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2000 | 52 | 1 | 119-132
Tytuł artykułu

Large time behaviour of a class of solutions of second order conservation laws

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
% We study the large time behaviour of entropy solutions of the Cauchy problem for a possibly degenerate nonlinear diffusion equation with a nonlinear convection term. The initial function is assumed to have bounded total variation. We prove the convergence of the solution to the entropy solution of a Riemann problem for the corresponding first order conservation law.
Słowa kluczowe
Rocznik
Tom
52
Numer
1
Strony
119-132
Opis fizyczny
Daty
wydano
2000
Twórcy
  • Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
  • Analyse Numérique et EDP, CNRS et Université de Paris-Sud, 91405 Orsay Cedex, France
Bibliografia
  • [BGH] M. Bertsch, J. Goncerzewicz and D. Hilhorst, Large time behaviour of solutions of scalar viscous and nonviscous conservation laws, Appl. Math. Lett. 12 (1999), 83-87
  • [BT] Ph. Benilan and H. Touré, Sur l'équation générale $u_t = φ(u)_xx - ψ(u)_x + v$, C. R. Acad. Sc. Paris 299 (1984), 919-922.
  • [dB] E. Di Benedetto, Partial Differential Equations, Birkhäuser, 1995
  • [vDdG] C. J. van Duijn and J. M. de Graaf, Large time behaviour of solutions of the porous medium equation with convection, J. Differential Equations 84 (1990), 183-203.
  • [EG] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press 1992.
  • [GM] G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modèles Non Linéaires de l'Ingénierie Pétrolière, Springer-Verlag 1996.
  • [GR] E. Godlewski and P. A. Raviart, Hyperbolic Systems of Conservation Laws, SMAI 3/4, Ellipses-Edition Marketing, Paris 1991.
  • [IO1] A. M. Il'in and O. A. Oleinik, Behaviour of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time, Dokl. Akad. Nauk S.S.S.R. 120 (1958), 25-28; Am. Math. Soc. Transl. 42 (1964), 19-23.
  • [IO2] A. M. Il'in and O. A. Oleinik, Asymptotic behaviour of solutions of the Cauchy problem for some quasilinear equations for large values of time, Matem. Sb. 51 (1960), 191-216 (in Russian).
  • [K] S. N. Kruzhkov, First order quasi-linear equations in several independent variables, Mat. USSR Sbornik 10 (1970), 217-243. Translation of: Mat. Sb. 81 (1970), 228-255.
  • [LSU] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, Amer. Math. Soc., Providence, RI., 1968.
  • [MNRR] J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-valued Solutions to Evolutionary PDEs, Chapman & Hall, 1996.
  • [MT] M. Maliki and H. Touré, Solution généralisée locale d'une équation parabolique quasilinéaire dégénérée du second ordre, Ann. Fac. Sci. Toulouse 7 (1998), 113-133.
  • [M] P. Marcati, Weak solutions to a nonlinear partial differential equation of mixed type, Differential Integral Equations 9 (1996), 827-848.
  • [Se] D. Serre, Systèmes de Lois de Conservation. I (Hyperbolicité, entropies, ondes de choc), Diderot Editeur, 1996.
  • [Si] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. CXLVI (1987), 65-96.
  • [W] H. F. Weinberger, Long-time behaviour for a regularized scalar conservation law in the absence of genuine nonlinearity, Ann. Inst. Henri Poincaré 7 (1990), 407-425.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv52z1p119bwm
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