Inviscid mixing by fluid flows. Optimal stirring with cellular and radial velocity fields

We study the problem of optimal mixing of a passive scalar $\rho$ advected by an incompressible flow on $\mathbb{R}^d$. The scalar $\rho$ solves the transport equation $\partial_t\rho(t,x) + u(t,x)\cdot\nabla_x\rho(t,x) =0$ with a divergence-free velocity field $u$. 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 a "fixed palenstrophy constraint" (i.e. a uniform-in-time bound on the homogeneous Sobolev semi-norm $\dot{W}^{s,p}$, where $s>1$ and $1< p \leq \infty$), it is known that the decay of both mixing scales cannot be faster than exponentially fast. We analyze velocity fields of cellular type, which is a special localized structure often used in constructions of explicit analytical examples of mixing flows. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially and thus this structure slows down the mixing process in a significant way. We further construct an explicit example of an autonomous, smooth velocity field $u$ that solves the Euler equation and mixes polynomially fast.