Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Hyperbolic Conservation Laws

  • Alberto BressanEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_278-2


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.


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 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....


Weak Solution Space Dimension Hyperbolic System Riemann Problem Piecewise Constant Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.


  1. Ambrosio L, Bouchut F, De Lellis C (2004) Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Commun Part Differ Equ 29:1635–1651CrossRefzbMATHGoogle Scholar
  2. Baiti P, Bressan A, Jenssen HK (2006) BV instability of the Godunov scheme. Commun Pure Appl Math 59:1604–1638MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bianchini S (2003a) On the Riemann problem for non-conservative hyperbolic systems. Arch Ration Mech Anal 166:1–26MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bianchini S (2003b) BV solutions of the semidiscrete upwind scheme. Arch Ration Mech Anal 167:1–81MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bianchini S (2006) Hyperbolic limit of the Jin-Xin relaxation model. Commun Pure Appl Math 59:688–753MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bianchini S, Bressan A (2005) Vanishing viscosity solutions to nonlinear hyperbolic systems. Ann Math 161:223–342MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bressan A (1992) Global solutions to systems of conservation laws by wave-front tracking. J Math Anal Appl 170:414–432MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bressan A (2000) Hyperbolic systems of conservation laws. The one dimensional Cauchy problem. Oxford University Press, OxfordzbMATHGoogle Scholar
  9. Bressan A (2003) An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend Sem Mat Univ Padova 110:103–117Google Scholar
  10. Bressan A, Marson A (1998) Error bounds for a deterministic version of the Glimm scheme. Arch Ration Mech Anal 142:155–176MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bressan A, Liu TP, Yang T (1999) L 1 stability estimates for n × n conservation laws. Arch Ration Mech Anal 149:1–22MathSciNetCrossRefzbMATHGoogle Scholar
  12. Chen G-Q (2011) Multidimensional conservation laws: overview, problems, and perspective. In: Nonlinear conservation laws and applications, vol 153, IMA volumes in mathematics and its applications. Springer, New York, pp 23–72CrossRefGoogle Scholar
  13. Chen G-Q, Feldman M (2010) Global solutions of shock reflection by large-angle wedges for potential flow. Ann Math 171:1067–1182MathSciNetCrossRefzbMATHGoogle Scholar
  14. Chen GQ, Zhang Y, Zhu D (2006) Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch Ration Mech Anal 181:261–310MathSciNetCrossRefzbMATHGoogle Scholar
  15. Chiodaroli E, Kreml O (2014) On the energy rissipation rate of solutions to the compressible isentropic Euler system. Arch Ration Mech Anal 214:1019–1049Google Scholar
  16. Cockburn B, Shu CW (1998) The local discontinuous Galerkin finite element method for convection diffusion systems. SIAM J Numer Anal 35:2440–2463MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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 54:545–581MathSciNetADSzbMATHGoogle Scholar
  18. Courant R, Friedrichs KO (1948) Supersonic flow and shock waves. Wiley-Interscience, New YorkzbMATHGoogle Scholar
  19. Crandall MG (1972) The semigroup approach to first-order quasilinear equations in several space variables. Isr J Math 12:108–132MathSciNetCrossRefzbMATHGoogle Scholar
  20. Dafermos C (1972) Polygonal approximations of solutions of the initial value problem for a conservation law. J Math Anal Appl 38:33–41MathSciNetCrossRefzbMATHGoogle Scholar
  21. Dafermos C (2010) Hyperbolic conservation laws in continuum physics, 3rd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  22. De Lellis C, Székelyhidi L (2010) On admissibility criteria for weak solutions of the Euler equations. Arch Ration Mech Anal 195:225–260MathSciNetCrossRefzbMATHGoogle Scholar
  23. De Lellis C, Székelyhidi L (2012) The h-principle and the equations of fluid dynamics. Bull Am Math Soc 49:347–375CrossRefzbMATHGoogle Scholar
  24. DiPerna RJ (1976) Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J Differ Equ 20:187–212MathSciNetCrossRefADSzbMATHGoogle Scholar
  25. DiPerna R (1983) Convergence of approximate solutions to conservation laws. Arch Ration Mech Anal 82:27–70MathSciNetCrossRefzbMATHGoogle Scholar
  26. Euler L (1755) Principes généraux du mouvement des fluides. Mém Acad Sci Berl 11:274–315Google Scholar
  27. Evans LC, Gariepy RF (1992) Measure theory and fine properties of functions. CRC Press, Boca RatonzbMATHGoogle Scholar
  28. Friedrichs KO, Lax P (1971) Systems of conservation laws with a convex extension. Proc Natl Acad Sci U S A 68:1686–1688MathSciNetCrossRefADSzbMATHGoogle Scholar
  29. Glimm J (1965) Solutions in the large for nonlinear hyperbolic systems of equations. Commun Pure Appl Math 18:697–715MathSciNetCrossRefzbMATHGoogle Scholar
  30. Glimm J, Lax P (1970) Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society 101. American Mathematical Society, ProvidenceGoogle Scholar
  31. Glimm J, Grove JW, Li XL, Shyue KM, Zeng Y, Zhang Q (1998) Three dimensional front tracking. SIAM J Sci Comput 19:703–727MathSciNetCrossRefzbMATHGoogle Scholar
  32. Glimm J, Li X, Liu Y (2002) Conservative front tracking in one space dimension. In: Fluid flow and transport in porous media: mathematical and numerical treatment, (South Hadley, MA, 2001), vol 295, Contemporary Mathematics. American Mathematical Society, Providence, pp 253–264CrossRefGoogle Scholar
  33. Goodman J, Xin Z (1992) Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch Ration Mech Anal 121:235–265MathSciNetCrossRefzbMATHGoogle Scholar
  34. Holden H, Risebro NH (2002) Front tracking for hyperbolic conservation laws. Springer, New YorkCrossRefzbMATHGoogle Scholar
  35. Jabin P, Perthame B (2002) Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim Calc Var 8:761–774MathSciNetCrossRefzbMATHGoogle Scholar
  36. Jenssen HK (2000) Blowup for systems of conservation laws. SIAM J Math Anal 31:894–908MathSciNetCrossRefzbMATHGoogle Scholar
  37. 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 35:2147–2168MathSciNetCrossRefzbMATHGoogle Scholar
  38. Jin S, Xin ZP (1995) The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun Pure Appl Math 48:235–276MathSciNetCrossRefzbMATHGoogle Scholar
  39. Kawashima S, Matsumura A (1994) Stability of shock profiles in viscoelasticity with non-convex constitutive relations. Commun Pure Appl Math 47:1547–1569MathSciNetCrossRefzbMATHGoogle Scholar
  40. Keyfitz B (2004) Self-similar solutions of two-dimensional conservation laws. J Hyperbol Differ Equ 1:445–492MathSciNetCrossRefzbMATHGoogle Scholar
  41. Kruzhkov S (1970) First-order quasilinear equations with several space variables. Math USSR Sb 10:217–273CrossRefzbMATHGoogle Scholar
  42. Kuznetsov NN (1976) Accuracy of some approximate methods for computing the weak solution of a first order quasilinear equation. USSR Comp Math Math Phys 16:105–119CrossRefGoogle Scholar
  43. Lax P (1957) Hyperbolic systems of conservation laws II. Commun Pure Appl Math 10:537–566MathSciNetCrossRefzbMATHGoogle Scholar
  44. Lax P (1971) Shock waves and entropy. In: Zarantonello E (ed) Contributions to nonlinear functional analysis. Academic, New York, pp 603–634Google Scholar
  45. Leveque RJ (1990) Numerical methods for conservation laws, Lectures in mathematics. Birkhöuser, BaselCrossRefzbMATHGoogle Scholar
  46. Leveque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  47. Li J, Zhang T, Yang S (1998) The two dimensional problem in gas dynamics. Pitman/Longman, EssexzbMATHGoogle Scholar
  48. Lighthill MJ, Whitham GB (1955) On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc Roy Soc Lond A 229:317–345MathSciNetCrossRefADSzbMATHGoogle Scholar
  49. Lions PL, Perthame E, Tadmor E (1994) A kinetic formulation of multidimensional scalar conservation laws and related equations. J Am Math Soc 7:169–191MathSciNetCrossRefzbMATHGoogle Scholar
  50. Liu TP (1977) Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Commun Pure Appl Math 30:767–796CrossRefADSzbMATHGoogle Scholar
  51. Liu TP (1981) Admissible solutions of hyperbolic conservation laws, Memoirs of the American Mathematical Society 240. American Mathematical Society, ProvidenceGoogle Scholar
  52. Liu TP (1985) Nonlinear stability of shock waves for viscous conservation laws. Memoirs Amer. Math. Soc. 328, American Mathematical Society, Providence.Google Scholar
  53. Liu TP (1987) Hyperbolic conservation laws with relaxation. Commun Math Phys 108:153–175CrossRefADSzbMATHGoogle Scholar
  54. Majda A (1984) Compressible fluid flow and systems of conservation laws in several space variables. Springer, New YorkCrossRefzbMATHGoogle Scholar
  55. Metivier G (2001) Stability of multidimensional shocks. In: Advances in the theory of shock waves. Birkhäuser, Boston, pp 25–103CrossRefGoogle Scholar
  56. Morawetz CS (1994) Potential theory for regular and mach reflection of a shock at a wedge. Commun Pure Appl Math 47:593–624MathSciNetCrossRefzbMATHGoogle Scholar
  57. Nessyahu H, Tadmor E (1990) Non-oscillatory central differencing for hyperbolic conservation laws. J Comp Phys 87:408–463MathSciNetCrossRefADSzbMATHGoogle Scholar
  58. Perthame B (2002) Kinetic formulation of conservation laws. Oxford University Press, OxfordzbMATHGoogle Scholar
  59. Rauch J (1986) BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one. Commun Math Phys 106:481–484MathSciNetCrossRefADSzbMATHGoogle Scholar
  60. Riemann B (1860) Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Gött Abh Math Cl 8:43–65Google Scholar
  61. Shu CW (1998) Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997), Lecture notes in mathematics 1697. Springer, Berlin, pp 325–432CrossRefGoogle Scholar
  62. Smoller J (1994) Shock waves and reaction–diffusion equations, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
  63. Tadmor E (1998) Approximate solutions of nonlinear conservation laws. In: Advanced numerical approximation of nonlinear hyperbolic equations. Lecture notes in mathematics 1697, 1997 C.I.M.E. course in Cetraro. Springer, Berlin, pp 1–149Google Scholar
  64. Volpert AI (1967) The spaces BV and quasilinear equations. Math USSR Sb 2:225–267CrossRefGoogle Scholar
  65. Young R (2003) Isentropic gas dynamics with large data. In: Hyperbolic problems: theory, numerics, applications. Springer, Berlin, pp 929–939CrossRefGoogle Scholar
  66. Zheng Y (2001) Systems of conservation laws. Two-dimensional Riemann problems. Birkhäuser, BostonCrossRefzbMATHGoogle Scholar
  67. Zumbrun K, With an appendix by Jenssen HK, Lyng G (2004) Stability of large-amplitude shock waves of compressible Navier–Stokes equations. In: Handbook of mathematical fluid dynamics, vol III. North-Holland, Amsterdam, pp 311–533Google Scholar

