论文标题
在时间上全球的Smoluchowski方程的自相似解决方案和乘法内核
On global in time self-similar solutions of Smoluchowski equation with multiplicative kernel
论文作者
论文摘要
我们研究了Smoluchowski凝结方程的相似性解决方案（SS），其乘积内核$ k（x，y）=（xy）^{s} $ for $ s <\ frac {1} {2} {2} $。当$ s <0 $％时，SS由三个具有不同渐近行为的区域组成。适当的匹配产生了由伽马分布尾部组成的溶液的全局描述，该溶液是由对数正态分布描述的中间区域以及溶液在原点附近零衰变的非常快速衰减的区域。当$ s \ in \ left（0，\ frac {1} {2} \ right）$时，ss在原点上是无限的。它还提出了三个区域：一个伽马分布尾，一个类似功率（或帕累托分布）衰减的中间区域以及靠近发生奇异性的原点的区域。最后，对Smoluchowski方程的完整数值模拟有助于验证我们的理论结果，并显示解决方案与自相似性制度的收敛性。
We study the similarity solutions (SS) of Smoluchowski coagulation equation with multiplicative kernel $K(x,y)=(xy)^{s}$ for $s<\frac{1}{2}$. When $s<0$% , the SS consists of three regions with distinct asymptotic behaviours. The appropriate matching yields a global description of the solution consisting of a Gamma distribution tail, an intermediate region described by a lognormal distribution and a region of very fast decay of the solutions to zero near the origin. When $s\in \left( 0,\frac{1}{2}\right) $, the SS is unbounded at the origin. It also presents three regions: a Gamma distribution tail, an intermediate region of power-like (or Pareto distribution) decay and the region close to the origin where a singularity occurs. Finally, full numerical simulations of Smoluchowski equation serve to verify our theoretical results and show the convergence of solutions to the selfsimilar regime.
