Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T06:59:03.395Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of solutions of hyperbolic conservation laws ut + (um)x = 0 as m → ∞ with inconsistent initial values

Published online by Cambridge University Press:  14 November 2011

Xiangsheng Xu
Xiangsheng Xu
Affiliation:
Department of Mathematics, Texas A& M University, College Station, TX 77843, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study the behaviour of solutions u = um of ut, + (um)x = 0 for t > 0, x ∊ R, u(x, 0) = u0(x), u0 ≧0, u0 ∊ L1(R) as m → ∞. This is a singular perturbation problem about m = ∞ if u0 > 1 on a set of positive measure. It is shown that the limit exists and satisfies the stationary equation

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 61 - 71
Copyright
Copyright © Royal Society of Edinburgh 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bénilan, Ph. and Crandall, M. G.. Regularizing Effects of Homogeneous Evolution Equations (Madison: University of Wisconsin, Madison Mathematics Research Center, Technical Summary Report 2076).Google Scholar
2Bénilan, Ph. and Crandall, M.. The continuous dependence on ø of solutions of u t, − Δ ø(u) = 0. Indiana Univ. Math. J. 30 (1981), 161–177.CrossRefGoogle Scholar
3Caffarelli, L. and Friedman, A.. Asymptotic behaviour of solutions of u t, = Δu m as m → ∞. Indiana Univ. Math. J. 36 (1987), 711–728.CrossRefGoogle Scholar
4Crandall, M. G.. The semigroup approach to first order quasilinear equations in several space variables. Israel J. Math. 12 (1972), 108–132.CrossRefGoogle Scholar
5Crandall, M. and Liggett, T.. Generation of semigroups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971), 265–298.CrossRefGoogle Scholar
6Dafermos, C. M.. Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J. 26 (1977), 1097–1119.CrossRefGoogle Scholar
7Dafermos, C. M.. Characteristics in hyperbolic conservation laws. In Nonlinear Analysis and Mechanics, ed. Knops, R. J., Research Notes in Math. No. 17 (London: Pitman, 1977).Google Scholar
8Filippov, A. F.. Differential equations with discontinuous right-hand side. Mat. Sb. (N.S.) 51 (1960), 99–128. English translation: Amer. Math. Soc. Transl. 42 (1964), 199–231.Google Scholar
9Friedman, A. and Huang, S.. Asymptotic Behavior of Solutions of ut, = Δφm(u) as m→ ∞ with Inconsistent Initial Values (Lafayette, IN; Purdue University, Center for Applied Mathematics, Technical Report 44).Google Scholar
10Ladyzhenskaya, A.. On the construction of discontinuous solutions of quasilinear hyperbolic equations when the viscosity coefficient is approaching zero. Dokl. Akad. Nauk SSSR. 3 (1956), 291–295.Google Scholar
11Lax, P. D.. Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10 (1957), 537–666.CrossRefGoogle Scholar
12Showalter, R. E. and Xu, X.. An approximate scalar conservation law from dynamics of gas absorptions. J. Differential Equations (to appear).Google Scholar