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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.