B. Niethammer | Searcharxiv

arXiv2013

Uniqueness of self-similar solutions to Smoluchowski's coagulation equations for kernels that are close to constant

We consider self-similar solutions to Smoluchowski's coagulation equation for kernels $K=K(x,y)$ that are homogeneous of degree zero and close to constant in the sense that \[ -\eps \leq K(x,y)-2 \leq \eps \Big(\Big(\frac{x}{y}\Big)^α + \Big(\frac{y}{x}\Big)^α\Big) \] for $α\in [0,1)$. We prove that self-similar solutions with given mass are unique if $\eps$ is sufficiently small which is the first such uniqueness result for kernels that are not solvable. Our proof relies on a contraction argument in a norm that measures the distance of solutions with respect to the weak topology of measures.

math.AP↗

arXiv2011

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

We consider mass-conserving self-similar solutions for Smoluchowski's coagulation equation with kernel $K(ξ,η)= (ξη)^λ$ with $λ\in (0,1/2)$. It is known that such self-similar solutions $g(x)$ satisfy that $x^{-1+2λ} g(x)$ 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 $h(x)=h_λ x^{-1+2λ} g(x)$ in the limit $λ\to 0$. It turns out that $h \sim 1+ C x^{λ/2} \cos(\sqrtλ \log x)$ as $x \to 0$. 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 $x \to \infty$. Our analysis is based on different approximations of a nonlocal operator, that reduces the original equation in certain regimes to a system of ODE.

math.AP↗