Cellular mixing with bounded palenstrophy

Abstract

We study the problem of optimal mixing of a passive scalar \(\rho\) advected by an incompressible flow on the two dimensional unit square. The scalar \(\rho\) solves the continuity equation with a divergence-free velocity field \(u\) with uniform-in-time bounds on the homogeneous Sobolev semi-norm \(\dot{W}^{s,p}\), where \(s>1\) and \(1< p \leq \infty\). We measure the degree of mixedness of the tracer \(\rho\) via the two different notions of mixing scale commonly used in this setting, namely the functional and the geometric mixing scale. For velocity fields with the above constraint, it is known that the decay of both mixing scales cannot be faster than exponential. Numerical simulations suggest that this exponential lower bound is in fact sharp, but so far there is no explicit analytical example which matches this result. We analyze velocity fields of cellular type, which is a special localized structure often used in constructions of explicit analytical examples of mixing flows and can be viewed as a generalization of the self-similar construction by Alberti, Crippa and Mazzucato. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially.