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12:00 pm Friday, September 16, 2011 Geometry-Analysis Seminar: Relaxation Enhacement by Fluid Flow and Quantum Dynamicsby
Alexander Kiselev (University of Wisconsin at Madison) in HB 427- 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 result is 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 $A >> 1$. The proof employs ideas from quantum dynamics, in particular a generalization of RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian.
Submitted by damanik@rice.edu |