Divergence-free vector fields in ℝ2

In this survey we describe some well-posedness results that are available in the two-dimensional case. Due to the special structure of the problem, which admits a Hamiltonian function conserved (at least formally) by the flow, the assumptions needed for the uniqueness are dramatically weaker than those needed for general L∞ vector fields in RN , with bounded divergence. also mention a first result obtained with Colombini and Rauch [9] which goes beyond the divergence-free assumption. The rest of the chapter is devoted to the presentation of a work in progress in collaboration with Alberti and Bianchini [2], in which sharp well-posedness results in the two-dimensional case are obtained. We present here just the basic case of a bounded divergence-free vector field, while some variations are possible. The uniqueness holds under an additional assumption: we must require the weak Sard property (3.1), which turns out to be necessary in view of the counterexamples contained in [2].