Time-asymptotic behaviour of a~solution of the Cauchy initial-value problem for a~conservation law with non-linear divergent viscosity

A. V. Gasnikov

Izvestiya. Mathematics , Tome 73 (2009) no. 6, pp. 1111-1148.

Résumé

We study the time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity. We shall prove that when a bounded measurable initial function has limits at $\pm\infty$, a solution of the Cauchy initial-value problem converges uniformly to a system of waves consisting of travelling waves and rarefaction waves, where the phase shifts of the travelling waves are allowed to depend on time. The rate of convergence is estimated under additional conditions on the initial function.

Détail

Keywords: conservation law with non-linear divergent viscosity, equation of Burgers type, asymptotics of solutions, convergence on the phase plane, travelling wave, rarefaction wave, system of waves, maximum principle, comparison principle (on the phase plane), inequality of Kolmogorov type.
Mots-clés : convergence in form

Bibliographie

[1] Ya. B. Zeldovich, Yu. P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Vol. I, II, Academic Press, New York, 1966 | Zbl

[2] R. Courant, K. O. Friedrichs, Supersonic flow and shock waves, Intersci. Publ., New York, 1948 | MR | Zbl

[3] G. B. Whitham, Linear and nonlinear waves, Wiley, New York–London–Sydney, 1974 | MR | MR | Zbl

[4] L. A. Peletier, “The porous media equation”, Applications of nonlinear analysis in the physical sciences (Bielefeld, 1979), Surveys Reference Works Math., 6, Pitman, Boston, MA–London, 1981, 229–241 | MR | Zbl

[5] A. Harten, J. M. Hyman, P. D. Lax, B. Keyfitz, “On finite-difference approximations and entropy conditions for shocks”, Comm. Pure Appl. Math., 29:3 (1976), 297–322 | DOI | MR | Zbl

[6] V. M. Polterovich, G. M. Khenkin, “Matematicheskii analiz ekonomicheskikh modelei. Evolyutsionnaya model vzaimodeistviya protsessov sozdaniya i zaimstvovaniya tekhnologii”, Ekonomika i mat. metody, 24:6 (1988), 1071–1083 | MR

[7] G. M. Henkin, V. M. Polterovich, “Schumpeterian dynamics as a non-linear wave theory”, J. Math. Econom., 20:6 (1991), 551–590 | DOI | MR | Zbl

[8] L. M. Gelman, M. I. Levin, V. M. Polterovich, V. A. Spivak, “Modelirovanie dinamiki raspredeleniya predpriyatii otrasli po urovnyam effektivnosti (na primere chernoi metallurgii)”, Ekonomika i mat. metody, 29:3 (1993), 460–469

[9] G. M. Henkin, V. M. Polterovich, “A difference-differential analogue of the burgers equation: Stability of the two-wave behavior”, J. Nonlinear Sci., 4:1 (1994), 497–517 | DOI | MR | Zbl

[10] G. M. Henkin, V. M. Polterovich, “A difference-differential analogue of the Burgers equations and some models of economic development”, Discrete Contin. Dynam. Systems, 5:4 (1999), 697–728 | DOI | MR | Zbl

[11] I. M. Gel'fand, “Some problems in the theory of quasilinear equations”, Amer. Math. Soc. Transl. (2), 29 (1963), 295–381 | MR | MR | Zbl | Zbl

[12] A. M. Ilin, O. A. Oleinik, “Asimptoticheskoe povedenie reshenii zadachi Koshi dlya nekotorykh kvazilineinykh uravnenii pri bolshikh znacheniyakh vremeni”, Matem. sb., 51(93):2 (1960), 191–216 | MR | Zbl

[13] H. F. Weinberger, “Long-time behavior for a regularized scalar conservation law in the absence of genuine nonlinearity”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7:5 (1990), 407–425 | MR | Zbl

