In this paper, we resolve an important long-standing question of Alberti (Rend Lincei 23:477–491, 2012) that asks whether or not if there is a continuous vector field with bounded divergence and of class (W^{1, p}) for some (p ge 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 (varvec{u}) and a corresponding flow map (X^{varvec{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^{varvec{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^{varvec{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.
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