Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff
Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff
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DOI：
10.1016/j.matpur.2007.03.003
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发表时间：
2006-07
期刊：
影响因子：
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通讯作者：
C. Mouhot;Robert M. Strain
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文献类型：
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作者：
C. Mouhot;Robert M. Strain
在本文中，我们证明了新的建设性的玻尔兹曼碰撞算子无截止，即长程相互作用的不稳定性估计。特别是，我们给出了一个广义的充分条件的谱隙的存在，其中涉及的增长行为的碰撞内核在大的相对速度和它的奇异行为在掠和正面碰撞。它特别提供了一个谱隙的存在性，并对来自硬势的相互作用进行了估计，其中硬势是<$（r）=r−（s−1），s <$5，而软势是<$（r）=r−（s−1），3<s<5（没有角截断）。特别地，本文（用构造性的方法）恢复、改进和推广了Pao [Y. P. Pao，Boltzmann碰撞算符与反幂次分子间势的结果. I，Comm. Pure Appl.Math.27（1974）407-428; Y. P. Pao，具有反幂次分子间势的玻尔兹曼碰撞算子。II，Comm. Pure Appl.Math.27（1974）559-581]。对于[Y. Guo，The朗道equation in a periodic box，Comm.Math.Phys.231（2002）391-434]，我们提出了一个关于Boltzmann和朗道线性化碰撞算子存在谱隙的统一充要条件的猜想.
In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for long-range interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials ϕ(r)=r−(s−1), s⩾5, or the so-called moderately soft potentials ϕ(r)=r−(s−1), 3<s<5 (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao [Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I, Comm. Pure Appl. Math. 27 (1974) 407–428; Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II, Comm. Pure Appl. Math. 27 (1974) 559–581]. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in [Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002) 391–434] and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators.