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Equazioni alle Derivate Parziali nella Dinamica dei Fluidi

A uniqueness result for the decomposition of vector fields in $\mathbb R{\MakeLowercase{d}}$

speaker: Stefano Bianchini (SISSA, Trieste)

abstract: Given a vector field $\rho (1,\mathbf b) \in L1{\mathrm loc}(\mathbb R+\times \mathbb R{d},\R{d+1})$ 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 $\gamma : (t-\gamma,t+\gamma) \to \mathbb Rd$, $\dot \gamma (t)= \mathbf b(t,\gamma(t))$. 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 \begin{equation} \mathrm{div}{t,x} \big( u \rho (1,\mathbf b) \big) \in \mathcal M(\mathbb R{d+1}), \qquad u \in L\infty(\mathrm R+\times \mathrm R{d}), \end{equation} 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. We finally show that $\mathbf b \in L1t(\mathrm{BV}x)\mathrm{loc}$ satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible $\mathrm{BV}$ vector fields.

Mon 5 Feb, 9:05 - 9:40, Aula Dini
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