Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient F. Bouchut1, F. James2, S. Mancini3 1 DMA, Ecole Normale Superieure et CNRS 45 rue d'Ulm 75230 Paris cedex 05, France e-mail: 2 Laboratoire MAPMO, UMR 6628 Universite d'Orleans 45067 Orleans cedex 2, France e-mail: 3 Laboratoire J.-L. Lions, UMR 7598 Universite Pierre et Marie Curie, BP 187 4 place Jussieu, 75252 Paris cedex 05, France e-mail: Abstract The Cauchy problem for a multidimensional linear transport equa- tion with discontinuous coefficient is investigated. Provided the coef- ficient 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. Spe- cific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the intro- duction 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. Keywords. Linear transport equations, discontinuous coefficients, reversible solutions, generalized flows, weak stability.
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- nonlinear context
- problem
- spe- cific uniqueness criteria
- linear transport
- uniqueness
- transport flows
- stability results