We provide an informal overview on the theory of transport equa-
tions with non smooth velocity fields, and on some applications of this theory
to the well-posedness of hyperbolic systems of conservation laws.
We consider the class of integer rectifiable currents without boundary in $\R^n\times\R$
satisfying a positivity condition.
We establish that these currents can be written as a linear superposition of graphs of
finitely many functions with bounded variation.
We consider the Cauchy problem for the repulsive Vlasov-Poisson system in the three dimensional space, where the initial datum is the sum of a diffuse density, assumed to be bounded and integrable, and a point charge. Under some decay assumptions for the diffuse density close to the point charge, under bounds on the total energy, and assuming that the initial total diffuse charge is strictly less than one, we prove existence of global Lagrangian solutions. Our result extends the Eulerian theory of , proving that solutions are transported by the flow trajectories. The proof is based on the ODE theory developed in  in the setting of vector fields with anisotropic regularity, where some components of the gradient of the vector field is a singular integral of a measure.
We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second result provides uniform continuity and is restricted to Hölder continuous solutions.
The vortex-wave system is a version of the vorticity equation governing the motion of 2D incompressible fluids in which vorticity is split into a finite sum of Diracs, evolved
through an ODE, plus an $L^p$ part, evolved through an active scalar transport equation. Existence of a weak solution for this system
was recently proved by Lopes Filho, Miot and Nussenzveig Lopes, for $p>2$, but their result left open the existence and basic properties
of the underlying Lagrangian flow. In this article we study existence, uniqueness and the qualitative properties of the (Lagrangian flow for the)
linear transport problem associated to the vortex-wave system. To this end, we study the flow associated to
a two-dimensional vector field which is singular at a moving point. We first observe that existence and uniqueness of the regular Lagrangian flow are ensured by combining previous results by Ambrosio and by Lacave and Miot. In addition we prove that, generically, the Lagrangian trajectories do not collide with the point singularity. In the second part we present an approximation scheme for the flow, with explicit error estimates obtained by adapting results by Crippa and De Lellis for Sobolev vector fields.
The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper  concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coefficient. Second, we show how the ideas in  can be used to provide an alternative proof of the result in [6,9,12], where the usual requirement of boundedness of the divergence of the vector field has been relaxed to various settings of exponentially integrable functions.