Skip to main content
SpringerLink
Log in

Hyperbolic Conservation Laws

  • Reference work entry
Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Examples of Conservation Laws

Shocks and Weak Solutions

Hyperbolic Systems in One Space Dimension

Entropy Admissibility Conditions

The Riemann Problem

Global Solutions

Hyperbolic Systems in Several Space Dimensions

Numerical Methods

Future Directions

Bibliography

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 600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 329.99
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

Abbreviations

Conservation law :

Several physical laws state that certain basic quantities such as mass, energy, or electric charge, are globally conserved. A conservation law is a mathematical equation describing how the density of a conserved quantity varies in time. It is formulated as a partial differential equation having divergence form.

Flux function :

The flux of a conserved quantity is a vector field, describing how much of the given quantity moves across any surface, at a given time.

Shock :

Solutions to conservation laws often develop shocks, i. e. surfaces across which the basic physical fields are discontinuous. Knowing the two limiting values of a field on opposite sides of a shock, one can determine the speed of propagation of a shock in terms of the Rankine–Hugoniot equations.

Entropy :

An entropy is an additional quantity which is globally conserved for every smooth solution to a system of conservation laws. In general, however, entropies are not conserved by solutions containing shocks. Imposing that certain entropies increase (or decrease) in correspondence to a shock, one can determine a unique physically admissible solution to the mathematical equations.

Bibliography

Primary Literature

  1. Ambrosio L, Bouchut F, De Lellis C (2004) Well‐posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Comm Part Diff Equat 29:1635–1651

    Article  MATH  Google Scholar 

  2. Baiti P, Bressan A, Jenssen HK (2006) An instability of the Godunov scheme. Comm Pure Appl Math 59:1604–1638

    Article  MathSciNet  MATH  Google Scholar 

  3. Bianchini S (2003) On the Riemann problem for non‐conservative hyperbolic systems. Arch Rat Mach Anal 166:1–26

    MathSciNet  MATH  Google Scholar 

  4. Bianchini S (2003) BV solutions of the semidiscrete upwind scheme. Arch Ration Mech Anal 167:1–81

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Bianchini S, Bressan A (2005) Vanishing viscosity solutions to nonlinear hyperbolic systems. Ann Math 161:223–342

    Article  MathSciNet  MATH  Google Scholar 

  7. Bressan A (1992) Global solutions to systems of conservation laws by wave-front tracking. J Math Anal Appl 170:414–432

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  9. Bressan A (2003) An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend Sem Mat Univ Padova 110:103–117

    MathSciNet  MATH  Google Scholar 

  10. Bressan A, Liu TP, Yang T (1999) L 1 stability estimates for \({n\times n}\) conservation laws. Arch Ration Mech Anal 149:1–22

    Article  MathSciNet  MATH  Google Scholar 

  11. Bressan A, Marson A (1998) Error bounds for a deterministic version of the Glimm scheme. Arch Rat Mech Anal 142:155–176

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen GQ, Zhang Y, Zhu D (2006) Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch Ration Mech Anal 181:261–310

    Article  MathSciNet  MATH  Google Scholar 

  13. Cockburn B, Shu CW (1998) The local discontinuous Galerkin finite element method for convection diffusion systems. SIAM J Numer Anal 35:2440–2463

    Article  MathSciNet  MATH  Google Scholar 

  14. Cockburn B, Hou S, Shu C-W (1990) The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math Comput 54:545–581

    MathSciNet  MATH  Google Scholar 

  15. Courant R, Friedrichs KO (1948) Supersonic Flow and Shock Waves. Wiley Interscience, New York

    MATH  Google Scholar 

  16. Crandall MG (1972) The semigroup approach to first-order quasilinear equations in several space variables. Israel J Math 12:108–132

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Dafermos C (2005) Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin, pp 325–626

    Google Scholar 

  19. DiPerna RJ (1976) Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J Differ Equ 20:187–212

    Article  MathSciNet  MATH  Google Scholar 

  20. DiPerna R (1983) Convergence of approximate solutions to conservation laws. Arch Ration Mech Anal 82:27–70

    Article  MathSciNet  MATH  Google Scholar 

  21. Euler L (1755) Principes généraux du mouvement des fluides. Mém Acad Sci Berlin 11:274–315

    Google Scholar 

  22. Evans LC, Gariepy RF (1992) Measure Theory and Fine Properties of Functions. C.R.C. Press, Boca Raton, pp viii–268

    Google Scholar 

  23. Friedrichs KO, Lax P (1971) Systems of conservation laws with a convex extension. Proc Nat Acad Sci USA 68:1686–1688

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Glimm J, Lax P (1970) Decay of solutions of systems of nonlinear hyperbolic conservation laws. Am Math Soc Memoir 101:xvii–112

