Small-Mass Solutions in the Two-Dimensional Keller--Segel System Coupled to the Navier--Stokes Equations

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.