TAILIEUCHUNG - Đề tài " Di_usion and mixing in uid ow "

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. . | Annals of Mathematics Di_usion and mixing in uid ow By P. Constantin A. Kiselev L. Ryzhik and A. Zlato_s Annals of Mathematics 168 2008 643 674 Diffusion and mixing in fluid flow By P. Constantin a. Kiselev L. Ryzhik and A. Zlatos Abstract 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 r iAL with a negative unbounded self-adjoint operator r 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-diffusion equations are also considered. 1. Introduction Let M be a smooth compact d-dimensional Riemannian manifold. The main objective of this paper is the study of the effect of a strong incompressible flow on diffusion on M. Namely we consider solutions of the passive scalar equation A x t Au -VỘA x t AộA x t 0 ộA x 0 0 x . Here A is the Laplace-Beltrami operator on M u is a divergence free vector field V is the covariant derivative and A 2 R is a parameter regulating the strength of the flow. We are interested in the behavior of solutions of for A 1 at a fixed time T 0. 644 P. CONSTANTIN A. KISELEV L. RYZHIK AND A. ZLATOS It is well known that as time tends to infinity the solution A x t .

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