Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$

Tarek M. Elgindi; Kyle Liss; Jonathan C. Mattingly

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arXiv · 2304.05374

Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$

Tarek M. Elgindi·Kyle Liss·Jonathan C. Mattingly

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Abstract

We consider the advection-diffusion equation on $\mathbb{T}^2$ with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale $|\log \nu|$, where $\nu$ is the diffusivity parameter. This is the optimal decay rate as $\nu \to 0$ for uniformly-in-time Lipschitz velocity fields. We also establish exponential mixing for the $\nu = 0$ problem.

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Tarek M. Elgindi, Kyle Liss, Jonathan C. Mattingly. 2023-04-11. Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$. https://arxiv.org/abs/2304.05374

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