CHEMOTAXIS(-FLUID) SYSTEMS WITH LOGARITHMIC SENSITIVITY AND SLOW CONSUMPTION: GLOBAL GENERALIZED SOLUTIONS AND EVENTUAL SMOOTHNESS

We consider the system ⎧ ⎨ ⎪ ⎪⎩ nt +u center dot Vn= Delta n -chi V center dot (ncVc), ct + u center dot Vc = Delta c - nf (c), ut + (u center dot V)u = Delta u + VP + nV0, V center dot u = 0, in smooth bounded domains omega C RN, N E N, for given f >= 0, 0 and comple-mented with initial and homogeneous Neumann-Neumann-Dirichlet boundary conditions, which models aerobic bacteria in a fluid drop. We assume f (0) = 0 and f'(0) = 0, that is, that f decays slower than linearly near 0, and construct global generalized solutions provided that either N = 2 or N > 2 and no fluid is present. If additionally N = 2, we next prove that this solution eventually becomes smooth and stabilizes in the large-time limit. We emphasize that these results require smallness neither of chi nor of the initial data.