We give an example of a bounded divergence free autonomous vector field in R-3 (and of a nonautonomous bounded divergence free vector field in R-2) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.
Smoothing does not give a selection principle for transport equations with bounded autonomous fields
De Lellis, C.;
2022-01-01
Abstract
We give an example of a bounded divergence free autonomous vector field in R-3 (and of a nonautonomous bounded divergence free vector field in R-2) and of a smooth initial data for which the Cauchy problem for the corresponding transport equation has 2 distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.File in questo prodotto:
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