Skip to main content
SpringerLink
Log in

Open Questions in the Theory of One Dimensional Hyperbolic Conservation Laws

  • Conference paper
  • First Online:
Nonlinear Conservation Laws and Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 153))

Abstract

These notes are intended to provide an overview of the basic theory of one-dimensional hyperbolic systems of conservation laws, focusing on some major remaining open problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Amadori and R.M. Colombo, Continuous dependence for 2 × 2 conservation laws with boundary, J. Differential Equations 138 (1997), 229–266.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Amadori, L. Gosse, and G. Guerra, Global BV entropy solutions and unique-ness for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal. 162 (2002), 327–366.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Baiti and H.K. Jenssen, On the front tracking algorithm, J. Math. Anal. Appl. 217 (1998), 395–404.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Baiti and H.K. Jenssen, Blow-up in L1 for a class of genuinely nonlinear systems of conservation laws, Discr. Cont. Dynam. Syst. 7 (2001), 837–853.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Baiti, A. Bressan, and H.K. Jenssen, Instability of travelling wave profiles for the Lax-Friedrichs scheme, Discr. Cont. Dynam. Syst. 13 (2005), 877–899.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Baiti, A. Bressan, and H.K. Jenssen, BV instability of the Godunov scheme, Comm. Pure Appl. Math. 59 (2006), 1604–1638.

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Bianchini, A note on singular limits to hyperbolic systems of conservation laws, Comm. Pure Appl. Analysis 2 (2003), 51–64.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Bianchini, A note on the Riemann problem for non-conservative hyperbolic systems, Arch. Rational Mech. Anal. 166 (2003), 1–26.

    MATH  MathSciNet  Google Scholar 

  9. S. Bianchini, BV solutions of the semidiscrete upwind scheme, Arch. Rational Mech. Anal. 167 (2003), 1–81.

    MATH  MathSciNet  Google Scholar 

  10. S. Bianchini, Hyperbolic limit of the Jin-Xin relaxation model, Comm. Pure Appl. Math. 59 (2006), 688–753.

    Article  MATH  MathSciNet  Google Scholar 

  11. S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyper-bolic systems, Annals of Mathematics 161 (2005), 223–342.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Bianchini and L. Spinolo, The boundary Riemann solver coming from the real vanishing viscosity approximation, Arch. Ration. Mech. Anal. 191 (2009), 1–96.

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Bianchini and L.. Spinolo, Invariant manifolds for a singular ordinary differ-ential equation, to appear in J. Differential Equations.

    Google Scholar 

  14. A. Bressan, Global solutions to systems of conservation laws by wave-front track-ing, J. Math. Anal. Appl. 170 (1992), 414–432.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995), 205–230.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Bressan, Hyperbolic Systems of Conservation Laws, The One Dimensional Cauchy Problem, Oxford University Press, 2000.

    Google Scholar 

  17. A. Bressan, BV solutions to systems of conservation laws by vanishing viscosity, C.I.M.E. course in Cetraro, 2003, P.Marcati ed., Springer Lecture Notes in Math. 1911 (2007), 1–78.

    Google Scholar 

  18. A. Bressan, A tutorial on the Center Manifold Theorem, C.I.M.E. course in Ce-traro, 2003, P.Marcati ed., Springer Lecture Notes in Math. 1911 (2007), 327–344.

    Google Scholar 

  19. A. Bressan and R.M. Colombo, The semigroup generated by 2×2 conservation laws, Arch. Rational Mech. Anal. 133 (1995), 1–75.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Bressan, G. Crasta, and B. Piccoli, Well posedness of the Cauchy problem for n × n conservation laws, Amer. Math. Soc. Memoir 694 (2000).

    Google Scholar 

  21. A. Bressan and P. Goatin, Oleinik type estimates and uniqueness for n × n conservation laws, J. Diff. Equat. 156 (1999), 26–49.

