arXiv subjects

Lucas Huysmans

Publications and source records attributed to Lucas Huysmans.

3 recordsLinked to original sources

arXiv2025

Mixing Estimates for Passive Scalar Transport by $BV$ Vector Fields

We prove a quantitative mixing estimate for the Cauchy problem for transport along divergence-free vector fields with bounded variation. By developing a framework that quantifies Ambrosio's regularisation scheme, we derive the first explicit bounds on the mixing rate for general $BV$ vector fields. Our analysis reveals that tetration (repeated exponentiation) emerges in the mixing rate from the local nature of Ambrosio's regularisation.

math.AP↗

arXiv2025

Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation

We study selection by vanishing viscosity for the transport of a passive scalar $f(x,t)\in\mathbb{R}$ advected by a bounded, divergence-free vector field $u(x,t)\in\mathbb{R}^2$. This is described by the initial value problem to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) = 0$, or with positive viscosity/diffusivity $ν>0$, to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -νΔf = 0$. We demonstrate the failure of the vanishing viscosity limit to select (a) unique solutions or (b) physically admissible solutions in the sense of non-increasing energy/entropy.

math.AP↗

arXiv2024

A Unified Approach to Mixing and Regularity for Passive Scalar Transport by Sobolev Vector Fields

We develop a new framework for quantitative estimates of passive scalar transport along Sobolev vector fields in $W^{1,p}$, when $p>1$. Our approach is based on Christ-Journ\'{e} singular integral estimates. We show (i) a new stability estimate which quantifies the dependence of the solution on specific frequencies of the initial data; (ii) a new exponential mixing bound in the full DiPerna-Lions well-posedness class; (iii) propagation of logarithmic Fourier regularity of the solution; (iv) quantitative convergence rates for the vanishing diffusivity and mollification limits; and (v) a logarithmic decay rate for the standard DiPerna-Lions commutator.

math.AP↗