Mild solutions to the Cauchy problem for time-space fractional Keller-Segel-Navier-Stokes system

TL;DR

This paper studies the existence, regularity, and properties of mild solutions to a complex fractional PDE system modeling chemotaxis and fluid flow, incorporating memory effects and Lévy processes.

Contribution

It establishes local and global existence of solutions, improves regularity results, and analyzes key properties like stability and decay for the fractional Keller-Segel-Navier-Stokes system.

Findings

Existence of local and global mild solutions.

Enhanced regularity in fractional Sobolev spaces.

Properties such as mass conservation and decay estimates.

Abstract

This paper investigates the Cauchy problem of the time-space fractional Keller-Segel-Navier- Stokes model, which can describe both memory effect and L\'evy process of the system. The local existence and global existence in Lebesgue space are obtained by means of Banach fixed point theorem and Banach implicit function theorem, respectively. In addition, the regularities of local and global mild solutions are improved in fractional homogeneous Sobolev spaces. Furthermore, some properties of mild solutions including mass conservation, decay estimates, stability and self-similarity are established.