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Existence and Uniqueness Results for the Continuity Equation and Applications to the Chromatography System

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Nonlinear Conservation Laws and Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 153))

Abstract

We discuss some new well-posedness results for the continuity equation in arbitrary space dimension and we then illustrate applications to a system of conservation laws in one space dimension known as the chromatography system. In the last section, we discuss some related open problems.

AMS(MOS) subject classifications. 35F10, 35L65.

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References

  1. L. Ambrosio, The flow associated to weakly differentiable vector fields: re- cent results and open problems, in Nonlinear Conservation Laws and Ap- plications, IMA Vol. Math. Appl., Springer. To appear. Available at http://cvgmt.sns.it/cgi/get.cgi/papers/amb09/.

  2. ––––, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), pp. 227–260.

    Google Scholar 

  3. ––––, Transport equation and Cauchy problem for non-smooth vector fields, in Calculus of variations and nonlinear partial differential equations, Vol. 1927 of Lecture Notes in Math., Springer, Berlin, 2008, pp. 1–41.

    Google Scholar 

  4. L. Ambrosio, F. Bouchut, and C. De Lellis, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions, Comm. Partial Differential Equations, 29 (2004), pp. 1635–1651.

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Ambrosio and G. Crippa, Existence, uniqueness, stability and differentia- bility properties of the flow associated to weakly differentiable vector fields, In: Transport Equations and Multi-D Hyperbolic Conservation Laws, Lecture Notes of the Unione Matematica Italiana, 5 (2008).

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  8. L. Ambrosio, C. De Lellis, and J. Malý, On the chain rule for the divergence of BV-like vector fields: applications, partial results, open problems, in Perspec- tives in nonlinear partial differential equations, Vol. 446 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, pp. 31–67.

    Google Scholar 

  9. P. Baiti and A. Bressan, The semigroup generated by a Temple class system with large data, Differential Integral Equations, 10 (1997), pp. 401–418.

    MATH  MathSciNet  Google Scholar 

  10. S. Bianchini, Stability of L∞ solutions for hyperbolic systems with coinciding shocks and rarefactions, SIAM J. Math. Anal., 33 (2001), pp. 959–981 (elec- tronic).

    Article  MATH  MathSciNet  Google Scholar 

  11. F. Bouchut and G. Crippa, Uniqueness, renormalization, and smooth approx- imations for linear transport equations, SIAM J. Math. Anal., 38 (2006), pp. 1316–1328 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Bressan, Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem, Vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000.

    MATH  Google Scholar 

  13. ––––, An ill posed Cauchy problem for a hyperbolic system in two space dimen- sions, Rend. Sem. Mat. Univ. Padova, 110 (2003), pp. 103–117.

    Google Scholar 

  14. A. Bressan and P. Goatin, Stability of L∞ solutions of Temple class systems, Differential Integral Equations, 13 (2000), pp. 1503–1528.

    MATH  MathSciNet  Google Scholar 

  15. A. Bressan and W. Shen, Uniqueness for discontinuous ODE and conservation laws, Nonlinear Anal., 34 (1998), pp. 637–652.

    Article  MATH  MathSciNet  Google Scholar 

  16. G.-Q. Chen and M. Rascle, Initial layers and uniqueness of weak entropy solu- tions to hyperbolic conservation laws, Arch. Ration. Mech. Anal., 153 (2000), pp. 205–220.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Crippa, The flow associated to weakly differentiable vector fields, Vol. 12 of Tesi. Scuola Normale Superiore di Pisa (Nuova Serie) [Theses of Scuola Nor- male Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009.

    MATH  Google Scholar 

  18. G. Crippa and C. De Lellis, Oscillatory solutions to transport equations, Indiana Univ. Math. J., 55 (2006), pp. 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  19. ––––, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), pp. 15–46.

    Google Scholar 

  20. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer- Verlag, Berlin, Second ed., 2005.

    MATH  Google Scholar 

  21. C. De Lellis, Notes on hyperbolic systems of conservation laws and trans- port equations, Handbook of Differential Equations: Evolutionary Differential Equations, III (2006), pp. 277–383.

    Google Scholar 

  22. N. Depauw, Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d’un hyperplan, C. R. Math. Acad. Sci. Paris, 337 (2003), pp. 249–252.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Freistühler, On the Cauchy problem for a class of hyperbolic systems of conservation laws, J. Differential Equations, 112 (1994), pp. 170–178.

    Article  MATH  MathSciNet  Google Scholar 

  25. B.L. Keyfitz and H.C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal., 72 (1979/80), pp. 219–241.

    Article  MathSciNet  Google Scholar 

  26. S.N. Kružkov, First order quasilinear equations with several independent vari- ables., Mat. Sb. (N.S.), 81 (123) (1970), pp. 228–255.

    MathSciNet  Google Scholar 

  27. O.A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2), 26 (1963), pp. 95–172.

    MathSciNet  Google Scholar 

  28. E.Y. Panov, On the theory of generalized entropy solutions of the cauchy problem for a class of non-strictly hyperbolic systems of conservation laws, SB MATH, 191 (2000), pp. 121–150.

    Article  MathSciNet  Google Scholar 

  29. E.Y. Panov, Generalized Solutions of the Cauchy Problem for a Transport Equa- tion with Discontinuous Coefficients, in Instability in Models Connected with Fluid Flows II. Edited by Claude Bardos and Andrei Fursikov, Springer New York, 2008, pp. 23–84.

    Book  Google Scholar 

  30. D. Serre, Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation, J. Differential Equations, 68 (1987), pp. 137–168.

    Article  MATH  MathSciNet  Google Scholar 

  31. B. Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), pp. 781–795.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy

    Luigi Ambrosio

  2. Dipartimento di Matematica, Università degli Studi di Parma, viale G.P. Usberti 53/A (Campus), 43100, Parma, Italy

    Gianluca Crippa

  3. Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, Texas, 78712-1082, USA

    Alessio Figalli

  4. Centro De Giorgi, Collegio Puteano, Scuola Normale Superiore, Piazza dei Cavalieri 3, 56126, Pisa, Italy

    Laura V. Spinolo

Authors
  1. Luigi Ambrosio
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  2. Gianluca Crippa
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  3. Alessio Figalli
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  4. Laura V. Spinolo
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Corresponding author

Correspondence to Luigi Ambrosio .

Editor information

Editors and Affiliations

  1. McAllister Bldg 201, University Park, 16802-6401, Pennsylvania, USA

    Alberto Bressan

  2. , Oxford Centre for Nonlinear PDE, Mathema, University of Oxford, St Giles 24–29, Oxford, OX1 3LB, Montserrat

    Gui-Qiang G. Chen

  3. , School of Mathematics, University of Minnesota, Church Street S.E, 206, Minneapolis, 55455, Minnesota, USA

    Marta Lewicka

  4. , Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260, Pennsylvania, USA

    Dehua Wang

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Ambrosio, L., Crippa, G., Figalli, A., Spinolo, L.V. (2011). Existence and Uniqueness Results for the Continuity Equation and Applications to the Chromatography System. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_8

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