Two-loop calculation of anomalous kinetics of the reaction $A+A\to \mathrm{\varnothing }$ in randomly stirred fluid

The single-species annihilation reaction $A+A\to \mathrm{\varnothing }$ is studied in the presence of a random velocity field generated by the stochastic Navier-Stokes equation. The renormalization group is used to analyze the combined influence of the density and velocity fluctuations on the long-time behavior of the system. The direct effect of velocity fluctuations on the reaction constant appears only from the two- loop order, therefore all stable fixed points of the renormalization group and their regions of stability are calculated in the two-loop approximation in the two-parameter $(\epsilon, \Delta)$ expansion. A renormalized integro- differential equation for the number density is put forward which takes into account the effect of density and velocity fluctuations at next-to-leading order. Solution of this equation in perturbation theory is calculated in a homogeneous system.