Well-posedness for chemotaxis-fluid models in arbitrary dimensions

TL;DR

This paper establishes well-posedness results for chemotaxis-fluid models in any dimension, introducing a new function space framework that ensures existence, uniqueness, and key properties of solutions for low regularity data.

Contribution

It introduces a novel function space, nd, for analyzing chemotaxis-Navier-Stokes systems, proving existence, uniqueness, and optimality of solutions in higher dimensions.

Findings

Existence of local and global solutions with low regularity data

Solutions preserve mass and nonnegativity

Optimality of the new function space nd

Abstract

We study the Cauchy problem for the chemotaxis Navier-Stokes equations and the Keller-Segel-Navier-Stokes system. Local-in-time and global-in-time solutions satisfying fundamental properties such as mass conservation and nonnegativity preservation are constructed for low regularity data in $2$ and higher dimensions under suitable conditions. Our initial data classes involve a new scale of function space, that is $\Y(\rn)$ which collects divergence of vector-fields with components in the square Campanato space $\mathscr{L}_{2,N-2}(\rn)$, $N>2$ (and can be identified with the homogeneous Besov space $\dot{B}^{-1}_{22}(\rn)$ when $N=2$) and are shown to be optimal in a certain sense. Moreover, uniqueness criterion for global solutions is obtained under certain limiting conditions.