Abstract
Abstract Keller-Segel chemotaxis model is described by a system of nonlinear partial differential equations: a convection diffusion equation for the cell density coupled with a reaction-diffusion equation for chemoattractant concentration. In this work, we study the phenomenon of Keller-Segel model coupled with Boussinesq equations. The main objective of this work is to study the global existence and uniqueness and boundedness of the weak solution for the problem, which is carried out by the Galerkin method.
Highlights
The phenomenon of chemotaxis is one of the most important phenomena that have aroused the interest of many researchers, as chemotaxis is a biological process where cells move towards a more appropriate chemical direction
The Keller and Segel model, which describes chemotaxis, is considered one of the best studied models in mathematical biology; on the other hand, nature cells often live in a viscous fluid and transport cells and chemical substrates with the fluid, and in the meantime the fluid movement is subject to affecting factors
⎧nt + u∇n = δΔnm − ∇(χ(c)n∇c), ⎪ct + u∇c = μΔc − k(c)n, ⎨ut + ∇p = υΔu − n∇Φ, ⎩⎪∇u = 0, (x, t) ∈ Ω × +, and they studied a model arising from biology, consisting of chemotaxis equations coupled with viscous incompressible fluid equations through transport and external forcing and proved global existence of large data and solutions to the Cauchy problem investigated under certain conditions for the chemotaxis-NavierStokes system in two space dimensions, and we obtain precisely global existence of weak solutions for the chemotaxis-Stokes system with nonlinear diffusion for the cell density in three space dimensions
Summary
The phenomenon of chemotaxis is one of the most important phenomena that have aroused the interest of many researchers, as chemotaxis is a biological process where cells (bacteria) move towards a more appropriate chemical direction. ⎧nt + u∇n = δΔnm − ∇(χ(c)n∇c), ⎪ct + u∇c = μΔc − k(c)n, ⎨ut + ∇p = υΔu − n∇Φ, ⎩⎪∇u = 0, (x, t) ∈ Ω × +, and they studied a model arising from biology, consisting of chemotaxis equations coupled with viscous incompressible fluid equations through transport and external forcing and proved global existence of large data and solutions to the Cauchy problem investigated under certain conditions for the chemotaxis-NavierStokes system in two space dimensions, and we obtain precisely global existence of weak solutions for the chemotaxis-Stokes system with nonlinear diffusion for the cell density in three space dimensions. The main objective of this work is to study the problem Keller-Segel coupled with Boussinesq equations, and we demonstrate the global existence and uniqueness of a weak solution for KSB problem with the Dirichlet conditions and initial conditions defined as: 560 Ali Slimani et al.
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