Math @ Duke

Publications [#318454] of JianGuo Liu
Papers Published
 Huang, H; Liu, JG, Wellposedness for the kellersegel equation with fractional laplacian and the theory of propagation of chaos,
Kinetic and Related Models, vol. 9 no. 4
(January, 2016),
pp. 715748 [doi]
(last updated on 2019/06/16)
Abstract: © American Institute of Mathematical Sciences. This paper investigates the generalized KellerSegel (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 hypercontractive 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 selfconsistent stochastic process driven by the rotationally invariant αstable Lévy process Lα(t). Also, we prove the weak solution is L1 bounded uniformly in time. Lastly, we consider the Nparticle 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 cutoff parameter ε ~ (ln N)1/d, and show that the mean field limit equation is exactly the generalized KS equation.


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