Uniqueness of self-similar solutions to Smoluchowski's coagulation equations for kernels that are close to constant

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 \[ -\eps \leq K(x,y)-2 \leq \eps \Big(\Big(\frac{x}{y}\Big)^α + \Big(\frac{y}{x}\Big)^α\Big) \] for $\alpha \in [0,1)$. 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.

References in corpus (2)

• Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels
• Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients

Cited by in corpus (2)

• Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation with inverse power law kernels
• Precise tail behaviour of self-similar profiles with infinite mass for Smoluchowski's coagulation equation