Journal of Statistical Physics

, Volume 157, Issue 1, pp 158–181 | Cite as

Uniqueness of Self-similar Solutions to Smoluchowski’s Coagulation Equations for Kernels that are Close to Constant

  • B. NiethammerEmail author
  • J. J. L. Velázquez


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
$$\begin{aligned} -\varepsilon \le K(x,y)-2 \le \varepsilon \Big ( \Big (\frac{x}{y}\Big )^{\alpha } + \Big (\frac{y}{x}\Big )^{\alpha }\Big ) \end{aligned}$$
for \(\alpha \in [0,1)\). We prove that self-similar solutions with given mass are unique if \(\varepsilon \) 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.


Smoluchowski’s coagulation equations Self-similar solutions Uniqueness 



The authors acknowledge support through the CRC 1060 The mathematics of emergent effects at the University of Bonn, that is funded through the German Science Foundation (DFG).


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of Applied MathematicsUniversity of BonnBonnGermany

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