Blow-up Prevention by Saturated Chemotactic Sensitivity in a 2D Keller-Segel-Stokes System
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Blow-up Prevention by Saturated Chemotactic Sensitivity in a 2D Keller-Segel-Stokes System Pei Yu1
Received: 14 June 2019 / Accepted: 27 December 2019 © Springer Nature B.V. 2020
Abstract This paper deals with a two-chemical reaction type Keller-Segel system coupled with incompressible viscous fluid equations which models the dynamics of cells in fluid in a two dimensional bounded domain Ω ⎧ ∂t n + u · ∇n = n − ∇ · nχ (n, v, w, x)∇v , ⎪ ⎪ ⎨ ∂t v + u · ∇v = v − v + w, ∂t w + u · ∇w = w − w + n, ⎪ ⎪ ⎩ ∂t u + ∇P = u + n∇ψ, ∇ · u = 0. Here, χ (n, v, w, x) represents the saturated sensitivity. Our result suggests that suitable saturation can prevent the blow-up arising from the classical Keller-Segel type signal production mechanism without any smallness condition on initial mass. Keywords Keller-Segel model · Stokes equation · Indirect process · Global existence Mathematics Subject Classification 35K45 · 35K91 · 35Q92 · 76D07 · 92C17
1 Introduction Keller-Segel system Chemotaxis is a ubiquitous phenomenon in biology and ecology processes that cells or species move towards higher or lower concentration of chemical signal. For examples, hunting animals follow the smells of chemical of their prey, butterflies and bees find food sources by chemotaxis and almost all mammals attracted to the other gender by pheromoners. One of the best examples of chemotactic phenomena is the movement of D. discoideum which will stream together via chemotaxis to form a multicellular organism. In 1970s, Keller and
B P. Yu
[email protected]
1
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
P. Yu
Segel [16] used a two-equation system to describe this movement in space-time domain Ω × (0, T ) ∂t n = n − ∇ · nχ (n, v, x)∇v , (1.1) ∂t v = v − v + n, where n and v denote the population of cells and concentration of chemical signal, respectively. The known function χ (n, v, x) is chemotactic sensitivity function which may depend on n, v and the position x ∈ Ω. In this model, the chemical signal is produced by cells. This type of dynamics is called direct signal production. There are rich mathematical investigations on this original Keller-Segel system (1.1) in an unbounded or bounded domain. We can refer [10–12, 23, 25, 33, 43] to have a board view of these models which have been studied for decades. For instance, when χ = 1, it is well known from [12, 23, 25, 33, 43] that in a bounded domain with non-flux boundary condition, • if N = 1, all solutions exist globally and bounded; • if N = 2, – when the initial mass of cells m = n0 dx < 4π , all solutions are global exist and bounded, while – when m ∈ (4π, ∞)/{4kπ | k ∈ N}, there always exists solution that blows up in finite or infinite time; • if N = 3, for any small mass of cells, system (1.1) processes unbounded solutions. It is worth mentioning the results of saturated non-symmetric sensitivity
χ (n, v, x) ≤ Cχ (1 + n)−α ,
(1.2)
where | · | is the Frobenius norm. This saturation assumption on sensitiv
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