Abstract

This seminar will try to present 3 results.

1. A general theory for uniqueness of Lagrangian representations Given a vector field $\rho (1,{\mathbf b}) \in L^1_{\mathrm{loc}}({\mathbb R}^+\times {\mathbb R}^{d},{\mathbb R}^{d+1})$ such that ${\mathrm{div}}_{t,x} (\rho (1,{\mathbf b}))$ is a measure, we consider the problem of uniqueness of the representation $\eta$ of $\rho (1,{\mathbf b}) \mathcal L^{d+1}$ as a superposition of characteristics $\gamma : (t^-_\gamma,t^+_\gamma) \to {\mathbb R}^d$, $\dot \gamma (t)= \b(t,\gamma(t))$. We give conditions in terms of a local structure of the representation $\eta$ on suitable sets in order to prove that there is a partition of ${\mathbb R}^{d+1}$ into disjoint trajectories $\wp_{\mathfrak a}$, ${\mathfrak a} \in {\mathfrak A}$, such that the PDE $$ {\mathrm{div}}_{t,x} \big( u \rho (1,{\mathbf b}) \big) \in \mathcal M({\mathbb R}^{d+1}), \qquad u \in L^\infty({\mathbb R}^+\times {\mathbb R}^{d}), $$ can be disintegrated into a family of ODEs along $\wp_{\mathfrak a}$ with measure r.h.s.. The decomposition $\wp_{\mathfrak a}$ is essentially unique.
2. The application to BV vector fields We show that ${\mathbf b} \in L^1_t({\mathrm{BV}}_x)_{\mathrm{loc}}$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible ${\mathrm{BV}}$ vector fields.
3. Differentiability of the flow The unique flow generated in point 2 enjoys a weak differentiability property, and its weak derivative satisfies the equation for the Jacobian matrix in this weak setting.