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

Abstract: 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 $$L^{\infty_xL^1_{v}\cap} L<sup>\infty_{x,v}$$ 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 $L<sup>\infty$ norm under some smallness condition on $L<sup>1_xL<sup>\infty_v$ 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 $L<sup>\infty_{x,v}$ norm with explicit rates of convergence is also studied.