| Abstract: |
| We deal with the chemotaxis model under effect of Navier-Stokes fluids,
{\it i.e.}, incompressible viscous fluids. We show the existence of a local {\it mild solution} for large initial data and a global {\it mild solution} for small initial data in the scale invariant class demonstrating that $n_0 \in L^1(\re^2)$ and $u_0 \in L_{\sigma}^2(\re^2)$. Our method is based on a perturbation of the linearization together with the $L^p-L^q$-estimates of the heat semigroup. As a by-product of our method, we prove the smoothing effect and uniqueness of our {\it mild solution}. In addition, we explore a type of fluid which can be forced up or down depending on the flow, under a given force. From this, we construct a solution for our model using this type of fluid, which has an arbitrary threshold number (different from 8$\pi$) for the initial mass. Here, the threshold number determines whether or not the solution diverges in some norm. In the divergent case, we give the characteristics of the time at which our solution diverges. |
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