Exponential convergence to equilibrium of solutions of the Kac equation and homogeneous Boltzmann equation for Maxwellian without angular cut-off

Article type: Research Article

Authors: Meng, Fei^{a; *} | Yang, Xiao-Ping^b

Affiliations: [a] School of Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, P.R. China | [b] Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: For the Kac equation and homogeneous Boltzmann equation of Maxwellian without Grad’s angular cut-off, we prove an exponential convergence towards the equilibrium as t→∞ in a weak norm which is equivalent to the weak convergence of measures, extending results of Gabetta, Toscani and Wennberg (J. Stat. Phys. 81 (1995), 901–934) and Carlen, Gabetta and Toscani (Commun. Math. Phys. 199 (1999), 521–546) from the cut-off case to the non-cut-off case. We give quantitative estimates of the convergence rate, which are governed by the spectral gap of the linearized collision operator. We then prove a uniform bound in time on Sobolev norms of the solutions. The results are then combined with some interpolation inequalities, to obtain the rate of the exponential convergence in the strong L1 norm, as well as various Sobolev norms.

Keywords: Kac equation, Boltzmann equation, non-cut-off, Maxwellian, Fourier transform

DOI: 10.3233/ASY-171407

Journal: Asymptotic Analysis, vol. 101, no. 4, pp. 251-271, 2017