Boundedness in a three-dimensional chemotaxis–fluid system involving tensor-valued sensitivity with saturation

We consider the chemotaxis–fluid systemequation(CF){nt+u⋅∇n=∇⋅(D(n)∇n)−∇⋅(nS(x,n,c)⋅∇c)+an−bn2,x∈Ω,t>0,ct+u⋅∇c=Δc−c+n,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ+g(x,t),x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊆R3Ω⊆R3, where ϕ∈W1,∞(Ω)ϕ∈W1,∞(Ω), a≥0a≥0 and b>0b>0. Here g∈C1(Ω¯×[0,∞))∩L∞(Ω×(0,∞)), D(n)≥um−1D(n)≥um−1, |S(x,n,c)|≤(1+n)−α|S(x,n,c)|≤(1+n)−α, and the parameter α>0α>0. If m>max⁡{65−α,13}, then for all reasonably regular initial data, a corresponding initial–boundary value problem for (CF) possesses a globally defined weak solution through the Moser-type iteration.