Existence and uniqueness of global large-data solutions for the Chemotaxis-Navier-Stokes system in $\mathbb{R}^2$

TL;DR

This paper proves the global existence and uniqueness of smooth solutions for the Chemotaxis-Navier-Stokes system in two dimensions with large initial data, using entropy-energy estimates and bootstrap arguments.

Contribution

It establishes the first comprehensive results on global solutions for large initial data in the 2D Chemotaxis-Navier-Stokes system, combining entropy methods with energy estimates.

Findings

Global existence and uniqueness of solutions for large initial data

Development of entropy-energy estimates for low regularity data

Higher-order energy estimates via bootstrap argument

Abstract

This work investigates the Cauchy problem for the classical Chemotaxis-Navier-Stokes (CNS) system in $\mathbb{R}^2$. We establish the global existence and uniqueness of strong, classical, and arbitrarily smooth solutions under large initial data, which has not been addressed in the existing literature. The key idea is to first derive an entropy-energy estimate for initial data with low regularity, by leveraging the intrinsic entropy structure of the system. Building on this foundation, we then obtain higher-order energy estimates for smoother initial data via a bootstrap argument, in which the parabolic nature of the CNS system plays a crucial role in the iterative control of regularity.