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Publication 13-CNA-021

Lower Bounds on the Mix Norm of Passive Scalars Advected by Incompressible Enstrophy-Constrained Flows

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Alexander Kiselev
University of Wisconsin-Madison
Madison, WI 53706, USA

Xiaoqian Xu
University of Wisconsin-Madison
Madison, WI 53706, USA

Abstract: Consider a diffusion-free passive scalar $\theta$ being mixed by an in- compressible flow $u$ on the torus $\Bbb{T}^d$. Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field. Our main result shows that the mix-norm $(||\theta(t)||_{H^{-1}} )$ is bounded below by an exponential function of time. The exponential decay rate we obtain is not universal and depends on the size of the support of the initial data. We also perform numerical simulations and confirm that the numerically observed decay rate scales similarly to the rigorous lower bound, at least for a significant initial period of time. The main idea behind our proof is to use recent work of Crippa and DeLellis ('08) making progress towards the resolution of Bressan's rearrangement cost conjecture.

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