Absence of collapse into persistent Dirac-type singularities in a Keller-Segel-Navier-Stokes system involving local sensing. (English) Zbl 1520.35157
A parabolic system of chemotaxis-fluid type is considered in two-dimensional domains \(\Omega\). The specific structure is the equation for the density \(n\)
\[
n_t+u\cdot \nabla n=\Delta(n\phi(c))
\]
with a general function \(\phi\), i.e. chemotaxis with local sensing. Buoyancy effects are described by the coupling with the Navier-Stokes equation with the fluid velocity \(u\) and the gravitational potential. Under the conditions \(\xi\phi(\xi)\to\infty\) and \(\xi\phi'^2(\xi)/\phi(\xi)\to 0\) as \(\xi\to\infty\), suitable global-in-time weak solutions are constructed satisfying the relation \(\int_0^T\int_\Omega n\log(n+1)<\infty\) for each \(T<\infty\). This excludes formation of Dirac mass singularities that can occur for fluid-free model.
Reviewer: Piotr Biler (Wrocław)
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35Q30 | Navier-Stokes equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
chemotaxis-fluid system; local sensing; planar domains; absence of point mass singularitiesReferences:
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