Small-data solutions of chemotaxis-fluid system with indirect signal production

Abstract

The Keller-Segel-Navier-Stokes system with indirect signal production mechanism{nt+u⋅∇n=Δn−∇⋅(n∇v),x∈Ω,t>0,vt+u⋅∇v=Δv−v+w,x∈Ω,t>0,wt+u⋅∇w=Δw−w+n,x∈Ω,t>0,ut+(u⋅∇)u=Δu+∇P+n∇ϕ,∇⋅u=0,x∈Ω,t>0, is considered in a bounded domain Ω⊂R2 with smooth boundary, where ϕ∈W2,∞(Ω) is a given function. Under the homogeneous Neumann boundary conditions for n, v, w and Dirichlet boundary condition for u, it is shown that when the initial mass satisfies ∫Ωn0<8π, this problem possesses a global generalized solution for all reasonably regular initial data, and after a certain waiting time, this solution can solve the system in the classical sense. This result extends previous studies on global solvability for the associated chemotaxis-Stokes system obtained on neglecting the nonlinear convective term in the fluid equation.