In this paper, we investigate an initial-boundary value problem for the self-consistent chemotaxis-Navier-Stokes system { n t + u · ∇ n = ∆ n − ∇ · ( n ( 1 + n ) − α ∇ c ) + ∇ · ( n ∇ ϕ ) , x ∈ Ω , t > 0 , c t + u · ∇ c = ∆ c − c + n , x ∈ Ω , t > 0 , u t + κ ( u · ∇ ) u + ∇ P = ∆ u − n ∇ ϕ + n ( 1 + n ) − α ∇ c , x ∈ Ω , t > 0 , ∇ · u = 0 , x ∈ Ω , t > 0 , where Ω ⊂ R 2 is a bounded domain with smooth boundary, α>0, κ∈R and the gravitational potential function ϕ ∈ W 2 , ∞ ( Ω ) . The novelty of this work lies in the consideration of both the effect of gravity (potential force) on cells and the influence of chemotactic force on the fluid, resulting in a stronger coupling mechanism than that observed in the usual chemotaxis-Navier-Stokes model studied in most existing literatures. It is shown that if α > 1 4 , then for any sufficiently regular initial data, this system admits at least one global and bounded solution to this system under no-flux boundary conditions for n, c and homogeneous Dirichlet boundary condition for u. Our analytic approach is based on a new energy-like functional ∫ Ω n 1 + k 0 α + ∫ Ω | ∇ c | 2 with some integer k 0 > 3 α .

This paper establishes the global existence of classical solutions to the initial-boundary value problem for a system of four reaction-diffusion-chemotaxis equations in smooth bounded domains. The model is governed by the following equations { u t = ∆ u − ∇ · ( u ∇ v ) , x ∈ Ω , t > 0 , 0 = ∆ v − v + w , x ∈ Ω , t > 0 , w t = d ∆ w − w + ρuz k u + θu , x ∈ Ω , t > 0 , z t = ∆ z − uz k u + θu + w , x ∈ Ω , t > 0 , where d , ρ , k u , θ are positive constants, and Ω ⊂ R N ( N ≤ 5 ) denotes a bounded domain with smooth boundary. Under homogeneous Neumann boundary conditions, we establish the existence and uniqueness of global classical solution for this system. AMS subject classifications: 35B45, 35Q92, 92C17

In this paper we deal with the following attraction-repulsion chemotaxis system { u t = ∆ u − χ ∇ · ( u ∇ v )+ ξ ∇ · ( u ∇ w ) , x ∈ Ω , t > 0 , 0 = ∆ v − βv + αu , x ∈ Ω , t > 0 , 0 = ∆ w − δw + γu , x ∈ Ω , t > 0 , ∂u ∂ν = ∂v ∂ν = ∂w ∂ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 )= u 0 ( x ) , x ∈ Ω , under homogenous Neumann boundary conditions in a smoothly bounded domain Ω ⊂ R 4 , where χ, ξ, β, α, δ and γ are positive constants. In this paper, we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption ξγ= χα and ξ δ λ 0 γ ∫ Ω u 0 < 1 C GN , where C GN and λ 0 are some positive constants only depending on Ω. This result significantly improves or extends previous results of several authors (see Remark 1.1).