Remarks on the breakdown of smooth solutions for the 3-D Euler equations

The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches; equivalently, if the vorticity remains bounded, a smooth solution persists.Partially supported by O.N.R. Contract No. N00014-76-C-0316 and N.S.F. Grant No. MCS-81-01639Partially supported by N.S.F. Grant No. MCS-82-00171Partially supported by N.S.F. Grant No. MCS-81-02360     

Structure of the electromagnetic fields of loop radiators in magnetized plasmas over the whistler frequency range

The role of different sections of the spatial spectrum of whistler waves excited by a loop antenna in the formation of an electromagnetic field structure is investigated experimentally and theoretically. Primary attention is given to the intermediate range of distances between the source and the observation point.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 2, pp. 192–202, February, 1996.T. M. Zaboronkova, A. V. Kostrov, A. I. Smirnov, and A. A. Shaikin are grateful to The Russian Foundation for Fundamental Research (Grant No.94-02-05447a) and the George Soros International Foundation (Grant No. NOP 300) for financial support. A. V. Kudrin is grateful to the Russian Fund for Fundamental Research (Grant No. 96-02-18666) for financial support.     

Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit

Equivalence (allowing for convex combinations) of microcanonical, canonical and grandcanonical ensembles for states of classical systems is established under very mild assumptions on the limiting state. We introduce the notion of conditional equilibrium (C.E.), a property of states of infinite systems which characterizes convex combinations of limits of microcanonical ensembles. It is shown that C.E. states are, under quite general conditions, mixtures of Gibbs states.Supported in part by NSF Grant No. MCS 75-21684 A02Supported in part by NSF Grant No. MPS 72-04534Supported in part by NSF Grant No. Phy 77-22302     

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry. Research supported by NSERC grant # RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

On the plasma dynamics in solar magnetic tubes

On the basis of a known selfsimilar solution of the system of MHD equations describing plasma flows with homogeneous deformation, a dynamic model of a solar magnetic tube has been developed. A general form of the energy equation allowing for Joule heating, radiative cooling, and effects related to thermal conductivity and viscosity has been used in the construction of this model. Various dynamic regimes of magneticloop evolution are studied. Of great interest are the processes of flare-like fast compression and heating of plasma in the tube, the process of plasma compression accompanied by cooling, which corresponds to prominence formation in the tube, as well as various oscillatory regimes and processes of plasma decompression accompanied by both its cooling and heating. The model has been analytically and numerically studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 1, pp. 53–79, January, 1996.The author thanks V.V. Zaytsev for useful advice and discussions. This paper was partially supported by the Russian Foundation for Fundamental Research (grants No. 93-02-3008 and No. 95-02-04272-a) and by grants from the Southern-European Observatory (No. A-01-080) and the International Scientific Foundation (No. R92000, No. R92300).     

Decay of Solutions of the Wave Equation in the Kerr Geometry

We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in time in L ^∞ _loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable ω on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables. Research supported in part by the Deutsche Forschungsgemeinschaft. Research supported by NSERC grant #RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30. An erratum to this article is available at .     

An identification of the connections of Quillen and Beilinson-Schechtman

Given a family of Riemann surfaces and a holomorphic vector bundle Beilinson and Schechtman construct a canonical connection on the associated determinant bundle. We prove the conjecture which states that their connection coincides with the Quillen connection. This is done by reducing to the case where along fibers are invertible. Both connection forms become more accessible in this case.Supported in part by N.S.F. Grant No. DMS-9201022Supported in part by National Science Council of Republic of China Grant No. NSC 82-0208-M-002-125-T, and NSERC of Canada Grant No. OGP 0121883     

Smooth static solutions of the Einstein/Yang-Mills equations

We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions withSU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.Research supported in part by the NSF, Contract No. DMS 89-05205Research supported in part by the ONR, Contract No. DOD-C-N-00014-88-K-0082Research supported in part by the DOE, Grant No. DE-FG02-88ER25065Research supported in part by the U.K. Science and Engineering Research Council     

