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

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 $L^1$.

87 pages

References in corpus (2)

• Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity
• Propagation of chaos for interacting particles subject to environmental noise

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