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

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

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.

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

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

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

Planning for Complementarity: An Examination of the Roll and Opportunities of First-Tier and Second-Tier Cities Along the High-Speed Rail Network in California, Research Report 11-17

The coming of California High-Speed Rail (HSR) offers opportunities for positive urban transformations in both first-tier and second-tier cities. The research in this report explores the different but complementary roles that first-tier and second-tier cities along the HSR network can play in making California more sustainable and less dependent on fossil fuels while reducing mobile sources of greenhouse gas emissions and congestion at airports and on the state’s roadways. Drawing from case studies of cities in Northern and Southern California, the study develops recommendations for the planning, design, and programming of areas around California stations for the formation of transit-supportive density nodes