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Preprint 2006-005

On the Relations between Uniqueness for the Cauchy Problem and Existence of Smooth Approximations for Linear Transport Equations

François Bouchut and Gianluca Crippa

Abstract: 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, well-posedness 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.

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Francois Bouchut,
Gianluca Crippa,
Publishing information:
Submitted by:
; 2006-02-20.