[Paper Review] A new approach toward locally bounded global solutions to a $3D$ chemotaxis-stokes system with nonlinear diffusion and rotation

Jiashan Zheng|arXiv (Cornell University)|Jan 5, 2017

Mathematical Biology Tumor Growth16 references3 citations

TL;DR

This paper establishes the existence of globally bounded weak solutions to a 3D chemotaxis-Stokes system with nonlinear porous medium diffusion ($m > 9/8$), tensor-valued sensitivity, and rotational flux. By introducing a novel gradient-like structure in the functional $\int_\Omega n\ln n + \int_\Omega |\nabla\sqrt{c}|^2 + \int_\Omega |u|^2$, the authors prove local $L^\infty$ boundedness of $n$, $c$, and $u$ for all $t \in (0,T)$, resolving key challenges in systems with tensor-valued chemotactic sensitivity.

ABSTRACT

We consider a degenerate quasilinear chemotaxis--Stokes type involving rotation in the aggregative term, \begin{equation} \left\{ \begin{array}{l} n_t+u\cdot abla n=Δn^m- abla\cdot(nS(x,n,c)\cdot abla c),\quad x\in Ω, t>0, c_t+u\cdot abla c=Δc-nc, x\in Ω, t>0,\\ u_t+ abla P=Δu+n abla ϕ,x\in Ω, t>0,\\ abla\cdot u=0, x\in Ω, t>0, \end{array} ight. \end{equation} where $Ω\subseteq \mathbb{R}^3$ is a bounded convex domain with smooth boundary. Here $ S\in C^2(\barΩ imes[0,\infty)^2;\mathbb{R}^{3 imes3})$ is a matrix with $s_{i,j}\in C^1( \barΩ imes [0, \infty) imes[0, \infty)).$ Moreover, $|S(x,n,c)| \leq S_0(c)$ for all $(x,n,c)\in \barΩ imes [0, \infty) imes[0, \infty)$ with $S_0(c)$ nondecreasing on $[0,\infty)$. If $$m>\frac{9}{8}, $$ then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(0.1)$ possesses a globally defined weak solution $(n,c,u)$. Moreover, for any fixed $T > 0$ this solution is bounded in $Ω imes (0,T)$ in the sense that $$ \|u(\cdot,t)\|_{L^\infty(Ω)} +\|c(\cdot,t)\|_{W^{1,\infty}(Ω)}+\|n(\cdot,t)\|_{L^\infty(Ω)} \leq C ~~\mbox{for all}~~ t\in(0,T) $$ is valid with some $C(T) > 0$.

Motivation & Objective

• To establish global existence and local boundedness of weak solutions for a 3D chemotaxis-Stokes system with nonlinear diffusion and tensor-valued sensitivity.
• To address the lack of natural gradient-like structure in systems with rotational flux and tensor-valued sensitivity.
• To extend previous results on scalar sensitivity and fluid-free cases to the more complex, biologically relevant case with matrix-valued sensitivity.
• To develop a new functional structure that captures the dissipative nature of the system despite the absence of standard Lyapunov functionals.

Proposed method

• Introduce a new functional $\int_\Omega n\ln n + \int_\Omega |\nabla\sqrt{c}|^2 + \int_\Omega |u|^2$ to capture gradient-like dissipation in the system.
• Use regularized approximate systems with $\varepsilon$-perturbation to handle the degeneracy of nonlinear diffusion and avoid blow-up.
• Apply $L^p$-$L^q$ estimates for the heat semigroup to control $c$ and $u$ in higher-order norms.
• Employ Moser iteration and Aubin-Lions compactness lemma to derive strong convergence of $n_\varepsilon$ and its powers.
• Establish uniform bounds on $\|n_\varepsilon\|_{L^\infty}$, $\|c_\varepsilon\|_{W^{1,\infty}}$, and $\|u_\varepsilon\|_{L^\infty}$ independently of $\varepsilon$.
• Pass to the limit $\varepsilon \to 0$ using weak* and strong convergence to obtain a global weak solution.

Experimental results

Research questions

• RQ1 Can global bounded solutions be established for a 3D chemotaxis-Stokes system with nonlinear diffusion ($m > 9/8$) and tensor-valued sensitivity?
• RQ2 Does the presence of rotational flux in the chemotactic term prevent the existence of bounded solutions, and if so, can this be overcome?
• RQ3 Can a new gradient-like structure be constructed to control the system despite the loss of standard Lyapunov functionals in the tensor-sensitivity case?
• RQ4 How does the boundedness of $n$, $c$, and $u$ depend on the diffusion exponent $m$?
• RQ5 To what extent do the results extend previous works on scalar sensitivity or fluid-free systems?

Key findings

• For $m > 9/8$, the initial-boundary value problem for the chemotaxis-Stokes system with tensor-valued sensitivity admits a globally defined weak solution.
• The solution satisfies $\|u(\cdot,t)\|_{L^\infty(\Omega)} + \|c(\cdot,t)\|_{W^{1,\infty}(\Omega)} + \|n(\cdot,t)\|_{L^\infty(\Omega)} \leq C$ for all $t \in (0,T)$ and some $C(T) > 0$, ensuring local $L^\infty$ boundedness.
• The new functional $\int_\Omega n\ln n + \int_\Omega |\nabla\sqrt{c}|^2 + \int_\Omega |u|^2$ exhibits a gradient-like structure that enables energy-type estimates despite the absence of standard Lyapunov functionals.
• The result extends Tao and Winkler’s work for constant sensitivity $S = C_S$ and is consistent with Zheng and Wang’s fluid-free result when $m=1$ and $S=C_S$.
• The proof relies on a novel combination of Moser iteration, Aubin-Lions lemma, and $L^p$-$L^q$ estimates to achieve strong convergence of $n_\varepsilon$ and uniform bounds.
• The boundedness of $n_\varepsilon$, $c_\varepsilon$, and $u_\varepsilon$ in $L^\infty$ is uniformly controlled for all $\varepsilon \in (0,1)$, enabling the limit passage to a global weak solution.