An alternative approach to Shnirelman's inequality
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914316261785600 |
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| author | Zizza, Martina |
| author_facet | Zizza, Martina |
| contents | In this paper we examine the discrete Shnirelman's inequality [Shnirelman A., 1985], which relates the $L^2$-distance of two discrete configurations of a fluid to the $L^1_tL^2_x$-norm of the vector field connecting them. Our proof is inspired by [Shnirelman A., 1985], where it was obtained $α=\frac{1}{64}$ in dimension $ν=2$, while here we get $α\geq\frac{2}{7}$. Moreover we prove that $α\geq\frac{1}{ν+1}$ for any dimension $ν\geq 3$. We point out that, even if this does not improve the bound in the continuous version, where it was proved that $α\geq\frac{2}{4+ν}$, with $ν\geq 3$, our bound is the best one achieved for the $2$-dimensional case. Our method uses an alternative approach based on volume estimates of permutations, which count the number of maximum cubes that are moved by a permutation $P$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09377 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An alternative approach to Shnirelman's inequality Zizza, Martina Analysis of PDEs In this paper we examine the discrete Shnirelman's inequality [Shnirelman A., 1985], which relates the $L^2$-distance of two discrete configurations of a fluid to the $L^1_tL^2_x$-norm of the vector field connecting them. Our proof is inspired by [Shnirelman A., 1985], where it was obtained $α=\frac{1}{64}$ in dimension $ν=2$, while here we get $α\geq\frac{2}{7}$. Moreover we prove that $α\geq\frac{1}{ν+1}$ for any dimension $ν\geq 3$. We point out that, even if this does not improve the bound in the continuous version, where it was proved that $α\geq\frac{2}{4+ν}$, with $ν\geq 3$, our bound is the best one achieved for the $2$-dimensional case. Our method uses an alternative approach based on volume estimates of permutations, which count the number of maximum cubes that are moved by a permutation $P$. |
| title | An alternative approach to Shnirelman's inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.09377 |