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

We consider the system$ \begin{align*} \begin{cases} n_t + u \cdot \nabla n = \Delta - \chi (\frac{n}{c} c), \\ c_t c nf(c), u_t (u \nabla) P \phi, \quad 0, \end{cases} \end{align*} $in smooth bounded domains $ \Omega \subset \mathbb R^N $, N \in for given f \ge 0 \phi and complemented with initial homogeneous Neumann–Neumann–Dirichlet boundary conditions, which models aerobic bacteria in a fluid drop. assume f(0) f'(0) that is, decays slower than linearly near construct global generalized solutions provided either 2 or > no is present.If additionally we next prove this solution eventually becomes stabilizes large-time limit. emphasize these results require smallness neither of nor data.