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PDE Seminar

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