SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 2041-2080, January 2020.
The fully parabolic Keller--Segel system is coupled to the incompressible Navier--Stokes equations through transport and buoyancy. It is shown that when posed with no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded planar domains and along with appropriate assumptions on regularity of the initial data, under a smallness condition exclusively involving the total initial population mass $m$ an associated initial-boundary value problem admits a globally defined generalized solution; in particular, this hypothesis is fully explicit and independent of the initial size of further solution components. Moreover, the obtained solution is seen to enjoy a certain temporally averaged boundedness property which, inter alia, rules out any finite-time collapse into persistent Dirac-type measures, as well as convergence to such singular profiles in the large time limit.

中文翻译：

---

耦合Navier-Stokes方程的二维Keller-Segel系统中的小质量解

SIAM数学分析杂志，第52卷，第2期，第2041-2080页，2020年1月。
完全抛物线的Keller-Segel系统通过输运和浮力耦合到不可压缩的Navier-Stokes方程。结果表明，当在光滑边界域内，在无通量/无通量/ Dirichlet边界条件的情况下，加上对初始数据的规律性的适当假设，在仅涉及总初始种群质量$ m $ an的小条件下相关的初始边界值问题允许全局定义的广义解；特别是，该假设是完全明确的，并且与其他解决方案组件的初始大小无关。此外，可以看到，所获得的解具有一定的时间平均有界性，除其他外，它可以将任何有限时间崩溃排除为持久的狄拉克型测度，