Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mathematical modeling and analysis for the chemotactic diffusion in porous media with incompressible Navier-Stokes equations over bounded domain

Fugui Ma, Wenyi Tian, Weihua Deng

TL;DR

This work develops a time-fractional Keller-Segel-Navier-Stokes (TF-KSNS) model for chemotactic diffusion in porous media, derived from CTRW and coupled to incompressible fluid flow to capture memory effects and advection. It establishes local well-posedness and regularity of mild solutions in bounded domains via a Banach fixed point framework and Mittag-Leffler operator estimates, and analyzes finite-time blow-up through continuation arguments. The results provide a rigorous micro-to-macro foundation for anomalous diffusion in soil environments and lay groundwork for global existence analysis and numerical simulations in porous-media chemotaxis. The methodology integrates fractional calculus, stochastic microdynamics, and fluid-structure interactions to advance mathematical understanding of complex biological transport systems.

Abstract

Myxobacteria aggregate and generate fruiting bodies in the soil to survive under starvation conditions. Considering soil as a porous medium, the biological mechanism and dynamic behavior of myxobacteria and slime (chemoattractants) affected by favorable environments in the soil can not be well characterized by the classical full parabolic Keller-Segel system combined with the incompressible Navier-Stokes equations. In this work, we employ the continuous time random walk (CTRW) approach to characterize the diffusion behavior of myxobacteria and slime in porous media at the microscale, and develop a new macroscopic model named as the time-fractional Keller-Segel system. Then it is coupled with the incompressible Navier-Stokes equations through transport and buoyancy, resulting in the TF-KSNS system, which reveals the biological mechanism from micro to macro and then appropriately describes the dynamic behavior of the chemotactic diffusion of myxobacteria and slime in the soil. In addition, we demonstrate that the TF-KSNS system associated with initial and no-flux/no-flux/Dirichlet boundary conditions over smoothly bounded domain in $\mathbb{R}^{d}$ ($d\geq2$) admits a local well-posed mild solution, which continuously depends on the initial data with proper regularity under a small initial condition. Moreover, the blow-up of the mild solution is rigorously investigated.

Mathematical modeling and analysis for the chemotactic diffusion in porous media with incompressible Navier-Stokes equations over bounded domain

TL;DR

This work develops a time-fractional Keller-Segel-Navier-Stokes (TF-KSNS) model for chemotactic diffusion in porous media, derived from CTRW and coupled to incompressible fluid flow to capture memory effects and advection. It establishes local well-posedness and regularity of mild solutions in bounded domains via a Banach fixed point framework and Mittag-Leffler operator estimates, and analyzes finite-time blow-up through continuation arguments. The results provide a rigorous micro-to-macro foundation for anomalous diffusion in soil environments and lay groundwork for global existence analysis and numerical simulations in porous-media chemotaxis. The methodology integrates fractional calculus, stochastic microdynamics, and fluid-structure interactions to advance mathematical understanding of complex biological transport systems.

Abstract

Myxobacteria aggregate and generate fruiting bodies in the soil to survive under starvation conditions. Considering soil as a porous medium, the biological mechanism and dynamic behavior of myxobacteria and slime (chemoattractants) affected by favorable environments in the soil can not be well characterized by the classical full parabolic Keller-Segel system combined with the incompressible Navier-Stokes equations. In this work, we employ the continuous time random walk (CTRW) approach to characterize the diffusion behavior of myxobacteria and slime in porous media at the microscale, and develop a new macroscopic model named as the time-fractional Keller-Segel system. Then it is coupled with the incompressible Navier-Stokes equations through transport and buoyancy, resulting in the TF-KSNS system, which reveals the biological mechanism from micro to macro and then appropriately describes the dynamic behavior of the chemotactic diffusion of myxobacteria and slime in the soil. In addition, we demonstrate that the TF-KSNS system associated with initial and no-flux/no-flux/Dirichlet boundary conditions over smoothly bounded domain in () admits a local well-posed mild solution, which continuously depends on the initial data with proper regularity under a small initial condition. Moreover, the blow-up of the mild solution is rigorously investigated.

Paper Structure

This paper contains 15 sections, 13 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and previous works
  3. Major contributions
  4. Outline of the paper
  5. Mathematical modeling: From microscopic to macroscopic
  6. The biology
  7. Time-fractional chemotaxis diffusion equations with space- and time-dependent forces
  8. Main results
  9. Preliminaries
  10. Main results
  11. Analysis of local well-posedness and regularity
  12. The mild solution of the TF-KSNS system
  13. The proof of Theorem \ref{['thm:RegularThm']}
  14. Blow-up of the mild solution
  15. Conclusions

Key Result

Lemma 3.1

Let $(e^{t\Delta})_{t\geq0}$ be the heat semigroup under the homogenous Neumann boundary condition. Then there exist some positive constants $C_i$ depending on $\Omega$, $p$ and $q$ such that the following estimates hold.

Figures (1)

  • Figure 1: Illustration of the biological environment and the chemotactic diffusion: the myxobacteria (i.e., $\{P_j\}_{j\geq1, j\in\mathbb{N}_0}$) glide in a porous medium towards a relatively high concentration of chemical signals, where black round holes with different sizes refer to obstacles in porous media, and the shades of gray in the background indicate the concentration of the chemoattractant; the darker the color, the higher the concentration. Myxobacteria aggregate and form fruiting bodies that produce new slime for survival.

Theorems & Definitions (25)

  • Lemma 3.1: Celinski21Winkler10
  • Lemma 3.2
  • proof
  • Lemma 3.3: Strong continuity Andrade22Costa23
  • Lemma 3.4
  • proof
  • Theorem 3.1: Well-posedness and $L^q$-regularity
  • Theorem 3.2: Blow-up
  • Definition 4.1: $L^{q}(\Omega)$-mild solution
  • Lemma 4.1: Priori estimates in $L^{\rho q}(\Omega)$
  • ...and 15 more