On admissibility criteria for weak solutions of the Euler equations

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.Comment: 33 pages, 1 figure; v2: 35 pages, corrected typos, clarified proof

The “End of Times” and the Antichrist’s Arrival: The Orthodox Dogmas and Prophecies in the National-Patriotic Media in Post-Soviet Russia

Received 8 December 2020. Accepted 14 May 2021. Published online 9 July 2021.A return of the Orthodox religion and a renaissance of the Russian Orthodox Church gave a way for politically active movements of Orthodox fundamentalists and monarchists. They were obsessed with the idea of the “end of time” and argued that the Antichrist was at the door. The article focuses on several national-patriotic newspapers and their interest to Orthodox prophecies about the end of time, which can be traced from the turn of the 1990s. It is examined who exactly, in what way and for what goals developed and discussed eschatological ideas. The major themes, rhetorical means and key words are scrutinized, which helped consumers to disclose the “enemies of Russia” and to reveal their “perfidious plans” and “harmful actions” aimed at the destruction of Russia and its people. A relationship between this ideology and theological teaching of the end of time is analyzed.The research was supported by the Fundamental and Applied Studies Program of the Ministry of Education and Science of the Russian Federation “The Ethnocultural Diversity of Russian Society and Consolidation of An All-Russian Identity, 2020–2022”, within a project “The Ideological Basis and Practices of Radicalism and Extremism”

Scar Intensity Statistics in the Position Representation

We obtain general predictions for the distribution of wave function intensities in position space on the periodic orbits of chaotic ballistic systems. The expressions depend on effective system size N, instability exponent lambda of the periodic orbit, and proximity to a focal point of the orbit. Limiting expressions are obtained that include the asymptotic probability distribution of rare high-intensity events and a perturbative formula valid in the limit of weak scarring. For finite system sizes, a single scaling variable lambda N describes deviations from the semiclassical N -> infinity limit.Comment: To appear in Phys. Rev. E, 10 pages, including 4 figure

Deviations from Berry--Robnik Distribution Caused by Spectral Accumulation

By extending the Berry--Robnik approach for the nearly integrable quantum systems,\cite{[1]} we propose one possible scenario of the energy level spacing distribution that deviates from the Berry--Robnik distribution. The result described in this paper implies that deviations from the Berry--Robnik distribution would arise when energy level components show strong accumulation, and otherwise, the level spacing distribution agrees with the Berry--Robnik distribution.Comment: 4 page

Young Measures Generated by Ideal Incompressible Fluid Flows

In their seminal paper "Oscillations and concentrations in weak solutions of the incompressible fluid equations", R. DiPerna and A. Majda introduced the notion of measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.Comment: 35 pages. Final revised version. To appear in Arch. Ration. Mech. Ana

Berry's conjecture and information theory

It is shown that, by applying a principle of information theory, one obtains Berry's conjecture regarding the high-lying quantal energy eigenstates of classically chaotic systems.Comment: 8 pages, no figure

Two simple systems with cold atoms: quantum chaos tests and nonequilibrium dynamics

This article is an attempt to provide a link between the quantum nonequilibrium dynamics of cold gases and fifty years of progress in the lowdimensional quantum chaos. We identify two atomic systems lying on the interface: two interacting atoms in a harmonic multimode waveguide and an interacting two-component Bose-Bose mixture in a double-well potential. In particular, we study the level spacing distribution, the wavefunction statistics, the eigenstate thermalization, and the ability to thermalize in a relaxation process as such.Comment: 18 pages, 9 figure

Dissipative continuous Euler flows

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy

Evidence for the Validity of the Berry-Robnik Surmise in a Periodically Pulsed Spin System

We study the statistical properties of the spectrum of a quantum dynamical system whose classical counterpart has a mixed phase space structure consisting of two regular regions separated by a chaotical one. We make use of a simple symmetry of the system to separate the eigenstates of the time-evolution operator into two classes in agreement with the Percival classification scheme \cite{Per}. We then use a method firstly developed by Bohigas et. al. \cite{BoUlTo} to evaluate the fractional measure of states belonging to the regular class, and finally present the level spacings statistics for each class which confirm the validity of the Berry-Robnik surmise in our model.Comment: 15 pages, 9 figures available upon request, Latex fil

Quantum localization in rough billiards

We study the level spacing statistics p(s) and eigenfunction properties in a billiard with a rough boundary. Quantum effects lead to localization of classical diffusion in the angular momentum space and the Shnirelman peak in p(s) at small s. The ergodic regime with Wigner-Dyson statistics is identified as a function of roughness. Applications for the Q-spoiling in optical resonators are also discussed.Comment: revtex, 4 pages, 5 figure