• continuity equation
• convex integration
• ordinary differential equations
• Sobolev vector fields
• well-posedness

In this thesis we wish to discuss some old and new results related to the theory of the continuity equation with Sobolev vector fields and the associated system of ordinary differential equations. In the first part, we will review the classical aspects of the theory, mainly focusing on the key findings of Di Perna, Lions, and Ambrosio, which were developed between the late 1980s and the early 2000s. Subsequently, we will turn to the analysis of some fine issues related to the well-posedness of the equations, which have attracted the interest of many researchers in the last few years. In particular, we present the technique of convex integration, which is used to construct incompressible Sobolev vector fields admitting exotic solutions of the continuity equation in suitable spaces of unbounded densities. In this context, we also propose an original scheme of application of the method that leads to new results in appropriate integrability regimes.