Maximal mixing by incompressible fluid flows
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In this paper, the impossibility of perfect mixing in finite time for flows with finite viscous dissipation was proved for a model for mixing binary viscous fluids under an incompressible flow, and the authors derived rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time.Abstract:
We consider a model for mixing binary viscous fluids under an incompressible flow. We prove the impossibility of perfect mixing in finite time for flows with finite viscous dissipation. As measures of mixedness we consider a Monge–Kantorovich–Rubinstein transportation distance and, more classically, the H−1 norm. We derive rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time. The rate of the exponential decay is uniform in the initial data.read more
Citations
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Exponential self-similar mixing and loss of regularity for continuity equations
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Estimates and regularity results for the DiPerna-Lions flow
TL;DR: In this paper, simple estimates for ordinary differential equations with Sobolev coefficients were derived, which not only allow to recover some old and recent results in a simple direct way, but also have some new interesting corollaries.
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A multiscale measure for mixing
TL;DR: The Mix-Norm as discussed by the authors is a multiscale measure for mixing that is based on the concept of weak convergence and averages the mixedness of an advected scalar field at various scales.