Existence of weak and regular solutions for Keller–Segel system with degradation coupled to fluid equations

Abstract

We establish the global well-posedness for the following chemotaxis-fluid system{∂tn+u⋅∇n=Δn−∇⋅(n∇c)−μnq,∂tc+u⋅∇c=Δc−c+n,∂tu+κ(u⋅∇)u+∇P=Δu−n∇ϕ,∇⋅u=0, in Rd, d=2,3, where μ>0, q>2−1d and κ∈{0,1}. For either q≥2, (κ,d)=(1,2) or q>2, (κ,d)=(0,3), we prove the global existence of regular solutions. In case that q>2−1d and κ=0, very weak solutions are constructed as well.