×

zbMATH — the first resource for mathematics

Justification of nonlinear geometric optics for a system of conservation laws. (Justification de l’optique géométrique non linéaire pour un système de lois de conservation.) (French) Zbl 0914.35078
Systems of conservation laws \(u_t+f(u)_x=0\) with the initial condition of the type \[ u_i(0,x)=\varepsilon h_i\left( x,{\psi_i (x)\over \varepsilon}\right) \] are considered in particular in one space dimension. The author proves propagation theorems and estimates, deals with the conditions on the phase \(\psi\) and gives global existence theorem with the asymptotic estimate.
Reviewer: A.Doktor (Praha)

MSC:
35L65 Hyperbolic conservation laws
78A05 Geometric optics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Bressan, Global solutions of systems of conservation laws by wave-front tracking , J. Math. Anal. Appl. 170 (1992), no. 2, 414-432. · Zbl 0779.35067 · doi:10.1016/0022-247X(92)90027-B
[2] C. Cheverry, Justification de l’optique géométrique pour une loi de conservation , Fascicule D’Equations aux Dérivées Partielles, Publ. Inst. Rech. Math. Rennes, Rennes, 1992, pp. 55-84.
[3] C. Cheverry, Oscillations de faible amplitude pour les systèmes \(2\times 2\) de lois de conservation , Asymptotic Anal. 12 (1996), no. 1, 1-24. · Zbl 0852.35093
[4] C. Cheverry, The modulation equations of nonlinear geometric optics , Comm. Partial Differential Equations 21 (1996), no. 7-8, 1119-1140. · Zbl 0867.35061 · doi:10.1080/03605309608821220 · eudml:221878
[5] C. Cheverry, Systémes de lois de conservation et stabilité BV , prépublication 96-08, Prépubl. Inst. Rech. Math. Rennes, Rennes, 1996; soumis aux Ann. Inst. H. Poincaré Anal. Non Linéaire.
[6] R. J. DiPerna and A. Majda, The validity of nonlinear geometric optics for weak solutions of conservation laws , Comm. Math. Phys. 98 (1985), no. 3, 313-347. · Zbl 0582.35081 · doi:10.1007/BF01205786
[7] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations , Comm. Pure Appl. Math. 18 (1965), 697-715. · Zbl 0141.28902 · doi:10.1002/cpa.3160180408
[8] J. Hunter, Hyperbolic waves and nonlinear geometrical acoustics , Transactions of the Sixth Army Conference on Applied Mathematics and Computing (Boulder, CO, 1988), ARO Rep., vol. 89, U.S. Army Res. Office, Research Triangle Park, NC, 1989, pp. 527-570. · Zbl 0668.76075
[9] J. K. 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. · Zbl 0657.35084
[10] J. L. Joly, G. Métivier, and J. Rauch, Resonant one-dimensional nonlinear geometric optics , J. Funct. Anal. 114 (1993), no. 1, 106-231. · Zbl 0851.35023 · doi:10.1006/jfan.1993.1065
[11] J. L. Joly, G. Métivier, and J. Rauch, Remarques sur l’optique géométrique non linéaire multidimensionnelle , Séminaire sur les Équations aux Dérivées Partielles, 1990-1991, École Polytech., Palaiseau, 1991, Exp. No. I, 17. · Zbl 0749.35055 · numdam:SEDP_1990-1991____A1_0 · eudml:112012
[12] J. L. Joly, G. Métivier, and J. Rauch, Focusing and absorbtion of nonlinear oscillations , Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1993), École Polytech., Palaiseau, 1993, Exp. No. III, 11. · Zbl 0815.35071 · numdam:JEDP_1993____A3_0 · eudml:93272
[13] 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. · Zbl 0836.35087 · numdam:ASENS_1995_4_28_1_51_0 · eudml:82378
[14] J. L. Joly, G. Métivier, and J. Rauch, A nonlinear instability for \(3\times 3\) systems of conservation laws , Comm. Math. Phys. 162 (1994), no. 1, 47-59. · Zbl 0820.35093 · doi:10.1007/BF02105186
[15] S. Junca, Optique géométrique nonlinéaire: Chocs forts, Relaxation , Thése d’université, Nice, 1995.
[16] P. Lax, Hyperbolic systems of conservation laws. II , Comm. Pure Appl. Math. 10 (1957), 537-566. · Zbl 0081.08803 · doi:10.1002/cpa.3160100406
[17] A. Majda, Nonlinear geometric optics for hyperbolic systems of conservation laws , Oscillation theory, computation, and methods of compensated compactness (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., vol. 2, Springer, New York, 1986, pp. 115-165. · Zbl 0622.65117
[18] 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. · Zbl 0572.76066
[19] A. Majda, R. Rosales, and M. Schonbek, A canonical system of integrodifferential equations arising in resonant nonlinear acoustics , Stud. Appl. Math. 79 (1988), no. 3, 205-262. · Zbl 0669.76103
[20] R. L. Pego, Some explicit resonating waves in weakly nonlinear gas dynamics , Stud. Appl. Math. 79 (1988), no. 3, 263-270. · Zbl 0669.76104
[21] S. Schochet, Resonant nonlinear geometric optics for weak solutions of conservation laws , J. Differential Equations 113 (1994), no. 2, 473-504. · Zbl 0856.35080 · doi:10.1006/jdeq.1994.1133
[22] S. Schochet, Fast singular limits of hyperbolic partial differential equations , J. Differential Equations, à paraitre. · Zbl 0838.35071
[23] 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., 1979, pp. 136-212. · Zbl 0437.35004
[24] A. I. Vol’bert, The spaces BV and quasilinear equations , Math. USSR Sb. 2 (1968), 225-267. · Zbl 0168.07402
[25] E. Weinan and D. Serre, Correctors for the homogenization of conservation laws with oscillatory forcing terms , prépublication, École Normale Supérieure de Lyon, 1992. · Zbl 0766.35026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.