Propagation of singularities in nonlinear problems

我们的课题是研究非线性双曲型方程的柯西问题。我们感兴趣的是这个柯西问题的整体理论。但它在今天还没有完成。其中一个原因是经典解在大范围内不存在，也就是说在它们的解中出现了奇异性。此外，我们看到，“奇点”引起许多有趣的现象。要建立全球理论，必须研究以下两个问题。第一个问题是“描述经典解存在的区域”，第二个问题是“将解扩展到奇点之外”。奇点问题一直是数学的基本问题之一，特别是在代数几何中。代数几何中经常使用的方法是“奇点分解”，也就是说将曲面提升到高维空间，使奇点消失。我们将把这一思想应用于我们的问题。也就是说， 关于我们 将解曲面映射到余切空间，使奇点消失，并在那里构造整体解。然后我们将其投影到基空间上，并将其解作为定义在基空间上的函数。首先，我们从上述观点出发研究了单阶非线性偏微分方程。我们在这一主题上的成就总结在我们的专著出版于1999年，美国。见本报告中的出版物列表。其次，我们考虑了非线性二阶单双曲方程，以及双曲守恒律组。在这种情况下，奇点一般也出现在有限时间内。我们对整体理论感兴趣。因此，我们的问题是如何扩展的解决方案后，出现的奇性。为此，我们将解曲面提升到余切空间，使奇点消失。虽然这是一个困难的问题，但我们可以在某个情况下这样做。然后，由于提升解（称为“几何解”）没有奇点，我们可以将其推广到整个余切空间。最后将其投影到基空间，得到一个弱解.我们已将这个结果写在将发表在《越南数学学报》上的文稿中，我们的问题已部分解决。我们今后要做的工作是完善二阶非线性双曲型方程的几何理论，推广方程。到目前为止我们所讨论的方程有点太特殊了。少

Our subject is to study the Cauchy problem for nonlinear hyperbolic equations. We are interested in the global theory for this Cauchy problem. But it is not complete at today's point. One of the reasons is that classical solutions do not exist in the large, that is to say that singularities appear in their solutions. Moreover we see that "singularities" cause many interesting phenomena. To establish the global theory, we must study the following two problems. The first problem is "to describe the domain where classical solutions exist", and the second one is "to extend the solutions beyond the singularities". The problem of "singularities" has been one of fundamental problems of mathematics, especially in "Algebraic geometry". The method used very often in algebraic geometry is the "resolution of singularities", that is to say to lift the surfaces into higher dimensional space so that the singularities would disappear. We will apply this idea to our problems. That is to say, we lift th … More e solution surface to the cotangent space so that the singularities would disappear, and construct the global solution there. Next we will project it to the base space and get the solution as a function defined on the base space.First we have studied single first order nonlinear partial differential equations from the above point of view. Our achievements on this subject are summerized in our monograph published in 1999, USA.See the list of publications in this report. Next we have considered nonlinear second order single hyperbolic equations, and also hyperbolic systems of conservation laws. In this case also, singularities generally appear in finite time. We are interested in the global theory. Therefore our problem is how to extend the solution after the appearance of singularities. For this purpose, we have lifted the solution surface into cotangent space so that the singularities would disappear. Though this has been a difficult problem, we could do so for a certain case. Then, as the lifted solution, called a "geometric solution", have no singularity, we can extend it to the whole cotangent space. Finally we project it to the base space for getting a weak solution. We have written this result in our manuscript which would be published in "Acta Mathematica Vietnamica".Our problem has been partially solved. What we must do from now is to complete the geometric theory for second order nonlinear hyperbolic equations and to generalize the equations. The equations which we have treated until now are a little too special. Less