Comptes Rendus
Mathematical Analysis/Probability Theory
Almost everywhere well-posedness of continuity equations with measure initial data
[Existence « presque partout » des équations de continuité avec données initiales mesures]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 249-252.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.01.018

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/

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

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  • Victor Chabu Semiclassical analysis of the Schrödinger equation with conical singularities, Asymptotic Analysis, Volume 103 (2017) no. 4, p. 165 | DOI:10.3233/asy-171423
  • Luigi Ambrosio Well posedness of ODE’s and continuity equations with nonsmooth vector fields, and applications, Revista Matemática Complutense, Volume 30 (2017) no. 3, p. 427 | DOI:10.1007/s13163-017-0244-3
  • V.I. Bogachev; G. Da Prato; M. Röckner; S.V. Shaposhnikov On the uniqueness of solutions to continuity equations, Journal of Differential Equations, Volume 259 (2015) no. 8, p. 3854 | DOI:10.1016/j.jde.2015.05.003
  • Alexander V. Kolesnikov; Michael Röckner On continuity equations in infinite dimensions with non-Gaussian reference measure, Journal of Functional Analysis, Volume 266 (2014) no. 7, p. 4490 | DOI:10.1016/j.jfa.2014.01.010
  • Clotilde Fermanian-Kammerer; Patrick Gérard; Caroline Lasser Wigner Measure Propagation and Conical Singularity for General Initial Data, Archive for Rational Mechanics and Analysis, Volume 209 (2013) no. 1, p. 209 | DOI:10.1007/s00205-013-0622-z
  • Luigi Ambrosio; Alessio Figalli; Gero Friesecke; Johannes Giannoulis; Thierry Paul Corrigendum: Semiclassical Limit of Quantum Dynamics with Rough Potentials and Well‐Posedness of Transport Equations with Measure Initial Data, Communications on Pure and Applied Mathematics, Volume 66 (2013) no. 4, p. 646 | DOI:10.1002/cpa.21440
  • Luigi Ambrosio; Alessio Figalli; Gero Friesecke; Johannes Giannoulis; Thierry Paul Semiclassical limit of quantum dynamics with rough potentials and well‐posedness of transport equations with measure initial data, Communications on Pure and Applied Mathematics, Volume 64 (2011) no. 9, p. 1199 | DOI:10.1002/cpa.20371
  • Luigi Ambrosio The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems, Nonlinear Conservation Laws and Applications, Volume 153 (2011), p. 181 | DOI:10.1007/978-1-4419-9554-4_7

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