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

Hui  Huang; Jian-Guo Liu

doi:10.3934/krm.2016013

Article Contents

2016, Volume 9, Issue 4: 715-748. Doi: 10.3934/krm.2016013

This issue Previous Article A degenerate $p$-Laplacian Keller-Segel model Next Article Chaotic distributions for relativistic particles

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

Hui Huang^1, and
Jian-Guo Liu^2,

1.
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084
2.
Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708

Received: July 2015

Revised: February 2016

Published: December 2016

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Abstract

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 Lévy 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 Lévy 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.

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Mathematics Subject Classification: Primary: 65M75, 35K55; Secondary: 60J70.

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