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
gautam@math.cmu.edu
Alexander Kiselev
University of Wisconsin-Madison
Madison, WI 53706, USA
kiselev@math.wisc.edu
Xiaoqian Xu
University of Wisconsin-Madison
Madison, WI 53706, USA
xxu@math.wisc.edu
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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