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A note on positive solutions for conservation laws with singular source


Authors: D. Amadori and G. M. Coclite
Journal: Proc. Amer. Math. Soc. 141 (2013), 1613-1625
MSC (2010): Primary 35B25, 35B09, 35L65
DOI: https://doi.org/10.1090/S0002-9939-2012-11694-6
Published electronically: October 10, 2012
MathSciNet review: 3020849
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Cauchy problem for the scalar conservation law

$\displaystyle \partial _t u+\partial _x f(u) =\displaystyle \frac {1}{g(u)},\qquad t>0,\ x\in \mathbb{R},$    

with $ g\in C^1(\mathbb{R})$, $ g(0)=0$, $ g(u)>0$ for $ u>0$, and assume that the initial datum $ u_0$ is nonnegative.

We show the existence of entropy solutions that are positive a.e. by means of an approximation of the equation that preserves positive solutions and by passing to the limit using a monotonicity argument. The difficulty lies in handling the singularity of the right-hand side (the source term) as $ u$ possibly vanishes at the initial time. The source term is shown to be locally integrable.

Moreover, we prove a uniqueness and stability result for the above equation.


References [Enhancements On Off] (What's this?)

  • 1.
    D. Amadori, L. Gosse, and G. Guerra. Godunov-type approximation for a general resonant balance law with large data. J. Differential Equations 198 (2) (2004), 233-274. MR 2038581 (2004m:65114)
  • 2.
    A. Aw and M. Rascle.
    Resurrection of ``second order'' models of traffic flow.
    SIAM J. Appl. Math. 60 (3) (2000), 916-938. MR 1750085 (2001a:35111)
  • 3.
    A. Bressan.
    Hyperbolic Systems of Conservation Laws. The one-dimensional Cauchy problem.
    Oxford Lecture Series in Mathematics and its Applications 20. Oxford University Press, Oxford, 2000. MR 1816648 (2002d:35002)
  • 4.
    G. M. Coclite and M. M. Coclite.
    Conservation laws with singular nonlocal sources.
    J. Differential Equations 250 (10) (2011), 3831-3858. MR 2774070 (2012b:35190)
  • 5.
    C. M. Dafermos.
    Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften 325. Springer-Verlag, Berlin, 2010. MR 2574377 (2011i:35150)
  • 6.
    H. Holden and N. H. Risebro.
    Front tracking for hyperbolic conservation laws.
    Applied Mathematical Sciences 152. Springer-Verlag, New York, 2002. MR 1912206 (2003e:35001)
  • 7. S. N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (123) (1970), 228-255. MR 0267257 (42:2159)
  • 8.
    T. P. Liu and J. A. Smoller.
    On the vacuum state for the isentropic gas dynamics equations.
    Adv. in Appl. Math. 1(4) (1980), 345-359. MR 603135 (83a:35065)
  • 9.
    R. Natalini, C. Sinestrari, and A. Tesei.
    Incomplete blowup of solutions of quasilinear hyperbolic balance laws.
    Arch. Rational Mech. Anal. 135 (3) (1996), 259-296. MR 1418466 (98d:35135)
  • 10.
    M. E. Schonbek.
    Existence of solutions to singular conservation laws.
    SIAM J. Math. Anal. 15 (6) (1984), 1125-1139. MR 762969 (86c:35099)

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Additional Information

D. Amadori
Affiliation: Department of Pure & Applied Mathematics, University of L’Aquila, Via Vetoio 1, 67010 Coppito (L’Aquila), Italy
Email: amadori@univaq.it

G. M. Coclite
Affiliation: Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy
Email: coclitegm@dm.uniba.it

DOI: https://doi.org/10.1090/S0002-9939-2012-11694-6
Keywords: Singular nonlinear problems, positive solutions, conservation laws
Received by editor(s): August 26, 2011
Published electronically: October 10, 2012
Communicated by: Walter Craig
Article copyright: © Copyright 2012 American Mathematical Society

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