Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with indirect signal production

Abstract

This paper deals with the Keller-Segel-Navier-Stokes system with indirect signal production nt+u⋅∇n=Δnm−∇⋅(nS(x,n,v,w)⋅∇v); vt+u⋅∇v=Δv−v+w; wt+u⋅∇w=Δw−w+n; ut+(u⋅∇)u=Δu−∇P+n∇ϕ; ∇⋅u=0, x∈Ω, t>0 in a bounded and smooth domain Ω⊂R3 with no-flux/no-flux/no-flux/Dirichlet boundary conditions, where ϕ∈W1,∞(Ω), m>0, and the tensor-valued sensitivity S:Ω¯×[0,∞)2→R3×3 is given smooth function such that |S(x,n,v,w)|≤CS(1+n)−α for all (x,n,v,w)∈Ω¯×[0,∞)3 with some CS>0 and α≥0. It is shown that under the conditions m≥13, α≥0 and m+α≥43 and proper regularity assumptions on the initial data, the associated initial-boundary value problem possesses at least one global weak solution. It is worth mentioning that in the previously known results concerning the corresponding three-dimensional Keller-Segel-Navier-Stokes system with direct signal production, the global weak or very weak solution was obtained under some more restrictive conditions on m and α. Our result rigorously confirms that the indirect signal production mechanism authentically contributes to the global solvability of the three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion and tensor-valued sensitivity.