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On the Boltzmann Equation with Stochastic Kinetic Transport: Global Existence of Renormalized Martingale Solutions

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Abstract

This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in \({{L}^{1}}\).

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Authors and Affiliations

  1. University of Maryland, College Park, MD, 20742, USA

    Samuel Punshon-Smith

  2. Max-Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

    Scott Smith

Authors
  1. Samuel Punshon-Smith
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  2. Scott Smith
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Correspondence to Samuel Punshon-Smith.

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Communicated by L. Saint-Raymond

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Punshon-Smith, S., Smith, S. On the Boltzmann Equation with Stochastic Kinetic Transport: Global Existence of Renormalized Martingale Solutions. Arch Rational Mech Anal 229, 627–708 (2018). https://doi.org/10.1007/s00205-018-1225-5

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  • DOI: https://doi.org/10.1007/s00205-018-1225-5

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