# Well-posedness for the keller-segel equation with fractional laplacian and the theory of propagation of chaos

Published

Journal Article

© American Institute of Mathematical Sciences. 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 ∈ L1∩L d/α (ℝd) (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 ∈ L1 ∩ L2(ℝd) (d = 2, 3), the local existence is obtained. Thirdly, for ρ0 ∈ L1 (ℝd; (1 + |x|)dx ∩ L∞(ℝd)( 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 ∫ℝd |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.

### Full Text

### Duke Authors

### Cited Authors

- Huang, H; Liu, JG

### Published Date

- January 1, 2016

### Published In

### Volume / Issue

- 9 / 4

### Start / End Page

- 715 - 748

### Electronic International Standard Serial Number (EISSN)

- 1937-5077

### International Standard Serial Number (ISSN)

- 1937-5093

### Digital Object Identifier (DOI)

- 10.3934/krm.2016013

### Citation Source

- Scopus