Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules
Yoshinori Morimoto, Chao-Jiang Xu, ,Université de Rouen-Normandie, CNRS UMR 6085, Laboratoire de Mathématiques, 76801 Saint-Etienne du Rouvray, France
Abstract
We consider the Cauchy problem of the nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if $ H^r_x(L^2_v), r >3/2 $ norm of the initial perturbation is small enough, then the Cauchy problem of the nonlinear Landau equation admits a unique global solution which becomes analytic with respect to both position $ x $ and velocity $ v $ variables for any time $ t>0 $. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation. The method used here is microlocal analysis and energy estimates. The key point is adopting a time integral weight of exponential type associated with the kinetic transport operator.