Propagation of Sobolev regularity for a class of random kinetic models on the real line

A class of Kac-like kinetic equations on the real line is considered, with general smoothing transforms as collisional kernels. These equations have been introduced recently e.g. in the context of econophysics (Cordier et al 2005 J. Stat. Phys. 120 253-77) or as models for granular gases with a background heat bath (Carrillo et al 2009 Discrete Contin. Dyn. Syst. 24 59-81). We show that the stationary solutions to these equations are not smooth in general, and we characterize their (finite) Sobolev regularity in dependence of the properties of the collisional kernel. Moreover, we prove that any initial Sobolev regularity below a well-defined threshold is uniformly propagated in time by the transient weak solutions, implying their strong convergence to the steady state. The applied techniques differ from the classical ones developed for the Kac equation as the models at hand neither dissipate the entropy nor the Fisher information. Instead, the proof relies on direct estimates on the collisional operator.