非线性Boltzmann方程Cauchy问题解的整体存在性对位势U(r)=r^(-s),s>2

Global Solution of the Boltzmann Equation for Interparticle Potentials of the Form U (r)= r^(-5), s>2

本文的B、E、(Bolizmann方程)是这样的。 (1.1)是B、E的解。∈R^+×R^3×R^3 (1.2) (1.3)S>2 (1.4)(1.5) (1.6) (1.7)上式以为z轴有动量守恒。能量守恒 (1.8) (1.9) (1.10)β>0 (1.11) (1.12)对连续。 (1.13) (1.14)对l∈t(0.T)有一阶连续导数。t固定时对连续, (1.15) (1.16)本文是小初值的整体存在性定理,本文定理是定理1 设<1/(8A_3(s_9β)) (1.17)A_3(s_9β)≡A_1(β)(2^(1/8)x/3+A_2(S)) (1.18)A_1(β)≡(π/β)^(1/2) 2x integral from n=0 to π/2 β(θ)dθ (1.19) (1.20)则初值问题(1.1)和(1.10)有唯一整体解f∈C^1(0,∞;S_3^+)本文模仿[1]的证明。但对[1]作了推广,[1]的结果相当于这儿s=+∞的情形。这儿讨论的是s>2。我们主要利用了工具[2]。[3]。[4]等文章也讨论Cauchy问题的解。它们对要求受cxp-αζ~2,α>0或,p>1的限制,当然结果也就更好,本文对的假设追随[1],不同于[3],[4]。 [5]给出大初值的整体存在性,但本文初值函数不满足[5]中(2)。本文对软硬位势处理提供了一种方法,它可能有普遍意义。

In this paper, which is an extension of the paper [Commun. Math. Phys. 95, 217-226 (1984)] for s>2. Solutions of the Boltzmann equation are proved to exist, globally in time under conditions that include the interparticle potentials of the form U(r)=r^(-3). 3>2 when the initial datum is sufficiently small.