论文标题
希尔伯特扩展相对论兰道方程
Hilbert Expansion for the Relativistic Landau Equation
论文作者
论文摘要
在本文中，我们研究了相对论Landau方程的希尔伯特扩张的当地有效性。我们证明，相对论Landau方程的解决方案会融合到限制相对论Euler方程的小型古典解决方案中，因为在加权Sobolev空间中，Knudsen的数量缩小到零。关键困难来自当地麦克斯韦人的时间和空间衍生物，这些衍生物产生了动量增长项，并且基于标准的$ l^2 $的能量和耗散是无法控制的。我们引入了新型的时间依赖性重量功能，以生成其他耗散术语以抑制巨大的动量。该论点依赖于具有或没有权重的能量消散结构的层次结构。据作者所知，这是希尔伯特（Hilbert）扩展兰道型方程的第一个结果。
In this paper, we study the local-in-time validity of the Hilbert expansion for the relativistic Landau equation. We justify that solutions of the relativistic Landau equation converge to small classical solutions of the limiting relativistic Euler equations as the Knudsen number shrinks to zero in a weighted Sobolev space. The key difficulty comes from the temporal and spatial derivatives of the local Maxwellian, which produce momentum growth terms and are uncontrollable by the standard $L^2$-based energy and dissipation. We introduce novel time-dependent weight functions to generate additional dissipation terms to suppress the large momentum. The argument relies on a hierarchy of energy-dissipation structures with or without weights. As far as the authors are aware of, this is the first result of the Hilbert expansion for the Landau-type equation.