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Bressan and H.K. Jenssen, On the convergence of Godunov scheme for non-linear hyperbolic systems, Chinese Ann. Math. B-21 (2000), 1–16.

    MathSciNet  Google Scholar 

  23. A. Bressan and P. LeFloch, Uniqueness of weak solutions to systems of conser-vation laws, Arch. Rat. Mech. Anal. 140 (1997), 301–317.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Bressan and M. Lewicka, A uniqueness condition for hyperbolic systems of conservation laws, Discr. Cont. Dynam. Syst. 6 (2000), 673–682.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Bressan, T.P. Liu, and T. Yang, L1 stability estimates for n×n conservation laws, Arch. Rational Mech. Anal. 149 (1999), 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  26. A. Bressan and W. Shen, Uniqueness for discontinuous O.D.E. and conservation laws, Nonlinear Analysis, T.M.A. 34 (1998), 637–652.

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Bressan and T. Yang, On the convergence rate of vanishing viscosity approx- imations, Comm. Pure Appl. Math 57 (2004), 1075–1109.

    Article  MATH  MathSciNet  Google Scholar 

  28. G.Q. Chen and J. Glimm, Global solutions to the compressible Euler equations with geometrical structure, Comm. Math. Phys. 180 (1996), 153–193.

    Article  MATH  MathSciNet  Google Scholar 

  29. C. Cheverry, Systèmes de lois de conservation et stabilité BV, Mémoire Soc. Math. France 75 (1998).

    Google Scholar 

  30. M.G. Crandall, The semigroup approach to first-order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108–132.

    Article  MATH  MathSciNet  Google Scholar 

  31. C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41.

    Article  MATH  MathSciNet  Google Scholar 

  32. C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Dafermos, Constantine M. Hyperbolic conservation laws in continuum physics. Second edition. Springer-Verlag, Berlin, 2005.

    Google Scholar 

  33. X. Ding, G.Q. Chen, and P. Luo, Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dy-namics, Comm. Math. Phys. 121 (1989), 63–84.

    Article  MATH  MathSciNet  Google Scholar 

  34. R. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of con-servation laws, J. Diff. Equat. 20 (1976), 187–212.

    Article  MATH  MathSciNet  Google Scholar 

  35. R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), 27–70.

    Article  MATH  MathSciNet  Google Scholar 

  36. C. Donadello and A. Marson, Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws, Nonlin. Diff. Equat. Appl. 14 (2007), 569–592.

    Article  MATH  MathSciNet  Google Scholar 

  37. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715.

    Article  MATH  MathSciNet  Google Scholar 

  38. J. Glimm and P. Lax, Decay of solutions of systems of nonlinear hyperbolic con-servation laws, Amer. Math. Soc. Memoir 101 (1970).

    Google Scholar 

  39. H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer Verlag, New York 2002.

    MATH  Google Scholar 

  40. H.K. Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal. 31 (2000), 894–908.

    Article  MATH  MathSciNet  Google Scholar 

  41. H.K. Jenssen and R. Young, Gradient driven and singular flux blowup of smooth solutions to hyperbolic systems of conservation laws, J. Hyp. Diff. Eq. 1 (2004), 627–641.

    Article  MATH  MathSciNet  Google Scholar 

  42. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1955), 235–277.

    Article  MathSciNet  Google Scholar 

  43. P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537–566.

    Article  MATH  MathSciNet  Google Scholar 

  44. P.G. LeFloch and M. Westdickenberg, Finite energy solutions to the isen-tropic Euler equations with geometric effects, J. Math. Pures Appl. 88 (2007), 389–429.

    Article  MATH  MathSciNet  Google Scholar 

  45. R.J. LeVeque and B. Temple, Stability of Godunov’s method for a class of 2×2 systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985), 115–123.

    MATH  MathSciNet  Google Scholar 

  46. M. Lewicka, Well-posedness for hyperbolic systems of conservation laws with large BV data, Arch. Rational Mech. Anal. 173 (2004), 415–445.

