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

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.

### Duke Scholars

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*Kinetic and Related Models*,

*9*(4), 715–748. https://doi.org/10.3934/krm.2016013

*Kinetic and Related Models*9, no. 4 (January 1, 2016): 715–48. https://doi.org/10.3934/krm.2016013.

*Kinetic and Related Models*, vol. 9, no. 4, Jan. 2016, pp. 715–48.

*Scopus*, doi:10.3934/krm.2016013.

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Applied Mathematics