# Some remarks on the continuity equation

Patrick Bernard^{[1]}

- [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 (2008-2009)

- Volume: 2008-2009, page 1-12

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topBernard, Patrick. "Some remarks on the continuity equation." Séminaire Équations aux dérivées partielles 2008-2009 (2008-2009): 1-12. <http://eudml.org/doc/11202>.

@article{Bernard2008-2009,

abstract = {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.},

affiliation = {Université Paris-Dauphine, CEREMADE, UMR CNRS 7534 Pl. du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France},

author = {Bernard, Patrick},

journal = {Séminaire Équations aux dérivées partielles},

keywords = {Borel map; generic uniqueness},

language = {eng},

pages = {1-12},

publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {Some remarks on the continuity equation},

url = {http://eudml.org/doc/11202},

volume = {2008-2009},

year = {2008-2009},

}

TY - JOUR

AU - Bernard, Patrick

TI - Some remarks on the continuity equation

JO - Séminaire Équations aux dérivées partielles

PY - 2008-2009

PB - Centre de mathématiques Laurent Schwartz, École polytechnique

VL - 2008-2009

SP - 1

EP - 12

AB - 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.

LA - eng

KW - Borel map; generic uniqueness

UR - http://eudml.org/doc/11202

ER -

## References

top- L. Ambrosio : Transport equation and Cauchy problem for BV vector fields. Invent. Math.158 (2004), no. 2, 227–260. Zbl1075.35087MR2096794
- 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. Zbl1159.35041MR2408257
- 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. Zbl1155.35313
- L. Ambrosio, N. Gigli and G. Savaré : Gradient flows, Lectures in Math. ETH Zürich, Birkhäuser (2005). Zbl1090.35002MR2129498
- 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. Zbl1204.35069MR2439520
- 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. Zbl0821.76012MR1288809
- P. Bernard : Notes on measurable vector-valued functions.
- P. Bernard : Young measures, superposition and transport, Indiana Univ. Math. Journal, 57 (2008) no. 1, 247-276. Zbl1239.49059MR2400257
- P. Bernard and B. Buffoni : Optimal mass transportation and Mather theory, J. E. M. S.9 (2007) no. 1, 85–121. Zbl1241.49025MR2283105
- R.J. Di Perna, P.L. Lions : Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math.98 (1989), 511–548. Zbl0696.34049MR1022305
- A. F. Filippov : Differential equation with discontinuous right hand side, Am. Math. Soc. translation ser. 2 42 (1960) 199-231. Zbl0148.33002
- L. Hörmander : Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques et Applications 26 (1996), Springer. Zbl0881.35001MR1466700
- C. Kuratowski : Topology.
- 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. Zbl0919.34028MR1648524
- S. Maniglia : Probabilistic representation and uniqueness results for measure-valued solutions of transport equations. J. Math. Pures Appl.87 (2007), 601–626. Zbl1123.60048MR2335089
- K. R. Parthasarathy : Probability measures on metric spaces, Academic Press (1967). Zbl0153.19101MR226684
- 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. Zbl0832.49024MR1246427

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