Abstract. In these lectures I will describe some results and open problems on the (linear) transport equation associated to a divergence-free velocity field $u=u(t,x)$, namely $\rho_t + \nabla\cdot(\rho u) = 0$. I will focus in particular on the following issues: uniqueness, mixing, loss of regularity of the solution.These results have been obtained in collaboration with Stefano Bianchini (SISSA, Trieste), Gianluca Crippa (University of Basel), Anna Mazzuccato (Pennsylvania State University). A tentative program is the following:
Uniqueness: The uniqueness solution for the transport equation with given initial datum is easy to prove of the velocity field $u$ is sufficiently regular, e.g., Lipschitz in space. However, this issue turns out to be quite delicate if we consider less regular velocity fields. If the space dimension is two and $u$ is bounded and autonomous ($u=u(x)$), then it is possible to give an exact characterization of those $u$ for which uniqueness holds, and, interestingly enough, this characterization is not expressed in terms of function spaces, but by a "qualitative" property of $u$, namely a suitable weak formulation of the Sard property (by comparison, all existing results in general dimension require that $u$ belongs to some Sobolev class in space---cf. Di Perna-Lions, Ambrosio, etc.)
Mixing: The starting point is a conjecture by A. Bressan which states that, under certain assumptions, the "mixing scale" of the flow associated to a the velocity field $u$ decays at most exponentially in time, a statement that can be formulated in various ways in terms of the solutions of the associated transport equation. Despite the fact that this conjecture has already been proved in some relevant cases (see the work of G. Crippa and C. De Lellis) there are relatively few examples of flows which actually exhibit such exponential decay. I will illustrate some examples of this phenomenon, and in particular given by a smooth velocity field ("smooth exponential mixer").
Loss of regularity: Consider a solution $\rho$ of the transport equation above with initial datum $\rho_0$. If $u$ is sufficiently regular, e.g., Lipschitz in space uniformly in time, then $\rho$ is given by the composition of $\rho_0$ and the flow associated to $u$, and therefore (some) regularity of $\rho_0$ is preserved for all positive times. This result may fail completely if we $u$ belongs only to some Sobolev space that does not embed in the space of Lipschitz functions. In particular $\rho_0$ may be smooth and compactly supported, while $\rho(t,\cdot)$ does not belong to any fractional Sobolev space with order $s>0$ for any $t>0$.