Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff
Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff
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DOI：
10.1007/s00220-011-1242-9
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发表时间：
2009-12
影响因子：
2.4
通讯作者：
Radjesvarane Alexandre;Y. Morimoto;S. Ukai;Chao-Jiang Xu;Tong Yang
中科院分区：
文献类型：
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作者：
Radjesvarane Alexandre;Y. Morimoto;S. Ukai;Chao-Jiang Xu;Tong Yang
我们证明了在一些 Sobolev 空间中没有角截止的玻尔兹曼方程平衡经典解的全局存在性和唯一性。此外，由此获得的解在任何正时间的所有变量中都显示为非负且 C∞。在本文中，我们研究了具有温和奇点的麦克斯韦分子型碰撞算子。关键的观察结果之一是引入了与碰撞算子中横截面的奇异行为相关的新的重要规范。该范数捕捉了奇点的基本属性，并通过著名的 H 定理精确地产生了线性化碰撞算子的耗散。
We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative andC∞in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.