Title: Exponential Lower Bounds for Shear-Driven Advection–Diffusion
Abstract:
We study the advection–diffusion equation on the flat two-dimensional torus
[
\partial_t \rho + \nabla \cdot (u \rho) = \mu \Delta \rho.
]
This model describes the evolution of a scalar quantity—such as temperature, pollutant concentration, or chemical density—transported by an incompressible fluid. Two fundamental mechanisms govern the dynamics. The first is filamentation, in which stretching and folding of the flow generate increasingly fine structures in the scalar field. The second is homogenization, where diffusion drives the scalar toward its spatial average.
An important objective is to understand the interplay between filamentation and homogenization. Research on enhanced dissipation shows that filamentation can accelerate homogenization. In this talk, we investigate the advection–diffusion equation with general shear drifts and establish an exponential lower bound for the decay of solutions. This bound shows that, in a relatively general setting, homogenization can in fact suppress filamentation.
This is joint work with Xiaoqian Xu.