Analytic smoothing effect for the nonlinear Landau equation for the nonlinear Landau

We consider the Cauchy problem of the full nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if the initial perturbation is small enough in Sobolev space, then the Cauchy problem of the nonlinear Landau equation admits a unique solution which becomes analytic with respect to both position and velocity variables for any positive time. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation.