Calculus of Variations and Geometric Measure Theory

C. De Lellis - V. Giri

Smoothing does not give a selection principle for transport equations with bounded autonomous fields

created by delellis on 01 Dec 2020
modified on 09 Mar 2021


Accepted Paper

Inserted: 1 dec 2020
Last Updated: 9 mar 2021

Journal: To appear in Annales Mathe'matiques du Que'bec
Year: 2020

In honor of Sasha Shnirelman on occasion of his 75th birthday


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.