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.

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Maxwellian分子的非线性Landau方程的解析平滑效果

我们考虑了麦克斯韦分子的非线性朗道方程的柯西问题，在摄动框架下达到了全局平衡。我们证明，如果$ H ^ r_x（L ^ 2_v），初始扰动的r> 3/2 $范数足够小，那么非线性Landau方程的Cauchy问题就可以得到一个唯一的全局解，该解对于两个位置$ x $和速度$ v $任何时间$ t> 0 $的变量。这是空间非均匀非线性动力学方程解析平滑效果的第一个结果。这里使用的方法是微局部分析和能量估计。关键是采用与动力传递算子相关的指数型时间积分权重。