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It is well known that given a regular velocity field $b: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ (say, continuous in time and uniformly Lipshitz in space), the flux $X$ associated to $b$ is a …

Consider a map $\Phi: [0,T] \times \mathbb{R}^d \to \mathbb{R}^d$, and assume that there exists a set $U \subset \mathbb{R}^d$ such that $\Phi\rvert_{[0,T] \times U}$ is $L$-Lipschitz. It is well know …

I post an answer to expand on the comment. Considering $\nabla _x X (t, X(t,x)= [\nabla _x X(t,x)]^{-1}$, we can bound from above its operator norm. Assume for simplicity that the velocity $b$ is dive …

I am reading about the uniqueness problem for the continuity equation $\partial_t \mu_t + div_x (b \mu_t)=0$ in the lecture notes by Ambrosio (here: https://warwick.ac.uk/fac/sci/maths/research/events …

I am reading the article, and I am more or less halfway through it. I have some questions though on some parts I am not understanding, so I wanted to ask about these here. I apologize for listing the …