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Archive for Rational Mechanics and Analysis

, Volume 225, Issue 1, pp 375–424 | Cite as

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

  • Renjun Duan
  • Feimin Huang
  • Yong Wang
  • Tong Yang
Article

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Renjun Duan
    • 1
  • Feimin Huang
    • 2
    • 3
  • Yong Wang
    • 3
  • Tong Yang
    • 4
  1. 1.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.Institute of Applied MathematicsAMSS, CASBeijingPeople’s Republic of China
  4. 4.Department of MathematicsCity University of Hong KongKowloonHong Kong

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