Divergence-free drifts decrease concentration
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908872566898688 |
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| author | Hess-Childs, Elias Raquépas, Renaud Rowan, Keefer |
| author_facet | Hess-Childs, Elias Raquépas, Renaud Rowan, Keefer |
| contents | We show that bounded divergence-free vector fields $u : [0,\infty) \times \mathbb{R}^d \to\mathbb{R}^d$ decrease the ''concentration'', quantified by the modulus of absolute continuity with respect to the Lebesgue measure, of solutions to the associated advection-diffusion equation when compared to solutions to the heat equation. In particular, for symmetric decreasing initial data, the solution to the advection-diffusion equation has (without a prefactor constant) larger variance, larger entropy, and smaller $L^p$ norms for all $p \in [1,\infty]$ than the solution to the heat equation. We also note that the same is not true on $\mathbb{T}^d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Divergence-free drifts decrease concentration Hess-Childs, Elias Raquépas, Renaud Rowan, Keefer Analysis of PDEs 35K15, 35Q35, 47D07, 60J60, 76R50 We show that bounded divergence-free vector fields $u : [0,\infty) \times \mathbb{R}^d \to\mathbb{R}^d$ decrease the ''concentration'', quantified by the modulus of absolute continuity with respect to the Lebesgue measure, of solutions to the associated advection-diffusion equation when compared to solutions to the heat equation. In particular, for symmetric decreasing initial data, the solution to the advection-diffusion equation has (without a prefactor constant) larger variance, larger entropy, and smaller $L^p$ norms for all $p \in [1,\infty]$ than the solution to the heat equation. We also note that the same is not true on $\mathbb{T}^d$. |
| title | Divergence-free drifts decrease concentration |
| topic | Analysis of PDEs 35K15, 35Q35, 47D07, 60J60, 76R50 |
| url | https://arxiv.org/abs/2503.16723 |