Recent progress in velocity averaging
Recent progress in velocity averaging
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速度平均的最新进展
DOI：
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发表时间：
2015
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通讯作者：
Diogo Arsénio
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文献类型：
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作者：
Diogo Arsénio
动力学理论中的一个经典结果是：如果f（x，v）和v ·<$xf（x，v）都属于L2（Rx × Rv），则对任意紧集K <$Rv，<$Kfdv ∈ H1 2（Rx）.这样的正则性陈述被称为速度平均引理，在动力学方程的分析中具有重要意义。在[2]中有人问，速度平均的其他设置是否可以产生半个导数的规律性的类似最大增益。这个问题是由皮埃尔-埃马纽埃尔·杰宾（Pierre-Emmanuel Jabin）和路易斯·维加（Luis Vega）[17]关于这个问题的早期著作所激发的，结果是令人惊讶的丰富和困难，而且目前还远远没有被完全理解。本文在回顾了该领域的一些经典结果后，综述了文[2]中速度平均引理的新设置的最新进展。我们还制定了一些apturtures，主要来自于量纲分析和类比与已知的结果，从而界定了其他新的设置速度平均的可能性。速度平均引理是关于动力学迁移方程（t，x，v）f（t，x，v）= g（t，x，v），（0.1），其中（t，x，v）∈ R× R × R，或它的定态对应方程v ·xf（x，v）= g（x，v），（0.2），其中（x，v）∈ R × R，n ≥ 1.上述等式的变体也是相关的。事实上，不同的空间和速度域，以及非线性速度场（考虑相对论的情况下），有时被研究。然而，为了简单起见，我们将主要关注欧几里得定态设置（0.2），我们相信它抓住了动力学输运的基本特征（至少就速度平均而言）。然而，我们将在下面的第2节中简要地提到非平稳情况（0.1），而不扩展这个主题。在第1、2和3节中描述了关于经典速度平均的现代观点之后，我们打算在本文的其余部分中综述关于这个问题的最新定理和理论。这里给出的大多数结果直接取自[2]，我们将系统地概述所断言的结果的证明。尽管如此，我们还是请读者参阅[2]及其参考文献，以了解更多详情和完整的依据。1.经典的速度平均，希尔伯特情形经典的希尔伯特情形的速度平均包含在下面的结果中。它是动力学输运方程正则性理论的起点，并已在[12]中首次建立。然而，请注意，这样的正则性结果已经在[1，13]中以较弱的形式提出。
A classical result in kinetic theory establishes that if f(x, v) and v ·∇xf(x, v) both belong to L2 (Rx × Rv ), then ∫ K fdv ∈ H 1 2 (Rx), for any compact set K ⊂ Rv . Such regularity statements are known as velocity averaging lemmas and have important implications in the analysis of kinetic equations. It was asked in [2] whether other settings of velocity averaging could produce a similar maximal gain of regularity of half a derivative. This question, motivated by an earlier work of Pierre-Emmanuel Jabin and Luis Vega [17] on the subject, turns out to be surprisingly rich and difficult, and it is, for the moment, far from being fully understood. In this article, after recalling some classical results in the field, we survey the recent developments from [2], where new settings of velocity averaging lemmas were investigated. We also formulate a few conjectures, mainly derived from a dimensional analysis and by analogy with known results, thus delimiting the possibilities for other new settings of velocity averaging. Velocity averaging lemmas concern the regularity theory of solutions to the kinetic transport equation (∂t + v · ∇x) f(t, x, v) = g(t, x, v), (0.1) where (t, x, v) ∈ R× R × R, or its stationary counterpart v · ∇xf(x, v) = g(x, v), (0.2) where (x, v) ∈ R × R, with n ≥ 1. Variants of the above equations are also relevant. Indeed, different spatial and velocity domains, as well as non-linear velocity fields (consider the relativistic case), are sometimes studied. Nevertheless, for the sake of simplicity, we will mainly focus on the Euclidean stationary setting (0.2), which, we believe, captures the essential features of kinetic transport (at least as far as velocity averaging is concerned). We will nevertheless make brief references to the non-stationary case (0.1) in Section 2 below without expanding on the subject. After describing a modern viewpoint on classical velocity averaging in Sections 1, 2 and 3, we intend to survey, in the remainder of this text, recent theorems and conjectures on the subject. Most results presented here are taken directly from [2] and we will systematically sketch proofs of the asserted results. Nevertheless, we refer the reader to [2] and the references therein for further details and complete justifications. 1. Classical velocity averaging, the Hilbertian case The classical Hilbertian case of velocity averaging is contained in the following result. It is the starting point of the regularity theory of kinetic transport equations and has been established first in [12]. Note, however, that such regularity results had already been suggested in weaker forms in [1, 13].