Time global existence and finite time blow-up criterion for solutions to the Keller-Segel system coupled with the Navier-Stokes fluid

We will deal with the chemotaxis model under the effect of the Navier-Stokes fluid, i.e., the incompressible viscous fluid. We shall show the existence of a local mild solution for large initial data and a global mild solution for small initial data in the scale invariant class demonstrating that n₀∈L¹(R²) and u₀∈L_σ ²(R²). Our method is based on the perturbation of linearization together with the L^p−L^q-estimates of the heat semigroup. As a by-product of our method, we shall prove the smoothing effect and uniqueness of our mild solution. In addition, we shall show a blow-up criterion which almost covers the well-known threshold number 8π of the size ‖n₀‖_L¹(R²) under the rest state of the fluid motion. Furthermore, the blow-up rate will be also discussed.