Mathematical Sciences: Existence and Blow-Up of Solutions to Systems of Nonlinear Wave Equations

Lindblad的研究涉及非线性波动方程系统。基本的数学问题是：(1)在某一类中解是否具有局部存在性和唯一性？（ii）是否有放大的解决方案？（iii）溶液的长期行为是什么？使用的一些主要技术是（iv）伪黎曼或洛伦兹几何，(v)傅立叶变换方法（vi）能量方法。Lindblad建议针对这些一般性问题开展两个项目，结合上述技术：(1)回答一些关于低规律性解和小数据的问题。（二）研究自然界中存在的相对论性欧拉方程的自由边界问题的规律性。这些问题的解决将有助于我们了解物理学的基本方程。一方面，我们正在结合不同的技术(iv)-(vi)，通常情况下，有趣和有用的数学起源于不同数学领域之间以及数学与其他科学之间的相互作用。另一方面，对爱因斯坦方程的研究在很大程度上被数学家们忽视了。例如，人们可以用引力波作为观察宇宙的一种手段。为了解决问题（II），我们正在开发全新的技术，我们相信它也将有助于研究许多其他问题，例如两种流体之间的界面。这样的问题可以在工业上应用。Lindblad还在他的研究中使用计算机和数值计算，这使得应用科学家更容易获得。这项研究的实际动机是物理学中一些重要的方程可以写成非线性波动方程的系统。在经典场论和连续介质力学以及经典物理学中都有这样的例子，例如爱因斯坦方程，杨-米尔斯方程和非线性弹性方程。对于某些方程，例如Yang-Mills，我们期望解在任何时候都保持正则。另一方面，对于爱因斯坦方程和非线性弹性方程，解在某些数据下是不成立的。对于爱因斯坦的方程来说，爆炸发生在自然界（广义相对论中的黑洞），而对于非线性弹性方程来说，爆炸是由于方程不再准确地描述自然界中发生的事情。在任何一种情况下，理解存在与爆炸的机制都很重要。

Abstract Lindblad 9623207 Lindblad's research concerns systems of nonlinear wave equations. The basic mathematical questions are: (i) Does one have local existence and uniqueness of solutions in a certain class? (ii) Does one have blow-up of solutions? (iii) What is the long time behavior of solutions? Some of the main techniques to be used are (iv) Pseudo-Riemannian or Lorentzian geometry, (v) Fourier transform methods (vi) Energy-methods. Lindblad proposes to carry out two projects related to these general questions, combining the techniques mentioned: (I) To answer certain questions concerning low regularity solutions and small data. (II) To study regularity of a free-boundary problem, that occur in nature, for the relativistic Euler equation. Solution to these questions would contribute to our knowledge of fundamental equations from physics. On the one hand we are combining the different techniques (iv)-(vi), and it has usually been the case that interesting and useful mathematics originates from interaction between different fields of mathematics as well as between mathematics and other sciences. On the other hand, the study of Einstein's equations have to a large extent been neglected by mathematicians. It is e.g. perceivable that one could use gravitational waves as a means of observing the universe. To solve problem (II) we are developing completely new techniques, and we believe that it will be useful for studying many other problems as well, e.g. an interface between two fluids. Such a problem would have industrial applications. Lindblad is also using computers and numerical calculations in his research, which makes it more accessible for applied scientists. The practical motivation for this research is that several of the important equations in physics can be written as a system of nonlinear wave equations. There are examples of this in classical field theory and continuum mechanics as well as in classical physics, e.g. Einstein's equations, the Yang-Mills equatio n and the equations of nonlinear elasticity. For some equations, for example Yang-Mills, we expect the solutions to remain regular for all times. On the other hand, for Einstein's equations and the equations of nonlinear elasticity, the solutions blow-up for certain data. For Einstein's equation the blow-up occurs in nature (black holes in general relativity) whereas for the equation of nonlinear elasticity the blow-up is due to the fact that the equations no longer describe what is happening in nature accurately. In either case it is important to understand the mechanism of existence versus blow-up.