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Global Well-Posedness of the Boltzmann Equation with Large Amplitude Initial Data

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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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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Renjun Duan

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Feimin Huang

  3. Institute of Applied Mathematics, AMSS, CAS, Beijing, 100190, People’s Republic of China

    Feimin Huang & Yong Wang

  4. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Tong Yang

Authors
  1. Renjun Duan
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  2. Feimin Huang
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  3. Yong Wang
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  4. Tong Yang
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Corresponding author

Correspondence to Feimin Huang.

Additional information

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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  • DOI: https://doi.org/10.1007/s00205-017-1107-2

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