On the Cauchy problem for Boltzmann equations: global existence and weak stability
On the Cauchy problem for Boltzmann equations: global existence and weak stability
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DOI：
10.2307/1971423
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发表时间：
1989-09
影响因子：
4.9
通讯作者：
R. J. Diperna;P. Lions
中科院分区：
文献类型：
--
作者：
R. J. Diperna;P. Lions
研究了具有一般碰撞核的Boltzmann方程的大数据Cauchy问题。我们证明了仅满足物理上自然的先验界的解序列在L'中弱收敛到一个解。从这个稳定性的结果，我们推出整体存在的柯西问题的解决方案。我们的方法依赖于最近的紧凑性结果的速度平均值，一个新的玻尔兹曼方程，其中涉及非线性归一化和分析的下解和上解。它使我们能够克服缺乏强有力的先验估计，并定义一个有意义的碰撞算子的一般配置。
We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution. From this stability result we deduce global existence of a solution to the Cauchy problem. Our method relies upon recent compactness results for velocity averages, a new formulation of the Boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions. It allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations.