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.Comment: 19-03-200
