Asymptotics of self-similar solutions to coagulation equations with product kernel

Abstract

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(\xi ,\eta )=(\xi \eta {)}^{\lambda }$ with $\lambda \in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2\lambda }g(x)$ is bounded above and below as $x\to 0$. In this paper we describe in detail via formal asymptotics the qualitative behavior of a suitably rescaled function $h(x)=h_{\lambda }{x}^{-1+2\lambda }g(x)$ in the limit $\lambda \to 0$. It turns out that $h\sim 1+C{x}^{\lambda /2}\cos (\sqrt{\lambda }\log x)$ as $x\to 0$. As $x$ becomes larger $h$ develops peaks of height $1/\lambda$ that are separated by large regions where $h$ is small. Finally, $h$ converges to zero exponentially fast as $x\to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.

Cited by in corpus (3)

• Weak compactness techniques and coagulation equations
• Instabilities and oscillations in coagulation equations with kernels of homogeneity one
• Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation with inverse power law kernels