Propagation of singularities in nonlinear problems

我们的主题是研究非线性双曲型方程的柯西问题。我们对这个柯西问题的整体理论很感兴趣。但在今天的时间点上，它还不完整。其中一个原因是，经典解并不普遍存在，也就是说，它们的解中出现了奇点。此外，我们还看到“奇点”导致了许多有趣的现象。要建立全球理论，我们必须研究以下两个问题。第一个问题是“描述存在经典解的区域”，第二个问题是“将解推广到奇点之外”。“奇点”问题一直是数学的基本问题之一，在“代数几何”中更是如此。在代数几何中经常使用的方法是“奇点分解”，即将曲面提升到更高维空间，使奇点消失。我们将把这个想法应用到我们的问题上。这就是说，我们抬起了这个…更多的解面到余切空间，这样奇点就会消失，并在那里构造整体解。接下来，我们将其投影到基空间，并将其作为定义在基空间上的函数来求解。首先，我们从上述角度研究了一阶非线性偏微分方程解。在1999年出版的专著《美国》中总结了我们在这方面的成就，见本报告的出版物清单。接下来，我们考虑了二阶非线性单双曲型方程，以及双曲型守恒律组。在这种情况下，奇点通常在有限时间内出现。我们对全球理论感兴趣。因此，我们的问题是如何在奇点出现后将解推广。为此，我们将解曲面提升到余切空间，这样奇点就会消失。虽然这一直是一个难题，但我们可以针对某个案例这样做。然后，由于提升的解，称为“几何解”，没有奇点，我们可以把它推广到整个余切空间。最后将其投影到基空间，得到弱解。我们已经把这个结果写在我们的手稿中，并将发表在《越南数学学报》上。我们的问题已经部分解决了。从现在起，我们要做的是完善二阶非线性双曲型方程的几何理论，并将其推广。我们到目前为止所处理的方程有点太特殊了。较少

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