Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T11:19:34.051Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of the 1D IBVP for a non autonomous scalar conservation law

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  26 December 2018

Rinaldo M. Colombo  and
Elena Rossi
Rinaldo M. Colombo
Affiliation:
INDAM Unit, University of Brescia, Italy (rinaldo.colombo@unibs.it; elena.rossi@unibs.it;)
Elena Rossi
Affiliation:
INDAM Unit, University of Brescia, Italy (rinaldo.colombo@unibs.it; elena.rossi@unibs.it;)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the stability with respect to the flux of solutions to initial – boundary value problems for scalar non autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions.

Keywords

conservation lawsboundary valueproblems for conservation laws

MSC classification

Primary: 35L65: Conservation laws
Secondary: 35L04: Initial-boundary value problems for first-order hyperbolic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 561 - 592
Copyright
Copyright © Royal Society of Edinburgh 2018 

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

1Amadori, D. and Colombo, R. M.. Continuous dependence for 2×2 conservation laws with boundary. J. Differ. Equ. 138(2) (1997), 229–266.Google Scholar
2Ammar, K., Wittbold, P. and Carrillo, J.. Scalar conservation laws with general boundary condition and continuous flux function. J. Differ. Equ. 228(1) (2006), 111–139.Google Scholar
3Ancona, F. and Marson, A.. Scalar non-linear conservation laws with integrable boundary data. Nonlinear Anal. 35(6) (1999), 687–710, Ser. A: Theory Methods.Google Scholar
4Bardos, C., le Roux, A. Y. and Nédélec, J.-C.. First order quasilinear equations with boundary conditions. Comm. Partial. Differ. Equ. 4(9) (1979), 1017–1034.Google Scholar
5Bianchini, S. and Colombo, R. M.. On the stability of the Standard Riemann Semigroup. Proc. Amer. Math. Soc. 130(7) (2002), 1961–1973 (electronic).Google Scholar
6Bressan, A.. Hyperbolic systems of conservation laws, volume 20 of Oxford Lecture Series in Mathematics and its Applications. The one-dimensional Cauchy problem (Oxford: Oxford University Press, 2000).Google Scholar
7Colombo, R. M., Mercier, M. and Rosini, M. D.. Stability and total variation estimates on general scalar balance laws. Commun. Math. Sci. 7(1) (2009), 37–65.Google Scholar
8Colombo, R. M. and Rossi, E.. Rigorous estimates on balance laws in bounded domains. Acta Math. Sci. Ser. B Engl. Ed. 35(4) (2015), 906–944.Google Scholar
9Dafermos, C. M.. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn (Berlin: Springer-Verlag, 2010).Google Scholar
10Dubois, F. and Le Floch, P.. Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equ. 71(1) (1988), 93–122.Google Scholar
11Evans, L. C. and Gariepy, R. F.. Measure theory and fine properties of functions. Studies in Advanced Mathematics (Boca Raton, FL: CRC Press, 1992).Google Scholar
12Goodman, J.. Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws. PhD thesis, California University (1982).Google Scholar
13Holden, H. and Risebro, N. H.. Front tracking for hyperbolic conservation laws, volume 152 of Applied Mathematical Sciences. First softcover corrected printing of the 2002 original (New York: Springer, 2011)Google Scholar
14Kružkov, S. N.. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123) (1970), 228–255.Google Scholar
15Málek, J., Nečas, J., Rokyta, M. and Růžička, M.. Weak and measure-valued solutions to evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation (London: Chapman & Hall, 1996).Google Scholar
16Martin, S.. First order quasilinear equations with boundary conditions in the L ∞ framework. J. Differ. Equ. 236(2) (2007), 375–406.Google Scholar
17Otto, F.. Ein Randwertproblem für skalare Erhaltugssätze. PhD thesis, Universität Bonn (1993).Google Scholar
18Otto, F.. Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8) (1996), 729–734.Google Scholar
19Serre, D.. Systems of conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems, Translated from the 1996 French original by I. N. Sneddon (Cambridge: Cambridge University Press, 2000).Google Scholar
20Vovelle, J.. Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains. Numer. Math. 90(3) (2002), 563–596.Google Scholar