A uniqueness result for self-similar profiles to Smoluchowski's coagulation equation revisited
In this note, the authors correct the proof of a previous result [the first and the third author, J. Stat. Phys. 157, No. 1, 158--181 (2014; Zbl 1302.82136)] on the uniqueness question of self-similar profiles of the Smoluchowski's coagulation equation. This is an integro-differential evolution equation, derived from a mean-field model, that describes a system of coagulating particles via their density function, where the variables of this function are the time and the size of the particles. The gap in the previous uniqueness proof was a wrongly used contraction argument for a very weak norm. They explain why this argument is wrong and why new ideas are needed. Here, the authors point out the need of some additional assumptions on the kernel in the integral operator, in order to obtain the uniqueness result for self-similar solutions with given mass (the result which was stated in the above-mentioned reference). Actually, this note only explains the new strategy how to fill the gap in the previous paper. The complete details of the proof are contained in a third paper [the authors, ``A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant'', Preprint, \url{arXiv:1510.03361}].
scientific article; zbMATH DE number 6629231
scientific article