    Google Scholar 

  26. Glimm J, Li X, Liu Y (2001) Conservative Front Tracking in One Space Dimension. In: Fluid flow and transport in porous media: mathematical and numerical treatment, (South Hadley, 2001), pp 253–264. Contemp Math 295, Amer Math Soc, Providence

    Google Scholar 

  27. Glimm J, Grove JW, Li XL, Shyue KM, Zeng Y, Zhang Q (1998) Three dimensional front tracking. SIAM J Sci Comput 19:703–727

    Article  MathSciNet  MATH  Google Scholar 

  28. Goodman J, Xin Z (1992) Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch Ration Mech Anal 121:235–265

    Article  MathSciNet  MATH  Google Scholar 

  29. Holden H, Risebro NH (2002) Front tracking for hyperbolic conservation laws. Springer, New York

    Book  MATH  Google Scholar 

  30. Jabin P, Perthame B (2002) Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim Calc Var 8:761–774

    Article  MathSciNet  MATH  Google Scholar 

  31. Jenssen HK (2000) Blowup for systems of conservation laws. SIAM J Math Anal 31:894–908

    Article  MathSciNet  MATH  Google Scholar 

  32. Jiang G-S, Levy D, Lin C-T, Osher S, Tadmor E (1998) High‐resolution non‐oscillatory central schemes with non‐staggered grids for hyperbolic conservation laws. SIAM J Numer Anal 35:2147–2168

    Article  MathSciNet  MATH  Google Scholar 

  33. Jin S, Xin ZP (1995) The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm Pure Appl Math 48:235–276

    Article  MathSciNet  MATH  Google Scholar 

  34. Kawashima S, Matsumura A (1994) Stability of shock profiles in viscoelasticity with non‐convex constitutive relations. Comm Pure Appl Math 47:1547–1569

    Article  MathSciNet  MATH  Google Scholar 

  35. Keyfitz B (2004) Self‐similar solutions of two‐dimensional conservation laws. J Hyperbolic Differ Equ 1:445–492

    Article  MathSciNet  MATH  Google Scholar 

  36. Kruzhkov S (1970) First-order quasilinear equations with several space variables. Math USSR Sb 10:217–273

    Article  MATH  Google Scholar 

  37. Kuznetsov NN (1976) Accuracy of some approximate methods for computing the weak solution of a first order quasilinear equation. USSR Comp Math Math Phys 16:105–119

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Lax P (1971) Shock waves and entropy. In: Zarantonello E (ed) Contributions to Nonlinear Functional Analysis. Academic Press, New York, pp 603–634

    Google Scholar 

  40. Leveque RJ (1990) Numerical methods for conservation laws. Lectures in Mathematics. Birkhäuser, Basel

    Google Scholar 

  41. Leveque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  42. Li J, Zhang T, Yang S (1998) The two dimensional problem in gas dynamics. Pitman, Longman Essex

    MATH  Google Scholar 

  43. Lighthill MJ, Whitham GB (1955) On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc Roy Soc Lond A 229:317–345

    Article  MathSciNet  MATH  Google Scholar 

  44. Lions PL, Perthame E, Tadmor E (1994) A kinetic formulation of multidimensional scalar conservation laws and related equations. JAMS 7:169–191

    MathSciNet  MATH  Google Scholar 

  45. Liu TP (1977) Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Comm Pure Appl Math 30:767–796

    Article  MathSciNet  MATH  Google Scholar 

  46. Liu TP (1981) Admissible solutions of hyperbolic conservation laws. Mem Am Math Soc 30(240):iv–78

    Google Scholar 

  47. Liu TP (1985) Nonlinear stability of shock waves for viscous conservation laws. Mem Am Math Soc 56(328):v–108

    Google Scholar 

  48. Liu TP (1987) Hyperbolic conservation laws with relaxation. Comm Math Pys 108:153–175

    Article  MATH  Google Scholar 

  49. Majda A (1984) Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York

    Book  MATH  Google Scholar 

  50. Metivier G (2001) Stability of multidimensional shocks. Advances in the theory of shock waves. Birkhäuser, Boston, pp 25–103

    Book  Google Scholar 

  51. Morawetz CS (1994) Potential theory for regular and Mach reflection of a shock at a wedge. Comm Pure Appl Math 47:593–624

    Article  MathSciNet  MATH  Google Scholar 

  52. Nessyahu H, Tadmor E (1990) Non‐oscillatory central differencing for hyperbolic conservation laws J Comp Phys 87:408–463

    Article  MathSciNet  MATH  Google Scholar 

  53. Perthame B (2002) Kinetic Formulation of Conservation Laws. Oxford Univ. Press, Oxford

    MATH  Google Scholar 

  54. Rauch J (1986) BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one. Comm Math Phys 106:481–484

    Article  MathSciNet  MATH  Google Scholar 

  55. Riemann B (1860) Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Gött Abh Math Cl 8:43–65

    Google Scholar 

  56. Shu CW (1998) Essentially non‐oscillatory and weighted essentially non‐oscillatory schemes for hyperbolic conservation laws. Advanced numerical approximation of nonlinear hyperbolic equations. (Cetraro, 1997), pp 325–432. Lecture Notes in Mathematics, vol 1697. Springer, Berlin

