Abstract
This chapter aims to provide a unified presentation of some recent studies (see [CGT99], [CCG00], [BGR05], [CCG05] and [GR05]) on the convergence to equilibrium of the solution of Boltzmann’s equation for Maxwellian pseudomolecules and, in particular, of the solution of Kac’s analog of Boltzmann’s equation. The main feature of these researches, with respect to the other ones, is the pursuit of the optimal rate of exponential convergence both in a weighted X-metric (see for example [GTW95]) and in the total variation metric for probability measures. For the definition of these metrics see the next section. Now, recall the following basic facts about the aforesaid equations.
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Gabetta, E. (2007). Results on optimal rate of convergence to equilibrium for spatially homogeneous Maxwellian gases. In: Cercignani, C., Gabetta, E. (eds) Transport Phenomena and Kinetic Theory. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4554-0_2
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