Incompressible Navier-Stokes-Fourier Limit Of The Two Dimensional Steady Boltzmann Equation

In this paper,we study incompressible Navier–Stokes–Fourier(INSF for short)limit of the two dimensional steady Boltzmann equation.Indeed,despite the importance of the steady INSF equation in applied subjects,the derivation of INSF from the steady Boltzmann equation is important and has been an outstanding open problem.Since a solution of boundary value problem of the Boltzmann equation has no high order regularity,high order energy method falls to treat this problem.We have to use a recent quantitative L~2-L~∞ framework,introduced by[17],to overcome this difficulty.With a new L~4estimate of the hydrodynamic part,we get a uniform estimate of the linear equation.Then we obtain existence and uniqueness of positive solution to the steady Boltzmann equation and prove its convergence limit by constructing a positive-preserving scheme of solution.