Global well-posedness of the Boltzmann equation with large amplitude initial data

The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new $L_x^{\mathrm{\infty }}{L}_v^1\cap L_{x,v}^{\mathrm{\infty }}$ approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted $L^{\mathrm{\infty }}$ norm under some smallness condition on $L_x^1{L}_v^{\mathrm{\infty }}$ norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in $L_{x,v}^{\mathrm{\infty }}$ norm with explicit rates of convergence is also studied.