Non Uniqueness for the Transport Equation with Sobolev Vector Fields

Explore a lecture on the non-uniqueness of solutions for the transport equation with Sobolev vector fields. Delve into the counterintuitive result that even for incompressible, well-behaved Sobolev vector fields, uniqueness of solutions can fail dramatically. Examine this finding as a counterpart to DiPerna and Lions' well-posedness theorem. Follow the speaker's journey through the introduction, the central question, proof of uniqueness, Sobolev vector fields, and the concept of non-uniqueness. Investigate special cases, convex integration, solid vector fields, and concentration. Conclude with a Lagrangian construction to solidify understanding of this complex topic in the theory of linear transport equations.