Inventiones mathematicae

, Volume 158, Issue 2, pp 227–260 | Cite as

Transport equation and Cauchy problem for BV vector fields

  • Luigi Ambrosio


Vector Field Cauchy Problem Transport Equation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aizenman, M.: On vector fields as generators of flows: a counterexample to Nelson’s conjecture. Ann. Math. 107, 287–296 (1978)zbMATHGoogle Scholar
  2. 2.
    Alberti, G.: Rank-one properties for derivatives of functions with bounded variation. Proc. R. Soc. Edinb., Sect. A, Math. 123, 239–274 (1993)Google Scholar
  3. 3.
    Alberti, G., Ambrosio, L.: A geometric approach to monotone functions in ℝn. Math. Z. 230, 259–316 (1999)zbMATHGoogle Scholar
  4. 4.
    Alberti, G., Müller, S.: A new approach to variational problems with multiple scales. Commun. Pure Appl. Math. 54, 761–825 (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Alberti, G.: Personal communicationGoogle Scholar
  6. 6.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs 2000Google Scholar
  7. 7.
    Ambrosio, L., Kirchheim, B., Lecumberry, M., Riviere, T.: On the rectifiability of defect measures arising in a micromagnetics model. Nonlinear Problems in Mathematical Physics and related topics II, in honor of O.A. Ladyzhenskaya, ed. by M.S. Birman, S. Hildebrandt, V.A. Solonnikov and N. Uraltseva pp. 29–60. International Mathematical Series, Kluwer/Plenum 2002Google Scholar
  8. 8.
    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)CrossRefzbMATHGoogle Scholar
  9. 9.
    Ambrosio, L., Bouchut, F., De Lellis, C.: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Preprint 2003 (submitted to Comm. PDE and available at Scholar
  10. 10.
    Benamou, J.-D., Brenier, Y.: Weak solutions for the semigeostrophic equation formulated as a couples Monge-Ampere transport problem. SIAM J. Appl. Math. 58, 1450–1461 (1998)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bouchut, F., James, F.: One dimensional transport equation with discontinuous coefficients. Nonlinear Anal. 32, 891–933 (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Bouchut, F.: Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Ration. Mech. Anal. 157, 75–90 (2001)zbMATHGoogle Scholar
  13. 13.
    Bressan, A.: An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Preprint 2003Google Scholar
  14. 14.
    Capuzzo Dolcetta, I., Perthame, B.: On some analogy between different approaches to first order PDE’s with nonsmooth coefficients. Adv. Math. Sci. Appl. 6, 689–703 (1996)Google Scholar
  15. 15.
    Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. Lect. Notes Math. 580. Berlin: Springer 1977Google Scholar
  16. 16.
    Cellina, A.: On uniqueness almost everywhere for monotonic differential inclusions. Nonlinear Anal., Theory Methods Appl. 25, 899–903 (1995)Google Scholar
  17. 17.
    Cellina, A., Vornicescu, M.: On gradient flows. J. Differ. Equations 145, 489–501 (1998)CrossRefzbMATHGoogle Scholar
  18. 18.
    Colombini, F., Lerner, N.: Uniqueness of continuous solutions for BV vector fields. Duke Math. J. 111, 357–384 (2002)CrossRefzbMATHGoogle Scholar
  19. 19.
    Colombini, F., Lerner, N.: Uniqueness of L solutions for a class of conormal BV vector fields. Preprint 2003Google Scholar
  20. 20.
    Colombini, F., Rauch, J.: Unicity and nonunicity for nonsmooth divergence free transport. Preprint 2003Google Scholar
  21. 21.
    Cullen, M., Gangbo, W.: A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Ration. Mech. Anal. 156, 241–273 (2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    Cullen, M., Feldman, M.: Lagrangian solutions of semigeostrophic equations in physical space. To appearGoogle Scholar
  23. 23.
    De Lellis, C., Otto, F.: Structure of entropy solutions to the eikonal equation. J. Eur. Math. Soc. (JEMS) 5, 107–145 (2003)CrossRefzbMATHGoogle Scholar
  24. 24.
    De Lellis, C., Otto, F., Westdickenberg, M.: Structure of entropy solutions for multi-dimensional conservation laws. Arch. Ration. Mech. Anal. 170, 137–184 (2003)CrossRefzbMATHGoogle Scholar
  25. 25.
    De Pauw, N.: Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d’un hyperplan. C.R. Math. Sci. Acad. Paris 337, 249–252 (2003)Google Scholar
  26. 26.
    Di Perna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Evans, L.C., Gariepy, R.F.: Lecture notes on measure theory and fine properties of functions. CRC Press 1992Google Scholar
  28. 28.
    Federer, H.: Geometric measure theory. Springer 1969Google Scholar
  29. 29.
    Hauray, M.: On Liouville transport equation with potential in BVloc. To appear on Commun. Partial Differ. Equations (2003)Google Scholar
  30. 30.
    Hauray, M.: On two-dimensional Hamiltonian transport equations with Lploc coefficients. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, 625–644 (2003)Google Scholar
  31. 31.
    Lions, P.L.: Sur les équations différentielles ordinaires et les équations de transport. C. R. Acad. Sci., Paris, Sér. I, Math. 326, 833–838 (1998)Google Scholar
  32. 32.
    Petrova, G., Popov, B.: Linear transport equation with discontinuous coefficients. Commun. Partial Differ. Equations 24, 1849–1873 (1999)zbMATHGoogle Scholar
  33. 33.
    Poupaud, F., Rascle, M.: Measure solutions to the liner multidimensional transport equation with non-smooth coefficients. Commun. Partial Differ. Equations 22, 337–358 (1997)zbMATHGoogle Scholar
  34. 34.
    Young, L.C.: Lectures on the calculus of variations and optimal control theory. Saunders 1969Google Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly

Personalised recommendations