TSALLIS-BGK MODEL NEAR A GLOBAL EQUILIBRIUM

Abstract

We study global-in-time well-posedness and asymptotic behavior of a classical solution to the generalized BGK model, which will be referred as the Tsallis--BGK model, whose relaxation distribution is the Tsallis q-Maxwellian distribution. The Tsallis distribution is a generalization of the classical Maxwellian distribution toward heavy-tailed and to light-tailed distributions depending on the choice of the nonextensivity index q. To establish well-posedness of the Tsallis-- BGK model, we linearize it near the global q-Maxwellian and use a nonlinear energy estimate to obtain local well-posedness, which can be extended to the global-in-time by using the coercivity of the linearization operator. The asymptotic behavior follows from the energy estimate. We also verify numerically that such linearization argument breaks down when q < 1 due to the discrepancy of the velocity support.