Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations
zbMATH Open document ID: 1088.35058
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2005-05-04T00:00:00Z
review text: The authors study the asymptotics of solutions of the initial value problem for a class of Burgers-type equations, and for a related differential-difference analogue, in terms of shifted wave-trains solutions. These equations are \(f_t+\varphi(f)f_x=\varepsilon f_{xx}\) and \({dF\over dt}+\varphi(F){F(x,t)- F(x-\varepsilon,t)\over\varepsilon}=0\), respectively; the initial conditions in each case are \(f(x,0)=\begin{cases} 0, & x<x^-,\\ f^0(x), & x^-\leq x\leq x^+, \\ 1, & x>x^+,\end{cases}\) where \(0<x^-<x^+<\infty\), \(0\leq f^0(x)\leq 1\). In this paper the authors prove their results for the second of these equations; proofs for the first are left to a further paper. Further improvements (for both equations) are to be given in a joint work with A. E. Tumanov. The problem studied is one that goes back to Gelfand, and of which particular cases have been studied over the years by various authors, for example, for \(\varphi(f)\) linear (Burgers equation) or monotonic, as well as for other special cases. Analogous results have been given for the second equation by G.~M.~Henkin and V.~M.~Polterovich. Here results are obtained for a large class of functions \(\varphi\). The equations considered have a variety of physical applications, including to the displacement of oil by water in a porous medium, and to economics.
reviewed by: Andrew Pickering
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35R10
zbMATH DE Number: 2164953
zbMATH Keywords: Burgers equation
zbMATH Keywords: intermediate asymptotics
zbMATH Keywords: training waves
zbMATH Keywords: nonlinear diffusions
author: Gennadi Henkin
author: Alexander A. Shananin
MaRDI profile type: Publication
full work available at URL: j.matpur.2004.09.009
OpenAlex ID: W2101715856
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DOI: 10.1016/J.MATPUR.2004.09.009
description: scientific article; zbMATH DE number 2164953