Geodesic
Diffusion and mixing in fluid flow
Peter Constantin1 ; Alexander Kiselev2 ; Lenya Ryzhik3 ; Andrej Zlatoš3
1 Department of Mathematics<br/>University of Chicago<br/>Chicago, IL 60637<br/>United States
2 Department of Mathematics<br/>University of Wisconsin<br/>Madison, WI 53706<br/>United States
3 Department of Mathematics<br/>University of Chicago<br/>Chicago, IL 60637<br/>United States
Annals of mathematics, Tome 168 (2008) no. 2, pp. 643-674.

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We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance dissipation in this sense. The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form $\Gamma+iAL$ with a negative unbounded self-adjoint operator $\Gamma$, a self-adjoint operator $L$, and parameter $A\gg 1$. In particular, they employ the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (related to a classical theorem of Wiener on Fourier transforms of measures). Applications to quenching in reaction-diffusion equations are also considered.
DOI : 10.4007/annals.2008.168.643

Peter Constantin 1 ; Alexander Kiselev 2 ; Lenya Ryzhik 3 ; Andrej Zlatoš 3

1 Department of Mathematics<br/>University of Chicago<br/>Chicago, IL 60637<br/>United States
2 Department of Mathematics<br/>University of Wisconsin<br/>Madison, WI 53706<br/>United States
3 Department of Mathematics<br/>University of Chicago<br/>Chicago, IL 60637<br/>United States 
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Peter Constantin; Alexander Kiselev; Lenya Ryzhik; Andrej Zlatoš. Diffusion and mixing in fluid flow. Annals of mathematics, Tome 168 (2008) no. 2, pp. 643-674. doi : 10.4007/annals.2008.168.643. http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.643/

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