Relaxation Enhancement by Time-Periodic Flows

We study enhancement of diffusive mixing by fast incompressible time-periodic flows. The class of relaxation-enhancing flows that are especially efficient in speeding up mixing has been introduced in [2]. The relaxation-enhancing property of a flow has been shown to be intimately related to the properties of the dynamical system it generates. In particular, time-independent flows $u$ such that the operator $u \cdot \nabla$ has sufficiently smooth eigenfunctions are not relaxation-enhancing. Here we extend results of [2] to time-periodic flows $u(x,t)$ and in particular show that there exist flows such that for each fixed time the flow is Hamiltonian, but the resulting time-dependent flow is relaxation-enhancing. Thus we confirm the physical intuition that time dependence of a flow may aid mixing. We also provide an extension of our results to the case of a nonlinear diffusion model. The proofs are based on a general criterion for the decay of a semigroup generated by an operator of the form $\Gamma+iAL(t)$ with a negative unbounded self-adjoint operator $\Gamma$, a time-periodic self-adjoint operator-valued function $L(t)$, and a parameter $A>>1$.