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

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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.