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arXiv · 2506.10578

On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow

Shikun Cui·Lili Wang·Wendong Wang·Juncheng Wei
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Abstract

As is well-known, the solution of the Patlak-Keller-Segel system in 3D may blow up in finite time regardless of any initial cell mass. In this paper, we are interested in the suppression of blow-up and the critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via the Couette flow $(Ay, 0, 0)$. It is proved that if the Couette flow is sufficiently strong ($A$ is large enough), then the solutions for the system are global in time in the periodic domain $(x,y,z)\in\mathbb{T}^{3}$ as long as the initial cell mass is less than $16π^{2}$. This result seems to be sharp, since the zero-mode function (the mean value in $x-$direction) of the three dimensional density is a complication of the two-dimensional Keller-Segel equations, whose critical mass in 2D is $8π$. One new observation is the dissipative decay of $(\widetilde{u}_{2,0},\widetilde{u}_{3,0})$ (see Lemma 4.3 for more details), then we combine the quasi-linear method proposed by Wei-Zhang (Comm. Pure Appl. Math., 2021) with the zero-mode estimate of the density by the logarithmic Hardy-Littlewood-Sobolev inequality as Bedrossian-He (SIAM J. Math. Anal., 2017) or He (Nonlinearity, 2025) to obtain the bounded-ness of the density and the velocity.

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Shikun Cui, Lili Wang, Wendong Wang, Juncheng Wei. 2025-06-12. On the sharp critical mass threshold for the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow. https://arxiv.org/abs/2506.10578

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