Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields

Abstract

In the theory of DiPerna-Lions for Sobolev vector fields W 1,p , an important question was whether the uniqueness of regular Lagrangian flow could be implied by proving almost everywhere uniqueness of trajectories. In this work, we construct an explicit example of divergence-free vector fields in W 1,p with p < d such that the set of initial conditions for which trajectories are not unique is a set of full measure. To prove this, we build a vector field u and a corresponding flow map X u such that after finite time T > 0, the flow map takes the whole domain T d to a Cantor set CΦ, i.e., X u (T, T d ) = CΦ and the Hausdorff dimension of this Cantor set is strictly less than d. The flow map X u constructed as such is not a regular Lagrangian flow. The nonuniqueness of trajectories on a full measure set is then deduced from the existence of the regular Lagrangian flow in the DiPerna-Lions theory.