论文标题
具有对数灵敏度和缓慢消耗的趋化（-Fluid）系统：全球通用解决方案和最终平滑度
Chemotaxis(-fluid) systems with logarithmic sensitivity and slow consumption: global generalized solutions and eventual smoothness
论文作者
论文摘要
我们考虑系统\ begin {align*} \ begin {case} n_t + u \ cdot \ nabla n =Δn -χ\ nabla \ cdot（\ frac {n} {c} {c} {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 $Ω\subset \mathbb R^N$, $N \in \mathbb N$, for given $f \ge 0$, $ϕ$ and complemented with initial and homogeneous Neumann--Neumann--Dirichlet boundary conditions, which models aerobic bacteria in a fluid drop.我们假设$ f（0）= 0 $和$ f'（0）= 0 $，也就是说，$ f $衰减比线性接近$ 0 $慢，并且只要$ n = 2 $ n = 2 $或$ n> 2 $且没有液体，则构建全球通用解决方案。 如果另外为$ n = 2 $，我们接下来证明该解决方案最终会变得平滑并稳定在较大的限制中。我们强调，这些结果不需要小$χ$，也不需要初始数据。
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 $Ω\subset \mathbb R^N$, $N \in \mathbb N$, for given $f \ge 0$, $ϕ$ 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 $f'(0) = 0$, 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 $χ$ nor of the initial data.
