Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Focusing at a point and absorption of nonlinear oscillations

Authors: J.-L. Joly, G. Métivier and J. Rauch
Journal: Trans. Amer. Math. Soc. 347 (1995), 3921-3969
MSC: Primary 35B40; Secondary 35B05, 35C20, 35L70, 35Q99
MathSciNet review: 1297533
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Several recent papers give rigorous justifications of weakly nonlinear geometric optics. All of them consider oscillating wave trains on domains where focusing phenomena do not exist, either because the space dimension is equal to one, or thanks to a coherence assumption on the phases. This paper is devoted to a study of some nonlinear effects of focusing. In a previous paper, the authors have given a variety of examples which show how focusing in nonlinear equations can spoil even local existence in the sense that the domain of existence shrinks to zero as the wavelength decreases to zero. On the other hand, there are many problems for which global existence is known and in those cases it is natural to ask what happens to oscillations as they pass through a focus. The main goal of this paper is to present such a study for some strongly dissipative semilinear wave equations and spherical wavefronts which focus at the origin. We show that the strongly nonlinear phenomenon which is produced is that oscillations are killed by the simultaneous action of focusing and dissipation. Our study relies on the analysis of Young measures and two-scale Young measures associated to sequences of solutions. The main step is to prove that these measures satisfy appropriate transport equations. Then, their variances are shown to satisfy differential inequalities which imply a propagation result for their support.

References [Enhancements On Off] (What's this?)

  • [D] J. M. Delort, Oscillations semi-linéaires multiphasées compatibles en dimension deux ou trois d'espace, Comm. Partial Differential Equations 16 (1991), 845-872. MR 1113110 (92g:35138)
  • [DP1] R. J. Di Perna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292 (1985), 383-420. MR 808729 (87g:35148)
  • [DP2] -, Measure valued solutions of conservation laws, Arch. Rational Mech. Anal. 8 (1985), 223-270. MR 775191 (86g:35121)
  • [DP-M] R. J. Di Perna and A. Majda, Oscillations and concentration in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), 667-689. MR 877643 (88a:35187)
  • [E] W. E, Homogeneization of linear and nonlinear transport equations, Comm. Pure Appl. Math. 45 (1992), 301-326. MR 1151269 (92k:35026)
  • [ES] W. E and D. Serre, Correctors for the homogenization of conservation laws with oscillatory forcing terms, preprint, 1992. MR 1157236 (92m:35170)
  • [Ev] L. C. Evans, Weak convergence methods for non linear partial differential equations, Amer. Math. Soc., Providence RI, 1990. MR 1034481 (91a:35009)
  • [G1] O. Guès, Développements asymptotiques de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal. 6 (1993), 241-270.
  • [G2] -, Ondes multidimensionnelles $ \varepsilon $-statifiées et oscillations, Duke Math. J. 68 (1992), 401-446. MR 1194948 (94a:35011)
  • [JMR1] J.-L. Joly, G. Metivier, and J. Rauch, Remarques sur l'optique géométrique non linéaire multidimensionelle, Séminaire Equations aux Dérivées Partielles de l'Ecole Polytechnique 1990-1991, Exposé no. 1.
  • [JMR2] -, Coherent and focusing multidimensional nonlinear geometric optics, Ann. Sci. École Norm. Sup. 28 (1995), 51-113. MR 1305424 (95k:35035)
  • [JMR3] -, Generic rigourous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J. 70 (1993), 373-404. MR 1219817 (94c:35048)
  • [JMR4] -, Coherent nonlinear waves and the Wiener algebra, Ann. Inst. Fourier (Grenoble) 44 (1994), 167-196. MR 1262884 (95c:35163)
  • [JMR5] -, Nonlinear oscillations beyond caustics, preprint.
  • [JR] J-L. Joly and J. Rauch, Justification of multidimensional single phase semilinear geometric optics, Trans. Amer. Math. Soc. 330 (1992), 599-625. MR 1073774 (92f:35040)
  • [L] J. L. Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires, Dunod and Gauthiers-Villars, Paris, 1969. MR 0259693 (41:4326)
  • [LS] J. L. Lions and W. Strauss, Some nonlinear evolution equations, Bull. Soc. Math. France 93 (1965), 43-96. MR 0199519 (33:7663)
  • [N] G. Nguetseng, A general convergence result for a functional related to the theory of homogeneisation, SIAM J. Math. Anal. 20 (1989), 608-623. MR 990867 (90j:35030)
  • [RR1] J. Rauch and M. Reed, Striated solutions of semilinear two speed wave equations, Indiana Univ. Math. J. 34 (1985), 337-353. MR 783919 (86m:35111)
  • [RR2] -, Nonlinear superposition and absorption of delta waves in one space dimension, J. Funct. Anal. 73 (1987), 152-178. MR 890661 (89b:35085)
  • [S] S. Schochet, Fast singular limit of hyperbolic PDEs, J. Differential Equations 114 (1994), 476-512. MR 1303036 (95k:35131)
  • [T] L. Tartar, Compensated compactness and applications to partial differential equations, Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, vol. 4 (R. J. Knops, ed.), Pitman Press, New York, 1979. MR 584398 (81m:35014)
  • [Y] L. C. Young, Lectures on the calculus of variations and optimal control theory, Saunders, Philadelphia and London, 1969. MR 0259704 (41:4337)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35B40, 35B05, 35C20, 35L70, 35Q99

Retrieve articles in all journals with MSC: 35B40, 35B05, 35C20, 35L70, 35Q99

Additional Information

Keywords: Oscillation, focusing, dissipation, geometric optics, Young measure, transport equation, independence, absorption, nonlinear waves
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society