Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production

Abstract

This article provides sharp constructive upper and lower bound estimates for the non-linear Boltzmann collision operator with the full range of physical non cut-off collision kernels ($\gamma >-n$ and $s\in (0,1)$) in the trilinear $L^2({\mathbb{R}}^n)$ energy $<\mathcal{Q}(g,f),f>$. These new estimates prove that, for a very general class of $g(v)$, the global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works [2009, 2010, 2010 arXiv:1011.5441v1]. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space $L^2({\mathbb{R}}^n)$.

29 pages, updated file based on referee report; Advances in Mathematics (2011)

References in corpus (3)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Global Strong Solutions of the Boltzmann Equation without Angular Cut-off
• Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials

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• Stability of vacuum for the Boltzmann Equation with moderately soft potentials
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• Hypoelliptic Estimates for a Linear Model of the Boltzmann Equation without Angular Cutoff
• Global hypoelliptic estimates for a linear model of non-cutoff Boltzmann equation