Existence and stability of a multidimensional scalar conservation law with discontinuous flux

Previous Article
The coolest path problem
NHM Home
This Issue
Next Article

March 2010, 5(1): 163-188. doi: 10.3934/nhm.2010.5.163

Existence and stability of a multidimensional scalar conservation law with discontinuous flux

1.	Faculty of Mathematics, University of Montenegro, Cetinjski put bb, 81000 Podgorica, Montenegro

Received July 2009 Revised December 2009 Published February 2010

Abstract
Related Papers

We prove the existence and stability of an entropy solution to a multidimensional scalar conservation law with discontinuous flux with no genuine nonlinearity assumptions. The proof is based on the corresponding kinetic formulation of the equation under consideration and a "smart" change of an unknown function.

Keywords: multidimensional scalar conservation law; discontinuous flux; existence and uniqueness; kinetic formulation..

Mathematics Subject Classification: Primary: 35L6.

Citation: Darko Mitrovic. Existence and stability of a multidimensional scalar conservation law with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (1) : 163-188. doi: 10.3934/nhm.2010.5.163

[1]	Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520
[2]	Raimund Bürger, Stefan Diehl, María Carmen Martí. A conservation law with multiply discontinuous flux modelling a flotation column. Networks & Heterogeneous Media, 2018, 13 (2) : 339-371. doi: 10.3934/nhm.2018015
[3]	Darko Mitrovic. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1191-1210. doi: 10.3934/dcds.2011.30.1191
[4]	Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[5]	. Adimurthi, Siddhartha Mishra, G.D. Veerappa Gowda. Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes. Networks & Heterogeneous Media, 2007, 2 (1) : 127-157. doi: 10.3934/nhm.2007.2.127
[6]	Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[7]	Adimurthi , Shyam Sundar Ghoshal, G. D. Veerappa Gowda. Exact controllability of scalar conservation laws with strict convex flux. Mathematical Control & Related Fields, 2014, 4 (4) : 401-449. doi: 10.3934/mcrf.2014.4.401
[8]	Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints. Networks & Heterogeneous Media, 2017, 12 (2) : 245-258. doi: 10.3934/nhm.2017010
[9]	Giuseppe Maria Coclite, Lorenzo di Ruvo, Jan Ernest, Siddhartha Mishra. Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. Networks & Heterogeneous Media, 2013, 8 (4) : 969-984. doi: 10.3934/nhm.2013.8.969
[10]	Vicent Caselles. An existence and uniqueness result for flux limited diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1151-1195. doi: 10.3934/dcds.2011.31.1151
[11]	Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052
[12]	Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2981-2990. doi: 10.3934/dcds.2016.36.2981
[13]	Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic & Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901
[14]	Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic & Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205
[15]	Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419
[16]	Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[17]	Luís Balsa Bicho, António Ornelas. Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 439-451. doi: 10.3934/dcds.2011.29.439
[18]	Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[19]	F. R. Guarguaglini, R. Natalini. Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology. Communications on Pure & Applied Analysis, 2007, 6 (1) : 287-309. doi: 10.3934/cpaa.2007.6.287
[20]	Marta Strani. Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1653-1667. doi: 10.3934/cpaa.2014.13.1653

2018 Impact Factor: 0.871

American Institute of Mathematical Sciences