LAUSR

Abstract In this paper, we study the initial-boundary value problem for the coupled chemotaxis(-Navier)-Stokes system with indirect signal production { ∂ t n + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n ∇ c ) + r n − μ n 2 , ( x , t ) ∈ Ω × ( 0 , ∞ ) , ∂ t c + u ⋅ ∇ c = Δ c − c + v , ( x , t ) ∈ Ω × ( 0 , ∞ ) , ∂ t v + u ⋅ ∇ v = Δ v − v + n , ( x , t ) ∈ Ω × ( 0 , ∞ ) , ∂ t u + κ ( u ⋅ ∇ ) u + ∇ P = Δ u + n ∇ Φ , ∇ ⋅ u = 0 , ( x , t ) ∈ Ω × ( 0 , ∞ ) in a smooth bounded domain Ω ⊂ R d ( d = 2 , 3 ) , where κ ∈ { 0 , 1 } , r ≥ 0 and μ > 0 are given constants. It is shown that when posed with no-flux/no-flux/no-flux/Dirichlet boundary condition and along with appropriate assumption on regularity of the initial data ( n 0 , c 0 , v 0 , u 0 ) , the chemotaxis-Stokes system (i.e. κ = 0 ) admits globally bounded classical solution in Ω ⊂ R 3 ; the chemotaxis-Navier-Stokes system (i.e. κ = 1 ) possesses globally bounded classical solution in Ω ⊂ R 2 .