arXiv subjects

Kyle L. Liss

Publications and source records attributed to Kyle L. Liss.

4 recordsLinked to original sources

arXiv2026

Uniform-in-diffusivity mixing by shear flows: stochastic and dynamical perspectives

We study passive scalar mixing by parallel shear flows in the presence of weak molecular diffusion. We recover the sharp uniform-in-diffusivity mixing rate for shear flows with finitely many critical points, recently proven in [1]. Our approach is based on the stochastic representation formula of the associated advection-diffusion equation and yields two short proofs. The first uses a stochastic integration-by-parts argument and gives optimal mixing under the weakest regularity assumption required in the zero-diffusion case, answering Question II in [1, Section 4]. The second adopts a dynamical systems perspective and provides a proof of shear-induced mixing that, to our knowledge, is new even in the zero-diffusivity setting.

math.AP↗

arXiv2026

The Batchelor spectrum for a deterministically driven passive scalar

We study the long-time behavior of a passive scalar transported by an incompressible flow in the presence of smooth, deterministic forcing. For a specific spatially Lipschitz and time-periodic velocity field, we prove that all sufficiently smooth initial data is attracted to a limiting solution that satisfies a cumulative form of Batchelor's law. To our knowledge, this provides the first example for which a version of Batchelor's law can be established with deterministic forcing.

math.AP↗

arXiv2025

Quantitative positivity of transition densities for random perturbations of Hamiltonian systems

We study a class of diffusion processes arising from random perturbations of conservative Hamiltonian systems. Under a set of abstract hypotheses -- including basic structural assumptions on the Hamiltonian, a weak Lyapunov structure, and a quantitative notion of hypoellipticity -- we prove that transition densities satisfy a sharp, uniform pointwise lower bound over Hamiltonian sublevel sets in the small noise limit $ε\to 0$. By applying our general theorem, we obtain quantitative minorization estimates for a variety of models including Langevin dynamics, chains of oscillators coupled to heat bathes at different temperatures, and finite-dimensional fluid models such as stochastically forced Galerkin truncations of the Navier-Stokes equations and the Lorenz '96 system. As a corollary, assuming a stronger Lyapunov structure, our main result yields a sharp exponential rate of convergence to equilibrium for $0 < ε\ll 1$ in a weighted total variation norm. A central feature of our approach is that it does not require knowledge of the explicit form of the invariant measure, nor even its existence, and hence is broadly applicable to deduce minorization for physically relevant systems where invariant measures are inaccessible.

math.PR↗

arXiv2024

A pathwise approach to the enhanced dissipation of passive scalars advected by shear flows

We develop a framework for studying the enhanced dissipation of passive scalars advected by shear flows based on analyzing the particle trajectories of the stochastic differential equation associated with the governing drift-diffusion equation. We consider both shear flows on $\mathbb{T}^2$ and radially symmetric shears on $\mathbb{R}^2$ or the unit disk. Using our probabilistic approach, we are able to recover the well-known enhanced dissipation timescale for smooth shear flows on $\mathbb{T}^2$ with finite-order vanishing critical points [1, 5, 34, 36] and a generalized version of the results for radially symmetric shear flows from [42]. We also obtain results for shear flows with singularities and critical points where the derivative vanishes to infinite order. The proofs are all based on using Girsanov's theorem to reduce enhanced dissipation to a quantitative control problem that can be solved by leveraging the shearing across streamlines. Our method also has the feature that it is local in space, which allows us also to obtain estimates on the precise decay rate of solutions along each streamline in terms of the local shear profile.

math.AP↗