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 (Rend Lincei 23:477–491, 2012) that asks whether or not if there is a continuous vector field with bounded divergence and of class \(W^{1, p}\) $W^{1,p}$ for some \(p \ge 1\) $p\ge 1$ 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 \(W^{1, p}\) $W^{1,p}$ . In this work, we design a divergence-free vector field in \(W^{1, p}\) $W^{1,p}$ with \(p < d\) $p<d$ 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 \(\varvec{u}\) $\mathit{u}$ and a corresponding flow map \(X^{\varvec{u}}\) $X^{\mathit{u}}$ such that after finite time \(T > 0\) $T>0$ , the flow map takes the whole domain \(\mathbb {T}^d\) ${\mathbb{T}}^d$ to a Cantor set \(\mathcal {C}_\Phi \) ${\mathcal{C}}_{\mathrm{\Phi }}$ , i.e., \(X^{\varvec{u}}(T, \mathbb {T}^d) = \mathcal {C}_\Phi \) $X^{\mathit{u}}(T,{\mathbb{T}}^d)={\mathcal{C}}_{\mathrm{\Phi }}$ and the Hausdorff dimension of this Cantor set is strictly less than d. The flow map \(X^{\varvec{u}}\) $X^{\mathit{u}}$ 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.