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