Contents 
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 locallyintime 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 illposedness of the linearized model when the energy constraint on the velocity is taken as the kinetic energy, and the wellposedness when this energy constraint is the viscous dissipation energy. 
