Analytic Regularity Of Solutions To Spatially Homogeneous Landau Equation

In this paper,we main study the spatially homogeneous Landau equation which is not dependent on the location variable x but only rely on the velocity variable v.Firstly,we describe,under the condition of the Maxwellian molecule case which corresponds to ?=0,the ultra-analytic of solutions of Cauchy problem for the spatially homogeneous Landau equation,i.e.,when f0 ?L2(R)d ? L21(Rd),0<T ? +?,if f(t,x)>0,f?L?([0,T];L2(R)d ? L21((Rd))is a weak solution of the Cauchy problem for the spatially homogeneous Landau equation,then for any 0<t<T,we have f(t,·)? A1/2(Rd),and moreover,for any 0<T0?<T and 0<t<T0,there exists c0>0 such that ||e-c0t?vf(t,·)||L2(Rd)?e2/d)t ||f0||L2(Rd).Secondly,we give a estimate under the hard potential case which corresponds to??(0,1],i.e.,when f(t,v)is any solution of the Cauchy problem for the spatially homogeneous Landau equation,for all t in the interval[0,T]with T being an arbitrary nonnegative constant,there exist a constant A such that the following estimatesup t|?||(?)?f(t,v)||L2+(?0T t2|?|||?v(?)?f(t)||L2?2dt)1/2?A|?|+1[(|?|-2)!]t?[0,T]holds for any multi-indices ??N3.At last,according to the explicit dependence on the time of the estimate,we obtaian the analytic smoothness effect of solutions of Cauchy problem for the Landau equation.So far,we prove the analytic regularity of solutions of Cauchy problem for the spatially homogeneous Landau equation.