Well-posedness of Smoluchowski's coagulation equation for a class of homogeneous kernels

Abstract

The uniqueness and existence of measure-valued solutions to Smoluchowski's coagulation equation are considered for a class of homogeneous kernels. Denoting by λ ∈ ( - ∞ , 2 ] ⧹ { 0 } the degree of homogeneity of the coagulation kernel a , measure-valued solutions are shown to be unique under the sole assumption that the moment of order λ of the initial datum is finite. A similar result was already available for the kernels a ( x , y ) = 2 , x + y and xy, and is extended here to a much wider class of kernels by a different approach. The uniqueness result presented herein also seems to improve previous results for several explicit kernels. Furthermore, a comparison principle and a contraction property are obtained for the constant kernel.