论文标题
全球适应性，用于二维Keller-Segel-Euler的消费类型
Global well-posedness for a two-dimensional Keller-Segel-Euler system of consumption type
论文作者
论文摘要
我们考虑了凯勒 - 塞格消费类型系统的库奇问题，并在$ \ mathbb {r}^2 $中与不可压缩的Euler方程相连。该耦合系统描述了一种生物学现象，其中生活在略微粘性液（例如水）中的有氧细菌向更高的氧气浓度移动以生存。首先，我们证明了用于任意平滑初始数据的平滑解决方案的局部存在。然后，我们证明，如果氧的初始密度足够小，则可以在全球范围内扩展这些平滑溶液。证明中的主要成分是$ w^{1，q} $ - 能量估计$（q> 2）$由\ cite {c06}中部分无关的二维boussinesq系统动机。我们的结果改善了二维Keller-segel消费类型系统的全球著名全球良好性以及不可压缩的Navier-Stokes方程。
We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in $\mathbb{R}^2$. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the $W^{1,q}$-energy estimate $(q>2)$ motivated by the partially inviscid two-dimensional Boussinesq system in \cite{C06}. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations.
