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Suppression of blow-up for the 3D Patlak-Keller-Segel-Navier-Stokes system via the Couette flow
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abstract
As is well known, for the 3D Patlak-Keller-Segel system, regardless of whether they are parabolic-elliptic or parabolic-parabolic forms, finite-time blow-up may occur for arbitrarily small values of the initial mass. In this paper, it is proved for the first time that one can prevent the finite-time blow-up when the initial mass is less than a certain critical threshold via the stabilizing effect of the moving Navier-Stokes flows. In details, we investigate the nonlinear stability of the Couette flow $(Ay, 0, 0)$ in the Patlak-Keller-Segel-Navier-Stokes system and show that if the Couette flow is sufficiently strong (A is large enough), then the solutions for Patlak-Keller-Segel-Navier-Stokes system are global in time provided that the initial velocity is sufficiently small and the initial cell mass is less than $\frac{24}{5} \pi^2$.
Forward citations
Cited by 2 Pith papers
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Stability threshold of Couette flow for 3D Boussinesq system in Sobolev spaces
For the 3D Boussinesq system near Couette flow with constant background temperature, H2 perturbations satisfying velocity and temperature smallness bounds of order ν and ν² respectively stay global in time.
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On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow
For the 3D Patlak-Keller-Segel-Navier-Stokes system near a strong Couette flow, global regularity is proved for initial cell mass below 16π².
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