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Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions

Author(s): S. Martin; J. Vovelle
Journal: Quart. Appl. Math. 65 (2007), 425-450.
MSC (2000): Primary 35L65, 35B40
Posted: July 11, 2007
MathSciNet review: 2354881
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Abstract | References | Similar articles | Additional information

Abstract: We study the large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions. Under structural hypotheses on the flux of the equation, we describe the stationary solutions and show the convergence of the entropy solution to a stationary one. Numerical tests illustrate the theoretical results.

References:

[BLN79]
C. Bardos, A. Y. LeRoux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017-1034. MR 542510 (81b:35052)

[BMV05]
G. Bayada, S. Martin, and C. Vazquez, About a generalized Buckley-Leverett equation and lubrication multifluid flow, European J. Appl. Math. 17, no. 5, 491-524 (2006). MR 2296026

[BC82]
G. Bayada, and M. Chambat, Analysis of a free boundary problem in partial lubrication, Quart. Appl. Math. 40 (1982/83), no. 4, 369-375. MR 693872 (84f:76031)

[CR00]
G.-Q. Chen and M. Rascle, Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws, Arch. Ration. Mech. Anal. 153 (2000), no. 3, 205-220. MR 1771520 (2001f:35250)

[Car99]
J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999), no. 4, 269-361. MR 1709116 (2000m:35132)

[Daf85]
C. M. Dafermos, Regularity and large time behaviour of solutions of a conservation law without convexity, Proc. Roy. Soc. Edinburgh Sect. A 99 (1985), no. 3-4, 201-239. MR 785530 (86j:35107)

[DiP75]
R.J. DiPerna, Decay and asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws, Indiana Univ. Math. J. 24 (1974/75), no. 11, 1047-1071. MR 0410110 (53:13860)

[Do63]
G.I. Dowson, Cavitation of a viscous fluid in narrow passages, J. Fluid Mech. Math. 16 (1963), 595-619.

[EA75]
H.G. Elrod and M.L Adams, A computer program for cavitation, Cavitation and related phenomena in lubrication - Proceedings - Mech. Eng. Publ. Ltd (1975), 37-42.

[EGH00]
R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, North-Holland, Amsterdam, 2000, pp. 713-1020. MR 1804748 (2002e:65138)

[EGV03]
R. Eymard, T. Gallouët, and J. Vovelle, Limit boundary conditions for finite volume approximations of some physical problems, J. Comput. Appl. Math. 161 (2003), no. 2, 349-369. MR 2017019 (2005k:76083)

[FS01]
H. Freistühler and D. Serre, The $ L\sp 1$-stability of boundary layers for scalar viscous conservation laws, J. Dynam. Differential Equations 13 (2001), no. 4, 745-755. MR 1860284 (2002j:35206)

[IO60]
A. M. Il$ '$in and O. A. Oleinik, Asymptotic behavior of solutions of the Cauchy problem for some quasi-linear equations for large values of the time, Mat. Sb. (N.S.) 51 (93) (1960), 191-216. MR 0120469 (22:11222)

[Kaa99]
E. F. Kaasschieter, Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium, Comput. Geosci. 3 (1999), no. 1, 23-48. MR 1696184 (2000b:76103)

[Kim03]
Y.J. Kim, Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to $ N$-waves, J. Differential Equations 192 (2003), no. 1, 202-224. MR 1987091 (2004e:35147)

[KM85]
S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), no. 1, 97-127. MR 814544 (87h:35035)

[Lax57]
P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 0093653 (20:176)

[LN97]
T.-P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations 133 (1997), no. 2, 296-320. MR 1427855 (97j:35092)

[LP84]
T.-P. Liu and M. Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations 51 (1984), no. 3, 419-441. MR 735207 (85i:35094)

[Mar05]
S. Martin, First order quasilinear equations with boundary conditions in the $ L^\infty$ framework, J. Diff. Equations 236, no. 2, 375-406 (2007).

[MN94]
A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys. 165 (1994), no. 1, 83-96. MR 1298944 (95g:35120)

[MNRR96]
J. Málek, J. Necas, M. Rokyta, and M. Ruzicka, Weak and measure-valued solutions to evolutionary PDEs, Chapman & Hall, London, 1996.

[MT99]
C. Mascia and A. Terracina, Large-time behavior for conservation laws with source in a bounded domain, J. Differential Equations 159 (1999), no. 2, 485-514. MR 1730729 (2001b:35206)

[NPD97]
A. Nouri, F. Poupaud, and Y. Demay, An existence theorem for the multi-fluid Stokes problem, Quart. Appl. Math. 55 (1997), no. 3, 421-435. MR 1466141 (98h:35190)

[Ott93]
F. Otto, Initial-boundary value problem for a scalar conservation law, Ph. D. Thesis (1993).

[Ott96]
F. Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 8, 729-734. MR 1387428 (97c:35132)

[Pao03]
L. Paoli, Asymptotic behavior of a two fluid flow in a thin domain: from Stokes equations to Buckley-Leverett equation and Reynolds law, Asymptot. Anal. 34 (2003), no. 2, 93-120. MR 1992280 (2004j:35234)

[Ser96]
D. Serre, Systèmes de lois de conservation. II, Diderot Editeur, Paris, 1996. MR 1459989 (99e:35144)

[Ser04]
D. Serre, $ L\sp 1$-stability of nonlinear waves in scalar conservation laws, Evolutionary equations. Vol. I, Handb. Differ. Eq., North-Holland, Amsterdam, 2004, pp. 473-553. MR 2103702 (2006f:35177)

[Vas01]
A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal. 160 (2001), no. 3, 181-193. MR 1869441 (2002h:35186)

[Vov02]
J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains, Num. Math. 90 (2002), no. 3, 563-596. MR 1884231 (2002k:65158)

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Additional Information:

S. Martin
Affiliation: INSA de Lyon Institut Camille Jordan, CNRS UMR 5208, Bât. L. de Vinci, 21 av. Jean Capelle, F-69621 Villeurbanne cedex, France
Email: sebastien.martin@insa-lyon.fr

J. Vovelle
Affiliation: ENS Cachan Antenne de Bretagne IRMAR, CNRS UMR 6625, Avenue Robert Schuman, Campus de Ker Lann, F-35170 Bruz, France
Email: julien.vovelle@bretagne.ens-cachan.fr
PII: S0033-569X-07-01061-7
Received by editor(s): February 1, 2006
Posted: July 11, 2007
Copyright of article: Copyright 2007, Brown University
The copyright for this article reverts to public domain after 28 years from publication.



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