The Cauchy Problem for the Einstein Equations

Various aspects of the Cauchy problem for the Einstein equations are surveyed, with the emphasis on local solutions of the evolution equations. Particular attention is payed to giving a clear explanation of conceptual issues which arise in this context. The question of producing reduced systems of equations which are hyperbolic is examined in detail and some new results on that subject are presented. Relevant background from the theory of partial differential equations is also explained at some lengthComment: 98 page

Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed

Viscoacoustic squeeze-film force on a rigid disk undergoing small axial oscillations

This paper investigates the air flow induced by a rigid circular disk or piston vibrating harmonically along its axis of symmetry in the immediate vicinity of a parallel surface. Previous attempts to characterize these so-called 'squeeze-film' systems largely relied on simplifications afforded by neglecting either fluid acceleration or viscous forces inside the thin enclosed gas layer. The present viscoacoustic analysis employs the asymptotic limit of small vibration amplitudes to investigate the flow by systematic reduction of the Navier-Stokes equations in two distinct flow regions, namely, the inner gaseous film where streamlines are nearly parallel to the confining walls and the near-edge region of non-slender flow that features gas exchange with the surrounding stagnant atmosphere. The flow in the gaseous film depends on the relevant Stokes number, defined as the ratio of the characteristic viscous time across the film to the characteristic oscillation time, and on a compressibility parameter, defined as the square of the ratio of the acoustic time for radial pressure equilibration to the oscillation time. A Strouhal number based on the local residence time emerges as an additional governing parameter for the near-edge region, which is incompressible at leading order. The method of matched asymptotic expansions is used to describe the solution in both regions, across which the time-averaged pressure exhibits comparable variations that give opposing contributions to the resulting time-averaged force experienced by the disk or piston. A diagram structured with the Stokes number and compressibility parameter as coordinates reveals that this steady squeeze-film force, typically repulsive for small values of the Stokes number, alternates to attraction across a critical separation contour in the parametric domain that exists for all Strouhal numbers. This analysis provides, for the first time, a unifying viscoacoustic theory of axisymmetric squeeze films, which yields a reduced parametric description for the time-averaged repulsion/attraction force that is potentially useful in applications including non-contact fluid bearings and robot locomotion

Time correlation functions of equilibrium and nonequilibrium Langevin dynamics: Derivations and numerics using random numbers

We study the time correlation functions of coupled linear Langevin dynamics without and with inertia effects, both analytically and numerically. The model equation represents the physical behavior of a harmonic oscillator in two or three dimensions in the presence of friction, noise, and an external field with both rotational and deformational components. This simple model plays pivotal roles in understanding more complicated processes. The presented analytical solution serves as a test of numerical integration schemes, its derivation is presented in a fashion that allows to be repeated directly in a classroom. While the results in the absence of fields (equilibrium) or confinement (free particle) are omnipresent in the literature, we write down, apparently for the first time, the full nonequilibrium results that may correspond, e.g., to a Hookean dumbbell embedded in a macroscopically homogeneous shear or mixed flow field. We demonstrate how the inertia results reduce to their noninertia counterparts in the nontrivial limit of vanishing mass. While the results are derived using basic integrations over Dirac delta distributions, we mention its relationship with alternative approaches involving (i) Fourier transforms, that seems advantageous only if the measured quantities also reside in Fourier space, and (ii) a Fokker--Planck equation and the moments of the probability distribution. The results, verified by numerical experiments, provide additional means of measuring the performance of numerical methods for such systems. It should be emphasized that this manuscript provides specific details regarding the derivations of the time correlation functions as well as the implementations of various numerical methods, so that it can serve as a standalone piece as part of education in the framework of stochastic differential equations and calculus.Comment: 35 pages, 5 figure

Critical review of the trailing edge condition in steady and unsteady flow. Blade flutter in compressors and fans: Numerical simulation of the aerodynamic loading

Existing interpretations of the trailing edge condition, addressing both theoretical and experimental works in steady, as well as unsteady flows are critically reviewed. The work of Kutta and Joukowski on the trailing edge condition in steady flow is reviewed. It is shown that for most practical airfoils and blades (as in the case of most turbomachine blades), this condition is violated due to rounded trailing edges and high frequency effects, the flow dynamics in the trailing edge region being dominated by viscous forces; therefore, any meaningful modelling must include viscous effects. The question of to what extent the trailing edge condition affects acoustic radiation from the edge is raised; it is found that violation of the trailing edge condition leads to significant sound diffraction at the tailing edge, which is related to the problem of noise generation. Finally, various trailing edge conditions in unsteady flow are discussed, with emphasis on high reduced frequencies

Kelvin Waves, Klein-Kramers and Kolmogorov Equations, Path-Dependent Financial Instruments: Survey and New Results

We discover several surprising relationships between large classes of seemingly unrelated foundational problems of financial engineering and fundamental problems of hydrodynamics and molecular physics. Solutions in all these domains can be reduced to solving affine differential equations commonly used in various mathematical and scientific disciplines to model dynamic systems. We have identified connections in these seemingly disparate areas as we link together small wave-like perturbations of linear flows in ideal and viscous fluids described in hydrodynamics by Kevin waves to motions of free and harmonically bound particles described in molecular physics by Klein-Kramers and Kolmogorov equations to Gaussian and non-Gaussian affine processes, e.g., Ornstein-Uhlenbeck and Feller, arising in financial engineering. To further emphasize the parallels between these diverse fields, we build a coherent mathematical framework using Kevin waves to construct transition probability density functions for problems in hydrodynamics, molecular physics, and financial engineering. As one of the outcomes of our analysis, we discover that the original solution of the Kolmogorov equation contains an error, which we subsequently correct. We apply our interdisciplinary approach to advance the understanding of various financial engineering topics, such as pricing of Asian options, volatility and variance swaps, options on stocks with path-dependent volatility, bonds, and bond options. We also discuss further applications to other exciting problems of financial engineering.Comment: 76 page