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.
@article{SEDP_2008-2009____A6_0,
author = {Bernard, Patrick},
title = {Some remarks on the continuity equation},
journal = {S\'eminaire Goulaouic-Schwartz},
note = {talk:6},
pages = {1--12},
year = {2008-2009},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
language = {en},
url = {https://www.numdam.org/item/SEDP_2008-2009____A6_0/}
}
TY - JOUR AU - Bernard, Patrick TI - Some remarks on the continuity equation JO - Séminaire Goulaouic-Schwartz N1 - talk:6 PY - 2008-2009 SP - 1 EP - 12 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://www.numdam.org/item/SEDP_2008-2009____A6_0/ LA - en ID - SEDP_2008-2009____A6_0 ER -
Bernard, Patrick. Some remarks on the continuity equation. Séminaire Goulaouic-Schwartz (2008-2009), Exposé no. 6, 12 p.. https://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. | Zbl | MR
[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. | Zbl | MR
[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. | Zbl | MR | EuDML
[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. | Zbl | MR
[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. | Zbl | MR
[15] S. Maniglia : Probabilistic representation and uniqueness results for measure-valued solutions of transport equations. J. Math. Pures Appl. 87 (2007), 601–626. | Zbl | MR
[16] K. R. Parthasarathy : Probability measures on metric spaces, Academic Press (1967). | Zbl | MR
[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. | Zbl | MR
