Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient

By François James, Simona Mancini, François Bouchut. 2004

80 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/79bd0016d9d9d87027b33dcacefa6bfb03c3c4de

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In Applied Mathematics

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Citations in the corpus

Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data (2011, background): “Works that are related to the framework of measures are [7, 8, 6, 14, 5].”
Transport Equation and Cauchy Problem for Non-Smooth Vector Fields (2008, methodology): “This duality method is a classical tool to prove uniqueness in a sufficiently smooth setting (but see also [28], [ 31 ]).”