We consider the situation where two species are involved in an absorbing reaction.
To improve reaction rates, good mixing is important. In some situations, the mixing process may be aided by chemotaxis, where one of the densities is chemically attracted to the other.
Some of the relevant settings include monocytes fighting off an infection, or reproduction processes. We analyze a system of PDE which may serve as a model of such process, with a focus on two spacial dimensions. We obtain an estimate of how important the chemotaxis effect in reaction enhancement can be. The mathematical tools developed to treat this problem involve sharp estimates on the rates of convergence to equilibrium for a class of Fokker-Planck operators. They are based on a new weighted Poincare inequality which may be useful in other applications.
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