Sublinear signal production in a two-dimensional Keller–Segel–Stokes system. (October 2016)

• Record Type: Journal Article
• Authors: Black, Tobias
• Is Part Of: Nonlinear analysis. Volume 31(2016)
• Page Start: 593
• Page End: 609
• Publication Date: 2016-10
• Subjects: Chemotaxis -- Keller–Segel -- Stokes -- Chemotaxis–fluid interaction -- Global existence -- Boundedness; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2016.03.008
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: We study the chemotaxis–fluid system { n t = Δ n − ∇ ⋅ ( n ∇ c ) − u ⋅ ∇ n, x ∈ Ω, t > 0, c t = Δ c − c + f ( n ) − u ⋅ ∇ c, x ∈ Ω, t > 0, u t = Δ u + ∇ P + n ∇ ϕ, x ∈ Ω, t > 0, ∇ ⋅ u = 0, x ∈ Ω, t > 0, where Ω ⊂ R 2 is a bounded and convex domain with smooth boundary, ϕ ∈ W 2, ∞ ( Ω ) and f ∈ C 1 ( [ 0, ∞ ) ) satisfies 0 ≤ f ( s ) ≤ K 0 s α for all s ∈ [ 0, ∞ ), with K 0 > 0 and α ∈ ( 0, 1 ] . This system models the chemotactic movement of actively communicating cells in slow moving liquid. We will show that in the two-dimensional setting for any α ∈ ( 0, 1 ) the classical solution to this Keller–Segel–Stokes-system is global and remains bounded for all times.