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。我们对柯西问题的全局理论感兴趣。但在今天，它还没有完成。其中一个原因是经典解在大范围内不存在，也就是说奇点出现在它们的解中。此外，我们看到“奇点”引起了许多有趣的现象。为了建立全局理论，我们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，特别是代数几何中的基本问题之一。代数几何中经常使用的方法是“奇点解析”，也就是说，将曲面提升到高维空间，这样奇点就会消失。我们将把这个想法应用到我们的问题上。也就是说，我们将更多的解面提升到余切空间，使奇点消失，并在那里构造全局解。接下来我们将它投影到基空间并得到解作为一个定义在基空间上的函数。首先，我们从上述观点研究了单一阶非线性偏微分方程。1999年在美国出版的专著总结了我们在这方面的研究成果。请参阅本报告中的出版物列表。接下来我们考虑非线性二阶单双曲方程，以及双曲守恒律系统。在这种情况下，奇点通常在有限时间内出现。我们对全局理论感兴趣。因此我们的问题是如何在奇点出现后推广解。为此，我们将解面提升到余切空间，这样奇点就会消失。虽然这是一个难题，但我们可以在特定情况下这样做。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。最后我们将其投影到基空间以得到弱解。我们将这一结果写在了我们的手稿中，并将发表在《越南数学学报》上。我们的问题已经部分解决了。我们现在要做的是完善二阶非线性双曲型方程的几何理论，并对其进行推广。到目前为止我们处理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