Communications in Mathematical Physics

, Volume 313, Issue 1, pp 1–33 | Cite as

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one Space Dimension

  • Stefano Bianchini
  • Laura CaravennaEmail author


We prove that if \({t \mapsto u(t) \in BV(\mathbb{R})}\) is the entropy solution to a N × N strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields

$$u_t + f(u)_x = 0,$$
then up to a countable set of times \({\{t_n\}_{n \in \mathbb{N} }}\) the function u(t) is in SBV, i.e. its distributional derivative u x is a measure with no Cantorian part.

The proof is based on the decomposition of u x (t) into waves belonging to the characteristic families

$$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal{M}(\mathbb{R}), \, \tilde r_i(t) \in \mathbb{R}^N,$$
and the balance of the continuous/jump part of the measures v i in regions bounded by characteristics. To this aim, a new interaction measure μ i,jump is introduced, controlling the creation of atoms in the measure v i (t).

The main argument of the proof is that for all t where the Cantorian part of v i is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure μ i,jump is positive.


Shock Front Hyperbolic System Riemann Problem Entropy Solution Continuous Part 
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.


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  1. [ADL]
    Ambrosio L., De Lellis C.: A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton-Jacobi equations. J. Hyperbolic Diff. Eqs. 31(4), 813–826 (2004)MathSciNetCrossRefGoogle Scholar
  2. [AFP]
    Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford/Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  3. [AnN]
    Ancona, F., Nguyen, K.T.: SBV regularity of L solutions to genuinely nonlinear Temple systems of balance laws. In preparation Google Scholar
  4. [BDR]
    Bianchini S., De Lellis C., Robyr R.: SBV regularity for Hamilton-Jacobi equations in \({\mathbb{R}^{n}}\) . Arch. Ration. Mech. Anal. 200(3), 1003–1021 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [BT1]
    Bianchini, S., Tonon, D.: SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t, x). SIAM-SIMA (2012)Google Scholar
  6. [BT2]
    Bianchini S., Tonon D.: SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian. J. Math. Anal. Appl. 391, 190–208 (2012)zbMATHCrossRefGoogle Scholar
  7. [Bre]
    Bressan A.: Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem. Oxford University Press., Oxford (2000)zbMATHGoogle Scholar
  8. [BC]
    Bressan A., Colombo R.: Decay of positive waves in nonlinear systems of conservation laws. Ann. Sc. Norm. Super. Pisa Cl. Sci. 26(2), 133–160 (1998)MathSciNetzbMATHGoogle Scholar
  9. [BY]
    Bressan A., Yang T.: A sharp decay estimate for positive nonlinear waves. SIAM J. Math. Anal. 36(2), 659–677 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [CT]
    Christoforou C., Trivisa C.: Sharp decay estimates for hyperbolic balance laws. J. Diff. Eqs. 247(2), 401–423 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [CD]
    Dafermos C.M.: Hyperbolic Conservation Laws in Continuous Physics. Berlin-Heidelberg-NewYork, Springer (2000)Google Scholar
  12. [Daf]
    Dafermos C.M.: Wave fans are special. Acta Mathematicae Applicatae Sinica, English Series 24(3), 369–374 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [GL]
    Glimm J., Lax P.D.: Decay of solutions of systems of nonlinear hyperbolic conservation laws. Mem. Amer. Math. Soc. 101, 813–826 (1970)MathSciNetGoogle Scholar
  14. [L1]
    Liu T.-P.: Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws. Comm. Pure Appl. Math. 30, 585–610 (1977)zbMATHCrossRefGoogle Scholar
  15. [L2]
    Liu T.-P.: Admissible solutions of hyperbolic conservation laws. Mem. Amer. Math. Soc. 301, 240 (1981)Google Scholar
  16. [Ole]
    Oleinik, O.: Discontinuous solutions of nonlinear differential equations. Amer. Math. Soc. Transl. 26, 95–172 (1963), Translation of Uspehi Mat. Nauk 12, 3(75), 3–73 (1957)Google Scholar
  17. [Rog]
    Robyr R.: SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function. J. Hyperbolic Diff. Eqs. 5(2), 449–475 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Smo]
    Smoller J.A.: Shock Waves and Reaction Diffusion Equations. Springer-Verlag, New York (1982)Google Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Centro De Giorgi, Scuola Normale SuperiorePisaItaly

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