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 ±∞, 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.
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. Math., 73:6 (2009), 1111–1148
\Bibitem{Gas09}
\by A.~V.~Gasnikov
\paper Time-asymptotic behaviour of a~solution of the Cauchy initial-value problem for a~conservation law with non-linear divergent viscosity
\jour Izv. Math.
\yr 2009
\vol 73
\issue 6
\pages 1111--1148
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This publication is cited in the following 8 articles:
Fedor Bakharev, Aleksandr Enin, Yulia Petrova, Nikita Rastegaev, “Impact of dissipation ratio on vanishing viscosity solutions of the Riemann problem for chemical flooding model”, J. Hyper. Differential Equations, 20:02 (2023), 407
Henkin G.M., Shananin A.A., “Cauchy-Gelfand Problem For Quasilinear Conservation Law”, Bull. Sci. Math., 138:7 (2014), 783–804
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
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
Turanov Kh.T., Chuev N.P., “Chislennoe modelirovanie dvizheniya gruzovykh vagonov na mestakh neobschego polzovaniya”, Nauka i tekhnika transporta, 2012, no. 3, 8–18
Henkin G.M., “Burgers type equations, Gelfand's problem and Schumpeterian dynamics”, J. Fixed Point Theory Appl., 11:2 (2012), 199–223
A. V. Kazeykina, “Examples of the absence of a traveling wave for the generalized Korteweg-de Vries-Burgers equation”, MoscowUniv.Comput.Math.Cybern., 35:1 (2011), 14
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