Global existence in a two-dimensional chemotaxis-(Navier)-Stokes system with sub-logarithmic sensitivity

Abstract

This paper considers the following chemotaxis-(Navier)-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(c)∇c),ct+u⋅∇c=Δc−f(n)c,ut+κ(u⋅∇)u=Δu+∇P+n∇Φ in a smooth bounded domain Ω⊂R2 with no-flux/no-flux/Dirichlet boundary conditions, whereχ(c)=1cαandf(n)=nγ with α∈(0,1) and γ∈(0,1). We proved that if ‖c0‖L∞(Ω)<1, then for any κ∈R the problem possesses a global classical solution.