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Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

In this chapter we give some basic preliminaries about one-dimensional scalar conservation laws to highlight some of the fundamental issues and difficulties arising in the general theory of conservation laws. We also present some basic definitions and concepts which will be of constant use in this book.

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References

  1. Ambrosio, L., Caffarelli, L., Crandall, M., Evans, L., Fusco, N., Ambrosio, L.: Transport Equation and Cauchy Problem for Non-Smooth Vector Fields. In: Calculus of Variations and Nonlinear Partial Differential Equations. Lecture Notes in Mathematics, vol. 1927, pp. 1–41. Springer, Berlin (2008)

    Google Scholar 

  2. Bressan, A.: Hyperbolic systems of conservation laws. Oxford Lecture Series in Mathematics and its Applications, vol. 20. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  3. Burgers, J.M.: Application of a model system to illustrate some points of the statistical theory of free turbolence. Proc. Roy. Neth. Acad. Sci. Amsterdam 43, 2–12 (1940)

    MathSciNet  Google Scholar 

  4. Burgers, J.M.: The nonlinear diffusion equation. Dordrecht-Holland. D. Reidel Pub. Co., Boston (1974)

    MATH  Google Scholar 

  5. Colombo, R.M., Mercier, M., Rosini, M.D.: Stability and total variation estimates on general scalar balance laws. Commun. Math. Sci. 7(1), 37–65 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325. Springer, Berlin (2000)

    MATH  Google Scholar 

  7. Holden, H., Risebro, N.H.: Front tracking for hyperbolic conservation laws. Applied Mathematical Sciences, vol. 152. Springer, New York (2002)

    Book  MATH  Google Scholar 

  8. Hugoniot, H.: Sur un théorème général relatif à la propagation du mouvement dans le corps. C. R. Acad. Sci. Paris Sér. I Math. 102, 858–860 (1886)

    MATH  Google Scholar 

  9. Hugoniot, H.: Mémoire sur la propagation du mouvement dans le corps et spécialement dans le gaz parfaits. J. l’Ecoles Polytechn. 57, 3–97 (1887)

    Google Scholar 

  10. Hugoniot, H.: Mémoire sur la propagation du mouvement dans un fluid indéfini. J. Math. Pures Appl. 3, 477–492 (1887)

    Google Scholar 

  11. John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Communications on Pure and Applied Mathematics 27(3), 377–405 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karlsen, K.H., Towers, J.D.: Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux (2004)

    Google Scholar 

  13. Kružhkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228–255 (1970)

    MathSciNet  Google Scholar 

  14. Perthame, B.: Transport equations in biology. Frontiers in mathematics. Birkhäuser (2007)

    Google Scholar 

  15. Rankine, W.J.M.: On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance. Phil. Trans. Roy. Soc. 160, 277–288 (1870)

    Article  Google Scholar 

  16. Ritchmyer, R.D., Morton, K.W.: Difference methods for initial-value problems, 2nd edn. Interscience Tracts in Pure and Applied Mathematics, vol. 3. Interscience, New York (1967)

    Google Scholar 

  17. Whitham, G.B.: Linear and nonlinear waves. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York (1974)

    MATH  Google Scholar 

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Correspondence to Massimiliano Daniele Rosini .

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Rosini, M.D. (2013). One-Dimensional Scalar Conservation Laws. In: Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications. Understanding Complex Systems. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00155-5_3

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