Linear Hyperbolic Equations
Linear Hyperbolic Equations
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线性双曲方程
DOI：
10.1007/978-3-662-09207-1_2
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
V. Ivrii
中科院分区：
文献类型：
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作者：
V. Ivrii
本文主要研究线性双曲型方程和方程组。双曲型方程的概念最早出现在常系数二阶方程的情况下。证明了二次型是双曲的，即它的正子空间是1维的，n维−1的负子空间是1维的(反之亦然)。但现在这样的方程被称为严格的双曲型；双曲型方程可能有它的二次型a(·)退化(在旧术语中，抛物线退化)。在这种情况下，方程(0.1)具有运动(平面)波的类型u=v(<；x，ξ>；)的解，其中u=v是单变量的任意函数。当γ=1时，(0.1)的通解是两个移动波的线性组合，基于这一事实，我们可以很容易地用在任何非特征曲线上(即，在一条曲线上，如A(N)≠0，其中N是γ的法线)上规定的数据来求解柯西问题，甚至是合适的混合问题。此外，柯西问题是唯一可解的，并且存在一个依赖三角形；对于非齐次方程也是如此。当存在较低项时，柯西问题和混合问题至少在原则上可以用逐次逼近的方法来求解，就像常微分方程柯西问题的情形一样。类似的结构也适用于变系数的情况，同样，非特征柯西问题是唯一可解的，并且存在特征依赖三角(现在是曲线)。这些陈述不适用于其他类型的方程。因此，除了双曲性的代数定义外，还可以给出一个有意义的、封闭的解析定义：如果一个给定的方程(组)的某些非特征柯西问题对任何光滑的右端和初始数据是唯一可解的，并且如果存在依赖锥，则该方程(组)是双曲的。这一事实在上个世纪已经得到承认，从那时起，双曲性的代数定义被扩展到一阶系统、更高阶的方程和具有大量独立变量的系统，其方式仍然有效地接近解析定义。
The present survey is devoted to the linear hyperbolic equations and systems. The concept of a hyperbolic equation first appeared in the case of a second-order equationwith constant coefficients. It implied that the quadratic formis hyperbolic, that is, its positive subspaces are of dimension 1 and the negative subspaces of dimensionn− 1 (or vice-versa). But now such equations are referred to as strictly hyperbolic; a hyperbolic equation may have its quadratic forma(·) degenerate (parabolically degenerate, in old terminology). In this case, the equation (0.1) has solutions of the type of moving (plane) wavesu=v(<x,ξ>), whereandvis an arbitrary function of a single variable. Ifn= 1, the general solution of (0.1) is a linear combination of two moving waves, and on the basis of this fact we can easily solve the Cauchy problem with data prescribed on any non-characteristic curve (that is, on a curveγsuch thata(N) ≠ 0, whereNis the normal toγ), and even suitable mixed problems. Moreover, the Cauchy problem is uniquely solvable and there exists a triangle of dependence; the same is true for non-homogeneous equations. When lower terms are present, the Cauchy problem and the mixed problem can be solved, in principle at least, by the method of successive approximation, just as it is done in the case of the Cauchy problem for ordinary differential equations. Similar construction is available for the case of variable coefficients also, and again the non-characteristic Cauchy problem is uniquely solvable and there is a characteristic triangle of dependence (now curvilinear). These statements do not hold for equations of other types. Therefore, apart from the algebraic definition of hyperbolicity, it is possible to give a meaningful and close analytical definition: a given equation (system) is hyperbolic if some non-characteristic Cauchy problem for it is uniquely solvable for any smooth right-hand sides and initial data and if there exists a cone of dependence. This fact was already recognised in the last century, and from that time the algebraic definition of hyperbolicity was extended to systems of first order, to equations and systems of higher orders and with a large number of independent variables in such a way that it remained meaningfully close to the analytical definition.