Existence and large time behavior to coupled chemotaxis-fluid equations in Besov–Morrey spaces

Abstract

The paper deals with the Cauchy problem of coupled chemotaxis-fluid equations. By taking advantage of a coupling structure of the equations and using the semigroup approach, we show existence and asymptotic stability with small initial data and external force (u0,n0,∇c0,c0)∈N˙r1,λ,∞−β1×N˙r2,λ,∞−β2×N˙r3,λ,∞−β3×L∞,∇ϕ∈MN−λ,λ for certain technical assumptions. For the embedded relationship between function spaces, we show that our initial data class is larger than that of Kozono et al. (2016) [11] and covers physical cases of initial aggregation at points (Diracs) and on filaments. As an application, we obtain a class of asymptotically existence of a basin of attraction for each self-similar solutions with homogeneous initial data.