Abstract
The method of weakly nonlinear geometric optics is one of the main formal perturbation techniques used in analyzing nonlinear wave motion for hyperbolic systems. The tacit assumption in using such perturbation methods is that the corresponding solutions of the hyperbolic system remain smooth; since shock waves typically form in such solutions, these assumptions are rarely satisfied in practice. Nevertheless, in a variety of applied contexts, these methods give qualitatively reliable answers for discontinuous weak solutions. Here we give a rigorous proof for the validity of nonlinear geometric optics for general weak solutions of systems of hyperbolic conservation laws in a single space variable. The methods of proof do not mimic the formal construction of weakly nonlinear asymptotics but instead rely on structural symmetries of the approximating equations, stability estimates for intermediate asymptotic times, and the rapid decay in variation of weak solutions for large asymptotic times.
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References
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
DiPerna, R. J.: Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J.24, 1047–1071 (1975)
DiPerna, R. J.: Singularities of solutions of nonlinear hyperbolic systems of conservation laws. Arch. Rat. Mech. Anal.60, 75–100 (1975)
DiPerna, R. J.: Uniqueness of solutions of hyperbolic conservation laws, Indiana Univ. Math. J.28, 137–188 (1979)
Federer, H.: Geometric measure theory, Berlin, Heidelberg, New York: Springer, 1969
Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math.18, 697–715 (1965)
Glimm, J., Lax, P. D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Am. Math. Soc. Memoirs101 (1970)
Hunter, J. K., Keller, J. B.: Weak shock diffraction. (in press, Wave Motion)
Hunter, J. K., Majda, A., Rosales, R.: Resonantly interacting weakly nonlinear hyperbolic waves, II: several space variables. (in preparation)
Hunter, J. K., Keller, J. B.: Weakly nonlinear high frequency waves. Commun. Pure Appl. Math.36, 547–569 (1983)
Keyfitz, B.: Solutions with shocks, an example of anL 1-contractive semi-group. Commun. Pure Appl. Math,24, 125–132 (1971)
Kruzkov, N.: First order quasilinear equations in several independent variables. Math. USSR Sb.10, 127–243 (1970)
Landau, L. D.: On shock waves at large distances from their place of origin. J. Phys. USSR9, 495–500 (1945)
Lax, P. D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. CMBS Monograph No. 11, SIAM, 1973
Lax, P. D.: Shock waves and entropy. In: Contributions to nonlinear functional analysis, Zarantonello, E. ed. New York: Academic Press, 1971
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
Lighthill, M. J.: A method for rendering approximate solutions to physical problems uniformly valid. Phil. Mag.40, 1179–1201 (1949)
Liu, T.-P.: Admissible solutions to systems of conservation laws. Am. Math. Soc. Memoirs 1982
Liu, T.-P.: Decay toN-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Commun. Pure Appl. Math.30, 585–610 (1977)
Majda, A., Rosales, R.: Resonantly interaction weakly nonlinear hyperbolic waves, I: A single space variable, (in press Studies Appl. Math. 1984)
Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. Applied Mathematical Science Series. Berlin, Heidelber, New York: Springer, 1984
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)
Majda, A., Rosales, R.: Weakly nonlinear detonation waves. SIAM J. Appl. Math.43, 1086–1118 (1983)
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
Vol'pert, A. I.: The spaces BV and quasilinear equations. Math. USSR Sb.2, 257–267 (1967)
Whitham, G. R.: The flow pattern of a supersonic projectile. Commun. Pure Appl. Math.5, 301–348 (1952)
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Communicated by A. Jaffe
Partially supported by NSF Grant No. DMS-8301135
Partially supported by NSF Grant No. MCS-81-02360 and ARO Grant No. 483964-25530
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DiPerna, R.J., Majda, A. The validity of nonlinear geometric optics for weak solutions of conservation laws. Commun.Math. Phys. 98, 313–347 (1985). https://doi.org/10.1007/BF01205786
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DOI: https://doi.org/10.1007/BF01205786