Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation.

The Keller-Segel-Stokes system { nt + u ⋅ ∇n = Δn − ∇ ⋅ (n∇c) + ρn − μnα, ct + u ⋅ ∇c = Δc − c + n, ut = Δu + ∇P − n∇Λ , ∇ ⋅ u = 0 (*), is considered in a bounded domain Ω ⊂ ℝ3 with smooth boundary, with parameters ρ ≥ 0, μ > 0 and α > 1, and with a given gravitational potential Λ ∈ W2,∞(Ω). It is shown that in this general setting, when posed under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u, and for any suitably regular initial data, an associated initial value problem possesses at least one globally defined solution in an appropriate generalized sense. Since it is well-known that in the absence of absorption, already the corresponding fluid-free subsystem with u ≡ 0 and μ = 0 admits some solutions blowing up in finite time, this particularly indicates that any power-type superlinear degradation of the form in (*) goes along with some significant regularizing effect. [ABSTRACT FROM AUTHOR]