Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous non linear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.Comment: 22 pages, to appear in JD

Hypoelliptic operators of principal type with infinite degeneracy

Let P be a classical pseudodiccerential operator of order m. We assume P is of principal type, that is, the Hamilton vector field Hp of the principal symbol p of P is not parallel to the radial direction where the principal symbol ..