This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term $-\nu(-\Delta)^{\frac{\alpha}{2}}\rho~(1<\alpha<2)$. Firstly, the global existence of weak solutions is proved for the initial density $\rho_0\in L^1\cap L^{\frac{d}{\alpha}}(\mathbb{R}^d)~(d\geq2)$ with $\|\rho_0\|_{\frac {d}{\alpha}} < K$, where $K$ is a universal constant only depending on $d,\alpha,\nu$. Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in $L^r$ for any $1< r<\infty$. Secondly, for the more general initial data $\rho_0\in L^1\cap L^2(\mathbb{R}^d)$$~(d=2,3)$, the local existence is obtained. Thirdly, for $\rho_0\in L^1\big(\mathbb{R}^d,(1+|x|)dx\big)\cap L^\infty(\mathbb{R}^d)(~d\geq2)$ with $\|\rho_0\|_{\frac{d}{\alpha}} < K$, we prove the uniqueness and stability of weak solutions under Wasserstein metric through the method of associating the KS equation with a self-consistent stochastic process driven by the rotationally invariant $\alpha$-stable Levy process $L_{\alpha}(t)$. Also, we prove the weak solution is $L^\infty$ bounded uniformly in time. Lastly, we consider the $N$-particle interacting system with the Levy process $L_{\alpha}(t)$ and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment $\int_{\mathbb{R}^d}|x|^\gamma\rho_0dx$ for some $1<\gamma<\alpha$ is below a universal constant $K_\gamma$ and $\nu$ is also below a universal constant. Meanwhile, we prove the propagation of chaos as $N\rightarrow\infty$ for the interacting particle system with a cut-off parameter $\varepsilon\sim(\ln N)^{-\frac{1}{d}}$, and show that the mean field limit equation is exactly the generalized KS equation.

中文翻译：

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带分数拉普拉斯算子的 Keller-Segel 方程的适定性和混沌传播理论

本文研究了具有非局部扩散项 $-\nu(-\Delta)^{\frac{\alpha}{2}}\rho~(1<\alpha<2)$ 的广义 Keller-Segel (KS) 系统. 首先证明了初始密度$\rho_0\in L^1\cap L^{\frac{d}{\alpha}}(\mathbb{R}^d)~(d\ geq2)$ 与 $\|\rho_0\|_{\frac {d}{\alpha}} < K$，其中 $K$ 是一个通用常数，仅取决于 $d,\alpha,\nu$。此外，质量守恒成立，弱解满足任何 $1< r<\infty$ 中 $L^r$ 中的一些超收缩和衰减估计。其次，对于更一般的初始数据$\rho_0\in L^1\cap L^2(\mathbb{R}^d)$$~(d=2,3)$，得到局部存在。第三，对于 $\rho_0\in L^1\big(\mathbb{R}^d,(1+|x|)dx\big)\cap L^\infty(\mathbb{R}^d)(~ d\geq2)$ 与 $\|\rho_0\|_{\frac{d}{\alpha}} < K$，我们通过将KS方程与由旋转不变$\alpha$-stable Levy过程$L_{\alpha}(t)$驱动的自洽随机过程相关联的方法，证明了Wasserstein度量下弱解的唯一性和稳定性. 此外，我们证明了弱解是 $L^\infty$ 在时间上均匀有界的。最后，我们考虑 $N$-粒子相互作用系统与 Levy 过程 $L_{\alpha}(t)$ 和牛顿势聚集，并证明粒子之间碰撞时间的期望低于通用常数，如果时刻 $ \int_{\mathbb{R}^d}|x|^\gamma\rho_0dx$ 对于某些 $1<\gamma<\alpha$ 低于通用常数 $K_\gamma$ 并且 $\nu$ 也低于通用常数不变。同时，