Transport by rough vector fields

Abstract

The Cauchy-Lipschitz theorem guarantees that the ODE $\dotγ= b(t,γ)$ is well-posed if $b$ is Lipschitz in space, an assumption too restrictive for applications such as fluid mechanics. In 1989, DiPerna and Lions extended the theory to Sobolev vector fields with bounded divergence. Well-posedness then holds not for individual trajectories but for regular Lagrangian flows: flows that solve the ODE for almost every initial point and compress volume by at most a bounded factor. Their method rests on renormalized solutions of the associated transport equation. This thesis is primarily expository: it presents the DiPerna-Lions theory with detailed proofs. We treat existence, uniqueness and stability of weak and renormalized solutions of the transport equation, deduce existence (for autonomous fields) and uniqueness of regular Lagrangian flows, and discuss applications to the two-dimensional Euler equations. Three counterexamples with divergence-free fields show that uniqueness can fail without a growth condition at infinity, if $b$ has only fractional Sobolev regularity (due to DiPerna and Lions), or for $L^1$ solutions when $\nabla b \in L^r$, $r<d$ (the simplest case of a recent construction by Bruè, Colombo and Kumar). Contributions are (1) smooth approximations of $b$ with uniformly bounded divergence for the existence proof; (2) a direct proof of the known uniqueness of regular Lagrangian flows from uniqueness for the continuity equation; (3) a construction of the auxiliary cube-translating field in the example of Bruè, Colombo and Kumar that works for all $d \ge 2$, replacing an extension from two dimensions that is not a vector field on $\mathbb R^d$ for $d \ge 3$.