| Abstract: |
| Let $b\colon \mathbb R \times \mathbb R^d \to \mathbb R^d$ be a bounded Borel vector field.
Consider the continuity equation
\begin{equation*}
\partial_t \mu_t + \mathop{\mathrm{div}} (b \mu_t) = 0
\end{equation*}
with respect to the measurable family $\{\mu_t\}_{t\in \mathbb R}$ of Borel measures on $\mathbb R^d$ (the equation is understood in the sense of distributions).
If the solution $\mu_t$ is non-negative, then the superposition principle holds: $\mu_t$ can be decomposed into measures concentrated on the integral curves of $b$. For smooth $b$ this result follows from the method of characteristics, and in the general case it was established by L. Ambrosio.
A partial extension of this result for signed measure valued solutions $\mu_t$ was obtained in a paper by L. Ambrosio and P. Bernard, where the following problem was proposed: does the superposition principle hold for signed measure-valued solutions in presence of unique flow of homeomorphisms solving the associated ordinary differential equation?
We will present some related results (and counterexamples) obtained jointly with P. Bonicatto. |
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