• ordinary differential equation
• regular Lagrangian flow
• Sobolev
• well-posedness

The aim of this thesis is to study the well-posedness of ordinary differential equations associated with velocity fields whose spatial regularity lies in a Sobolev space. We establish well-posedness for the associated regular Lagrangian flow (RLF) through a Lagrangian approach based on a priori estimates for certain related quantities, as in papers by Crippa-De Lellis and Bouchut-Crippa. We then address two nearly equivalent problems: the existence of non-Lagrangian flows and the almost everywhere uniqueness of trajectories. We prove that if the Sobolev exponent p exceeds the space dimension d, then almost everywhere uniqueness holds, as shown by Caravenna-Crippa. Conversely, for any p<d, we present an example, due to Kumar, of a vector field with spatial regularity in W^{1,p} admitting a flow different from the regular Lagrangian one.