[Unicité et propriété lagrangienne des solutions manquant d'intégrabilité de l'équation de continuité]
On étudie l'unicité des solutions au sens des distributions de l'équation de continuité avec des champs de vecteurs Sobolev et la propriété d'être une solution lagrangienne, c'est-à-dire une solution transportée par le flot de l'équation différentielle ordinaire associée au champ de vecteurs. On travaille dans un cadre où les solutions considérées manquent d'intégrabilité locale et où on ne peut pas appliquer la théorie classique de DiPerna–Lions d'unicité des solutions au sens des distributions et de la propriété d'être lagrangienne, parce que l'on n'a pas assez d'intégrabilité pour le commutateur. On introduit un principe général pour démontrer la propriété d'être une solution lagrangienne : notre technique se base sur une désintégration le long du flot unique et sur un lemme d'extension lipschitzienne directionnelle, qui nous permet de construire une vaste famille de fonctions tests pour la formulation lagrangienne au sens des distributions de l'équation de continuité.
We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, i.e. transported by a flow of the associated ordinary differential equation. We work in a framework of lack of local integrability of the solution, in which the classical DiPerna–Lions theory of uniqueness and Lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator. We introduce a general principle to prove that a solution is Lagrangian: we rely on a disintegration along the unique flow and on a new directional Lipschitz extension lemma, used to construct a large class of test functions in the Lagrangian distributional formulation of the continuity equation.
Accepté le :
Publié le :
Laura Caravenna 1 ; Gianluca Crippa 2
@article{CRMATH_2016__354_12_1168_0,
author = {Laura Caravenna and Gianluca Crippa},
title = {Uniqueness and {Lagrangianity} for solutions with lack of integrability of the continuity equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {1168--1173},
publisher = {Elsevier},
volume = {354},
number = {12},
year = {2016},
doi = {10.1016/j.crma.2016.10.009},
language = {en},
}
TY - JOUR AU - Laura Caravenna AU - Gianluca Crippa TI - Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation JO - Comptes Rendus. Mathématique PY - 2016 SP - 1168 EP - 1173 VL - 354 IS - 12 PB - Elsevier DO - 10.1016/j.crma.2016.10.009 LA - en ID - CRMATH_2016__354_12_1168_0 ER -
%0 Journal Article %A Laura Caravenna %A Gianluca Crippa %T Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation %J Comptes Rendus. Mathématique %D 2016 %P 1168-1173 %V 354 %N 12 %I Elsevier %R 10.1016/j.crma.2016.10.009 %G en %F CRMATH_2016__354_12_1168_0
Laura Caravenna; Gianluca Crippa. Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation. Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1168-1173. doi : 10.1016/j.crma.2016.10.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.10.009/
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