Global classical solutions in a chemotaxis(-Navier)-Stokes system with indirect signal production

Abstract

In this paper, we study the initial-boundary value problem for the coupled chemotaxis(-Navier)-Stokes system with indirect signal production{∂tn+u⋅∇n=Δn−∇⋅(n∇c)+rn−μn2,(x,t)∈Ω×(0,∞),∂tc+u⋅∇c=Δc−c+v,(x,t)∈Ω×(0,∞),∂tv+u⋅∇v=Δv−v+n,(x,t)∈Ω×(0,∞),∂tu+κ(u⋅∇)u+∇P=Δu+n∇Φ,∇⋅u=0,(x,t)∈Ω×(0,∞) in a smooth bounded domain Ω⊂Rd(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 assumptions on regularity of the initial data (n0,c0,v0,u0), the chemotaxis-Stokes system (i.e. κ=0) admits globally bounded classical solution in Ω⊂R3; the chemotaxis-Navier-Stokes system (i.e. κ=1) possesses globally bounded classical solution in Ω⊂R2.