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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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