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

Anuj Kumar

Paper

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

Abstract

In this paper, we resolve an important long-standing question of Alberti \cite{alberti2012generalized} that asks if there is a continuous vector field with bounded divergence and of class

for some

such that the ODE with this vector field has nonunique trajectories on a set of initial conditions with positive Lebesgue measure? This question belongs to the realm of well-known DiPerna--Lions theory for Sobolev vector fields

. In this work, we design a divergence-free vector field in

with

such that the set of initial conditions for which trajectories are not unique is a set of full measure. The construction in this paper is quite explicit; we can write down the expression of the vector field at any point in time and space. Moreover, our vector field construction is novel. We build a vector field

and a corresponding flow map

such that after finite time

, the flow map takes the whole domain

to a Cantor set

, i.e.,

and the Hausdorff dimension of this Cantor set is strictly less than

. The flow map

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.