[14] G. M. Henkin, A. A. Shananin, “Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations”, J. Math. Pures Appl. (9), 83:12 (2004), 1457–1500 | DOI | MR | Zbl

[15] G. M. Henkin, A. A. Shananin, A. E. Tumanov, “Estimates for solutions of Burgers type equations and some applications”, J. Math. Pures Appl. (9), 84:6 (2005), 717–752 | DOI | MR | Zbl

[16] G. M. Henkin, “Asymptotic structure for solutions of the Cauchy problem for Burgers type equations”, J. Fixed Point Theory Appl., 1:2 (2007), 239–291 | DOI | MR | Zbl

[17] A. N. Kolmogorov, I. G. Petrovskii, N. S. Piskunov, “Issledovanie uravneniya diffuzii, soedinennoi s vozrastaniem kolichestva veschestva, i ego primenenie k odnoi biologicheskoi probleme”, Byul. MGU. Matematika i mekhanika, 1:6 (1937), 1–26 | Zbl

[18] P. C. Fife, J. B. McLeod, “The approach of solutions of nonlinear diffusion equations to travelling front solutions”, Arch. Ration. Mech. Anal., 65:4 (1977), 335–361 | DOI | MR | Zbl

[19] A. I. Volpert, Vit. A. Volpert, Vl. A. Volpert, Traveling wave solutions of parabolic systems, Transl. Math. Monogr., 140, Amer. Math. Soc., Providence, RI, 2000 | MR | Zbl

[20] M. Mejai, Vit. Volpert, “Convergence to systems of waves for viscous scalar conservation laws”, Asymptot. Anal., 20:3–4 (1999), 351–366 | MR | Zbl

[21] Sh. Engelberg, S. Schochet, “Nonintegrable perturbations of scalar viscous shock profiles”, Asymptot. Anal., 48:1–2 (2006), 121–140 | MR | Zbl

[22] S. N. Kruzhkov, N. S. Petrosyan, “Asymptotic behaviour of the solutions of the Cauchy problem for non-linear first order equations”, Russian Math. Surveys, 42:5 (1987), 1–47 | DOI | MR | Zbl

[23] O. A. Oleinik, T. D. Venttsel, “Pervaya kraevaya zadacha i zadacha Koshi dlya kvazilineinykh uravnenii parabolicheskogo tipa”, Matem. sb., 41(83):1 (1957), 105–128 | MR | Zbl

[24] O. A. Olejnik, “Discontinuous solutions of non-linear differential equations”, Amer. Math. Soc. Transl. (2), 26 (1963), 95–172 | MR | MR | Zbl

[25] S. N. Bernstein, “Limitation des modules des dérivées successives des solutions des équations du type parabolique”, Dokl. AN SSSR, 18:7 (1938), 385–389 | Zbl

[26] O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Transl. Math. Monogr., 23, Amer. Math. Soc., Providence, RI, 1968 | MR | MR | Zbl | Zbl

[27] S. N. Kruzkov, “Nonlinear parabolic equations in two independent variables”, Trans. Moscow Math. Soc., 16 (1967), 355–373 | MR | Zbl | Zbl

[28] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, NJ, 1964 | MR | Zbl | Zbl

[29] D. Hoff, J. Smoller, “Solutions in the large for certain nonlinear parabolic systems”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 213–235 | MR | Zbl

[30] S. N. Kružkov, “First order quasilinear equations in several independent variables”, Math. USSR-Sb., 10:2 (1970), 217–243 | DOI | MR | Zbl | Zbl

[31] M. G. Crandall, L. Tartar, “Some relations between nonexpansive and order preserving mappings”, Proc. Amer. Math. Soc., 78:3 (1980), 385–390 | DOI | MR | Zbl

[32] D. Serre, “$L^1$-stability of nonlinear waves in scalar conservation laws”, Evolutionary equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, 473–553 | DOI | MR | Zbl

[33] C. M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren Math. Wiss., 325, Springer-Verlag, Berlin–Heidelberg, 2005 | MR | Zbl

