Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels

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• 作者：B. Niethammer¹ ; ^{niethammer@iam.uni-bonn.de" class="auth_mail" title="E-mail the corresponding author} ; S. Throm ; ^{throm@iam.uni-bonn.de" class="auth_mail" title="E-mail the corresponding author} ; J.J.L. Velá ; zquez² ; ^{velazquez@iam.uni-bonn.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Smoluchowski's coagulation equation ; Self-similar solution ; Singular kernel ; Fat tail ; Dual problem
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：September-October 2016
• 年：2016
• 卷：33
• 期：5
• 页码：1223-1257
• 全文大小：509 K

文摘

We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying C₁(x^−ay^b+x^by^−a)≤K(x,y)≤C₂(x^−ay^b+x^by^−a)$C_1(x^{-a}{y}^b+x^b{y}^{-a})\le K(x,y)\le C_2(x^{-a}{y}^b+x^b{y}^{-a})$ with a>0$a>0$ and b<1$b<1$. This covers especially the case of Smoluchowski's classical kernel K(x,y)=(x^1/3+y^1/3)(x^−1/3+y^−1/3)$K(x,y)=(x^{1/3}+y^{1/3})(x^{-1/3}+y^{-1/3})$.

For the proof of existence we take a self-similar solution h_ε$h_{\epsilon }$ for a regularized kernel K_ε$K_{\epsilon }$ and pass to the limit ε→0$\epsilon \to 0$ to obtain a solution for the original kernel K  . The main difficulty is to establish a uniform lower bound on h_ε$h_{\epsilon }$. The basic idea for this is to consider the time-dependent problem and to choose a special test function that solves the dual problem.