Seminar Analysis and Mathematical Physics: Jules Pitcho (University of Zurich)

Almost everywhere non-uniqueness for integral curves of Sobolev vector fields

The recent work of Brué, Colombo and De Lellis has established that, for 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 Sobolev vector fields for which “most” integral curves are degenerate. More precisely, we produce Sobolev vector fields which have any finite number of integral curves starting almost everywhere. This is a joint work with Massimo Sorella. 

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