A Harris theorem for enhanced dissipation, and an example of Pierrehumbert
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916605208821760 |
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| author | Cooperman, William Iyer, Gautam Son, Seungjae |
| author_facet | Cooperman, William Iyer, Gautam Son, Seungjae |
| contents | In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium -- a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If $κ$ denotes the strength of the diffusion, then we show that with probability at least $1 - o(κ^N)$ enhanced dissipation occurs on time scales of order $|\ln κ|$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert '94. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19858 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Harris theorem for enhanced dissipation, and an example of Pierrehumbert Cooperman, William Iyer, Gautam Son, Seungjae Dynamical Systems Analysis of PDEs Probability 37A25 (Primary) 60J05, 76R99 (Secondary) In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium -- a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If $κ$ denotes the strength of the diffusion, then we show that with probability at least $1 - o(κ^N)$ enhanced dissipation occurs on time scales of order $|\ln κ|$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert '94. |
| title | A Harris theorem for enhanced dissipation, and an example of Pierrehumbert |
| topic | Dynamical Systems Analysis of PDEs Probability 37A25 (Primary) 60J05, 76R99 (Secondary) |
| url | https://arxiv.org/abs/2403.19858 |