We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
WELL-POSEDNESS FOR THE KELLER-SEGEL EQUATION WITH FRACTIONAL LAPLACIAN AND THE THEORY OF PROPAGATION OF CHAOS.
- Authors
HUI HUANG; JIAN-GUO LIU
- Abstract
This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term −ν(−Δ)α/2ρ (1<α<2). Firstly, the global existence of weak solutions is proved for the initial density ρ0∈L1∩Ldα(Rd) (d≥2) with ∥ρ0∥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 ∥ρ0∥dα<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 L∞ 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 Kγ 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∼(lnN)−1d, and show that the mean field limit equation is exactly the generalized KS equation.
- Subjects
LAPLACIAN operator; CHAOS theory; HEAT equation; UNIQUENESS (Mathematics); STOCHASTIC processes; LEVY processes; MEAN field theory
- Publication
Kinetic & Related Models, 2016, Vol 9, Issue 4, p715
- ISSN
1937-5093
- Publication type
Article
- DOI
10.3934/krm.2016013