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

Abstract

Considering soil as a porous medium, the biological mechanism and dynamic behavior of myxobacteria and slime affected by favorable environments in the soil cannot be well characterized by the classical Keller-Segel-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 appropriately describes 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 a smoothly bounded domain in $R^d$ ($d\ge 2$) admits a local well-posed mild solution, which continuously depends on the initial data. Moreover, the blow-up of the mild solution is rigorously investigated.