Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term

Abstract

The coupled chemotaxis fluid system(⋆){nt=Δn−∇⋅(nS(x,n,c)⋅∇c)−u⋅∇n,(x,t)∈Ω×(0,T),ct=Δc−nc−u⋅∇c,(x,t)∈Ω×(0,T),ut=Δu−κ(u⋅∇)u+∇P+n∇ϕ,(x,t)∈Ω×(0,T),∇⋅u=0,(x,t)∈Ω×(0,T), is considered under the no-flux boundary conditions for n,c and the Dirichlet boundary condition for u on a bounded smooth domain Ω⊂RN (N=2,3), κ∈{0,1}. We assume that S(x,n,c) is a matrix-valued sensitivity under a mild assumption such that |S(x,n,c)|<S0(c0) with some non-decreasing function S0∈C2((0,∞)). It contrasts with the related scalar sensitivity case that (⋆) does not possess the natural gradient-like functional structure. Associated estimates based on the natural functional seem no longer available. In the present work, a global classical solution is constructed under a smallness assumption on ‖c0‖L∞(Ω) and moreover we obtain boundedness and large time convergence for the solution, meaning that small initial concentration of chemical forces stabilization.