Reaction-Driven Relaxation in Three-Dimensional Keller–Segel–Navier–Stokes Interaction

The Keller–Segel–Navier–Stokes system n t + u · ∇ n = Δ n - χ ∇ · ( n ∇ c ) + ρ n - μ n 2 , c t + u · ∇ c = Δ c - c + n , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ ϕ + f ( x , t ) , ∇ · u = 0 , ( ⋆ ) is considered in a smoothly bounded convex domain Ω ⊂ R 3 , with ϕ ∈ W 2 , ∞ ( Ω ) and f ∈ C 1 ( Ω ¯ × [ 0 , ∞ ) ; R 3 ) , and with χ > 0 , ρ ∈ R and μ > 0 . As recent literature has shown, for all reasonably mild initial data a corresponding no-flux/no-flux/Dirichlet initial-boundary value problem possesses a global generalized solution, but the knowledge on its regularity properties has not yet exceeded some information on fairly basic integrability features. The present study reveals that whenever ω > 0 , requiring that ρ min { μ , μ 3 2 + ω } < η with some η = η ( ω ) > 0 , and that f satisfies a suitable assumption on ultimate smallness, is sufficient to ensure that each of these generalized solutions becomes eventually smooth and classical. Furthermore, under these hypotheses ( ⋆ ) is seen to admit an absorbing set with respect to the topology in L ∞ ( Ω ) . By trivially applying to the case when μ > 0 is arbitrary and ρ ≤ 0 , these results especially assert essentially unconditional statements on eventual regularity in taxis-reaction systems interacting with liquid environments, such as arising in contexts of models for broadcast spawning discussed in recent literature.