Enhanced Dissipation via time-modulated velocity fields
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Benthaus, Johannes Nobili, Camilla |
| author_facet | Benthaus, Johannes Nobili, Camilla |
| contents | Motivated by mixing processes in analytical laboratories, this work investigates enhanced dissipation in non-autonomous flows. We study the evolution of concentrations governed by the advection-diffusion equation, where the velocity field is modelled as the product of a shear flow and a time-dependent modulation function $ξ(t)$. The main objective of this paper is to derive quantitative estimates for the energy decay rates, which are shown to depend sensitively on the properties of $ξ$. We identify a class of time-dependent functions that are bounded by increasing functions, for which we demonstrate super-enhanced dissipation, characterized by energy decay rates faster than those observed in autonomous cases. Additionally, we explore the case of velocity fields that may be switched on and off over time. Here, the dissipation rates are comparable to those of autonomous flows. To illustrate our results, we analyse two prototypical flows of this class: one exhibiting a gradual turn-on and turn-off phase, and another that undergoes a significant acceleration following a slow initial activation phase. Both results are achieved through the application of the hypocoercivity framework, adapted to an augmented functional with time-dependent weights. These weights are designed to dynamically counteract the potential growth of $ξ$, ensuring robust decay estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16905 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enhanced Dissipation via time-modulated velocity fields Benthaus, Johannes Nobili, Camilla Analysis of PDEs Mathematical Physics Fluid Dynamics 76F25, 35B40, 35Q35 Motivated by mixing processes in analytical laboratories, this work investigates enhanced dissipation in non-autonomous flows. We study the evolution of concentrations governed by the advection-diffusion equation, where the velocity field is modelled as the product of a shear flow and a time-dependent modulation function $ξ(t)$. The main objective of this paper is to derive quantitative estimates for the energy decay rates, which are shown to depend sensitively on the properties of $ξ$. We identify a class of time-dependent functions that are bounded by increasing functions, for which we demonstrate super-enhanced dissipation, characterized by energy decay rates faster than those observed in autonomous cases. Additionally, we explore the case of velocity fields that may be switched on and off over time. Here, the dissipation rates are comparable to those of autonomous flows. To illustrate our results, we analyse two prototypical flows of this class: one exhibiting a gradual turn-on and turn-off phase, and another that undergoes a significant acceleration following a slow initial activation phase. Both results are achieved through the application of the hypocoercivity framework, adapted to an augmented functional with time-dependent weights. These weights are designed to dynamically counteract the potential growth of $ξ$, ensuring robust decay estimates. |
| title | Enhanced Dissipation via time-modulated velocity fields |
| topic | Analysis of PDEs Mathematical Physics Fluid Dynamics 76F25, 35B40, 35Q35 |
| url | https://arxiv.org/abs/2501.16905 |