SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 2925-2956, January 2021.
In this paper, we proposed a coupled Patlak--Keller--Segel--Navier--Stokes system, which has dissipative free energy. On the plane $\mathbb{R}^2$, if the total mass of the cells is strictly less than $8\pi$, classical solutions exist for any finite time, and their $H^s$-Sobolev norms are almost uniformly bounded in time. For the radially symmetric solutions, this $8\pi$-mass threshold is critical. On the torus $\mathbb{T}^2$, the solutions are uniformly bounded in time under the same mass constraint.

中文翻译：

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关于Patlak-Keller-Segel-Navier-Stokes系统的$ 8 \ pi $临界质量阈值

SIAM数学分析杂志，第53卷，第3期，第2925-2956页，2021
年1月。在本文中，我们提出了具有耗散自由能的耦合的Patlak-Keller-Segel-Navier-Stokes系统。在平面$ \ mathbb {R} ^ 2 $上，如果单元格的总质量严格小于$ 8 \ pi $，则经典解在任何有限时间内都存在，并且它们的$ H ^ s $ -Sobolev范数几乎是均匀的有时间限制。对于径向对称解，此$ 8 \ pi $质量阈值至关重要。在圆环\\ mathbb {T} ^ 2 $上，解在相同的质量约束下均匀地受时间限制。