Global existence of near-affine solutions to the compressible Euler equations

We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris found a class of expanding finite degree-of-freedom global-in-time affine solutions, obtained by solving nonlinear ODEs. In three space dimensions, the stability of these affine solutions, and hence global existence of solutions, was established by Had\v{z}i\'{c} \& Jang with the pressure-density relation $p = \rho^\gamma$ with the constraint that $1< \gamma\le {\frac{5}{3}}$. They asked if a different approach could go beyond the $\gamma > {\frac{5}{3}}$ threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all $\gamma >1$, thus also establishing global existence for the shallow water equations when $\gamma=2$.Comment: 51 pages, details added to Section 4.7, to appear in Arch. Rational Mech. Ana

Asymptotic bahavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions

We consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. Under the null condition for such systems, the global existence of small amplitude solutions is known. In this paper, we will show that the global solution is asymptotically free in the energy sense, by obtaining the asymptotic pointwise behavior of the derivatives of the solution. Nonetheless we can also show that the pointwise behavior of the solution itself may be quite different from that of the free solution. In connection with the above results, a theorem is also developed to characterize asymptotically free solutions for wave equations in arbitrary space dimensions.Comment: The final version. 30 page

Chaos and the continuum limit in the gravitational N-body problem. I. Integrable potentials

This paper summarises a numerical investigation of the statistical properties of orbits evolved in `frozen,' time-independent N-body realisations of smooth, time-independent density distributions, allowing for 10^(2.5)<N<10^(5.5). Two principal conclusions were reached: (1) In the limit of a nearly `unsoftened' two-body kernel, the value of the largest Lyapunov exponent does NOT appear to decrease systematically with increasing N. (2) Nevertheless, there is a clear quantifiable sense in which, on the average, as N increases chaotic orbits in the frozen-N systems come to more closely approximate characteristics in the smooth potential. When viewed in configuration or velocity space, or as probed by collisionless invariants like angular momentum, frozen-N orbits typically diverge from smooth potential characteristics as a power law in time on a time scale proportional to N^(1/2)t_D, with t_D a characteristic dynamical, or crossing, time

Optimum Currency Areas Structural Changes and the Endogeneity of the OCA Criteria: Evidence from Six New EU Member States

The present paper has two aims. The first aim is to test whether six new member states of the European Union (the six Central and Eastern European Countries) form an optimum currency area (OCA) with the eurozone, in an attempt to assess their readiness for euro adoption. The second aim is to examine whether the introduction of the euro in 1999 and the decision of the countries to seek to join the euro area created any forces fostering their convergence, evidence which would be in line with the theory on the endogeneity of the OCA criteria. Our findings indicate that the introduction of the euro did promote integration of the six new member states and that, at present, they are quite well aligned with the eurozone.EU enlargement; OCA; real exchange rates; cointegration; GPPP.

Real Exchange Rates over a Century: The Case of the Drachma/Sterling Rate, 1833-1939

Recent studies on real exchange rates advocate the use of long samples in order to reveal the low frequency properties of the processes. The present paper contributes to this strand of the literature by exploiting recently released time series for the drachma/sterling rate for the period 1833-1939. This is an interesting period as it covers different exchange rate regimes and the effects of important historical events. In the paper, the mean-reverting behaviour of the real drachma/sterling exchange rate is initially examined applying univariate unit root tests and then the validity of Purchasing Power Parity (PPP) is tested using cointegration analysis. The results provide support for a weak PPP relationship, which turns out to be robust across different sub-periods characterised by different exchange rate regimes. Adjustment to PPP is reached at a relatively high speed and occurs via movements of the nominal exchange rate.real exchange rates; cointegration; PPP

Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum

The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $\rm{GL}^+(3,\mathbb R)$. The evolution of the fluid domain is described by a family ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank $r=1$, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along $3-r$ of its principal axes. In the compressible case, the asymptotic limit has rank $r=3$, and asymptotic completeness holds, when the adiabatic index $\gamma$ satisfies $4/3<\gamma<2$. The number of possible degeneracies, $3-r$, increases with the value of the adiabatic index $\gamma$. In the incompressible case, affine motion reduces to geodesic flow in $\rm{SL}(3,\mathbb R)$ with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes

Promoting Intermodal Connectivity at California’s High Speed Rail Stations

High-speed rail (HSR) has emerged as one of the most revolutionary and transformative transportation technologies, having a profound impact on urban-regional accessibility and inter-city travel across Europe, Japan, and more recently China and other Asian countries. One of HSR’s biggest advantages over air travel is that it offers passengers a one-seat ride into the center of major cities, eliminating time-consuming airport transfers and wait times, and providing ample opportunities for intermodal transfers at these locales. Thus, HSR passengers are typically able to arrive at stations that are only a short walk away from central business districts and major tourist attractions, without experiencing any of the stress that car drivers often experience in negotiating such highly congested environments. Such an approach requires a high level of coordination and planning of the infrastructural and spatial aspects of the HSR service, and a high degree of intermodal connectivity. But what key elements can help the US high-speed rail system blend successfully with other existing rail and transit services? That question is critically important now that high-speed rail is under construction in California. The study seeks to understand the requirements for high levels of connectivity and spatial and operational integration of HSR stations and offer recommendations for seamless, and convenient integrated service in California intercity rail/HSR stations. The study draws data from a review of the literature on the connectivity, intermodality, and spatial and operational integration of transit systems; a survey of 26 high-speed rail experts from six different European countries; and an in-depth look of the German and Spanish HSR systems and some of their stations, which are deemed as exemplary models of station connectivity. The study offers recommendations on how to enhance both the spatial and the operational connectivity of high-speed rail systems giving emphasis on four spatial zones: the station, the station neighborhood, the municipality at large, and the region

Testing for Long-Run PPP in a System Context: Evidence for the US, Germany and Japan

The present paper tests for the validity of long-run purchasing power parity (PPP) for the three key currencies of the recent floating exchange rate period, the US dollar, the German mark and the Japanese yen. The novelty of the paper is that the validity of the PPP conditions relating the economies of the US, Germany and Japan is tested in a system framework, which allows for possible interactions in the determination of the exchange rates and the prices of the three economies. Some form of causality among the variables of the system is also assessed empirically with the aid of weak exogeneity tests. The results illustrate the importance of the multilateral testing. Positive evidence for PPP is found: long-run PPP is supported for the US and Germany but also for the US and Japan, in contrast to evidence of earlier empirical studies. In addition, causality is found running from the US prices to the exchange rates and German and Japanese prices.Money demand; PPP, cointegration, causality