Gelfand-Shilov And Gevrey Smoothing Effect Of The Cauchy Problem For Fokker-Planck Equation

In this paper,the Cauchy problem for Fokker-Planck equation with outer force is investigated,mainly focusing on theorem 1.2.1,and the proof of the theorem is divided into four chapters.In the first chapter,we introduce the problem itself,its background and the conclusion.Moreover,we introduce the relevant definitions.Finally,we explain the structure of the paper.In the second chapter,this paper mainly introduces the rudimentary knowledge and proves the related lemmas.Firstly,we prove five lemmas by using convolution,Plancherel theorem and Cauchy-Schwartz inequality,etc..Based on these lemmas,we applied them to the fourth chapter to prove the regularity estimates of the solution on Gelfand-Shilov and Gevrey.In the third chapter,we mainly discuss the first part of theorem 1.2.1,that is,study the well-posedness and regularity of solution of Fokker-Planck equation.Firstly,we prove the related lemmas by using the mathematical knowledge of fourier transform,Young inequality,convolution and Cauchy-Schwartz inequality.Finally,by using these lemmas and some mathematical knowledge such as Riesz and Hahn-Banach theorem,we prove the well-posedness and the regularity of the solution of the equation in（1.1.2）.In the fourth chapter,mainly discuss the second part of theorem 1.2.1,that is,prove Gelfand Shilov’s（S₁¹（³））regularity estimate of the solution in velocity variable and prove Gevrey’s(G^3/2（³）)regularity estimate of the solution in position variable.Firstly,we use the lemmas in chapter 2 to prove the regularity estimate in this chapter,and then the monotone convergence theorem is used to complete the proof of the regularity.