Difiusion and mixing in ∞uid ∞ow
Peter Constantin, Alexander Kiselev, Lenya Ryzhik, A. Zlato · 2008
We study enhancement of difiusive mixing on a compact Riemannian manifold by a fast incompressible ∞ow. Our main result is a sharp description of the class of ∞ows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the ∞ow amplitude is large enough. The necessary and su‐cient condition on such ∞ows is expressed naturally in terms of the spectral properties of the dynamical system associated with the ∞ow. In particular, we flnd that weakly mixing ∞ows 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 i + iAL with a negative unbounded self-adjoint operator i, a self-adjoint operator L, and parameter A ? 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-difiusion equations are also considered.