Hyperbolic Partial Differential Equations
Hyperbolic Partial Differential Equations
复制标题
DOI：
10.1007/978-0-387-87823-2
复制
发表时间：
2009-06
期刊：
影响因子：
--
通讯作者：
S. Alinhac
中科院分区：
文献类型：
--
作者：
S. Alinhac
这本书的目的是给出双曲偏分度？初等水平上的积分方程式。事实上，所要求的数学背景只是一门三年级的大学课程。多元函数的积分学。不需要泛函分析知识，也不需要任何分布理论(下面提到的冲击波除外)。K文中出现的所有解都是分段经典C解。在Simpli之外？首先，我们认为双曲型偏微分方程(PDE)的所有主要特征(柯西问题的适定性，？传播速度、确定域、能量不平等等。)可以在这种情况下展示。我们希望这本书本身能证明我们的信念。其次，在光滑函数的背景下建立的所有性质、解公式和不等式都可以很容易地推广到更一般的情况(索波列夫空间或温带分布中的解等)。通过泛函分析或分布理论的简单标准程序，它们是双曲型方程理论的“外部”：这些定理的深刻数学内容已经在本书的陈述和证明中找到。最后一个原因是：我们确实希望这本书的许多读者最终会在？在我们看来，这似乎是主题的自然延续：非线性双曲型系统(可压缩？UID、广义相对论等)。
The aim of this book is to present hyperbolic partial di? erential equations at an elementary level. In fact, the required mathematical background is only a third year university course on di? erential calculus for functions of several variables. No functional analysis knowledge is needed, nor any distribution theory (with the exception of shock waves mentioned below). k All solutions appearing in the text are piecewise classical C solutions. Beyond the simpli? cations it allows, there are several reasons for this choice: First, we believe that all main features of hyperbolic partial d-ferential equations (PDE)(well-posedness of the Cauchy problem,? nite speed of propagation, domains of determination, energy inequalities, etc.) canbedisplayedinthiscontext. Wehopethatthisbookitselfwillproveour belief. Second, allproperties, solutionformulas, andinequalitiesestablished here in the context of smooth functions can be readily extended to more general situations (solutions in Sobolev spaces or temperate distributions, etc.) by simple standard procedures of functional analysis or distribution theory, which are “external” to the theory of hyperbolic equations: The deep mathematical content of the theorems is already to be found in the statements and proofs of this book. The last reason is this: We do hope that many readers of this book will eventually do research in the? eld that seems to us the natural continuation of the subject: nonlinear hyp-bolic systems (compressible? uids, general relativity theory, etc.).