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

2014/11/06 by Barbara Niethammer, Niethammer, Barbara, Sebastian Throm +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1411.1602

openalex publication_date 2014/11/06 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We show the existence of self-similar solutions with fat tails for\nSmoluchowski's coagulation equation for homogeneous kernels satisfying C₁\n\(x^-ay^b+x^by^-a\)\≤ K\(x,y\)\≤\nC₂\(x^-ay^b+x^by^-a\) with a>0 and b<1. This covers\nespecially the case of Smoluchowski's classical kernel K(x,y)=(x^1/3 +\ny^1/3)(x^-1/3 + y^-1/3).\n For the proof of existence we first consider some regularized kernel\nK_\ε for which we construct a sequence of solutions h_\ε.\nIn a second step we pass to the limit \ε\→ 0 to obtain a solution for\nthe original kernel K. The main difficulty is to establish a uniform lower\nbound on h_\ε. The basic idea for this is to consider the\ntime-dependent problem and choosing a special test function that solves the\ndual problem.\n

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