Sharp uniform-in-diffusivity mixing rates for passive scalars in parallel shear flows
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| Format: | Preprint |
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2025
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| _version_ | 1866909919935987712 |
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| author | Albritton, Dallas Beekie, Rajendra |
| author_facet | Albritton, Dallas Beekie, Rajendra |
| contents | We consider the advection-diffusion equation describing the evolution of a passive scalar in a background shear flow. We prove the optimal uniform-in-diffusivity mixing rate $\| f \|_{H^{-1}} \lesssim \langle t \rangle^{-1/(N+1)}$, $t \geq 0$, where $N$ is the maximal order of vanishing of the derivative $b'(y)$ of the shear profile, e.g., $N=1$ for plane Pouseille flow. Our proof is based on the description of the solution in terms of resolvents and involves pointwise estimates on the resolvent kernel. In the non-degenerate case, we further give a rigorous asymptotic description of generic solutions in terms of shear layers localized around the critical points. This verifies formal asymptotics in [McLaughlin-Camassa-Viotti, \textit{Physics of Fluids}, 22(11), 2010]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_18536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp uniform-in-diffusivity mixing rates for passive scalars in parallel shear flows Albritton, Dallas Beekie, Rajendra Analysis of PDEs Fluid Dynamics We consider the advection-diffusion equation describing the evolution of a passive scalar in a background shear flow. We prove the optimal uniform-in-diffusivity mixing rate $\| f \|_{H^{-1}} \lesssim \langle t \rangle^{-1/(N+1)}$, $t \geq 0$, where $N$ is the maximal order of vanishing of the derivative $b'(y)$ of the shear profile, e.g., $N=1$ for plane Pouseille flow. Our proof is based on the description of the solution in terms of resolvents and involves pointwise estimates on the resolvent kernel. In the non-degenerate case, we further give a rigorous asymptotic description of generic solutions in terms of shear layers localized around the critical points. This verifies formal asymptotics in [McLaughlin-Camassa-Viotti, \textit{Physics of Fluids}, 22(11), 2010]. |
| title | Sharp uniform-in-diffusivity mixing rates for passive scalars in parallel shear flows |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2511.18536 |