[34] M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984 | MR | Zbl

[35] H. Freistühler, D. Serre, “$L^1$ stability of shock waves in scalar viscous conservation laws”, Comm. Pure Appl. Math., 51:3 (1998), 291–301 | 3.0.CO;2-5 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[36] H. Matano, “Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29:2 (1982), 401–441 | MR | Zbl

[37] I. P. Natanson, Theory of functions of real variable, Frederick Ungar, New York, 1955 | MR | MR | Zbl | Zbl

[38] O. A. Olejnik, “Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation”, Amer. Math. Soc. Transl. (2), 33 (1963), 285–290 | MR | Zbl | Zbl

[39] S. Osher, J. Ralston, “$L^1$ stability of travelling waves with applications to convective porous media flow”, Comm. Pure Appl. Math., 35:6 (1982), 737–749 | DOI | MR | Zbl

[40] P. D. Lax, “Hyperbolic systems of conservation laws. II”, Comm. Pure and Appl. Math., 10:4 (1957), 537–566 | DOI | MR | Zbl

[41] D. Serre, “Stabilité des ondes de choc de viscosité qui peuvent être caractéristiques”, Nonlinear partial differential equations and their applications (Paris, 1997–1998), Stud. Math. Appl., 31, North-Holland, Amsterdam, 2002, 647–654 | MR | Zbl

[42] G. G. Magaril-Il'yaev, V. M. Tikhomirov, Convex analysis: theory and applications, Transl. Math. Monogr., 222, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[43] V. N. Gabušin, “Inequalities between derivatives in $L_p$-metrics for $0

\le\infty$”, Math. USSR-Izv., 10:4 (1976), 823–844 | DOI | MR | Zbl

[44] V. N. Gabušin, “Inequalities between derivatives in $L_p$-metrics for $0

\le\infty$”, Differential Equations, 24:10 (1988), 1088–1094 | MR | Zbl | Zbl

[45] V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems”, Russian Math. Surveys, 51:6 (1996), 1093–1126 | DOI | MR | Zbl

[46] Ch. Abourjaily, Ph. Benilan, “Symmetrization of quasilinear parabolic problems”, Rev. Un. Mat. Argentina, 41:1 (1998), 1–13 | MR | Zbl

[47] A. V. Gasnikov, “On the intermediate asymptotic of the solution to the Cauchy problem for a quasilinear equation of parabolic type with a Monotone Initial Condition”, J. Comput. Systems Sci. Internat., 47:3 (2008), 475–484 | DOI

[48] A. V. Gasnikov, “Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves”, Comput. Math. Math. Phys., 48:8 (2008), 1376–1405 | DOI | MR

[49] B. L. Roždestvenskiǐ, N. N. Janenko, Systems of quasilinear equations and their applications to gas dynamics, Transl. Math. Monogr., 55, Amer. Math. Soc., Providence, RI, 1983 | MR | MR | Zbl | Zbl

[50] D. Serre, System of conservation laws. Vol. I, II, Cambridge Univ. Press, Cambridge, 1999, 2000 | MR | MR | Zbl

[51] T. P. Liu, “Admissible solutions of hyperbolic conservation laws”, Mem. Amer. Math. Soc., 30:240 (1981), 1–78 | MR | Zbl

[52] K.-S. Cheng, “Asymptotic behavior of solutions of a conservation law without convexity conditions”, J. Differential Equations, 40:3 (1981), 343–376 | DOI | MR | Zbl

[53] N. S. Petrosyan, “The asymptotic behaviour of the solution to the Cauchy problem for a quasilinear first-order equation with a non-convex state function”, Russian Math. Surveys, 38:2 (1983), 194–195 | DOI | MR | Zbl

[54] N. S. Petrosyan, “Asymptotic properties of a solution of Cauchy's problem for a first-order quasilinear equation”, Differential Equations, 20:3 (1984), 390–395 | MR | Zbl