Convergence of approximate solutions for Kac's model of the Boltzmann equation (Q1062236)

Kac's one-dimensional model of the Boltzmann equation reads as follows: \[ (1)\quad \partial_ tF=-v\partial_ xF+Q(F,F),\quad F(0,x,v)=F_ 0(x,v), \] (t,x,v)\(\in [0,\infty)\times R\times R\) where \(F=F(t,x,v)\) is a distribution function of particles with velocity v at time t and at position x, and Q is a collision operator. Substituting \(F=g+g^{1/2}f\), with \(g(v)=\exp (-v^ 2/2)\sqrt{2\pi}\) in (1) yields \((2)\quad \partial_ tf=-v\partial_ xf+Lf+\Gamma (f,f),\quad f(0,x,v)=f_ 0(x,v),\) where \(Lf=2g^{-1/2}Q(g,g^{1/2}f),\) and \(\Gamma (f,f)=g^{- 1/2}Q(g^{1/2}f,g^{1/2}f).\) The author shows that the solution of (2) can be approached by a sequence of approximate solutions, provided that the initial value is small enough.