Almost everywhere well-posedness of continuity equations with measure initial data

Luigi Ambrosio; Alessio Figalli

doi:10.1016/j.crma.2010.01.018

Mathematical Analysis/Probability Theory

Almost everywhere well-posedness of continuity equations with measure initial data

Presented by: Pierre-Louis Lions

Luigi Ambrosio ¹; Alessio Figalli ²

¹ Scuoli Normale Superiore, piazza Cavalieri 7, 56126 Pisa, Italy
² Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1082, USA

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 249-252.

Abstract
Résumé

The aim of this Note is to present some new results concerning “almost everywhere” well-posedness and stability of continuity equations with measure initial data. The proofs of all such results can be found in Ambrosio et al. [4], together with some application to the semiclassical limit of the Schrödinger equation.

Dans cette Note, nous présentons des nouveaux résultats concernant l'existence, l'unicité (au sens « presque partout ») et la stabilité pour des équations de continuité avec données initiales mesures. Les preuves de tous ces résultats sont données dans Ambrosio et al. [4], avec aussi des applications à la limite semiclassique pour l'équation de Schrödinger.

Received: 2009-10-14
Accepted: 2010-01-13
Published online: 2010-02-21

DOI: 10.1016/j.crma.2010.01.018

Author's affiliations:

Luigi Ambrosio ¹; Alessio Figalli ²

¹ Scuoli Normale Superiore, piazza Cavalieri 7, 56126 Pisa, Italy
² Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1082, USA

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Luigi Ambrosio; Alessio Figalli. Almost everywhere well-posedness of continuity equations with measure initial data. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 249-252. doi : 10.1016/j.crma.2010.01.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.018/

References
Cited by

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[2] L. Ambrosio Transport equation and Cauchy problem for non-smooth vector fields, CIME Series, Cetraro, 2005 (B. Dacorogna; P. Marcellini, eds.) (Lecture Notes in Mathematics), Volume vol. 1927 (2008), pp. 2-41

[3] L. Ambrosio, G. Friesecke, J. Giannoulis, Passage from quantum to classical molecular dynamics in the presence of Coulomb interactions, Comm. PDE, in press

[4] L. Ambrosio, A. Figalli, G. Friesecke, J. Giannoulis, Well posedness of transport equations with measure initial data and convergence of Wigner measures, work in preparation

[5] V. Bogachev Measure Theory, vols. I and II, Springer, 2007

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[7] F. Colombini; N. Lerner Uniqueness of continuous solutions for BV vector fields, Duke Math. J., Volume 111 (2002) no. 2, pp. 357-384

[8] R.J. DiPerna; P.L. Lions Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989), pp. 511-547

[9] A. Figalli, T. Paul, work in preparation

[10] P. Gérard, Mesures semi-classiques et ondes de Bloch, in: Seminaire sur les Équations aux Dérivées Partielles, 1990–1991. Exp. No. XVI, 19 pp., École Polytechnique, Palaiseau, 1991

[11] P.L. Lions; T. Paul Sur les mesures de Wigner, Rev. Mat. Iberoamericana, Volume 9 (1993), pp. 553-618

[12] P.L. Lions Mathematical Topics in Fluid Mechanics, vol. I: Incompressible Models, Oxford Lecture Series in Mathematics and Its Applications, vol. 3, Oxford University Press, 1996

[13] P.L. Lions Mathematical Topics in Fluid Mechanics, vol. II: Compressible Models, Oxford Lecture Series in Mathematics and Its Applications, vol. 10, Oxford University Press, 1998

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