Uniqueness, Renormalization, and Smooth Approximations for Linear Transport Equations

Transport equations arise in various areas of fluid mechanics, but the precise conditions on the vector field for them to be well-posed are still not fully understood. The renormalized theory of DiPerna and Lions for linear transport equations with unsmooth coefficient uses the tools of approximation of an arbitrary weak solution by smooth functions, and the renormalization property, that is to say to write down an equation on a nonlinear function of the solution. Under some W 1,1 regularity assumption on the coefficient, wellposedness holds. In this paper, we establish that these properties are indeed equivalent to the uniqueness of weak solutions to the Cauchy problem, without any regularity assumption on the coefficient. Coefficients with unbounded divergence but with bounded compression are also considered. Key-words: linear transport equations with unsmooth coefficient – renormalized solutions – approximation by smooth functions – coefficients of bounded compression Mathematics Subject Classification: 35R05, 35F99, 35L99