Global solvability in a two-dimensional singular chemotaxis-fluid system with indirect signal production or consumption

Abstract

This paper deals with the singular chemotaxis-fluid system with indirect signal production or consumption nt+u⋅∇n=Δn−χ∇⋅(nv∇v); vt+u⋅∇v=Δv+g(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⊂R2 with no-flux/no-flux/no-flux/Dirichlet boundary conditions, where Φ∈W2,∞(Ω). In the cases signal production (i.e., g(v,w)=−v+w) and consumption (i.e., g(v,w)=−vw), it is proved that for any appropriately regular initial data, the associated initial-boundary value problem possesses a global classical solution for any χ>0. It is worth mentioning that the global existence of classical solution for χ∈(0,1) or generalized solutions for arbitrary χ>0 was obtained in the previously known results on the corresponding system with direct signal production, and that only the global generalized solutions for arbitrary χ>0 were established in a precedent concerning the corresponding system with direct signal consumption. In comparison to these results for the case of direct signal production or consumption, our results rigorously confirm that both the indirect signal production and consumption mechanisms authentically contribute to the global solvability of the two-dimensional singular chemotaxis-fluid system.