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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 \({L^\infty_xL^1_{v}\cap L^\infty_{x,v}}\) 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 the \({L^1_xL^\infty_v}\) 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 the \({L^\infty_{x,v}}\) norm with explicit rates of convergence are also studied.

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Correspondence to Feimin Huang.

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Communicated by P.-L. Lions

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Duan, R., Huang, F., Wang, Y. et al. Global Well-Posedness of the Boltzmann Equation with Large Amplitude Initial Data. Arch Rational Mech Anal 225, 375–424 (2017). https://doi.org/10.1007/s00205-017-1107-2

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