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 $L_x^{\infty }{L}_v^1\cap L_{x,v}^{\infty }$ 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 $L^{\infty }$ norm under some smallness condition on $L_x^1{L}_v^{\infty }$ 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 $L_{x,v}^{\infty }$ norm with explicit rates of convergence is also studied.

34 pages. Typos corrected

References in corpus (1)

• Asymptotic Stability of the Boltzmann Equation with Maxwell Boundary Conditions

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