On the $8\pi$-critical mass threshold of a Patlak-Keller-Segel-Navier-Stokes system

Yishu Gong; Siming He

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arXiv · 2003.12174

On the $8\pi$-critical mass threshold of a Patlak-Keller-Segel-Navier-Stokes system

Yishu Gong·Siming He

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Abstract

In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane $\rr^2$, 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.

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Yishu Gong, Siming He. 2020-03-26. On the $8\pi$-critical mass threshold of a Patlak-Keller-Segel-Navier-Stokes system. https://arxiv.org/abs/2003.12174

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