A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant

Abstract

In this article we correct the proof of a uniqueness result for 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 \begin{equation*} -\varepsilon \leq K(x,y)-2 \leq \varepsilon \left( \Big(\frac{x}{y}\Big)^α + \Big(\frac{y}{x}\Big)^α\right) \end{equation*} for $\alpha \in [0,\frac{1}{2})$. Assuming in addition that $K$ has an analytic extension to $\mathbb{C}\setminus (-\infty ,0]$ and prescribing the precise asymptotic behaviour of $K$ at the origin, we prove that self-similar solutions with given mass are unique if $\epsilon$ is sufficiently small.

v2: correction of the statement and proof of Proposition 2.2, revision of several proofs affected by this change, a few typos have been corrected v3: correction of the proofs of Proposition 10.1 and Theorem 1.3 for α=0; several minor corrections, contact details updated

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• Instabilities and oscillations in coagulation equations with kernels of homogeneity one