• Cauchy problem
• functions of bounded variation
• transport equation
• functions of bounded deformation
• normal traces of vector fields
• renormalized solutions
• fine properties of functions
• chain-rule for traces of vector fields
• DiPerna-Lions theory
• Young measures

"Transport equation and Cauchy problem for weakly differentiable vector fields".

In this thesis we study the well-posedness of the Cauchy problem and of the transport equation, assuming a very low regularity on the vector field.

At first we give an overview of the problem and illustrate the classical setup. Then we summarize the classical results of DiPerna and Lions about vector fields with Sobolev regularity and the recent results of Ambrosio about vector fields with bounded variation, focussing on the notion of renormalized solution.

We also describe a recent paper (joint work with Ambrosio and Maniglia) in which we prove the renormalization property for special vector fields with bounded deformation. The proof is obtained studying the fine properties of the normal trace of vector fields with measure divergence, for which we show an important chain-rule formula.

In the last part of the thesis we illustrate some counterexamples to the uniqueness for the transport equation and we address some open problems.