论文标题
在亚临界和关键案例中，凯勒 - 塞格模型的经验度量的弱收敛性较弱
Weak convergence of the empirical measure for the Keller-Segel model in both subcritical and critical cases
论文作者
论文摘要
我们在一般初始条件下，显示了在亚临界和关键情况下，在亚临界和关键情况下，弱的粒子系统的经验度量量的弱收敛性。该粒子系统由$ N $ Planar Brownian运动组成，该动作通过库仑的吸引力相互作用，这是非常单数的。在亚临界情况下，Bresch-Jabin-Wang \ cite {bjw}以两种简化的价格建立了更强的结果：整个空间$ \ rr^2 $由圆环代替，并假定初始条件是规则的。在亚临界情况下，我们的证明相当简单：我们使用{\ it两个粒子}时刻参数，该参数表明粒子在有限的时间内不汇总，在粒子数量中均匀地汇总。关键案例需要更多的工作。
We show the weak convergence, up to extraction of a subsequence, of the empirical measure for the Keller-Segel system of particles in both subcritical and critical cases, for general initial conditions. This particle system consists of $N$ planar Brownian motions interacting through a Coulombian attractive force, which is quite singular. In the subcritical case, a stronger result has been established by Bresch-Jabin-Wang \cite{bjw} at the price of two simplifications: the whole space $\rr^2$ is replaced by a torus and the initial condition is assumed to be regular. In the subcritical case, our proof is fairly straightforward: we use a {\it two particles} moment argument, which shows that particles do not aggregate in finite time, uniformly in the number of particles. The critical case requires more work.
