Chemotaxis(-fluid) systems with logarithmic sensitivity and slow consumption: global generalized solutions and eventual smoothness

Abstract

We consider the system \begin{align*} \begin{cases} n_t + u \cdot \nabla n = Δn - χ\nabla \cdot (\frac{n}{c} \nabla c), \\ c_t + u \cdot \nabla c = Δc - nf(c), \\ u_t + (u \cdot \nabla) u = Δu + \nabla P + n \nabla ϕ, \quad \nabla \cdot u = 0, \end{cases} \end{align*} in smooth bounded domains $\Omega \subset {\mathbb{R}}^N$, $N\in \mathbb{N}$, for given $f\ge 0$, $\varphi$ and complemented with initial and homogeneous Neumann--Neumann--Dirichlet boundary conditions, which models aerobic bacteria in a fluid drop. We assume $f(0)=0$ and $f^{\prime }(0)=0$, 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.

25 pages

Cited by in corpus (2)

• Very mild diffusion enhancement and singular sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier--Stokes system
• Uniform $L^p$ estimates for solutions to the inhomogeneous 2D Navier-Stokes equations and application to a chemotaxis-fluid system with local sensing