Measure solutions for the Smoluchowski coagulation-diffusion equation

Abstract: A notion of measure solution is formulated for a coagulation-diffusion equation, which is the natural counterpart of Smoluchowski's coagulation equation in a spatially inhomogeneous setting. Some general properties of such solutions are established. Sufficient conditions are identified on the diffusivity, coagulation rates and initial data for existence, uniqueness and mass conservation of solutions. These conditions impose no form of monotonicity on the coagulation kernel, which may depend on complex characteristics of the particles. They also allow singular behaviour in both diffusivity and coagulation rates for small particles. The general results apply to the Einstein-Smoluchowski model for colloidal particles suspended in a fluid.