## Young Researchers Workshop: Multiscale phenomena: modeling, analysis and computation

### Mixing of passive scalars advected by incompressible enstrophy-constrained flows

Xiaoqian Xu

Carnegie Mellon University

Abstract:

Consider a diffusion-free passive scalar $\theta$ being mixed by an incompressible flow $u$ on the torus $\mathbb{T}^d$. Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field. Our main result shows that the mix-norm ($||\theta(t)||_{H^{-1}}$) is bounded below by an exponential function of time. We will also perform numerical simulations and confirm that the numerically observed decay rate scales similarly to the rigorous lower bound, at least for a significant initial period of time. This is the joint work with Gautam Iyer and Alexander Kiselev.