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

2004/03/23 by Francois James, Simona Mancini, Francois Bouchut · 3 citations
Mathematics · #math.AP #msc:35F10 #msc:34A36 #msc:35D05 #msc:35B35

paper · pdf

published as Annali della Scuola Normale Superiore di Pisa, Cl. Sci. (5) IV (2005) 1-25 · 19-03-2004

arxiv created 2004/03/23 · arxiv updated 2009/12/01

Abstract

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.

Cited by

• Well-Posedness of the Cauchy Problem for First-order Quasilinear Equations with Non-Lipschitz Source Terms and Its Applications
• Well-posedness of an optical flow based optimal control formulation for image registration
• Approximation of viscous transport and conservative equations with one sided Lipschitz velocity fields

Related

• Transport equations and flows with one-sided Lipschitz velocity fields
• On one criterion of the uniqueness of generalized solutions for linear transport equations with discontinuous coefficients
• A uniqueness lemma with applications to regularization and incompressible fluid mechanics
• Non-uniqueness of the transport equation at high spacial integrability
• Nonuniqueness of weak solutions for the transport equation at critical space regularity