Second-order hyperbolic Fuchsian systems. Gowdy spacetimes and the Fuchsian numerical algorithm

This is the second part of a series devoted to the singular initial value problem for second-order hyperbolic Fuchsian systems. In the first part, we defined and investigated this general class of systems, and we established a well-posedness theory in weighted Sobolev spaces. This theory is applied here to the vacuum Einstein equations for Gowdy spacetimes admitting, by definition, two Killing fields satisfying certain geometric conditions. We recover, by more direct and simpler arguments, the well-posedness results established earlier by Rendall and collaborators. In addition, in this paper we introduce a natural approximation scheme, which we refer to as the Fuchsian numerical algorithm and is directly motivated by our general theory. This algorithm provides highly accurate, numerical approximations of the solution to the singular initial value problem. In particular, for the class of Gowdy spacetimes under consideration, various numerical experiments are presented which show the interest and efficiency of the proposed method. Finally, as an application, we numerically construct Gowdy spacetimes containing a smooth, incomplete, non-compact Cauchy horizon.Comment: 22 pages. A shortened version is included in: F. Beyer and P.G. LeFloch, Second-order hyperbolic Fuchsian systems and applications, Class. Quantum Grav. 27 (2010), 24501

Self-gravitating fluid flows with Gowdy symmetry near cosmological singularities

We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus $T^3$ and, in this class, we solve the singular initial value problem for the Einstein-Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the notion of sub-critical, critical, and super-critical regimes. Solutions to the Einstein-Euler systems when the fluid is governed by a linear equation of state are constructed in the first two regimes, while additional difficulties arise in the latter one. All previous studies on inhomogeneous spacetimes concerned vacuum cosmological spacetimes only.Comment: 41 page

Proximity effect in planar Superconductor/Semiconductor junction

We have measured the very low temperature (down to 30 mK) subgap resistance of Titanium Nitride (Superconductor, Tc = 4.6 K)/highly doped Silicon (Semiconductor) SIN junction (the insulating layer stands for the Schottky barrier). As the temperature is lowered, the resistance increases as expected in SIN junction. Around 300 mK, the resistance shows a maximum and decreases at lower temperature. This observed behavior is due to coherent backscattering towards the interface by disorder in Silicon ("Reflectionless tunneling"). This effect is also observed in the voltage dependence of the resistance (Zero Bias Anomaly) at low temperature (T<300 mK). The overall resistance behavior (in both its temperature and voltage dependence) is compared to existing theories and values for the depairing rate, the barrier resistance and the effective carrier temperature are extracted.Comment: Submitted to LT22, Helsinki - August 1999, phbauth.cls include