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

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 $W^{1, p}$ for some $p \geq 1$ 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 $W^{1, p}$. In this work, we design a divergence-free vector field 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. 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 $\boldsymbol{u}$ and a corresponding flow map $X^{\boldsymbol{u}}$ such that after finite time $T > 0$, the flow map takes the whole domain $\mathbb{T}^d$ to a Cantor set $\mathcal{C}_\Phi$, i.e., $X^{\boldsymbol{u}}(T, \mathbb{T}^d) = \mathcal{C}_\Phi$ and the Hausdorff dimension of this Cantor set is strictly less than $d$. The flow map $X^{\boldsymbol{u}}$ 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.