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

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

2010-07-08

arxiv.org/abs/1007.1276v3

Mathematics

Analysis of PDEs

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

Scientific paper

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 $L^2(\R^n)$ 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 $L^2(\R^n)$.

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