Cauchy problems for time-space fractional coupled chemotaxis-fluid equations in Besov-Morrey spaces

In this paper, we consider the Cauchy problems for the time-space fractional coupled chemotaxis-fluid equations, which is a generalized form of the coupled chemotaxis-fluid equations studied in Yang et al. [Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces. J Differ Equ. 2019;266:5867–5894]. In contrast to Yang et al. [Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces. J Differ Equ. 2019;266:5867–5894], the solution operator of the system does not satisfy the semigroup effect, which makes the approach of Yang et al. [Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces. J Differ Equ. 2019;266:5867–5894] inapplicable. Based on the theory of harmonic analysis, using techniques such as real interpolation, embedding in Besov-Morrey spaces, the multiplier theorem, and the Hardy-Littelwood inequality in Morrey spaces, we establish global existence. As an application, we analyze the asymptotic behavior of the solutions. Compared to method [Yang et al. Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces. J Differ Equ. 2019;266:5867–5894], our approach exhibits greater applicability, as the solution operator in our case does not possess the semigroup property. Furthermore, the class of initial conditions we consider encompasses physical scenarios involving initial aggregations at points (Diracs) and on filaments.