We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case $s=1$ and $1 \leq p \leq \infty$ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.

中文翻译：

---

不可压缩流的指数自相似混合

我们研究了二维空间不可压缩流动作用下被动标量的最优混合问题。标量求解具有无发散速度场的连续性方程，该方程满足 Sobolev 空间 $W^{s,p}$ 中的边界，其中 $s \geq 0$ 和 $1\leq p\leq \infty$。混合特性以特征长度标度表示，称为混合标度。我们考虑混合尺度的两种概念，一种是函数式的，用齐次 Sobolev 范数 $\dot H^{-1}$ 表示，另一种是几何的，与集合的重排有关。我们研究了自相似混合下两个尺度的时间衰减率。对于 $s=1$ 和 $1 \leq p \leq \infty$ 的情况（包括 Lipschitz 连续速度的情况，以及对熵约束流的物理兴趣的情况），我们展示了速度场和标量初始配置的例子，这些标量在两个尺度的时间衰减上使指数下界饱和，在以前的工作中建立。我们还介绍了与 Sobolev 速度场相关的规则拉格朗日流的几何形状的几个结果。