The validity of nonlinear geometric optics for weak solutions of conservation laws. (English) Zbl 0582.35081

It is shown that the solution u(x,t) to the genuinely nonlinear hyperbolic system of n conservation laws \[ u_ t+f(u)_ x=0,\quad u(x,0)=u_ 0+\epsilon v(x) \] in one space dimension satisfies \[ u(x,t)=u_ 0+\epsilon \sum^{n}_{j=1}\sigma^ j(\phi^ j,\tau)r_ j(u_ 0)+O(\epsilon^ 2)\quad for\quad \epsilon \to 0, \] where \(r_ j\) are right eigenvectors of the matrix \(f'(u_ 0)\), and the scalar functions \(\sigma^ j\) are solutions of an equation of Burgers’ type. It is shown that for initial data v with compact support and bounded variation the terms \(O(\epsilon^ 2)\) is uniformly small for all \(t>0\) in the \(L_ 1\)-norm. Similar estimates are given for periodic initial data v.
Reviewer: H.-D.Alber


35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35A35 Theoretical approximation in context of PDEs
78A05 Geometric optics
Full Text: DOI


[1] Dafermos, C. M.: Characteristics in hyperbolic conservation laws. A Study of the structure and asymptotic behaviour of solutions. In: Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. I, Knops, R. J., ed. Pitman Research Notes in Mathematics #17
[2] DiPerna, R. J.: Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J.24, 1047-1071 (1975) · Zbl 0309.35050 · doi:10.1512/iumj.1975.24.24088
[3] DiPerna, R. J.: Singularities of solutions of nonlinear hyperbolic systems of conservation laws. Arch. Rat. Mech. Anal.60, 75-100 (1975) · Zbl 0324.35062 · doi:10.1007/BF00281470
[4] DiPerna, R. J.: Uniqueness of solutions of hyperbolic conservation laws, Indiana Univ. Math. J.28, 137-188 (1979) · Zbl 0409.35057 · doi:10.1512/iumj.1979.28.28011
[5] Federer, H.: Geometric measure theory, Berlin, Heidelberg, New York: Springer, 1969 · Zbl 0176.00801
[6] Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math.18, 697-715 (1965) · Zbl 0141.28902 · doi:10.1002/cpa.3160180408
[7] Glimm, J., Lax, P. D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Am. Math. Soc. Memoirs101 (1970) · Zbl 0204.11304
[8] Hunter, J. K., Keller, J. B.: Weak shock diffraction. (in press, Wave Motion) · Zbl 0524.73027
[9] Hunter, J. K., Majda, A., Rosales, R.: Resonantly interacting weakly nonlinear hyperbolic waves, II: several space variables. (in preparation) · Zbl 0657.35084
[10] Hunter, J. K., Keller, J. B.: Weakly nonlinear high frequency waves. Commun. Pure Appl. Math.36, 547-569 (1983) · Zbl 0547.35070 · doi:10.1002/cpa.3160360502
[11] Keyfitz, B.: Solutions with shocks, an example of anL 1-contractive semi-group. Commun. Pure Appl. Math,24, 125-132 (1971) · Zbl 0209.12401 · doi:10.1002/cpa.3160240203
[12] Kruzkov, N.: First order quasilinear equations in several independent variables. Math. USSR Sb.10, 127-243 (1970) · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156
[13] Landau, L. D.: On shock waves at large distances from their place of origin. J. Phys. USSR9, 495-500 (1945)
[14] Lax, P. D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. CMBS Monograph No. 11, SIAM, 1973 · Zbl 0268.35062
[15] Lax, P. D.: Shock waves and entropy. In: Contributions to nonlinear functional analysis, Zarantonello, E. ed. New York: Academic Press, 1971 · Zbl 0268.35014
[16] Lax, P. D.: Accuracy and resolution in the computation of solutions of linear and nonlinear equations. In: Recent advances in numerical analysis, pp. 107-118. DeBoor, C., Golub, G. ed. New York: Academic Press, 1978
[17] Lighthill, M. J.: A method for rendering approximate solutions to physical problems uniformly valid. Phil. Mag.40, 1179-1201 (1949) · Zbl 0035.20504
[18] Liu, T.-P.: Admissible solutions to systems of conservation laws. Am. Math. Soc. Memoirs 1982
[19] Liu, T.-P.: Decay toN-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Commun. Pure Appl. Math.30, 585-610 (1977) · Zbl 0357.35059 · doi:10.1002/cpa.3160300505
[20] Majda, A., Rosales, R.: Resonantly interaction weakly nonlinear hyperbolic waves, I: A single space variable, (in press Studies Appl. Math. 1984) · Zbl 0572.76066
[21] Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Science Series. Berlin, Heidelber, New York: Springer, 1984 · Zbl 0537.76001
[22] Majda, A., Rosales, R.: A theory for spontaneous Mach stem formation in reacting shock fronts, I: the basic perturbation analysis. SIAM J. Appl. Math.43, 1310-1334 (1983) · Zbl 0544.76135 · doi:10.1137/0143088
[23] Majda, A., Rosales, R.: Weakly nonlinear detonation waves. SIAM J. Appl. Math.43, 1086-1118 (1983) · Zbl 0544.76135 · doi:10.1137/0143088
[24] Nayfeh, A. H.: A comparison of perturbation methods for nonlinear hyperbolic waves. In: Singular perturbations and asymptotics, pp. 223-276. Meyer, R., Parter, S. eds. New York: Academic Press, 1980 · Zbl 0495.35010
[25] Vol’pert, A. I.: The spaces BV and quasilinear equations. Math. USSR Sb.2, 257-267 (1967) · Zbl 0168.07402 · doi:10.1070/SM1967v002n02ABEH002340
[26] Whitham, G. R.: The flow pattern of a supersonic projectile. Commun. Pure Appl. Math.5, 301-348 (1952) · Zbl 0047.19106 · doi:10.1002/cpa.3160050305
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