Books and Reviews

  1. Benzoni-Gavage S, Serre D (2007) Multidimensional hyperbolic partial differential equations. First-order systems and applications, Oxford mathematical monographs. Clarendon/Oxford University Press, OxfordGoogle Scholar
  2. Boillat G (1996) Nonlinear hyperbolic fields and waves. In: Recent mathematical methods in nonlinear wave propagation (Montecatini Terme, 1994), Lecture notes in mathematics 1640. Springer, Berlin, pp 1–47CrossRefGoogle 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 421–543CrossRefGoogle Scholar
  4. Courant R, Hilbert D (1962) Methods of mathematical physics, vol II. Wiley-Interscience, New YorkzbMATHGoogle Scholar
  5. Garavello M, Piccoli B (2006) Traffic flow on networks. American Institute of Mathematical Sciences, SpringfieldzbMATHGoogle Scholar
  6. Godlewski E, Raviart PA (1996) Numerical approximation of hyperbolic systems of conservation laws. Springer, New YorkCrossRefzbMATHGoogle Scholar
  7. Gurtin ME (1981) An introduction to continuum mechanics. Academic Press, New YorkzbMATHGoogle Scholar
  8. Jeffrey A (1976) Quasilinear hyperbolic systems and waves. Pitman, LondonzbMATHGoogle Scholar
  9. Hörmander L (1997) Lectures on nonlinear hyperbolic differential equations. Springer, BerlinzbMATHGoogle Scholar
  10. Kreiss HO, Lorenz J (1989) Initial-boundary value problems and the Navier–Stokes equations. Academic, San DiegozbMATHGoogle Scholar
  11. Kröner D (1997) Numerical schemes for conservation laws, Wiley-Teubner series advances in numerical mathematics. Wiley, ChichesterzbMATHGoogle Scholar
  12. Landau LD, Lifshitz EM (1959) Fluid mechanics. Pergamon, LondonGoogle Scholar
  13. Li T-T (1994) Global classical solutions for quasilinear hyperbolic systems. Wiley, ChichesterzbMATHGoogle Scholar
  14. Li T-T, Yu W-C (1985) Boundary value problems for quasilinear hyperbolic systems, vol 5, Mathematics series. Duke University, DurhamzbMATHGoogle Scholar
  15. Lu Y (2003) Hyperbolic conservation laws and the compensated compactness method. Chapman & Hall/CRC Press, Boca RatonzbMATHGoogle Scholar
  16. Morawetz CS (1981) Lecture notes on nonlinear waves and shocks. Tata Institute of Fundamental Research, BombayGoogle 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, 1983Google Scholar
  18. Serre D (2000) Systems of conservation laws I, II. Cambridge University Press, CambridgeGoogle Scholar
  19. Whitham GB (1999) Linear and nonlinear waves. Wiley-Interscience, New YorkCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsPenn State UniversityUniversity ParkUSA