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arXiv · 1902.08084

Smooth approximation is not a selection principle for the transport equation with rough vector field

Gennaro Ciampa·Gianluca Crippa·Stefano Spirito
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Abstract

In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation.

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Gennaro Ciampa, Gianluca Crippa, Stefano Spirito. 2019-02-21. Smooth approximation is not a selection principle for the transport equation with rough vector field. https://doi.org/10.1007/s00526-019-1659-0

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