[Existence « presque partout » des équations de continuité avec données initiales mesures]
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.
Accepté le :
Publié le :
Ambrosio, Luigi 1 ; Figalli, Alessio 2
@article{CRMATH_2010__348_5-6_249_0,
author = {Ambrosio, Luigi and Figalli, Alessio},
title = {Almost everywhere well-posedness of continuity equations with measure initial data},
journal = {Comptes Rendus. Math\'ematique},
pages = {249--252},
year = {2010},
publisher = {Elsevier},
volume = {348},
number = {5-6},
doi = {10.1016/j.crma.2010.01.018},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.crma.2010.01.018/}
}
TY - JOUR AU - Ambrosio, Luigi AU - Figalli, Alessio TI - Almost everywhere well-posedness of continuity equations with measure initial data JO - Comptes Rendus. Mathématique PY - 2010 SP - 249 EP - 252 VL - 348 IS - 5-6 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.crma.2010.01.018/ DO - 10.1016/j.crma.2010.01.018 LA - en ID - CRMATH_2010__348_5-6_249_0 ER -
%0 Journal Article %A Ambrosio, Luigi %A Figalli, Alessio %T Almost everywhere well-posedness of continuity equations with measure initial data %J Comptes Rendus. Mathématique %D 2010 %P 249-252 %V 348 %N 5-6 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.crma.2010.01.018/ %R 10.1016/j.crma.2010.01.018 %G en %F CRMATH_2010__348_5-6_249_0
Ambrosio, Luigi; Figalli, Alessio. Almost everywhere well-posedness of continuity equations with measure initial data. Comptes Rendus. Mathématique, Tome 348 (2010) no. 5-6, pp. 249-252. doi: 10.1016/j.crma.2010.01.018
[1] Transport equation and Cauchy problem for BV vector fields, Invent. Math., Volume 158 (2004), pp. 227-260
[2] Transport equation and Cauchy problem for non-smooth vector fields, CIME Series, Cetraro, 2005 (Dacorogna, B.; Marcellini, P., 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] Measure Theory, vols. I and II, Springer, 2007
[6] Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal., Volume 157 (2001), pp. 75-90
[7] Uniqueness of continuous solutions for BV vector fields, Duke Math. J., Volume 111 (2002) no. 2, pp. 357-384
[8] 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] Sur les mesures de Wigner, Rev. Mat. Iberoamericana, Volume 9 (1993), pp. 553-618
[12] Mathematical Topics in Fluid Mechanics, vol. I: Incompressible Models, Oxford Lecture Series in Mathematics and Its Applications, vol. 3, Oxford University Press, 1996
[13] Mathematical Topics in Fluid Mechanics, vol. II: Compressible Models, Oxford Lecture Series in Mathematics and Its Applications, vol. 10, Oxford University Press, 1998
Cité par Sources :
