Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos

Hui Huang Tsinghua University Jian-Guo Liu Duke University

Analysis of PDEs mathscidoc:1702.03023

Kinetic and Related Models, 9, (4), 715 - 748, 2016.12
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.
Newtonian potential aggregation, log-Lipschitz continuity, uniqueness of the weak solutions, stability in Wasserstein metric, rotationally invariant α-stable Lévy process, interacting particle system, collision between particles.
[ Download ] [ 2017-02-07 21:34:44 uploaded by jianguo ] [ 881 downloads ] [ 0 comments ]
@inproceedings{hui2016well-posedness,
  title={Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos},
  author={Hui Huang, and Jian-Guo Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207213444197834266},
  booktitle={Kinetic and Related Models},
  volume={9},
  number={4},
  pages={715 - 748},
  year={2016},
}
Hui Huang, and Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. 2016. Vol. 9. In Kinetic and Related Models. pp.715 - 748. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207213444197834266.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved