Boltzmann Equation and Hard Sphere Gas with Singular Initial Data: Well-posedness, Kinetic Limit, and Weak Stability
par
Amphithéâtre Léon Motchane
IHES
Séminaire d'Analyse
We develop a local well-posedness theory for the spatially inhomogeneous 3-d Boltzmann equation with hard sphere cross-section, accommodating a class of large initial data. This class is an interpolation between the space $L_{x}^{\infty}L_v^{1}$ and $L_{v}^1L_{x}^{\infty}$ (with weight in $v$). It is notable for including data that can have infinite $L_{x,v}^\infty$-norm, entropy density. The core idea of the proof is a homogenization method based on collision geometries. As a key application, we prove the kinetic limit of hard sphere gases with initial data in this space, having certain extra conditions. This is the first proof on the kinetic limit of hard sphere gases with singular initial data. We also derive weak stability estimates of Boltzmann equation and hard sphere gas within the same functional spaces.
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