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Steady nearly incompressible vector fields in 2D: chain rule and renormalization
Given bounded vector field , scalar field and a smooth function we study the characterization of the distribution
in terms of and . In the case of vector fields (and under some further assumptions)
such characterization was obtained by L. Ambrosio, C. De Lellis and J. Mal\'y,
up to an error term which is a measure concentrated on so-called
\emph{tangential set} of . We answer some questions posed in their paper
concerning the properties of this term. In particular we construct a nearly
incompressible vector field and a bounded function for which this
term is nonzero.
For steady nearly incompressible vector fields (and under some further
assumptions) in case when we provide complete characterization of
in terms of and . Our approach relies on the structure of level sets of Lipschitz functions
on obtained by G. Alberti, S. Bianchini and G. Crippa.
Extending our technique we obtain new sufficient conditions when any bounded
weak solution of is
\emph{renormalized}, i.e. also solves for any smooth function . As a
consequence we obtain new uniqueness result for this equation.Comment: 50 pages, 8 figure
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