Skip to main content
SpringerLink
Log in

Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We establish existence and uniqueness results for initial-boundary value problems with nearly incompressible vector fields. We then apply our results to establish well-posedness of the initial-boundary value problem for the Keyfitz and Kranzer system of conservation laws in several space dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158, 227–260 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Bouchut, F., De Lellis, C.: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Commun. Partial Differ. Equ. 29(9–10), 1635–1651 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Crippa, G.: Continuity equations and ODE flows with non-smooth velocity. Proc. R. Soc. Edinb. Sect. A Math. 6, 1191–1244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambrosio, L., Crippa, G., Figalli, A., Spinolo, L.V.: Some new well-posedness results for continuity and transport equations, and applications to the chromatography system. SIAM J. Math. Anal. 41(5), 1890–1920 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ambrosio, L., Crippa, G., Maniglia, S.: Traces and fine properties of a BD class of vector fields and applications. Ann. Fac. Sci. Toulouse 14(6), 527–561 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ambrosio, L., De Lellis, C.: Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. Int. Math. Res. Not. 41, 2205–2220 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. The Clarendon Press, New York (2000)

    MATH  Google Scholar 

  8. Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura. Appl. 135(4), 293–318 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bardos, C.: Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation application à l’équation de transport. Ann. Sci. Éc. Norm. Supér. 3(4), 185–233 (1970)

    Article  MATH  Google Scholar 

  10. Bardos, C., Le Roux, A.-Y., Nedelec, J.C.: First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4, 1017–1034 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boyer, F.: Trace theorems and spatial continuity properties for the solutions of the transport equation. Differ. Integral Equ. 18, 891–934 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Chen, G.-Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147, 89–118 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, G.-Q., Torres, M., Ziemer, W.P.: Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Commun. Pure. Appl. Math. 62, 242–304 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Colombo, R.M., Rossi, E.: Rigorous estimates on balance laws in bounded domains. Acta Math. Sci. Ser. B Engl. Ed. 35(4), 906–944 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Crippa, G., Donadello, C., Spinolo, L.V.: Initial-boundary value problems for continuity equations with BV coefficients. J. Math. Pures Appl. 102(1), 79–98 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Crippa, G., Donadello, C., Spinolo, L.V.: A note on the initial-boundary value problem for continuity equations with rough coefficients, hyperbolic problems: theory, numerics and applications. AIMS Ser. Appl. Math. 8, 957–966 (2014)

    MATH  Google Scholar 

  18. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Mathematischen Wissenschaften, vol. 325, 3rd edn. Springer, Berlin (2010)

    Book  Google Scholar 

  19. De Lellis, C.: Notes on hyperbolic systems of conservation laws and transport equations. In: Dafermos, C.M., Porkorny, M. (eds.) Handbook of Differential Equations: Evolutionary Equations, vol. III, pp. 277–382. Elsevier, Amsterdam (2007)

    Chapter  Google Scholar 

  20. De Lellis, C.: Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions], Asterisque, vol. 2006/2007, pp. 175–203 (Exp. No. 972). Seminaire Bourbaki (2008)

  21. DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Girault, V., Scott, L.R.: On a time-dependent transport equation in a Lipschitz domain. SIAM J. Math. Anal. 42, 1721–1731 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Godlewski, E., Raviart, P.A.: Numerical approximation of hyperbolic systems of conservation laws. In: Applied Mathematical Sciences, vol. 118, p. 509. Springer, New York (1996)

  24. Keyfitz, B.L., Kranzer, H.C.: A system of non-strictly hyperbolic conservation laws arising in elasticity theory. Arch. Ration. Mech. Anal. 72, 219–241 (1980)

    Article  MATH  Google Scholar 

  25. Kruzhkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR 10, 217–243 (1970)

    Article  MATH  Google Scholar 

  26. Mischler, S.: On the trace problem for solutions of the Vlasov equation. Commun. Partial Differ. Equ. 25, 1415–1443 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Serre, D.: Systems of Conservation Laws, I. Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  28. Serre, D.: Systems of Conservation Laws, II. Geometric Structures, Oscillations, and Initial-Boundary Value Problems. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051, Basel, Switzerland

    Anupam Pal Choudhury & Gianluca Crippa

  2. IMATI-CNR, via Ferrata 5, 27100, Pavia, Italy

    Laura V. Spinolo

Authors
  1. Anupam Pal Choudhury
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianluca Crippa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Laura V. Spinolo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gianluca Crippa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Choudhury, A.P., Crippa, G. & Spinolo, L.V. Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system. Z. Angew. Math. Phys. 68, 138 (2017). https://doi.org/10.1007/s00033-017-0883-8

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-017-0883-8

Mathematics Subject Classification

Keywords

Search

Navigation