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| Optimal stirring is an important issue in chemical engineering. The underlying optimization problem is the following: given a color function $c$ transported with a solenoidal velocity, what is the velocity $(t,X) \rightarrow V(t,X)$ that ensures the quickest mixing ? Of course, the answer depends on :
- The definition of the mixing criterion
- The energy constraint on the velocity
Recent works of Mathew et. al. have shown that a good criterion for measuring the mixing of two fluids in the periodic case is the $H^{-\frac{1}{2}}$ norm. An explicit locally-in-time optimal mixing scheme has been suggested in subsequent works from Lin et. al.
In this talk we will investigate this mixing scheme both numerically and through a linear stability analysis, in a framework that is as general as possible.
In particular we will show the ill-posedness of the linearized model when the energy constraint on the velocity is taken as the kinetic energy, and the well-posedness when this energy constraint is the viscous dissipation energy. |
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