Geodesic
Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient
Bouchut, Francois1 ; James, Francois2 ; Mancini, Simona3
1 DMA, Ecole Normale Supérieure et CNRS 45 rue d’Ulm 75230 Paris cedex 05, France
2 Laboratoire MAPMO, UMR 6628 Université d’Orléans 45067 Orléans cedex 2, France
3 Laboratoire J.-L. Lions, UMR 7598 Université Pierre et Marie Curie, BP 187 4 place Jussieu 75252 Paris cedex 05, France current address: Laboratoire MAPMO, UMR 6628 Université d’Orléans 45067 Orléans cedex 2, France
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 1-25

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The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.

Classification : 35F10, 34A36, 35D05, 35B35

Bouchut, Francois 1 ; James, Francois 2 ; Mancini, Simona 3

1 DMA, Ecole Normale Supérieure et CNRS 45 rue d’Ulm 75230 Paris cedex 05, France
2 Laboratoire MAPMO, UMR 6628 Université d’Orléans 45067 Orléans cedex 2, France
3 Laboratoire J.-L. Lions, UMR 7598 Université Pierre et Marie Curie, BP 187 4 place Jussieu 75252 Paris cedex 05, France current address: Laboratoire MAPMO, UMR 6628 Université d’Orléans 45067 Orléans cedex 2, France 
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     title = {Uniqueness and weak stability for multi-dimensional transport equations with one-sided {Lipschitz} coefficient},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     publisher = {Scuola Normale Superiore, Pisa},
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Bouchut, Francois; James, Francois; Mancini, Simona. Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 1, pp. 1-25. http://geodesic.mathdoc.fr/item/ASNSP_2005_5_4_1_1_0/