    Article  MATH  MathSciNet  Google Scholar 

  47. L. Lin, On the vacuum state for the equations of isentropic gas dynamics, J. Math. Anal. Appl. 121 (1987), 406–425.

    Article  MATH  MathSciNet  Google Scholar 

  48. L. Lin, Survey of the vacuum theory and conjecture, AMS/IP Stud. Adv. Math. 3 (1997), 293–305.

    Google Scholar 

  49. T.P. Liu, The Riemann problem for general systems of conservation laws, J. Dif-ferential Equations 18 (1975), 218–234.

    Article  MATH  Google Scholar 

  50. T.P. Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), 78–88.

    Article  MATH  MathSciNet  Google Scholar 

  51. T.P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135–148.

    Article  MATH  MathSciNet  Google Scholar 

  52. T. P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 767–796.

    Article  MATH  MathSciNet  Google Scholar 

  53. T.P. Liu, Asymptotic behavior of solutions of general system of nonlinear hyper-bolic conservation laws, Indiana Univ. Math. J. 27 (1978), 211–253.

    Article  MATH  MathSciNet  Google Scholar 

  54. T.P. Liu, Admissible solutions of hyperbolic conservation laws, Amer. Math. Soc. Memoir 240 (1981).

    Google Scholar 

  55. T.P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), 153–175.

    Article  MATH  MathSciNet  Google Scholar 

  56. T.P. Liu and J. Smoller, On the vacuum state for the isentropic gas dynamics equations, Advances Pure Appl. Math. 1 (1980), 345–359.

    MATH  MathSciNet  Google Scholar 

  57. T.P. Liu and S.H. Yu, Continuum shock profiles for discrete conservation laws I. Construction, Comm. Pure Appl. Math. 52 (1999), 85–127.

    Article  MATH  MathSciNet  Google Scholar 

  58. A. Majda and J. Ralston, Discrete shock profiles for systems of conservation laws, Comm. Pure Appl. Math. 32 (1979), 445–482.

    Article  MATH  MathSciNet  Google Scholar 

  59. S. Schochet, Sufficient conditions for local existence via Glimm’s scheme for large BV data, J. Differential Equations 89 (1991), 317–354.

    Article  MATH  MathSciNet  Google Scholar 

  60. D. Serre, Remarks about the discrete profiles of shock waves, Mat. Contemp. 11 (1996), 153–170.

    MATH  MathSciNet  Google Scholar 

  61. D. Serre, Systems of Conservation Laws I, II, Cambridge University Press, 2000.

    Google Scholar 

  62. D. Serre, Discrete shock profiles: existence and stability, C.I.M.E. course in Ce-traro, 2003, P. Marcati ed., Springer Lecture Notes in Math. 1911 (2007), 79–158.

    Google Scholar 

  63. J. Smoller, ShockWaves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.

    Google Scholar 

  64. R. Young, Global wave interactions in isentropic gas dynamics, Preprint 2008.

    Google Scholar 

  65. R. Young, Composite wave interactions and the collapse of vacuums in gas dy-namics, Preprint 2009.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Alberto Bressan

Authors
  1. Alberto Bressan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Bressan .

Editor information

Editors and Affiliations

  1. McAllister Bldg 201, University Park, 16802-6401, Pennsylvania, USA

    Alberto Bressan

  2. , Oxford Centre for Nonlinear PDE, Mathema, University of Oxford, St Giles 24–29, Oxford, OX1 3LB, Montserrat

    Gui-Qiang G. Chen

  3. , School of Mathematics, University of Minnesota, Church Street S.E, 206, Minneapolis, 55455, Minnesota, USA

    Marta Lewicka

  4. , Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260, Pennsylvania, USA

    Dehua Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this paper

Cite this paper

Bressan, A. (2011). Open Questions in the Theory of One Dimensional Hyperbolic Conservation Laws. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_1

Download citation

Publish with us

Policies and ethics

Search

Navigation