  In this work we consider the Keller-Segel system coupled with Navier-Stokes
equations in $\mathbb{R}^{N}$ for $N\geq2$. We prove the global well-posedness
with small initial data in Besov-Morrey spaces. Our initial data class extends
previous ones found in the literature such as that obtained by
Kozono-Miura-Sugiyama (J. Funct. Anal. 2016). It allows to consider initial
cell density and fluid velocity concentrated on smooth curves or at points
depending on the spatial dimension. Self-similar solutions are obtained
depending on the homogeneity of the initial data and considering the case of
chemical attractant without degradation rate. Moreover, we analyze the
asymptotic stability of solutions at infinity and obtain a class of
asymptotically self-similar ones.
