Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules

Yoshinori Morimoto; Chao-Jiang Xu

doi:10.3934/krm.2020033

Article Contents

2020, Volume 13, Issue 5: 951-978. Doi: 10.3934/krm.2020033

This issue Previous Article On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states Next Article On the generic complete synchronization of the discrete Kuramoto model

Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules

Yoshinori Morimoto^1, , and
Chao-Jiang Xu^2,3,

1.
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan
2.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, P. R. China
3.
Université de Rouen-Normandie, CNRS UMR 6085, Laboratoire de Mathématiques, 76801 Saint-Etienne du Rouvray, France

^* Corresponding author: morimoto.yoshinori.74r@st.kyoto-u.ac.jp
^* Corresponding author: morimoto.yoshinori.74r@st.kyoto-u.ac.jp

Received: November 30, 2019

Revised: February 29, 2020

Published: August 2020

The research of the first author is supported by JSPS Kakenhi Grant No.17K05318. The research of the second author is supported partially by "The Fundamental Research Funds for Central Universities"

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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.

Keywords:

Landau equation,
Maxwellian molecules,
analytic smoothing effect,
microlocal analysis,
time integral weight of exponential type.

Mathematics Subject Classification: Primary: 35B65; Secondary: 35Q82, 35S05.

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