Gevrey Regularity for the Solution of the Spatially Homogeneous Landau Equation

This is a non-linear diffusion equation, and the coefficients āi j, c̄ depend on the solution f . Here we are mainly concerned with the Gevrey class regularity for the solution of the Landau equation. This equation is obtained as a limit of the Boltzmann equation when the collisions become grazing (see [8] and references therein). Recently, a lot of progress has been made on the study of the Sobolev regularizing property, cf. [6, 11, 13, 18, 19] and references therein, which shows that in some sense the Landau equation can be regarded as a non-linear and non-local analog of the hypo-elliptic Fokker-Planck equation. That means the weak solution, which constructed under rather weak hypothesis on the initial datum, will become smooth or, even more, rapidly decreasing in v at infinity. This behavior is quite similar to that of the spatially homogeneous Boltzmann equation without cut-off (see [2, 10] for more details).