Mathematics

The recent work of Brué, Colombo and De Lellis has established that, for divergence-free Sobolev vector fields, the continuity equation may be well-posed in a Lagrangian sense, yet trajectories of the associated ODE need not be unique. We describe how a convex integration scheme for the continuity equation reveals these degenerate integral curves; we modify this scheme to produce divergence-free Sobolev vector fields for which “most” integral curves are degenerate. More precisely, we produce divergence-free Sobolev vector fields which have any finite number of integral curves starting almost everywhere. We also discuss mixing properties of the corresponding generalised flows. Finally we explain why the case of divergence-free continuous Sobolev vector fields requires new ideas. This is a joint work with Massimo Sorella.