SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1357-1378, January 2021.
This paper studies convergence to equilibrium for the spatially inhomogeneous linear relaxation Boltzmann equation in Boltzmann entropy and related entropy functionals, the $p$-entropies. Villani [Hypocoercivity, in Memoirs of the American Mathematical Society, Vol. 202, American Mathematical Society, Providence, RI, 2006] proved entropic hypocoercivity for a class of PDEs in a Hörmander sum-of-squares form. It was an open question to prove such a result for an operator which does not share this form. We prove a closed entropy-entropy production inequality à la Villani which implies exponentially fast convergence to equilibrium for the linear Boltzmann equation with a quantitative rate. The key new idea appearing in our proof is the use of a total derivative of the entropy of a projection of our solution to compensate for an error term which appears when using nonlinear entropies. We also extend the proofs for hypocoercivity for the linear relaxation Boltzmann to the case of $\Phi$-entropy functionals.

中文翻译：

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环上线性弛豫Boltzmann方程的P熵的低矫顽力

SIAM数学分析杂志，第53卷，第2期，第1357-1378页，2021年1月。
本文研究Boltzmann熵中空间非均匀线性弛豫Boltzmann方程及其相关熵泛函$ p $-熵的收敛性。Villani [低矫顽力，在美国数学学会回忆录中，第 202，美国数学协会，普罗维登斯，罗德岛州，2006]证明了Hörmander平方和形式的一类PDE的熵矫顽力。对于不共享此表格的运营商来说，证明这样的结果是一个悬而未决的问题。我们证明了一个封闭的熵－熵产生不等式àla Villani，它暗示了线性Boltzmann方程具有定量速率的指数快速收敛到平衡。我们的证明中出现的关键新思想是使用解决方案投影的熵的全导数来补偿使用非线性熵时出现的误差项。我们还将线性松弛玻尔兹曼的矫顽力的证明扩展到$ \ Phi $-熵泛函的情况。