Propagation of singularities for nonlinear hyperbolic, equations

本文研究了非线性双曲型方程的柯西问题，特别是它的全局理论。这个问题的难点在于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异点之外。我们解释我们的方法。首先，我们将方程提升到高维空间，并将其改写为普氏问题。它们有时有光滑的解。我们称之为“几何解”。然后将它们投影到基空间，构造一个合理的“弱解”。这是我们的节目。因此我们本质上没有改变方程。我们期望用我们的方法把各种结果统一起来。虽然我们的程序对单一阶偏微分方程是正确的，但它对高阶偏微分方程和系统并不普遍正确。但是，由于我们的方法是非常自然的，我们用同样的方法继续我们的考虑。在此过程中，我们解出了许多显式的例子，并将我们的解与目前得到的其他结果进行了比较。然后我们对流体力学的数学公式有了一些疑问。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，解的存在域被一组特征曲线所覆盖。然后构造了一个例子，该例子的域边界是特征曲线的包络，我们不能将解扩展到边界之外。我们还研究了一个方程对p=(δu/δx_1, δu/δx_n)不凸的例子，并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.