|Title||Smoothing does not give a selection principle for transport equations with bounded autonomous fields|
|Publication Type||Journal Article|
|Authors||De Lellis C, Giri V|
|Journal||To appear in Annales Math'ematiques du Que'bec|
|Type of Article||transport equations|
We give an example of a bounded divergence free autonomous vector field in $\mathbb R^3$ (and of a nonautonomous bounded divergence free vector field in $\mathbb R^2$) and of a bounded 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.
To appear in Annales Mathe'matiques du Que'bec