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Journal of Mathematical Sciences

, Volume 233, Issue 6, pp 905–929 | Cite as

Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview

  • S. Modena
Article

Abstract

In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws
$$ \left\{\begin{array}{l}{u}_t+f{(u)}_x=0,\\ {}u\left(t=0\right)={u}_0(x),\end{array}\right. $$
where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form
$$ \sum \limits_{t_j\;\mathrm{interaction}\ \mathrm{time}}\frac{\left|\sigma \left({\alpha}_j\right)-\sigma \left({\alpha}_j^{\prime}\right)\right|\left|{\alpha}_j\right|\left|{\alpha}_j^{\prime}\right|}{\left|{\alpha}_j\right|+\left|{\alpha}_j^{\prime}\right|}\le C(f)\mathrm{Tot}.\mathrm{Var}.{\left({u}_0\right)}^2, $$

where αj and \( {\alpha}_j^{\prime } \) are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form).

The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which:

• all the main ideas of the construction are presented;

• all the technicalities of the proof in the general setting [8] are avoided.

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References

  1. 1.
    F. Ancona and A. Marson, “Sharp convergence rate of the Glimm scheme for general nonlinear hyperbolic systems,” Commun. Math. Phys., 302, 581–630 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    P. Baiti and A. Bressan, “The semigroup generated by a Temple class system with large data,” Differ. Integr. Equ., 10, 401–418 (1997).MathSciNetzbMATHGoogle Scholar
  3. 3.
    S. Bianchini, “The semigroup generated by a Temple class system with nonconvex flux function,” Differ. Integr. Equ., 13 (10–12), 1529–1550 (2000).zbMATHGoogle Scholar
  4. 4.
    S. Bianchini, “Interaction estimates and Glimm functional for general hyperbolic systems,” Discrete Contin. Dyn. Syst., 9, 133–166 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. Bianchini and A. Bressan, “Vanishing viscosity solutions of nonlinear hyperbolic systems,” Ann. Math. (2), 161, 223–342 (2005).Google Scholar
  6. 6.
    S. Bianchini and S. Modena, “On a quadratic functional for scalar conservation laws,” J. Hyperbolic Differ. Equ., 11 (2), 355–435 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Bianchini and S. Modena, “Quadratic interaction functional for systems of conservation laws: a case study,” Bull. Inst. Math. Acad. Sin. (N.S.), 9 (3), 487–546 (2014).Google Scholar
  8. 8.
    S. Bianchini and S. Modena, “Quadratic interaction functional for general systems of conservation laws,” Commun. Math. Phys., 338, No. 3, 1075–1152 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem, Oxford University Press (2000).Google Scholar
  10. 10.
    A. Bressan and A. Marson “Error bounds for a deterministic version of the Glimm scheme,” Arch. Ration. Mech. Anal., 142, 155–176 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    C. Dafermos, Hyberbolic Conservation Laws in Continuum Physics, Springer (2005).Google Scholar
  12. 12.
    J. Glimm, “Solutions in the large for nonlinear hyperbolic systems of equations,” Commun. Pure Appl. Math., 18, 697–715 (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. D. Lax, “Hyperbolic systems of conservation laws. II,” Commun. Pure Appl. Math., 10, 537–566 (1957).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. D. Lax, “Shock waves and entropy,” In: Contribution to Nonlinear Functional Analysis, Academic Press, New York (1971), pp. 603–634.Google Scholar
  15. 15.
    T. P. Liu, “The Riemann problem for general 2 × 2 conservation laws,” Trans. Am. Math. Soc., 199, 89–112 (1974).MathSciNetzbMATHGoogle Scholar
  16. 16.
    T. P. Liu, “The Riemann problem for general systems of conservation laws,” J. Differ. Equ., 18, 218–234 (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    T. P. Liu, “The deterministic version of the Glimm scheme,” Commun. Math. Phys., 57, 135–148 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    B. Temple, “Systems of conservation laws with invariant submanifolds,” Trans. Am. Math. Soc., 280, 781–795 (1983).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.S.I.S.S.A.TriesteItaly

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