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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 6, Pages 39–76 (Mi izv2753)  

This article is cited in 6 scientific papers (total in 6 papers)

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

A. V. Gasnikov

Moscow Institute of Physics and Technology

Abstract: 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.

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

DOI: https://doi.org/10.4213/im2753

Full text: PDF file (852 kB)
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English version:
Izvestiya: Mathematics, 2009, 73:6, 1111–1148

Bibliographic databases:

UDC: 519.633
MSC: 35K59, 35B40, 35B50, 35B51, 35K15
Received: 10.12.2007
Revised: 21.04.2008

Citation: A. V. Gasnikov, “Time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity”, Izv. RAN. Ser. Mat., 73:6 (2009), 39–76; Izv. Math., 73:6 (2009), 1111–1148

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Kazeikina A.V., “Primery otsutstviya beguschei volny dlya obobschennogo uravneniya Kortevega–de Friza–Byurgersa”, Vestn. Mosk. un-ta. Ser. 15. Vychislitelnaya matematika i kibernetika, 2011, no. 1, 17a–24  mathscinet  elib
    2. Buslaev A.P., Gasnikov A.V., Yashina M.V., “Selected mathematical problems of traffic flow theory”, Int. J. Comput. Math., 89:3 (2012), 409–432  crossref  mathscinet  zmath  isi  elib
    3. A. V. Gasnikov, “On the velocity of separation between two successive traveling waves in the asymptotics of the solution to the Cauchy problem for a Burgers-type equation”, Comput. Math. Math. Phys., 52:6 (2012), 937–939  mathnet  crossref  mathscinet  adsnasa  isi  elib  elib
    4. Turanov Kh.T., Chuev N.P., “Chislennoe modelirovanie dvizheniya gruzovykh vagonov na mestakh neobschego polzovaniya”, Nauka i tekhnika transporta, 2012, no. 3, 8–18  elib
    5. Henkin G.M., “Burgers type equations, Gelfand's problem and Schumpeterian dynamics”, J. Fixed Point Theory Appl., 11:2 (2012), 199–223  crossref  mathscinet  zmath  isi  elib
    6. Henkin G.M., Shananin A.A., “Cauchy-Gelfand Problem For Quasilinear Conservation Law”, Bull. Sci. Math., 138:7 (2014), 783–804  crossref  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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