Divergence-free vector fields in~$\mathbb R^2$
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 1, Tome 35 (2010), pp. 22-32.

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G. Alberti; S. Bianchini; G. Crippa. Divergence-free vector fields in~$\mathbb R^2$. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 1, Tome 35 (2010), pp. 22-32. http://geodesic.mathdoc.fr/item/CMFD_2010_35_a1/

[1] Aizenman M., “On vector fields as generators of flows: a counterexample to Nelson's conjecture”, Ann. Math., 107 (1978), 287–296 | DOI | MR | Zbl

[2] Alberti G., Bianchini S., Crippa G. (to appear)

[3] Ambrosio L., “Transport equation and Cauchy problem for $BV$ vector fields”, Invent. Math., 158 (2004), 227–260 | DOI | MR | Zbl

[4] Ambrosio L., Lecture notes on transport equation and Cauchy problem for $BV$ vector fields and applications, Preprint, 1976 http://cvgmt.sns.it

[5] Ambrosio L., “Transport equation and Cauchy problem for non-smooth vector fields and applications”, Lect. Notes Math., 1927, 2008, 2–41 | MR

[6] Ambrosio L., Crippa G., “Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields”, Lect. Notes Unione Mat. Ital., 5 (2008), 3–57 | DOI | MR | Zbl

[7] Ambrosio L., Fusco N., Pallara D., Functions of bounded variation and free discontinuity problems, Clarendon Press, Oxford Univ. Press, New York, 2000 | MR | Zbl

[8] Bouchut F., Desvillettes L., “On two-dimensional Hamiltonian transport equations with continuous coefficients”, Differ. Integr. Equations, 14 (2001), 1015–1024 | MR | Zbl

[9] Colombini F., Crippa G., Rauch J., “A note on two-dimensional transport with bounded divergence”, Comm. Partial Differ. Equations, 31 (2006), 1109–1115 | DOI | MR | Zbl

[10] Colombini F., Lerner N., Sur les champs de vecteurs peu réguliers, École Polytech., Palaiseau, 2001 | MR

[11] Colombini F., Lerner N., “Uniqueness of $L^\infty$ solutions for a class of conormal $BV$vector fields”, Contemp. Math., 368 (2005), 133–156 | MR | Zbl

[12] Colombini F., Rauch J., “Uniqueness in the Cauchy problem for transport in $\mathbb R^2$ and $\mathbb R^{1+2}$”, J. Differ. Equations, 211 (2005), 162–167 | DOI | MR | Zbl

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

[14] De Lellis C., “Notes on hyperbolic systems of conservation laws and transport equations”, Handbook on Differential Equations: Evolutionary Equations, 3, Elsevier, North-Holland, Amsterdam, 2006

[15] De Lellis C., “Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions]”, Sémin. Bourbaki 2006/2007, 317, 2008, No 972, 175–203 | MR | Zbl

[16] Depauw N., “Non unicité des solutions bornées pour un champ de vecteurs $BV$ en dehors d'un hyperplan”, C. R. Math. Sci. Acad. Paris, 337 (2003), 249–252 | MR | Zbl

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

[18] Engelking R., General topology, Heldermann-Verlag, Berlin, 1989 | MR | Zbl

[19] Evans L. C., Gariepy R. F., Measure theory and fine properties of functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992 | MR | Zbl

[20] Federer H., Geometric measure theory, Springer-Verlag, New York, 1969 | MR | Zbl

[21] Hauray M., “On two-dimensional Hamiltonian transport equations with $L^p_\mathrm{loc}$ coefficients”, Ann. Inst. Poincaré Anal. Non Linéaire, 20 (2003), 625–644 | DOI | MR | Zbl

[22] Ziemer W. P., Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Springer-Verlag, New York, 1989 | MR | Zbl