Dissipation enhancement of cellular flows in general advection diffusion equations

The main contribution of this paper is twofold: (1) Recently, Iyer, Xu, and Zlatoš studied the dissipation enhancement by cellular flows based on standard advection-diffusion equations via a stochastic method. We generalize their results to advective hyper-diffusion equations and advective nonlinear diffusion equations. (2) We prove there exist smooth incompressible flows that are relaxation enhancing to hyper-diffusion but not to standard diffusion.