Local and some type of large solutions for the chemotaxis-fluid equations with partial dissipation

We investigate the Cauchy problem for the chemotaxis-Navier–Stokes equations without dissipation on the chemical concentration. Firstly, we obtain the local well-posedness of the system in the critical space which has the lower regularity. The main difficulty is how to overcome the lower regularity in our framework, the lacks of dissipation effect of the chemical concentration and the smallness of the initial chemical concentration. To conquer this difficulty, we fully explore the properties of certain type weighted Besov spaces and find that the chemical concentration is small enough in the weighted spaces. Secondly, we show global well-posedness with the exponential type and the quadratic functional type initial data which allow the so-called “well-prepared” highly oscillating initial velocity. Especially, we provide an example of initial data satisfying nonlinear smallness condition, but whose norm is arbitrarily large in C−1. Our proof is based on the special coupling structure of the system and the localization technique in Fourier spaces.