The Asymptotic Behavior Of The Solutions Of Kinetics Model To The Bgk Or The Es With Energy Input

This paper is to study the distance estimation of the equations with BGK modelor ES model with energy input, especially the exponent asymptotic behavior of thistwo equations. The main maths tools are two kinds of probability distances, which isWasserstain metric and Fourier-based metric.To this master's artical, first we introduce the history and the current research ofthe kinetic theory which connects with this paper. The main discussion is the long-time asymptotic behaviorof the solution to the Boltzman equation with space evenly,one-dimensional Kac equation and Vlasov-Fokker-Planck equation. It is mainly basedon the probability distance, the specific methods and theory is in the document. Thenintroduce some maths tools, which the most important is the two probability distances,which is Wasserstain distance and Fourier-based metric. We discuss the definition,property and the relationship between them. Furthermore, we discuss the relationshipbetween them and other distance as‖·‖H~r,‖·‖L~2,‖·‖L~1 norm distance on Sobolevspace.The final part is the main part, which discusses the asymptotic behavior of thesolutions. Calulating the ds distance is the focal point, we can know the ds metric isexponent convergence, especially, when s = 2, we can get the d2 metric is also exponentcenvergence. Based on the relation between the ds metric and ?m, W1, W2 distance,we can get the solution of the equation is exponent convergence under the m?, W1,W2 distance. Use interpolation inequality, we can create the relation between Hr,‖L~2,‖·‖L~1 norm distance on Sobolev space and the ds distance. Then we can getthe equation which we discussed is also exponent convergence under the‖·‖H~r,‖·‖L~2,‖·‖L~1 norm metric.