Global well-posedness and asymptotic behavior in Besov-Morrey spaces for chemotaxis-Navier-Stokes fluids

Abstract

In this work, we consider the Keller-Segel system coupled with Navier-Stokes equations in RN for N ≥ 2. 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, and Sugiyama [J. Funct. Anal. 270(5), 1663–1683 (2016)]. It allows us 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 a chemical attractant without the degradation rate. Moreover, we analyze the asymptotic stability of solutions at infinity and obtain a class of asymptotically self-similar ones.