Pointwise convergence of the boundary layer of the Boltzmann equation for the cutoff hard potential (Q1013009)

Using the Boltzmann equation \[ f(x,v,t)/\partial t+ \,\partial f(x,v,t)/\partial x= Kn^{-1} J_{st}(f,f ),\;v> 0, \] the nonlinear stability of a boundary layer when the Mach number is greater then one is studied analytically (at the boundary layer \(f(x= 0,v> 0,t)= f_b(x,v>0)\) and \(f(x,v,t)\to f_M(v)\), when \(x\to\infty\) (\(f_M(v)\) is the a global Maxwellian state function). The model of the cutoff hard sphere potential is used for the collision operator of the Boltzmann equation. The Cauchy one-dimensional problem is considered using the Green's function and the weighted energy method. The main goal of the article is to proof the nonlinear stability. It is proved that the Boltzmann equation has a weak solution for the initial boundary problem. The estimates for the Green's function of the initial boundary problem are obtained. Lemmas and main theorems are proved.