[Paper Review] Global Existence and Asymptotic Behavior of Solutions to a Chemotaxis-Fluid System on General Bounded Domain

Jie Jiang, Hao Wu+1 more|arXiv (Cornell University)|Sep 1, 2014

Mathematical Biology Tumor Growth7 references4 citations

TL;DR

This paper establishes global existence and asymptotic convergence to equilibrium for a chemotaxis-fluid system on general bounded domains in 2D and 3D, extending prior results that required convex domains. By introducing a novel entropy-energy estimate using a lemma from Mizoguchi & Souplet, the authors prove unique global classical solutions in 2D and global weak solutions in 3D, with solutions converging to a constant steady state as time tends to infinity.

ABSTRACT

In this paper, we investigate an initial-boundary value problem for a chemotaxis-fluid system in a general bounded regular domain $Ω\subset \mathbb{R}^N$ ($N\in\{2,3\}$), not necessarily being convex. Thanks to the elementary lemma given by Mizoguchi & Souplet [10], we can derive a new type of entropy-energy estimate, which enables us to prove the following: (1) for $N=2$, there exists a unique global classical solution to the full chemotaxis-Navier-Stokes system, which converges to a constant steady state $(n_\infty, 0,0)$ as $t o+\infty$, and (2) for $N=3$, the existence of a global weak solution to the simplified chemotaxis-Stokes system. Our results generalize the recent work due to Winkler [15,16], in which the domain $Ω$ is essentially assumed to be convex.

Motivation & Objective

• To extend the global existence and asymptotic behavior results for chemotaxis-fluid systems from convex domains to general bounded regular domains.
• To establish the existence of global classical solutions in 2D and global weak solutions in 3D without requiring domain convexity.
• To prove convergence of solutions to a constant steady state in 2D as time approaches infinity.
• To derive a new type of entropy-energy estimate that is robust under general domain geometry.

Proposed method

• Derives a new entropy-energy inequality using an elementary lemma from Mizoguchi & Souplet (2012), enabling control of nonlinear terms without convexity assumptions.
• Applies Schauder’s fixed point theorem to establish local well-posedness of classical solutions for initial-boundary value problems on general bounded domains.
• Uses a priori estimates involving $ \int_\Omega \frac{|\nabla n|^2}{n} dx $, $ \|\nabla u\|^2 $, and $ \|\psi(c)\|^2 $ to control solution growth.
• Employs Sobolev embeddings and Poincaré-type inequalities to bound higher-order norms and ensure regularity.
• Adapts and refines techniques from Winkler (2012, 2013) by replacing the $ \|u\|_{L^4}^4 $ term with $ \|\nabla u\|^2 $, simplifying the asymptotic analysis.
• Establishes uniform bounds on $ \|n\|_{L^\infty} $, $ \|c\|_{W^{1,q}} $, and $ \|u\|_{L^\infty} $, leading to convergence in $ L^\infty $-norm.

Experimental results

Research questions

• RQ1 Can global classical solutions exist for the full chemotaxis-Navier-Stokes system on a general bounded domain without convexity?
• RQ2 Does the solution to the simplified chemotaxis-Stokes system in 3D exist globally in time on a general bounded domain?
• RQ3 Do solutions to the 2D chemotaxis-fluid system converge to a constant steady state as $ t \to \infty $ on a general bounded domain?
• RQ4 Can a new entropy-energy estimate be constructed that avoids dependence on domain convexity?

Key findings

• For $ N=2 $, the system admits a unique global classical solution that converges to the constant steady state $ (n_\infty, 0, 0) $ in $ L^\infty $-norm as $ t \to \infty $, where $ n_\infty = \frac{1}{|\Omega|} \int_\Omega n_0 \, dx $.
• For $ N=3 $, a global weak solution exists to the simplified chemotaxis-Stokes system on any bounded regular domain, without requiring convexity.
• A new entropy-energy estimate is derived that controls the $ L^2 $-norm of $ \nabla c $ and $ \nabla u $ via $ \|\nabla u\|^2 $ and $ \|\psi(c)\|^2 $, replacing the previous $ \|u\|_{L^4}^4 $ term.
• The asymptotic convergence result in 2D is preserved under general domain geometry due to the improved structure of the new entropy estimate.
• The a priori estimates are uniform in time and depend only on initial data and domain size, ensuring global existence.
• The method generalizes Winkler’s results (2012, 2013) from convex to general bounded domains by removing geometric assumptions on $ \Omega $.