Chemotaxis-consumption system with Robin boundary conditions coupled to the (Navier-)Stokes equations

Abstract

In this paper, we consider the chemotaxis-consumption system on a bounded smooth domain Omega subset of & Ropf;(n), n = 2, 3, with fluid coupling {p(t)+u center dot del p-del center dot(D(p)del p)=del center dot(pS(x,p,c)center dot del c) u.del c-Delta c=pc ut+k(u center dot del)u-Delta u+del pi=p del Phi, del center dot u=0. subject to the boundary conditions v center dot (D(p)del p + pS(x, p, c)del c)|partial derivative Omega = 0, (v center dot del c + c)partial derivative|Omega =gamma , and u|partial derivative Omega= 0. When (n, k) = (2, 1), we establish the global existence and uniform boundedness of classical solutions for all suitably regular initial data, under general structural conditions on the tensor-valued sensitivity S and a strictly positive lower bound on the diffusivity D. In case (n, k) = (3, 0), we show that the same result holds provided that D meets a certain diffusion enhancement condition depending on gamma. Moreover, we construct finite-time blow-up solutions for the radially symmetric, fluid-free system when n = 2, 3, D( ) less than or similar to (1 +xi )(m-1) with 0 < m < 2n and S equivalent to (n & times;n). We prove that, for any prescribed initial mass, blow-up occurs when gamma is sufficiently large.