American Institute of Mathematical Sciences

Advanced
January  2010, 9(1): 47-60. doi: 10.3934/cpaa.2010.9.47

Solutions with large total variation to nonconservative hyperbolic systems

Rinaldo M. Colombo 1, and  Francesca Monti 2,

1. 

Dipartimento di Matematica, Via Branze, 38 – 25123 Brescia, Italy
2. 

Dip. di Matematica e Applicazioni, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy

Received  December 2008 Revised  May 2009 Published  October 2009

In this note we prove the existence of solutions for a class of first order non-linear hyperbolic systems. Given a possibly non conservative straight line system, we prove that all hyperbolic systems sufficiently near to the given one also admit global solutions. The total variation of the initial data is not assumed to be small. Stability estimates are also provided.
Keywords: non-conservative systems, First order hyperbolic equations, conservation laws..
Mathematics Subject Classification: 35L45, 35L6.
Citation: Rinaldo M. Colombo, Francesca Monti. Solutions with large total variation to nonconservative hyperbolic systems. Communications on Pure & Applied Analysis, 2010, 9 (1) : 47-60. doi: 10.3934/cpaa.2010.9.47
[1]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[2]

Eitan Tadmor. Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4579-4598. doi: 10.3934/dcds.2016.36.4579
[3]

Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673
[4]

Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011
[5]

Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51
[6]

Fumioki Asakura, Andrea Corli. The path decomposition technique for systems of hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 15-32. doi: 10.3934/dcdss.2016.9.15
[7]

Christophe Chalons, Paola Goatin, Nicolas Seguin. General constrained conservation laws. Application to pedestrian flow modeling. Networks & Heterogeneous Media, 2013, 8 (2) : 433-463. doi: 10.3934/nhm.2013.8.433
[8]

Xuemei Li, Zaijiu Shang. On the existence of invariant tori in non-conservative dynamical systems with degeneracy and finite differentiability. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4225-4257. doi: 10.3934/dcds.2019171
[9]

Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121
[10]

Yu Zhang, Yanyan Zhang. Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1523-1545. doi: 10.3934/cpaa.2019073
[11]

Tatsien Li, Libin Wang. Global exact shock reconstruction for quasilinear hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 597-609. doi: 10.3934/dcds.2006.15.597
[12]

Yanning Li, Edward Canepa, Christian Claudel. Efficient robust control of first order scalar conservation laws using semi-analytical solutions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 525-542. doi: 10.3934/dcdss.2014.7.525
[13]

Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419
[14]

Kaili Zhuang, Tatsien Li, Bopeng Rao. Exact controllability for first order quasilinear hyperbolic systems with internal controls. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1105-1124. doi: 10.3934/dcds.2016.36.1105
[15]

Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717
[16]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185
[17]

Weishi Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 871-884. doi: 10.3934/dcds.2004.10.871
[18]

Paolo Baiti, Helge Kristian Jenssen. Blowup in $\mathbf{L^{\infty}}$ for a class of genuinely nonlinear hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 837-853. doi: 10.3934/dcds.2001.7.837
[19]

Steinar Evje, Huanyao Wen, Lei Yao. Global solutions to a one-dimensional non-conservative two-phase model. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1927-1955. doi: 10.3934/dcds.2016.36.1927
[20]

Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences