Abstract
In this paper, we consider the two dimensional Patlak-Keller-Segel-Navier-Stokes system near the Couette flow (Ay,0) in T×R. It is shown that if A is large enough, the solution to the system stays globally regular. Both the parabolic-parabolic case and the parabolic-elliptic case are investigated. In particular, for the parabolic-parabolic case, an extra smallness assumption on the initial chemical gradient ‖(∇cin)≠‖L2 is needed to control the mixing destabilizing effect.
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