    Google Scholar 

  57. Smoller J (1994) Shock Waves and Reaction‐Diffusion Equations, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

  58. Tadmor E (1998) Approximate solutions of nonlinear conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol 1697. (1997 C.I.M.E. course in Cetraro, Italy) Springer, Berlin, pp 1–149

    Google Scholar 

  59. Volpert AI (1967) The spaces BV and quasilinear equations. Math USSR Sb 2:225–267

    Article  Google Scholar 

  60. Young R (2003) Isentropic gas dynamics with large data. In: Hyperbolic problems: theory, numerics, applications. Springer, Berlin, pp 929–939

    Chapter  Google Scholar 

  61. Zheng Y (2001) Systems of Conservation Laws. Two‐dimensional Riemann problems. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  62. Zumbrun K (2004) Stability of large‐amplitude shock waves of compressible Navier–Stokes equations. With an appendix by Helge Kristian Jenssen and Gregory Lyng. Handbook of Mathematical Fluid Dynamics, vol III. North‐Holland, Amsterdam, pp 311–533

    Google Scholar 

Books and Reviews

  1. Benzoni‐Gavage S, Serre D (2007) Multidimensional hyperbolic partial differential equations. First-order systems and applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford

    Google Scholar 

  2. Boillat G (1996) Nonlinear hyperbolic fields and waves. In: Recent Mathematical Methods in Nonlinear Wave Propagation (Montecatini Terme, 1994), pp 1–47. Lecture Notes in Math, vol 1640. Springer, Berlin

    Google Scholar 

  3. Chen GQ, Wang D (2002) The Cauchy problem for the Euler equations for compressible fluids. In: Handbook of Mathematical Fluid Dynamics, vol I. North‐Holland, Amsterdam, pp 421–543

    Chapter  Google Scholar 

  4. Courant R, Hilbert D (1962) Methods of Mathematical Physics, vol II. John Wiley & Sons - Interscience, New York

    MATH  Google Scholar 

  5. Garavello M, Piccoli B (2006) Traffic Flow on Networks. American Institute of Mathematical Sciences, Springfield

    MATH  Google Scholar 

  6. Godlewski E, Raviart PA (1996) Numerical approximation of hyperbolic systems of conservation laws. Springer, New York

    MATH  Google Scholar 

  7. Gurtin ME (1981) An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, 158. Academic Press, New York, pp xi–265

    Google Scholar 

  8. Hörmander L (1997) Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Berlin

    Google Scholar 

  9. Jeffrey A (1976) Quasilinear Hyperbolic Systems and Waves. Research Notes in Mathmatics, vol 5. Pitman Publishing, London, pp vii–230

    Google Scholar 

  10. Kreiss HO, Lorenz J (1989) Initial‐boundary value problems and the Navier–Stokes equations. Academic Press, San Diego

    MATH  Google Scholar 

  11. Kröner D (1997) Numerical schemes for conservation laws. In: Wiley‐Teubner Series Advances in Numerical Mathematics. John Wiley, Chichester

    Google Scholar 

  12. Landau LD, Lifshitz EM (1959) Fluid Mechanics. translated form the Russian by Sykes JB, Reid WH, Course of Theoretical Physics, vol 6. Pergamon press, London, Addison-Wesley, Reading, pp xii–536

    Google Scholar 

  13. Li T-T, Yu W-C (1985) Boundary value problems for quasilinear hyperbolic systems. Duke University Math. Series, vol 5. Durham, pp vii–325

    Google Scholar 

  14. Li T-T (1994) Global classical solutions for quasilinear hyperbolic systems. Wiley, Chichester

    MATH  Google Scholar 

  15. Lu Y (2003) Hyperbolic conservation laws and the compensated compactness method. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  16. Morawetz CS (1981) Lecture Notes on Nonlinear Waves and Shocks. Tata Institute of Fundamental Research, Bombay

    Google Scholar 

  17. Rozhdestvenski BL, Yanenko NN (1978) Systems of quasilinear equations and their applications to gas dynamics. Nauka, Moscow. English translation: American Mathematical Society, Providence

    Google Scholar 

  18. Serre D (2000) Systems of Conservation Laws, I Geometric structures, oscillations, and initial-boundry value problem, II Hyperbolicity, entropies, shock waves. Cambridge University Press, pp xii–263

    Google Scholar 

  19. Whitham GB (1999) Linear and Nonlinear Waves. Wiley‐Interscience, New York

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Alberto Bressan

Authors
  1. Alberto Bressan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Bressan, A. (2012). Hyperbolic Conservation Laws. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_44

Download citation

Publish with us

Policies and ethics

Search

Navigation