Global well-posedness for the 2D chemotaxis-fluid system with logistic source

ABSTRACT

In this paper, the two-dimensional incompressible chemotaxis fluid with logical source is studied as following: $\begin{array}{rl}& n_t+u\cdot \mathrm{\nabla }n=\mathrm{\Delta }n-\mathrm{\nabla }\cdot \left(n\mathrm{\nabla }c\right)+n-n^2,\\ & c_t+u\cdot \mathrm{\nabla }c=\mathrm{\Delta }c-nc,\\ & u_t+u\cdot \mathrm{\nabla }u+\mathrm{\nabla }P=\mathrm{\Delta }u-n\mathrm{\nabla }\phi ,\\ & \mathrm{\nabla }\cdot u=0.\end{array}$ By taking advantage of a coupling structure of the equations and using a scale decomposition technique, the global existence and uniqueness of weak solutions to the above system for a large class of initial data is obtained.

KEYWORDS:

• Global well-posedness
• chemotaxis
• Navier–Stokes
• Zygmund spaces
• microlocal analysis

AMS SUBJECT CLASSIFICATIONS:

• 35K55
• 35Q92
• 35Q35
• 92C17

Acknowledgments

Q. Zhang was partially supported by the National Natural Science Foundation of China [grant number 11501160 and 11771423]; Natural Science Foundation of Hebei Province [grant number A2017201144 and A2018201109]; Young Talents Foundation of Hebei Education Department [grant number BJ2017058]; Outstanding Youth Foundation of Hebei University [grant number 2015JQ01]; Research and Practice of Reform in Higher Education and Teaching Foundation of Hebei Province [grant number 2018GJJG012]; the Second Batch of Young Talents of Hebei Province.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Q. Zhang was partially supported by the National Natural Science Foundation of China [Grant Number 11501160 and 11771423]; Natural Science Foundation of Hebei Province [Grant Number A2017201144 and A2018201109]; Young Talents Foundation of Hebei Education Department [Grant Number BJ2017058]; Outstanding Youth Foundation of Hebei University [Grant Number 2015JQ01]; Research and Practice of Reform in Higher Education and Teaching Foundation of Hebei Province [Grant Number 2018GJJG012]; the Second Batch of Young Talents of Hebei Province.