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 (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.