Calculus of Variations and Geometric Measure Theory
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F. Bouchut - G. Crippa

Uniqueness, renormalization and smooth approximations for linear transport equations

created by crippa on 31 Jan 2006
modified on 17 Dec 2006

[BibTeX]

Published Paper

Inserted: 31 jan 2006
Last Updated: 17 dec 2006

Journal: SIAM J. Math. Anal.
Volume: 38
Number: 4
Pages: 1316-1328
Year: 2006

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.

Keywords: Renormalized solutions, Transport equation, Approximation by smooth functions, Coefficients of bounded compression


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