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Divergence-free vector fields in R2

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References

  1. M. Aizenman, “On vector fields as generators of flows: a counterexample to Nelson’s conjecture,” Ann. Math., 107, 287–296 (1978).

    Article  MathSciNet  Google Scholar 

  2. G. Alberti, S. Bianchini, and G. Crippa (work in preparation).

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

    Article  MATH  MathSciNet  Google Scholar 

  4. L. Ambrosio, Lecture Notes on Transport Equation and Cauchy Problem for BV Vector Fields and Applications, Preprint, (available at http://cvgmt.sns.it) (2004).

  5. L. Ambrosio,“Transport equation and Cauchy problem for non-smooth vector fields and applications,” Lecture Notes in Math., 1927, 1–41 (2008).

    Article  MathSciNet  Google Scholar 

  6. L. Ambrosio and G. Crippa, “Existence, uniqueness, stability and differentiability 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, Vol. 5, Springer (2008).

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

    MATH  Google Scholar 

  8. F. Bouchut and L. Desvillettes, “On two-dimensional Hamiltonian transport equations with continuous coefficients,” Differential Integral Equations, 14, 1015–1024 (2001).

    MATH  MathSciNet  Google Scholar 

  9. F. Colombini, G. Crippa, and J. Rauch, “A note on two-dimensional transport with bounded divergence” Comm. Partial Differential Equations, 31, 1109–1115 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  10. F. Colombini and N. Lerner, “Sur les champs de vecteurs peu réguliers,” Séminaire: Équations aux Dérivées Partielles, Exp. No. XIV, École Polytech., Palaiseau (2001).

  11. F. Colombini and N. Lerner, “Uniqueness of L solutions for a class of conormal BV vector fields,” Contemp. Math., 368, 133–156 (2005).

    MathSciNet  Google Scholar 

  12. F. Colombini and J. Rauch, “Uniqueness in the Cauchy problem for transport in \( {\mathbb{R}^2} \) and \( {\mathbb{R}^{1 + 2}} \),” J. Differential Equations, 211, 162–167 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  13. G. Crippa and C. De Lellis, “Estimates and regularity results for the DiPerna–Lions flow,” J. Reine Angew. Math., 616, 15–46 (2008).

    MATH  MathSciNet  Google Scholar 

  14. C. De Lellis, “Notes on hyperbolic systems of conservation laws and transport equations,” Handb. Differ. Equ.: Evolutionary Equations, Vol. III, Elsevier, Amsterdam (2006).

    Google Scholar 

  15. C. De Lellis, “Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio (d’après Ambrosio, DiPerna, Lions),” Séminaire Bourbaki, 2006-2007, 972 (2006).

  16. N. Depauw, “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).

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Engelking, “General Topology. Revised and completed edition,” Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin (1989).

    Google Scholar 

  19. L. C. Evans and R. F. Gariepy, Lecture notes on measure theory and fine properties of functions, studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992).

    Google Scholar 

  20. H. Federer, Geometric Measure Theory, Springer Berlin–Heidelberg–New York (1969).

    MATH  Google Scholar 

  21. M. Hauray, “On two-dimensional Hamiltonian transport equations with L loc p coefficients,” Ann. Inst. H. Poincaré Anal. Non Linéaire, 20, 625–644 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  22. W.P. Ziemer, “Weakly differentiable functions. Sobolev spaces and functions of bounded variation”, Grad. Texts in Math., 120, Springer-Verlag New York (1989).

    MATH  Google Scholar 

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Correspondence to G. Alberti.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 35, Proceedings of the Fifth International Conference on Differential and Functional Differential Equations. Part 1, 2010.

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Alberti, G., Bianchini, S. & Crippa, G. Divergence-free vector fields in R2 . J Math Sci 170, 283–293 (2010). https://doi.org/10.1007/s10958-010-0085-9

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