Propagation of singularities for nonlinear hyperbolic, equations

我们研究了非线性双曲型方程的柯西问题，特别是它的整体理论，该问题的难点是在有限时间内奇点的出现。我们的问题是如何将解扩展到奇点之外。我们解释我们的方法。首先，我们将方程提升到高维空间，并将其重写为Pfweenan问题。它们有时在总体上有平滑的解。我们称之为“几何解”。然后将它们投影到基空间，构造出一个合理的“弱解”。这是我们的计划。因此，我们基本上不改变方程。我们期望用我们的方法统一各种结果。虽然我们的程序对单个一阶偏微分方程是正确的，但对高阶偏微分方程和系统通常是不正确的。但是，由于我们的做法是非常自然的，我们继续用同样的方法进行审议。在这个过程中，我们已经解决了许多例子的显式形式，并比较我们的解决方案与其他结果，直到现在。然后我们对流体力学的数学公式提出了一些问题。请参阅我们的“研究报告”。另一方面，我们考虑了几个单一阶方程的未解问题，并认识到我们的程序是正确的。例如，解的存在域被一族特征曲线所覆盖。然后我们构造了一个例子，使得区域的边界作为特征曲线的包络而获得，并且我们不能将解延伸到边界之外。我们还研究了方程关于p=（δu/δx_1，δu/δx_n）非凸的一个例子，证明了我们的程序是正确的。

We have studied the Cauchy problem for nonlinear hyperbolic equations, especially the global theory for it. The difficulties of this problem is the appearance of singularities in finite time. Our problem is how to extend the solutions beyond the singularities. We explain our method. First we lift the equations into a higher dimensional space, and rewrite them as Pfaffian problems. They have sometimes smooth solutions in the large. We call them as "geometric solutions". Next we project them to the base space, and construct a reasonable "weak solution". This is our program. Therefore we do not change the equations essentially. We have expected that various kinds of results would be unified by our method. Though our program has been true for single first order partial differential equations, it has not generally been correct for higher order partial differential equations and systems. But, as our approach is very natural, we have continued our considerations by the same method. In this process we have solved many examples in explicit form and compared our solutions with another results obtained until now. Then we have had some question on the mathematical formulation of fluid mechanics. Refer to our "Research report". On the other hand we have considered several unsolved problems for single first order equations, and recognized that our program is correct. For example, an existence domain of solution is covered by a family of characteristic curves. Then we have constructed an example such that the boundary of the domain is obtained as an envelope of the characteristic curves, and that we can not extend the solution beyond the boundary. We have also studied an example where the equation is not convex with respect to p=(δu/δx_1, δu/δx_n) and showed that our program is correct.