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

摘要

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 -ε ≤ K(x, y) - 2 ≤ ε((x/y)~α+(y/x)~α) for α ∈ [0, 1). We prove that self-similar solutions with given mass are unique if ε 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.