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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**Debora Amadori, Laurent Gosse, and Graziano Guerra,*Godunov-type approximation for a general resonant balance law with large data*, J. Differential Equations**198**(2004), no. 2, 233–274. MR**2038581**, 10.1016/j.jde.2003.10.004**2.**A. Aw and M. Rascle,*Resurrection of “second order” models of traffic flow*, SIAM J. Appl. Math.**60**(2000), no. 3, 916–938 (electronic). MR**1750085**, 10.1137/S0036139997332099**3.**Alberto Bressan,*Hyperbolic systems of conservation laws*, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR**1816648****4.**G. M. Coclite and M. M. Coclite,*Conservation laws with singular nonlocal sources*, J. Differential Equations**250**(2011), no. 10, 3831–3858. MR**2774070**, 10.1016/j.jde.2010.12.001**5.**Constantine M. Dafermos,*Hyperbolic conservation laws in continuum physics*, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR**2574377****6.**Helge Holden and Nils Henrik Risebro,*Front tracking for hyperbolic conservation laws*, Applied Mathematical Sciences, vol. 152, Springer-Verlag, New York, 2002. MR**1912206****7.**S. N. Kružkov,*First order quasilinear equations with several independent variables.*, Mat. Sb. (N.S.)**81 (123)**(1970), 228–255 (Russian). MR**0267257****8.**T. P. Liu and J. A. Smoller,*On the vacuum state for the isentropic gas dynamics equations*, Adv. in Appl. Math.**1**(1980), no. 4, 345–359. MR**603135**, 10.1016/0196-8858(80)90016-0**9.**R. Natalini, C. Sinestrari, and A. Tesei,*Incomplete blowup of solutions of quasilinear hyperbolic balance laws*, Arch. Rational Mech. Anal.**135**(1996), no. 3, 259–296. MR**1418466**, 10.1007/BF02198141**10.**Maria Elena Schonbek,*Existence of solutions to singular conservation laws*, SIAM J. Math. Anal.**15**(1984), no. 6, 1125–1139. MR**762969**, 10.1137/0515088

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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:
http://dx.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