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

TL;DR

This paper investigates the global existence and asymptotic behavior of solutions to a complex system of time-space fractional chemotaxis-fluid equations within Besov-Morrey spaces, overcoming challenges posed by the lack of semigroup properties.

Contribution

It introduces a novel approach using harmonic analysis techniques to establish global solutions for a generalized chemotaxis-fluid system without semigroup effects.

Findings

Proved global existence of solutions in Besov-Morrey spaces.

Analyzed the asymptotic behavior of solutions.

Developed new analytical techniques for fractional PDEs.

Abstract

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 \cite{M.H. Yang}. In contrast to \cite{M.H. Yang}, the solution operator of the system does not satisfy the semigroup effect, which makes the approach of \cite{M.H. Yang} 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 analysis the asymptotic behavior of the solutions.