A multiscale measure for mixing

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Optimal stirring strategies for passive scalar mixing (2010, methodology): “The utility of the H −a norms (2.3) is indicated by the following theorem, an extension of Mathew et al. (2005).”
Frontiers of chaotic advection (2014, methodology): “Lin et al. (2011b) advocated the use of an H−1 norm, but H−1/2 seems to have better physical motivation, as it incorporates the idea of equal distributions in equal volumes for the perfect mixture.”
Exponential self-similar mixing and loss of regularity for continuity equations (2014, background): “The first one is based on homogeneous negative Sobolev norms (see for instance [10, 12]), the most common one being the Ḣ−1 norm, which should be viewed as a characteristic length of the mixing in the system (here and in the following we use the dot as in Ḣ−1 to denote the homogeneous Sobolev norm):”
Using multiscale norms to quantify mixing and transport (2011, methodology): “Another theorem from Mathew et al [59] now implies that the dynamics generated by u(x, t) are mixing in the sense of ergodic theory if and only if limt→∞ ‖θ(·, t)‖Ḣ q = 0, for any q < 0.”