[Paper Review] An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion

Jiashan Zheng|arXiv (Cornell University)|Jul 1, 2018

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TL;DR

This paper establishes the first optimal global existence and boundedness result for a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion, proving that solutions remain globally bounded for any $ m > \frac{4}{3} $, which matches the critical threshold from the fluid-free case. The analysis combines energy estimates, entropy-type inequalities, and compactness arguments to overcome the challenges of chemotaxis-fluid coupling and nonlinear diffusion in three dimensions.

ABSTRACT

This paper investigates the following quasilinear Keller-Segel-Navier-Stokes system $$\left\{ \begin{array}{l} n_t+u\cdot abla n=Δn^m- abla\cdot(n abla c),\quad x\in Ω, t>0, \\ c_t+u\cdot abla c=Δc-c+n,\quad x\in Ω, t>0,\\ u_t+ abla P=Δu+n abla ϕ,\quad x\in Ω, t>0,\\ abla\cdot u=0, \quad x\in Ω, t>0 \end{array} ight.$$ under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a three-dimensional bounded domains $Ω\subseteq \mathbb{R}^3$ with smooth boundary, where $ϕ\in W^{1,\infty}(Ω),m>0$. It is proved that if $m>\frac{4}{3}$, then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system $(KSF)$, which in view of the known results for the fluid-free system mentioned below (see Introduction) is an optimal restriction on $m$.

Motivation & Objective

• To establish global existence and boundedness of solutions for a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion.
• To determine the sharp threshold for the diffusion exponent $ m $ that ensures global boundedness, matching the critical exponent from the fluid-free case.
• To address the challenge of chemotaxis-fluid interaction in three dimensions, where nonlinear diffusion and fluid dynamics couple nontrivially.
• To extend known results on boundedness in lower dimensions and in the fluid-free setting to the full three-dimensional system with nonlinear diffusion.
• To prove that $ m > \frac{4}{3} $ is optimal, meaning that for $ m \leq \frac{4}{3} $, solutions may blow up, confirming the sharpness of the condition.

Proposed method

• Analyzes the Keller-Segel-Stokes system with nonlinear diffusion $ \Delta n^m $, chemotaxis $ -\nabla \cdot (n\nabla c) $, and fluid dynamics via the Stokes equations.
• Employs a regularized approximation scheme to handle the singularity in the nonlinear diffusion term $ n^m $, introducing a parameter $ \varepsilon \in (0,1) $.
• Derives uniform a priori estimates using energy-type inequalities and entropy-like functional estimates to control $ n $, $ c $, and $ u $.
• Applies the Aubin-Lions lemma to extract strong convergence of $ n_\varepsilon $ in $ L^p $ and weak-* convergence in $ L^\infty $, ensuring limit functions exist.
• Uses compactness and regularity arguments to pass to the limit in the approximate system, proving the existence of a global weak solution.
• Establishes equicontinuity and convergence in $ C^0_{\text{loc}} $ and $ L^\infty $ for $ n $, $ c $, $ \nabla c $, $ u $, and $ Du $, ensuring the limit satisfies the weak formulation.

Experimental results

Research questions

• RQ1 What is the sharp threshold for the diffusion exponent $ m $ that guarantees global existence and boundedness of solutions in the three-dimensional Keller-Segel-Stokes system with nonlinear diffusion?
• RQ2 Can global boundedness be achieved for $ m \leq \frac{4}{3} $, or does the system admit finite-time blow-up in this regime?
• RQ3 How does the coupling with the Stokes fluid system affect the critical exponent for boundedness compared to the fluid-free Keller-Segel model?
• RQ4 Is the condition $ m > \frac{4}{3} $ optimal in the sense that it matches the critical exponent $ 2 - \frac{2}{N} $ for $ N=3 $, known from the fluid-free case?
• RQ5 Can uniform bounds and compactness arguments be applied effectively in three dimensions to pass to the limit in a regularized approximation scheme?

Key findings

• For any $ m > \frac{4}{3} $, the system admits at least one global bounded weak solution in three dimensions.
• The condition $ m > \frac{4}{3} $ is optimal, as it matches the critical exponent $ 2 - \frac{2}{3} = \frac{4}{3} $ from the fluid-free Keller-Segel model.
• Uniform bounds on $ n $, $ c $, $ \nabla c $, $ u $, and $ Du $ are established via energy and entropy estimates, independent of the regularization parameter $ \varepsilon $.
• Strong convergence of $ n_\varepsilon $ to $ n $ a.e. in $ \Omega \times (0,\infty) $ and weak-* convergence in $ L^\infty $ are proven using the Aubin-Lions lemma.
• The limit functions $ n $, $ c $, and $ u $ satisfy the weak formulation of the system in $ \Omega \times (0,\infty) $, with $ n \in C^{0}_{\omega-*}([0,\infty); L^\infty(\Omega)) $.
• The result closes a gap in the literature by showing that the critical exponent $ m = \frac{4}{3} $ is sharp, and no better threshold exists for global boundedness.