Incompressible Navier-Stokes-Fourier Limit from The Boltzmann Equation: Classical Solutions

We first construct the global existence of the classical solutions near the absolute equilibrium to the Boltzmann equation with general collision kernels under the Navier-Stokes scaling in whole space. In particular, a global energy estimate uniform in the Knudsen number is derived. Then, by using the local conservation laws of this sequence of solutions to the Boltzmann equation, and by taking the zero Knudsen number limit, we obtain the global classical solution to the incompressible Navier-Stokes-Fourier equation with small initial data.