Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.01576 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Let $f$ and $g$ be two volume-preserving diffeomorphisms on the cube $Q=[0,1]^ν$, $ν\geq 3$. We show that there is a divergence-free vector field $v \in L^1((0,1);L^p(Q))$ such that $v$ connects $f$ and $g$ through the corresponding flow and $\Vert v \Vert_{L^1_t L^p_x} \leq C_{p,ν} \Vert f- g \Vert_{L^p_x}$. In particular we show Shnirelman's inequality, cf. [Shnirelman, Generalized fluid flows, their approximation and applications (1994)], for the optimal Hölder exponent $α=1$, thus proving that the metric on the group of volume-preserving diffeomorphisms of $Q$ is equivalent to the $L^2$-distance. To achieve this, we discretise our problem, use some results on flows in discrete networks and then construct a flow in non-discrete space-time out of the discrete solution.