On incompressible flows in discrete networks and Shnirelman's inequality
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929524300578816 |
|---|---|
| author | Schiffer, Stefan Zizza, Martina |
| author_facet | Schiffer, Stefan Zizza, Martina |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_01576 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On incompressible flows in discrete networks and Shnirelman's inequality Schiffer, Stefan Zizza, Martina Analysis of PDEs 76B75, 05C21, 35Q31 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. |
| title | On incompressible flows in discrete networks and Shnirelman's inequality |
| topic | Analysis of PDEs 76B75, 05C21, 35Q31 |
| url | https://arxiv.org/abs/2410.01576 |