Stationary Kolmogorov solutions of the Smoluchowski aggregation equation with a source term

In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation with a source term for arbitrary homogeneous coagulation kernel. The resulting power-law mass distributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. They are valid for masses much larger than the characteristic mass of the source. We derive a "locality criterion," expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissible solution. Whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the Kolmogorov spectrum. As an example, we compute the exact stationary state for the family of kernels, K(zeta) ( m(1), m(2) )= ( m(1) m(2) )(zeta/2) which includes both gelling and nongelling cases, reproducing the known solution in the case zeta=0. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.