Multiscale metrics such as negative Sobolev norms are effective for quantifying the degree of mixedness of a passive scalar field advected by an incompressible flow in the absence of diffusion. In this paper we introduce a mix norm that is motivated by Sobolev norm $H^{-1}$ for a general domain with a no-flux boundary. We then derive an explicit expression for the optimal flow that maximizes the instantaneous decay rate of the mix norm under fixed energy and enstrophy constraints. Numerical simulations indicate that the mix norm decays exponentially or faster for various initial conditions and geometries and the rate is closely related to the smallest non-zero eigenvalue of the Laplace operator. These results generalize previous findings restricted for a periodic domain for its analytical and numerical simplicity. Additionally, we observe that periodic boundaries tend to induce a faster decay in mix norm compared to no-flux conditions under the fixed energy constraint, while the comparison is reversed for the fixed enstrophy constraint. In the special case of even initial distributions, two types of boundary conditions yield the same optimal flow and mix norm decay.

翻译：多尺度度量（如负Sobolev范数）可有效量化无扩散条件下不可压缩流输运的被动标量场的混合程度。本文针对具有无通量边界的一般区域，引入一种受Sobolev范数$H^{-1}$启发而定义的混合范数。随后，在固定能量与涡量约束下，推导出使混合范数瞬时衰减率最大化的最优流场的显式表达式。数值模拟表明，对于多种初始条件与几何构型，混合范数以指数或更快的速率衰减，且其衰减率与拉普拉斯算子最小非零特征值密切相关。这些结果推广了先前仅适用于周期性域（因其解析与数值简便性）的发现。研究还观察到：在固定能量约束下，周期性边界条件相比无通量条件可诱导更快的混合范数衰减，而固定涡量约束下的结论则相反。当初始分布为偶函数时，两类边界条件的最优流场与混合范数衰减率完全相同。