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William Cooperman

Publications and source records attributed to William Cooperman.

12 recordsLinked to original sources
arXiv2025

Exponentially mixing flows with slow enhanced dissipation

Consider a passive scalar which is advected by an incompressible flow $u$ and has small molecular diffusivity $\kappa$. Previous results show that if $u$ is exponentially mixing and $C^1$, then the dissipation time is $O(|\log \kappa|^2)$. We produce a family of incompressible flows which are $C^0$ and exponentially mixing, uniformly in $\kappa$; however have a dissipation time of order $1/\kappa$ (i.e. exhibits no enhanced dissipation). We also estimate the dissipation time of mixing flows, and obtain improved bounds in terms of the mixing rate with explicit constants, and allow for a time inhomogeneous mixing rate which is typical for random constructions of mixing flows.

math.PR↗
arXiv2025

Residual Diffusivity for Expanding Bernoulli Maps

Consider a discrete time Markov process $X^\epsilon$ on $\mathbf R^d$ that makes a deterministic jump based on its current location, and then takes a small Gaussian step of variance $\epsilon^2$. We study the behavior of the asymptotic variance as $\epsilon \to 0$. In some situations (for instance if there were no jumps), then the asymptotic variance vanishes as $\epsilon \to 0$. When the jumps are "chaotic", however, the asymptotic variance may be bounded from above and bounded away from $0$, as $\epsilon \to 0$. This phenomenon is known as residual diffusivity, and we prove this occurs when the jumps are determined by certain expanding Bernoulli maps.

math.PR↗
arXiv2025

Fourier mass lower bounds for Batchelor-regime passive scalars

Batchelor predicted that a passive scalar $\psi^\nu$ with diffusivity $\nu$, advected by a smooth fluid velocity, should typically have Fourier mass distributed as $|\hat \psi^\nu|^2(k) \approx |k|^{-d}$ for $|k| \ll \nu^{-1/2}$. For a broad class of velocity fields, we give a quantitative lower bound for a version of this prediction summed over constant width annuli in Fourier space. This improves on previously known results, which require the prediction to be summed over the whole ball.

math.AP↗
arXiv2024

A Harris theorem for enhanced dissipation, and an example of Pierrehumbert

In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium -- a phenomenon known as enhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certain Harris conditions. If $\kappa$ denotes the strength of the diffusion, then we show that with probability at least $1 - o(\kappa^N)$ 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 surely independent of diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert '94.

math.DS↗
arXiv2023

Unique continuation on planar graphs

We show that a discrete harmonic function which is bounded on a large portion of a periodic planar graph is constant. A key ingredient is a new unique continuation result for the weighted graph Laplacian. The proof relies on the structure of level sets of discrete harmonic functions, using arguments as in Bou-Rabee--Cooperman--Dario (2023) which exploit the fact that, on a planar graph, the sub- and super-level sets cannot cross over each other. In the special case of the square lattice this yields a new, geometric proof of the Liouville theorem of Buhovsky--Logunov--Malinnikova--Sodin (2017).

math.AP↗
arXiv2023

Rigidity of harmonic functions on the supercritical percolation cluster

We use ideas from quantitative homogenization to show that nonconstant harmonic functions on the percolation cluster cannot satisfy certain structural constraints, for example, a Lipschitz bound. These unique-continuation-type results are false on the full lattice and hence the disorder is utilized in an essential way.

math.PR↗
arXiv2023

Slow periodic homogenization for Hamilton-Jacobi equations

Capuzzo-Dolcetta and Ishii proved that the rate of periodic homogenization for coercive Hamilton-Jacobi equations is $O(\varepsilon^{1/3})$. We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is $\Omega(\varepsilon^{1/2})$.

math.AP↗
arXiv2022

Exponential mixing by shear flows

We prove a version of Bressan's mixing conjecture where the advecting field is constrained to be a shear at each time. Also, inspired by recent work of Blumenthal, Coti Zelati and Gvalani, we construct a particularly simple example of a shear flow which mixes at the optimal rate. The constructed vector field alternates randomly in time between just two distinct shears.

math.AP↗
arXiv2022

On the random G equation with nonzero divergence

We prove a quantitative rate of homogenization for the G equation in a random setting with finite range of dependence and nonzero divergence, with explicit dependence of the constants on the Lipschitz norm of the environment. Inspired by work of Burago, Ivanov, and Novikov, the proof uses explicit bounds on the waiting time for the associated metric problem.

math.AP↗
arXiv2021

Quantitative stochastic homogenization of the G equation

We prove a quantitative rate of homogenization for the G equation in a random environment with finite range of dependence. Using ideas from percolation theory, the proof bootstraps a result of Cardaliaguet and Souganidis, who proved qualitative homogenization in a more general ergodic environment.

math.AP↗
arXiv2021

A near-optimal rate of periodic homogenization for convex Hamilton-Jacobi equations

We consider a Hamilton-Jacobi equation where the Hamiltonian is periodic in space and coercive and convex in momentum. Combining the representation formula from optimal control theory and a theorem of Alexander, originally proved in the context of first-passage percolation, we find a rate of homogenization which is within a log-factor of optimal and holds in all dimensions.

math.AP↗