SHOCK-FREE AND SHOCK-WAVE DYNAMCS in GENERAL RELATIVITY and CLASSICAL FLUIDS

在PI之前的资助期间，PI和合作者Robin Young完成了可压缩欧拉方程的非线性时间周期解的显式构造和存在性证明。 这些解描述了新的无冲击波，其通过振荡熵场传播而不破裂或耗散，从而为无耗散长距离信令的可能性提供了理论基础。 波根据一种全新的机制传输信息--压缩和稀疏在每个特性上都是沿着平衡的，波以一种新的速度传播（不同于冲击或声速），声波以与周期相称或不相称的速度穿过周期。 周期决定了波峰的速度（一种可观察到的群速度），但声波以更快的速度移动，通常的声速，这就像相速度。 本建议的第一部分涉及这些新波的数学研究和数值模拟。 特别是，数学表明，波可以计算的扰动系列，这是在封闭的形式给出从线性化的解决方案，显示的非线性波的结构，这是一个拟议的数值模拟的基础。 第二部分讨论了广义相对论（GR）中的冲击波传播问题。 这包括一个程序，以数值模拟一个物理上现实的宇宙冲击波，可能会产生从古斯的著名的暴胀时空。 这里的出发点是之前与PI的合作者Joel Smoller的联合工作，我们介绍了一个物理上可信的，数学上严格的宇宙学标准模型的冲击波细化。 我们建议通过将一个更现实的状态方程，从而解决二次波的问题，在数值上完善这个模型。 我们的目标是使用这种精确的模拟作为与观测数据进行比较的基线。 第二部分还涉及冲击波传播和黑洞形成的GR，和第三部分关注的共振相互作用在经典fluid.The可压缩气体动力学欧拉方程导致伦纳德欧拉在1752年成功地尝试完成牛顿的计划，给一个正确的版本的牛顿运动定律，适用于连续介质。 对欧拉来说，这些方程代表了声波的非线性理论。 欧拉的伟大成就是从物理原理中推导出非线性方程，然后证明在弱信号的极限下，该理论恢复了达朗贝尔原理，即声波在压力下近似为正弦振荡，并根据线性波动方程传播。 因此，我们的新建设的声波的完全非线性方程在第一部分解决了肯定的最基本的问题，人们会问的非线性理论的声波-做非线性方程支持振荡的解决方案，传播像正弦波的线性波动方程？ 有趣的是，这些解似乎无处不在，尽管两个世纪的工作表明，这种时间周期解在数学上可能是不可能的。PI认为，对这些新波的理解将产生重大影响，并可能改变我们对非线性双曲系统中基波传播机制的观点。 当爱因斯坦关于时间和空间的思想被纳入可压缩的欧拉方程时，人们就得到了广义相对论和理想流体的爱因斯坦方程。 在第二部分中，我们提出了一个可能的宇宙激波的数值模拟，是出于我们的想法，即二次（稀疏）波，这将是反射回来的冲击，在改进的模型，可能会考虑到异常加速的星系内的经典GR，没有宇宙学常数。 如果我们的模型是正确的，那么大爆炸后的宇宙开始于一个白色洞（黑洞的时间反转，其中的一切都向外爆炸），并最终演变成类似于经典的局部爆炸的后期阶段：一个有限范围的有限质量，在向外的冲击波后面演变，所有的东西都膨胀到一个空的空间。

In a major breakthrough during PI's prior funding period, the PI and collaborator Robin Young accomplished the explicit construction and proof of the existence of nonlinear time periodic solutions of the compressible Euler equations. These solutions describe new shock-free waves that propagate through an oscillating entropy field without breaking or dissipating, and thereby give a theoretical basis for the possibility of dissipation free, long- distance signaling. The waves transmit information according to a fundamentally new mechanism---compression and rarefaction are balanced along every characteristic, the waves propagate at a new speed, (different from a shock or sound speed), and sound waves move through periods at speeds that can be commensurate or incommensurate with the period. The period determines the speed of the wave crests, (a sort of observable group velocity), but the sound waves move at a faster speed, the usual speed of sound, and this is like a phase velocity. Part I of this proposal is concerned with both the mathematical investigation and numerical simulation of these new waves. In particular, the mathematics shows that the waves can be computed by a perturbation series which is given in closed form starting from linearized solutions that display the structure of the nonlinear waves, and this is the basis for a proposed numerical simulation. Part II addresses problems of shock-wave propagation in General Relativity (GR). This includes a program to numerically simulate a physically realistic cosmological shock-wave that could arise from Guth's well known inflationary spacetime. The point of departure here is prior joint work with PI's collaborator Joel Smoller in which we introduced a physically believable, mathematically rigorous shock-wave refinement of the Standard Model of Cosmology. We propose to numerically refine this model by incorporating a more realistic equation of state and thereby resolve secondary waves in the problem. Our goal is to use this accurate simulation as a baseline for comparison with observational data. Part II also addresses shock-wave propagation and black hole formation in GR, and Part III concerns resonant interactions in classical fluids.The compressible Euler equations of gas dynamics resulted from Leonard Euler's successful attempt in 1752 to complete Newton's program to give a correct version of the Newtonian laws of motion that apply to continuous media. For Euler, these equations represented the nonlinear theory of sound waves. Euler's great achievement was to derive the nonlinear equations from physical principles, and then to show that in the limit of weak signals the theory recovered D'Alembert's principle that sound waves were approximately sinusoidal oscillations in the pressure that propagate according to the linear wave equation. Thus our new construction of sound waves for the fully nonlinear equations in Part I resolves in the affirmative the most basic question one would ask about the nonlinear theory of sound waves---do the nonlinear equations support oscillatory solutions that propagate like the sinusoidal waves of the linear wave equation? Interestingly, these solutions appear to be ubiquitous, despite two centuries of work suggesting that such time periodic solutions might not be mathematically possible. The PI believes that an understanding of these new waves will have a significant impact, and may well change our point of view regarding the fundamental wave propagation mechanisms in nonlinear hyperbolic systems. When Einstein's ideas about time and space are incorporated into the compressible Euler equations, one is led to general relativity and the Einstein equations for a perfect fluid. In Part II, our proposed numerical simulation of a possible cosmic shock wave, is motivated by our idea that a secondary (rarefaction) wave, that would be reflected back from the shock in the refined model, might account for the anomalous acceleration of the galaxies within classical GR, without the cosmological constant. If our model is correct, then the universe after the Big Bang begins inside a White Hole, (the time reversal of a Black Hole in which everything is exploding outward), and eventually evolves into something similar to the late stages of a classical, localized explosion: a finite mass of bounded extent, evolving behind an outgoing shock wave, all expanding into an empty space beyond.