Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

本课题研究数学物理中有关非线性双曲型微分方程组的基本数学问题。这些包括经典场论和连续介质力学中的许多重要方程（例如，爱因斯坦的广义相对论方程，欧拉的流体方程）。基本问题是：(i)在某一类中解是否存在唯一性，是否对数据有连续依赖性？（ii）解会爆炸吗（例如广义相对论中的黑洞）？（iii）解决方案的长期行为是什么？更具体地说，首席研究员主要在两个领域工作。项目的一部分是研究爱因斯坦方程和其他相关方程的整体解的存在性问题。第一个目标是简化、推广和完善爱因斯坦方程的存在性结果。长期目标是研究广义相对论中黑洞等大解的稳定性。这与数学相对论中的一个中心问题有关，即彭罗斯的宇宙审查猜想。大解的稳定性或爆破问题也是目前非线性波动方程领域的主要问题。该项目的第二个组成部分涉及研究流体动力学和广义相对论中出现的一类问题，特别是证明真空中流体表面运动的自由边界问题的适定性。这个领域的第一个目标是证明本地存在。更长远的目标是研究天体物理体（如气态恒星）的长期行为，以及与流体和固体之间界面相关的其他问题。为了解决这些问题，首席研究员和他的合作者正在开发新的技术，这些技术也可以用于研究许多其他问题。特别是，他们用几何方法来研究双曲型微分方程。首席研究员和他的合作者最近极大地简化了爱因斯坦方程及其推广解的存在性证明。对构成当前项目一部分的这些想法的持续改进可能会产生重大影响。仅举一个例子，这种新方法应该会让研究生更容易学习数学相对论。此外，这些方法通过引入所谓的调和坐标所揭示的详细的渐近行为将对物理学和天文学界有用。物理学家正在建造大型引力波探测器来观测宇宙。为了让科学家们知道用这些仪器寻找什么，需要在爱因斯坦方程的基础上进行大规模的数值计算和模拟。迄今为止，唯一成功的尝试是借助谐波坐标。还可以想象，了解两种流体之间的界面的性质和控制可能具有工业应用。特别地，磁流体力学中有一个关于等离子体物理的版本。众所周知，控制等离子体的能力对核聚变反应堆的建造至关重要。

This project is concerned with basic mathematical questions about systems of nonlinear hyperbolic differential equations in mathematical physics. These include many important equations in classical field theory and continuum mechanics (e.g., Einstein's equations of general relativity, Euler's equations for fluids). The basic questions are: (i) Do we have existence and uniqueness of solutions, and continuous dependence on data, in a certain class? (ii) Can solutions blow up (e.g., black holes in general relativity)? (iii) What is the long-time behavior of solutions? More specifically, the principal investigator is working in two main areas. One part of the project is to study the problem of existence of global solutions of Einstein's equations and of other related equations. The first goal is to simplify, generalize, and refine the existence results for Einstein's equations. A long-term objective is to study the stability of large solutions like black holes in general relativity. This is related to one of the central problems in mathematical relativity, namely, the cosmic censorship conjecture of Penrose. The question of stability or blow-up of large solutions is also the main question now in the area of nonlinear wave equations. A second component of the project involves studying a class of problems that occur in fluid dynamics and general relativity, in particular, proving the well-posedness for the free boundary problem of the motion of the surface of a fluid in a vacuum. The first goal in this area is to prove local existence. A longer-range goal is to study the long-time behavior of astrophysical bodies such as gaseous stars, along with other problems related to the interfaces between fluids and solids. To solve these problems the principal investigator and his collaborators are developing new techniques that could be useful for studying many other problems as well. In particular, they are using geometric methods to study hyperbolic differential equations.The principal investigator and his collaborators have recently simplified greatly the existence proof for solutions to Einstein's equations and their generalizations. The continuing refinement of those ideas that constitute part of the current project could have a significant impact. To name just one, the new approach should make it much easier for graduate students to study mathematical relativity. Moreover, the detailed asymptotic behavior that these methods reveal through the introduction of so-called harmonic coordinates will be useful to the physics and astronomical communities. Physicists are in the process of constructing large gravitational wave detectors to observe the universe. In order for the scientists to know what to look for with these instruments, there is a need for a large-scale effort in doing numerical calculations and simulations based on Einstein's equations. The only successful attempts hitherto to do so have been with the aid of harmonic coordinates. It is also conceivable that understanding the properties of and controlling the interface between two fluids could have industrial applications. In particular, there is a version of the problem for plasma physics in magneto-hydrodynamics. As is well known, the ability to control a plasma is essential to the construction of fusion reactors.