The uniqueness of solutions to the free transport equation is classical when the transport is Lipschtzian, and still holds when the transport is  in Sobolev class due to Diperna-Lions, and in BV due to Ambrosio. The uniqueness theorems of Diperna-Lions and Ambrosio find various applications in fluids and kinetic equations.  In this talk, I will present an example of non-uniqueness of L-infinity solutions to the non-smooth divergence free transport,  joint with F. Colombini and J. Rauch. The construction is of geometric and intuitive nature.

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