Propagation of Gevrey Regularity for Solutions of Landau Equations

There are many papers concerning the propagation of regularity for the solution of the Boltzmann equation (cf. [5, 6, 8, 9, 13] and references therein). In these works, it has been shown that the Sobolev or Lebesgue regularity satisfied by the initial datum is propagated along the time variable. The solutions having the Gevrey regularity for a finite time have been constructed in [15] in which the initial data has the same Gevrey regularity. Recently, the uniform propagation in all time of the Gevrey regularity has been proved in [4] in the case of Maxwellian molecules, which was based on the Wild expansion and the characterization of the Gevrey regularity by the Fourier transform. In this paper, we study the propagation of Gevrey regularity for the solution of Landau equation, which is the limit of the Boltzmann equation when the collisions become grazing, see [3] for more details. Also we know that the Landau equation can be regarded as a non-linear and non-local analog of the hypo-elliptic Fokker-Planck equation, and if we choose a suitable orthogonal basis, the Landau equation in the Maxwellian molecules case will become a non-linear Fokker-Planck equation (cf. [17]). Recently, a lot of progress on the Sobolev regularity has been made for the spatially homogeneous and inhomogeneous Landau equations, cf. [2, 6, 7, 16] and references therein. On the other hand, in the Gevrey class frame, the local Gevrey regularity for all variables t, x, v is obtained in [1] for some semi-linear Fokker-Planck equations. Let us consider the following Cauchy problem for the spatially homogeneous Landau equation,