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 Levy 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 Levy 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.
Highlights
In this paper, we study the existence, uniqueness, stability and regularity for the following generalized Keller-Segel model with nonlocal diffusion term −ν (−∆) α 2 ρ (1 < α 2) in dimension d ∂tρ = −ν(−∆) −
We prove the weak solution is L∞ bounded uniformly in time
We study the existence, uniqueness, stability and regularity for the following generalized Keller-Segel model with nonlocal diffusion term
Summary
We study the existence, uniqueness, stability and regularity for the following generalized Keller-Segel model with nonlocal diffusion term. The fractional KS system was first studied by Escudero in [20], where the author prove that this model has blowing-up solutions for large initial conditions in dimensions d ≥ 2 He obtains the global existence with the initial data ρ0 ∈ L1 ∩ H1(R) in dimension d = 1, which is a subcritical case. In [7], authors study the conditions for local and global in time existence of positive weak solutions in dimensions d = 2, 3. In [8], authors deal with the socalled mild solutions based on applications of the linear analytic semigroup theory to quasi-linear evolution equations They prove the existence of local in time mild solutions and global mild solutions under the small initial data ρ0 q < ε in dimensions d ≥ 2.
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