Global Existence of Classical Solutions to the Boltzmann Equation with External Force for Hard Potentials

The Boltzmann equation with external force describes the time evolution of rarefied gas in an external field. In this paper, we construct the global existence of classical solutions to the Boltzmann equation for hard potentials with Grad's angular cutoff in the whole space when the initial datum is a perturbation of the local Maxwellian. The proof relies on the nonlinear energy method where the crucial step is to show the positivity of the linearized collision operator. Further, we introduce new energy functional to obtain the desired differential inequality.