda, and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves, II: Several space variables, Stud. Appl. Math. 75 (1986), no. 3, 187–226.
Mathematical Reviews (MathSciNet): MR89h:35199
Zentralblatt MATH: 0657.35084
[J] J.-L. Joly, Sur la propagation des oscillations par un système hyperbolique semi-linéaire en dimension $1$ d'espace, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 15, 669–672.
Mathematical Reviews (MathSciNet): MR84j:35112
Zentralblatt MATH: 0555.35081
[JMR1] J.-L. Joly, G. Metivier, and J. Rauch, Resonant one dimensional nonlinear geometric optics, J. Funct. Anal. 114 (1993), no. 1, 106–231.
Mathematical Reviews (MathSciNet): MR94i:35118
Zentralblatt MATH: 0851.35023
Digital Object Identifier: doi:10.1006/jfan.1993.1065
[JMR2] J.-L. Joly, G. Metivier, and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 1, 51–113.
Mathematical Reviews (MathSciNet): MR95k:35035
Zentralblatt MATH: 0836.35087
[JMR3] J.-L. Joly, G. Metivier, and J. Rauch, Focusing at a point and absorption of nonlinear oscillations, Trans. Amer. Math. Soc. 347 (1995), no. 10, 3921–3969.
Mathematical Reviews (MathSciNet): MR95m:35024
Zentralblatt MATH: 0857.35087
Digital Object Identifier: doi:10.2307/2155210
JSTOR: links.jstor.org
[JMR4] J.-L. Joly, G. Metivier, and J. Rauch, Trilinear compensated compactness and nonlinear geometric optics, Ann. of Math. (2) 142 (1995), no. 1, 121–169.
Mathematical Reviews (MathSciNet): MR96m:35194
Zentralblatt MATH: 0840.35013
Digital Object Identifier: doi:10.2307/2118612
JSTOR: links.jstor.org
[KK] M. Kline and I. W. Kay, Electromagnetic Theory and Geometrical Optics, Pure and Appl. Math., vol. 12, Wiley-Interscience, New York, 1965.
Mathematical Reviews (MathSciNet): MR31:4330
Zentralblatt MATH: 0123.23602
[L] P. D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24 (1957), 627–646.
Mathematical Reviews (MathSciNet): MR20:4096
Zentralblatt MATH: 0083.31801
Digital Object Identifier: doi:10.1215/S0012-7094-57-02471-7
Project Euclid: euclid.dmj/1077467631
[MR] A. Majda and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves. I. A single space variable, Stud. Appl. Math. 71 (1984), no. 2, 149–179.
Mathematical Reviews (MathSciNet): MR86e:35089
Zentralblatt MATH: 0572.76066
[MPT] D. McLaughlin, G. Papanicolaou, and L. Tartar, Weak limits of semilinear hyperbolic systems with oscillating data, Macroscopic Modelling of Turbulent Flows (Nice, 1984), Lecture Notes in Phys., vol. 230, Springer-Verlag, Berlin, 1985, pp. 277–289.
Mathematical Reviews (MathSciNet): MR815948
Zentralblatt MATH: 0588.76137
Digital Object Identifier: doi:10.1007/3-540-15644-5_21
[M] F. Murat, Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 69–102.
Mathematical Reviews (MathSciNet): MR82h:46051
Zentralblatt MATH: 0464.46034
[N] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), no. 3, 608–623.
Mathematical Reviews (MathSciNet): MR90j:35030
Zentralblatt MATH: 0688.35007
Digital Object Identifier: doi:10.1137/0520043
[P] H. Poincaré, Sur les surfaces de translation et les fonctions Abeliennes, Bull. Soc. Math. France 29 (1901), 61–86.
Zentralblatt MATH: 32.0459.04
Mathematical Reviews (MathSciNet): MR1504382
[S] S. Schochet, Resonant nonlinear geometric optics for weak solutions of conservation laws, J. Differential Equations 113 (1994), no. 2, 473–504.
Mathematical Reviews (MathSciNet): MR96c:35112
Zentralblatt MATH: 0856.35080
Digital Object Identifier: doi:10.1006/jdeq.1994.1133
[T1] L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV ed. R. J. Knopps, Res. Notes in Math., vol. 39, Pitman, Press, Boston, 1979, pp. 136–212.
Mathematical Reviews (MathSciNet): MR81m:35014
Zentralblatt MATH: 0437.35004
[T2] L. Tartar, Solutions oscillantes des équations de Carleman, Séminaire Goulaouic-Meyer-Schwartz (1980–81), XII, École Polytech., Palaiseau, 1981.
Mathematical Reviews (MathSciNet): MR83m:35016
Zentralblatt MATH: 0481.35010
[T3] L. Tartar, $H$ measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 3-4, 193–230.
Mathematical Reviews (MathSciNet): MR91h:35042
Zentralblatt MATH: 0774.35008
[W] A. Weil, L'integration dans les groupes topologiques et ses applications, Actual. Sci. Ind., no. 869, Hermann, Paris, 1940.
Mathematical Reviews (MathSciNet): MR3,198b
Zentralblatt MATH: 0063.08195
[Wh] G. Whitham, Linear and Nonlinear Waves, Pure and Appl. Math., Wiley-Interscience, New York, 1974.
Mathematical Reviews (MathSciNet): MR58:3905
Zentralblatt MATH: 0373.76001
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