A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier—Stokes System involving Dirichlet Boundary Conditions for the Signal.

The chemotaxis-Navier—Stokes system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n ∇ c) , c t + u ⋅ ∇ c = Δ c − n c , u t + (u ⋅ ∇) u = Δ u + ∇ P + n ∇ ϕ , ∇ ⋅ u = 0 is considered in a smoothly bounded planar domain Ω under the boundary conditions (∇ n − n ∇ c) ⋅ ν = 0 , c = c ∗ , u = 0 , x ∈ ∂ Ω , t > 0 , with a given nonnegative constant C✭. It is shown that if (n0, c0, u0) is sufficiently regular and such that the product ‖ n 0 ‖ L 1 (Ω) ‖ c 0 ‖ L ∞ (Ω) 2 is suitably small, an associated initial value problem possesses a bounded classical solution with (n, c, u)|t=0 = (n0, c0, u0). [ABSTRACT FROM AUTHOR]