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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
Article

Abstract

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.

Keywords

Smoluchowski’s coagulation equations Self-similar solutions Uniqueness 

Notes

Acknowledgments

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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of Applied MathematicsUniversity of BonnBonnGermany

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