Asymptotics of Self-similar Solutions to Coagulation Equations with Product Kernel

Journal article

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel K(ξ, η) = (ξη) λ with λ ∈ (0, 1/2). It is known that such self-similar solutions g(x) satisfy that x -1+2λg(x) is bounded above and below as x → 0. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function h(x) = h λx -1+2λg(x) in the limit λ → 0. It turns out that, as x → 0. As x becomes larger h develops peaks of height 1/λ that are separated by large regions where h is small. Finally, h converges to zero exponentially fast as x → ∞. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE. © 2011 Springer Science+Business Media, LLC.

Authors: McLeod, J; Niethammer, B; Velazquez, J

• Publication status: Published
• Journal: JOURNAL OF STATISTICAL PHYSICS
• Volume: 144
• Issue: 1
• Pages: 76-100
• Publication date: 2011-07-01
• DOI: 10.1007/s10955-011-0239-2
• EISSN: 1572-9613
• ISSN: 0022-4715
• Language: English
• Keywords: Coagulation equations; Product kernel; Self-similar solutions