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We consider Bose gases consisting of N particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of order N−1 (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as N→∞. Our results confirm Bogoliubov’s predictions.

In a recent paper we presented a model for a unified quantization of gravity with other fundamental sources of nature. After quantization we had to solve a Wheeler-DeWitt equation which was a hyperbolic equation in a fiber bundle which was equipped with a Lorentzian metric with a time function $t$ ranging from $0$ to infinity. The Lorentzian metric had a big bang singularity in $t=0$ and the coefficients of the hyperbolic operator also inherited this singularity. The solutions of the Wheeler-DeWitt equation were products of spatial and temporal eigenfunctions and in order to prove that these solutions also experience a big bang singularity in $t=0$ the temporal eigenfunctions $w(t)$ had to become unbounded if $t$ tends to $0$. In our former paper this was only proved in special cases where $w$ could be expressed with the help of Bessel functions but not in general. In the present paper we prove that also in the general case the temporal solutions become unbounded near $t=0$, or more precisely: $\lim_{t\ra 0} (\abs w^2+t^2 \abs{\dot w}^2)=\un$ and $\limsup_{t\ra 0}\abs w^2=\un$.

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We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field, which could be either vacuum or nonvacuum. The initial density is allowed to vanish, and the spatial measure of the set of vacuum can be arbitrarily large; in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states. © 2012 Wiley Periodicals, Inc