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

Abstract

top
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.

How to cite

top

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.