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
Published electronically: October 10, 2012
MathSciNet review: 3020849
Abstract: We consider the Cauchy problem for the scalar conservation law
with , , for , and assume that the initial datum 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 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.
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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)
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)
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)
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)
C. M. Dafermos.
Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften 325. Springer-Verlag, Berlin, 2010. MR 2574377 (2011i:35150)
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)
- S. N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (123) (1970), 228-255. MR 0267257 (42:2159)
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)
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)
M. E. Schonbek.
Existence of solutions to singular conservation laws.
SIAM J. Math. Anal. 15 (6) (1984), 1125-1139. MR 762969 (86c:35099)
Affiliation: Department of Pure & Applied Mathematics, University of L’Aquila, Via Vetoio 1, 67010 Coppito (L’Aquila), Italy
G. M. Coclite
Affiliation: Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy
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