Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows

Consider a diffusion-free passive scalar θ being mixed by an incompressible flow u on the torus ^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 (∥ θ (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 the recent work of Crippa and De Lellis (2008 J. Reine Angew. Math. 616 15-46) making progress towards the resolution of Bressan's rearrangement cost conjecture. © 2014 IOP Publishing Ltd & London Mathematical Society.

Duke Scholars

Author Xiaoqian Xu DKU Faculty

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