Global classical solutions of Keller-Segel-(Navier)-Stokes system with nonlinear motility functions

Abstract

This paper establishes the global existence of solutions to the following Keller-Segel-(Navier)-Stokes system with nonlinear motility functions(⁎){nt+u⋅∇n=∇⋅(γ(c)∇n−nφ(c)∇c),x∈Ω,t>0,ct+u⋅∇c=dΔc+n−c,x∈Ω,t>0,ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 in a bounded domain Ω⊂RN(2≤N≤3) with smooth boundary, where the density-dependent motility functions γ(c) and φ(c) denote the diffusive and chemotactic coefficients, respectively. The major technical difficulty in the analysis is the possible degeneracy of diffusion. Assume that N=2, κ≠0 or N=3,κ=0. Then for all reasonably regular initial data, an associated no-flux type initial-boundary value problem (⁎) admits a global classical solution when γ(c)>0 and φ(c)>0 are smooth on [0,∞) and satisfyinfc≥0⁡dγ(c)cφ(c)(cφ(c)+d−γ(c))+>N2.