Some remarks on the continuity equation
Bernard, Patrick1
1 Université Paris-Dauphine, CEREMADE, UMR CNRS 7534 Pl. du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 6, 12 p.

This text is the act of a talk given november 18 2008 at the seminar PDE of Ecole Polytechnique. The text is not completely faithfull to the oral exposition for I have taken this opportunity to present the proofs of some results that are not easy to find in the literature. On the other hand, I have been less precise on the material for which I found good references. Most of the novelties presented here come from a joined work with Luigi Ambrosio.

Bernard, Patrick 1

1 Université Paris-Dauphine, CEREMADE, UMR CNRS 7534 Pl. du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France 
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Bernard, Patrick. Some remarks on the continuity equation. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 6, 12 p. http://www.numdam.org/item/SEDP_2008-2009____A6_0/

[1] L. Ambrosio : Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158 (2004), no. 2, 227–260. | MR | Zbl

[2] L. Ambrosio : Transport equation and Cauchy problem for non-smooth vector fields. Lecture Notes in Mathematics “Calculus of Variations and Non-Linear Partial Differential Equations” (CIME Series, Cetraro, 2005) 1927, B. Dacorogna, P. Marcellini eds., 2–41, 2008. | MR

[3] L. Ambrosio, G. Crippa : Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. UMI Lecture Notes, Springer, in press.

[4] L. Ambrosio, N. Gigli and G. Savaré : Gradient flows, Lectures in Math. ETH Zürich, Birkhäuser (2005). | MR

[5] L. Ambrosio and P. Bernard : Uniqueness of signed measures solving the continuity equation for Osgood vector fields, Rendiconti Lincei - Mat. e App. (RLM) 19 (2008) no 3. , 237-245. | MR

[6] H. Bahouri, J.-Y. Chemin : Equations de transport relatives à des champs de vecteurs non-Lipschitziens et mécanique des fluides. Arch. Rat. Mech. Anal. 127 (1994) 159–181. | MR | Zbl

[7] P. Bernard : Notes on measurable vector-valued functions.

[8] P. Bernard : Young measures, superposition and transport, Indiana Univ. Math. Journal, 57 (2008) no. 1, 247-276. | MR

[9] P. Bernard and B. Buffoni : Optimal mass transportation and Mather theory, J. E. M. S. 9 (2007) no. 1, 85–121. | MR

[10] R.J. Di Perna, P.L. Lions : Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511–548. | EuDML | MR | Zbl

[11] A. F. Filippov : Differential equation with discontinuous right hand side, Am. Math. Soc. translation ser. 2 42 (1960) 199-231. | Zbl

[12] L. Hörmander : Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques et Applications 26 (1996), Springer. | MR | Zbl

[13] C. Kuratowski : Topology.

[14] P. L. Lions : Sur les équations différentielles ordinaires et les équations de transport. (French) [On ordinary differential equations and transport equations] C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 7, 833–838. | MR | Zbl

[15] S. Maniglia : Probabilistic representation and uniqueness results for measure-valued solutions of transport equations. J. Math. Pures Appl. 87 (2007), 601–626. | MR | Zbl

[16] K. R. Parthasarathy : Probability measures on metric spaces, Academic Press (1967). | MR | Zbl

[17] S. K. Smirnov : Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currrents, St. Petersbourg Math. J. 5 (1994), no 4, 841–867. | MR | Zbl