Semi-inhomogeneous solutions of the Kac model of Boltzmann equations

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Abstract
Constructs semi-inhomogeneous solutions of the Kac model, for which the spatial dependence is only present in the odd-velocity part of the distribution. At the macroscopic level, the local density, the local energy, and the associated current components are determined and discussed, while at the microscopic level there exist two different classes of distributions; as an illustration the author gives two solutions explicitly. For one class, the distributions relax towards a Maxwellian equilibrium solution and they correspond either to a contraction or an expansion, while for the other they go to zero when the time increases to infinity. The space variable appears linearly and, in order to maintain the positivity, stays inside a finite interval. Assuming that the distributions are zero outside such intervals, a physical interpretation can be obtained for the whole space axis, provided appropriate elastic walls, sinks and sources are introduced. These boundary conditions seem more natural for the class of distributions with Maxwellian relaxation.