Focusing at a point and absorption of nonlinear oscillations
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- by J.-L. Joly, G. Métivier and J. Rauch PDF
- Trans. Amer. Math. Soc. 347 (1995), 3921-3969 Request permission
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
- Jean-Marc Delort, Oscillations semi-linéaires multiphasées compatibles en dimension $2$ ou $3$ d’espace, Comm. Partial Differential Equations 16 (1991), no. 4-5, 845–872 (French). MR 1113110, DOI 10.1080/03605309108820781
- Ronald J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292 (1985), no. 2, 383–420. MR 808729, DOI 10.1090/S0002-9947-1985-0808729-4
- Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI 10.1007/BF00752112
- Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR 877643
- Weinan E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math. 45 (1992), no. 3, 301–326. MR 1151269, DOI 10.1002/cpa.3160450304
- Weinan E and Denis Serre, Correctors for the homogenization of conservation laws with oscillatory forcing terms, Asymptotic Anal. 5 (1992), no. 4, 311–316. MR 1157236
- Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481, DOI 10.1090/cbms/074 O. Guès, Développements asymptotiques de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal. 6 (1993), 241-270.
- Olivier Guès, Ondes multidimensionnelles $\epsilon$-stratifiées et oscillations, Duke Math. J. 68 (1992), no. 3, 401–446 (French). MR 1194948, DOI 10.1215/S0012-7094-92-06816-5 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.
- J.-L. Joly, G. Métivier, and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 1, 51–113. MR 1305424
- J.-L. Joly, G. Métivier, and J. Rauch, Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J. 70 (1993), no. 2, 373–404. MR 1219817, DOI 10.1215/S0012-7094-93-07007-X
- J.-L. Joly, G. Métivier, and J. Rauch, Coherent nonlinear waves and the Wiener algebra, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 1, 167–196 (English, with English and French summaries). MR 1262884 —, Nonlinear oscillations beyond caustics, preprint.
- Jean-Luc Joly and Jeffrey Rauch, Justification of multidimensional single phase semilinear geometric optics, Trans. Amer. Math. Soc. 330 (1992), no. 2, 599–623. MR 1073774, DOI 10.1090/S0002-9947-1992-1073774-7
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- Jacques-Louis Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France 93 (1965), 43–96. MR 199519
- Gabriel Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), no. 3, 608–623. MR 990867, DOI 10.1137/0520043
- Jeffrey Rauch and Michael Reed, Striated solutions of semilinear, two-speed wave equations, Indiana Univ. Math. J. 34 (1985), no. 2, 337–353. MR 783919, DOI 10.1512/iumj.1985.34.34020
- Jeffrey Rauch and Michael C. Reed, Nonlinear superposition and absorption of delta waves in one space dimension, J. Funct. Anal. 73 (1987), no. 1, 152–178. MR 890661, DOI 10.1016/0022-1236(87)90063-2
- Steven Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations 114 (1994), no. 2, 476–512. MR 1303036, DOI 10.1006/jdeq.1994.1157
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
- L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. Foreword by Wendell H. Fleming. MR 0259704
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 3921-3969
- MSC: Primary 35B40; Secondary 35B05, 35C20, 35L70, 35Q99
- DOI: https://doi.org/10.1090/S0002-9947-1995-1297533-8
- MathSciNet review: 1297533