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I was wondering about this fact. By Rademacher theorem, it is well known that a Lipschitz function in $\mathbb{R}^d$ is $\mathcal{L}^d$-a.e. differentiable. Of course, a $C^1$ function is everywhere d …

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 …

Given a set $A \subset \mathbb{R}^n$, this is called d-rectifiable if it can be covered by a countable union of images of lipshitz functions from $\mathbb{R}^d $ to $ \mathbb{R}^n $ and a $\mathcal{H} …

This is probably a simple question, maybe more suited for MSE. In the coarea formula, you have $$\int_{{\mathbb{R}}^n} g (x) |\nabla f(x)|\, dx= \int_\mathbb{R} \left(\int_{\{f=t\}} g d \mathcal{H}^{n …

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 …

Consider two smooth vector fields $v$ and $u$ in $\mathbb{R}^n$, and a smooth set $\Omega$. Consider the flow of $\Omega$ via $v$ and $u$ for a time $T$, namely let $$ \Omega_v =\{x(T, x_0) | x \text{ …

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 …

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 …

Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \to 0 …

$\DeclareMathOperator{\Mass}{Mass}$Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smoth map. Given a $2$-vector (in general a $k$-vector but let's stick to $2$) $v_1 \wedge v_2 \in \Lambda_2 (\mathbb{R} …

(I start by saying that the tags are probably not accurate but I didn't know what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit …

It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as $$ F= \nabla V+ \nabla \times R$$ with $V$ a potential and $R$ another vector field. These components …