Degree Thesis
Inserted: 2 dec 2025
Last Updated: 2 dec 2025
Year: 2025
Abstract:
This thesis investigates the well-posedness of ordinary differential equations, the continuity equation, and the transport equation in regimes of low regularity. After reviewing the classical theory for smooth vector fields, we move to the DiPerna–Lions framework, where vector fields belong to $L^1_{\text{loc}}(\mathbb{R}; W^{1,p}_{\text{loc}}(\mathbb{R}^d))$ and are divergence-free. In this setting, we discuss the notion of renormalized solutions and the link between PDE and ODE through Ambrosio’s superposition principle and the theory of regular Lagrangian flows.
The core contribution of the thesis is an extension of these results to vector fields with Osgood-type regularity, corresponding to having slightly less than one derivative in $L^p$. In this regime, we show that a regular Lagrangian flow still exists, and we identify conditions under which solutions to the transport equation remain renormalized. The proof relies on a Littlewood–Paley decomposition and a commutator estimate combining the regularity of the solution with that of the vector field. This final part is based on joint work with Guido De Philippis.
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