A note on positive solutions for conservation laws with singular source
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- by D. Amadori and G. M. Coclite PDF
- Proc. Amer. Math. Soc. 141 (2013), 1613-1625 Request permission
Abstract:
We consider the Cauchy problem for the scalar conservation law \begin{equation*} \partial _t u+\partial _x f(u) =\displaystyle \frac {1}{g(u)},\qquad t>0,\ x\in \mathbb {R}, \end{equation*} 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
- 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, DOI 10.1016/j.jde.2003.10.004
- A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math. 60 (2000), no. 3, 916–938. MR 1750085, DOI 10.1137/S0036139997332099
- 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
- G. M. Coclite and M. M. Coclite, Conservation laws with singular nonlocal sources, J. Differential Equations 250 (2011), no. 10, 3831–3858. MR 2774070, DOI 10.1016/j.jde.2010.12.001
- 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, DOI 10.1007/978-3-642-04048-1
- Helge Holden and Nils Henrik Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences, vol. 152, Springer-Verlag, New York, 2002. MR 1912206, DOI 10.1007/978-3-642-56139-9
- S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
- 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, DOI 10.1016/0196-8858(80)90016-0
- 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, DOI 10.1007/BF02198141
- Maria Elena Schonbek, Existence of solutions to singular conservation laws, SIAM J. Math. Anal. 15 (1984), no. 6, 1125–1139. MR 762969, DOI 10.1137/0515088
Additional Information
- D. Amadori
- Affiliation: Department of Pure & Applied Mathematics, University of L’Aquila, Via Vetoio 1, 67010 Coppito (L’Aquila), Italy
- MR Author ID: 352024
- 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
- Received by editor(s): August 26, 2011
- Published electronically: October 10, 2012
- Communicated by: Walter Craig
- © Copyright 2012 American Mathematical Society
- 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
- MathSciNet review: 3020849