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Abstract In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium—a phenomenon known asenhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certainHarris conditions. Ifκdenotes the strength of the diffusion, then we show that with probability at least $1-o\left({\kappa }^N\right)$enhanced dissipation occurs on time scales of order $|\ln \kappa |$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surelyindependentof diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert’94. 

We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is $$O(1/\epsilon^2)$$, where $$\epsilon$$ is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map~$$\varphi$$ the mixing time becomes $$O(\abs{\ln \epsilon})$$. We also study the \emph{dissipation time} of this process, and obtain~$$O(\abs{\ln \epsilon})$$ upper and lower bounds with explicit constants. 

We study the dissipation enhancement by cellular flows. Previous work by Iyer, Xu, and Zlato\v{s} produces a family of cellular flows that can enhance dissipation by an arbitrarily large amount. We improve this result by providing quantitative bounds on the dissipation enhancement in terms of the flow amplitude, cell size and diffusivity. Explicitly we show that the \emph{mixing time} is bounded by the exit time from one cell when the flow amplitude is large enough, and by the reciprocal of the effective diffusivity when the flow amplitude is small. This agrees with the optimal heuristics. We also prove a general result relating the \emph{dissipation time} of incompressible flows to the \emph{mixing time}. The main idea behind the proof is to study the dynamics probabilistically and construct a successful coupling. 

The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long time convergence of solutions to Bose–Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose–Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as 𝑡→∞, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in 𝐿1. None of these results impose a boundary condition at the zero-energy boundary. 

We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data ρ ¯ ∈ H loc 1 ( R d ) , d ≥ 2 , we construct a divergence-free advecting velocity field v (depending on ρ ¯ ) for which the unique weak solution to the transport equation does not belong to H loc 1 ( R d ) for any positive time. The velocity field v is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space W s , p that does not embed into the Lipschitz class. The velocity field v is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE , 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.