This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term -ν(-Δ) α/2 ρ (1 < α < 2). Firstly, the global existence of weak solutions is proved for the initial density ρ0 ∈ L01∩L d/α (60d) (d ≥ 2) with [norm of matrix]ρ0[norm of matrix] d/α < K, where K is a universal constant only depending on d, α, ν. Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in Lr for any 1 < r < ∞. Secondly, for the more general initial data ρ0 ∈ L01 ∩ L05(60d) (d = 2, 3), the local existence is obtained. Thirdly, for ρ0 ∈ L01 (60d; (1 + |x|)dx ∩ L∞(60d)( d ≥ 2) with [norm of matrix]ρ0[norm of matrix]d/α < 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 α-stable Lévy process Lα(t). Also, we prove the weak solution is L1 bounded uniformly in time. Lastly, we consider the N-particle interacting system with the Lévy process Lα(t) and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment ∫60d |x| γρ0dx for some 1 < γ < α is below a universal constant K γ and ν is also below a universal constant. Meanwhile, we prove the propagation of chaos as N → ∞ for the interacting particle system with a cut-off parameter ε ~ (ln N)-1/d, and show that the mean field limit equation is exactly the generalized KS equation.

where p > 1. 1 < p < 2 is called the fast p-Laplacian diffusion, while p > 2 is called the slow p-Laplacian diffusion. Especially, the p-Laplacian Keller-Segel model turns to the original model when p = 2.The Keller-Segel model was firstly presented in 1970 to describe the chemotaxis of cellular slime molds [13, 14]. The original model was considered in 2D,

The class of generating functions for completely monotone sequences (moments of finite positive measures on $[0,1]$) has an elegant characterization as the class of Pick functions analytic and positive on $(-\infty,1)$. We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on $[0,1]$, and we provide a simple analytic proof that for any real $p,r>0$, the Fuss-Catalan or Raney numbers $\frac{r}{pn+r}\binom{pn+r}{n}$, $n=0,1,\ldots$ are the moments of a probability distribution on some interval $[0,\tau]$ if and only if $p\ge1$ and $p\ge r$.

From a variational perspective, we derive a series of magnetization and quantum spin current systems coupled via an “s-d” potential term, including the Schrödinger--Landau--Lifshitz--Maxwell system, the Pauli--Landau--Lifshitz system, and the Schrödinger--Landau--Lifshitz system with successive simplifications. For the latter two systems, we propose using the time splitting spectral method for the quantum spin current and the Gauss--Seidel projection method for the magnetization. Accuracy of the time splitting spectral method applied to the Pauli equation is analyzed and verified by numerous examples. Moreover, behaviors of the Schrödinger--Landau--Lifshitz system in different “s-d” coupling regimes are explored numerically.

Abstract We investigate a system of N Brownian particles with the Coulomb interaction in any dimension \(d\ge 2\), and we assume that the initial data are independent and identically distributed with a common density \(\rho _0\) satisfying \(\int _{\mathbb {R}^{d}}\rho _0\ln \rho _0\,\hbox {d}x<\infty \) and \(\rho _0\in L^{\frac{2d}{d+2}} (\mathbb {R}^{d}) \cap L^1(\mathbb {R}^{d}, (1+|x|^2)\,\hbox {d}x)\). We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For \(d=2\), we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When \(N\rightarrow \infty \), the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson–Nernst–Planck equation of single component.