临界Keller-Segel方程崩溃的对数标度
Logarithmic scaling of the collapse in the critical Keller-Segel equation
摘要：
简化的Keller-Segel方程（RKSE）是一个描述细菌聚集和布朗粒子自引力气体坍缩的抛物-椭圆型偏微分方程组。我们认为RKSE在两个维度，其中解决方案有一个临界崩溃（爆破），如果细菌的总数超过一个临界值。研究了RKSE方程在爆破点附近的自相似解。在坍缩时刻$t=t_c$附近，临界坍缩由对数修正的$L\propto (t_c-t)^{1/2}$标度律表征，其中$L$是坍缩解的空间宽度.我们开发了一个渐近微扰理论，这些修改，并表明，所得的缩放同意与数值模拟。理论和模拟的定量比较需要考虑扰动级数的几个项。
英文摘要：
A reduced Keller-Segel equation (RKSE) is a parabolic-elliptic system of partial differential equations which describes bacterial aggregation and the collapse of a self-gravitating gas of brownian particles. We consider RKSE in two dimensions, where solution has a critical collapse (blow-up) if the total number of bacteria exceeds a critical value. We study the self-similar solutions of RKSE near the blow-up point. Near the collapse time, $t=t_c$, the critical collapse is characterized by the $L\propto (t_c-t)^{1/2}$ scaling law with logarithmic modification, where $L$ is the spatial width of collapsing solution. We develop an asymptotic perturbation theory for these modifications and show that the resulting scaling agrees well with numerical simulations. The quantitative comparison of the theory and simulations requires to take into account several terms of the perturbation series.
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