Influence of the intensity of chemical heat-release source on the heat exchange at a flow of pseudo-plastic liquid in the initial interval of coaxial duct

The critical regimes of heat exchange at a laminar steady flow of a pseudo-plastic liquid in the initial interval of a coaxial duct are investigated with regard for both dissipative and chemical sources in the Arrhenius representation [5] under the conditions of an insignificant variation of the concentrations of reacting substances. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 05-08-50043), FANI (State Contract No. 02.434.11.5009), and the Presidium of RAS (Program P-09).     

On the second-moment condition of Stillinger and Lovett

The Stillinger-Lovett second-moment condition of electrolyte solutions is derived rigorously and simply from only some reasonable (but apparently never proven rigorously) assumptions concerning the asymptotic form of the direct correlation function and the Ornstein-Zernike equation. The derivation suggests that this condition is not the first member of a hierarchy of moment conditions and that there exists no simple result for a fourth-moment condition.Alfred P. Sloan Fellow.This work was partially supported by a grant from the National Institutes of Health, RO1 GM 20800-03 (to D.A.M.), by the Research Foundation of the State University of New York (USAEC Contract No. AT (30-1) 3668B) and the USAFOSR under Grant No. 68-1416B (to J.G.), and by the Petroleum Research Fund (A.S.).     

On the probability properties of the density gradient of a randomly moving incompressible medium

We study the statistical properties of the density-field gradient of a passive admixture in a randomly moving medium. A expression for the relative gradient-variation field is derived. It is shown that the moments and practically all realizations of the relative gradient-variation field increase exponentially with time.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 5, pp. 597–606, May, 1996.This work was partially supported by Russian Foundation for Fundamental Research grant No. 95-05-14247.     

Measuring deformations of the heated liquid film by the fluorescence method

The fluorescence method was used to measure the instantaneous thickness field of the falling nonisothermal water film. The process of rivulet formation in a heated film was registered. Measurement averaging allowed determination of the degree of transverse deformation of a film. In the lower half of the heater within the interrivulet zone of the non-isothermal film, the wave amplitude decreases with a rise of the heat flux and reduction of the average thickness. Two zones of the heat flux effect on liquid film deformation were distinguished. At low heat fluxes, the film flow is weakly deformed. At high heat fluxes the thermal-capillary forces provide formation of rivulets and a thin film between them. The work was financially supported by the President of RF (NSh-6749.2006.8), Russian Foundation for Basic Research (Grants Nos. 05-08-33325-a, 06-01-00360-a), National Center on Science and Innovations (State contract No. 02.438.11.7002), and SB RAS (Interdisciplinary Integration Project No. 111).     

Uniform expansion of the transition rate in Kramers' problem

Kramers' model of diffusion over potential barriers, e.g., chemical reactions, based on the noise activated escape of a particle from a potential well, is considered. Kramers derived escape rates valid for intermediate and large damping, and in a separate analysis, for small damping. In the small damping limit, Kramers' intermediate result reduces to the transition state rate which does not agree with the small damping result. A new escape rate is derived that is uniformly valid for all values of the damping coefficient. The new rate reduces to Kramers' results in the appropriate limits and, in particular, connects Kramers' intermediate and small damping results.This work was partially supported by the Air Force Office of Scientific Research under Grant No. AFOSR-83-0086, U.S. Department of Energy under Grant No. DE-AC02-78ERO-4650, and the National Science Foundation under Grant No. MCS-83-00562. One of us (BJM) gratefully acknowledges the support of a John Simon Guggenheim Memorial Foundation Fellowship.     

Existence of conserved currents in the perturbative sine-Gordon and massive thirring models

The existence of infinitely many conserved currents in the quantized sine-Gordon and massive Thirring models is proved in renormalized perturbation theory.Research supported in part by National Science Foundation Grant No. PHY 74-22218 A03Research supported in part by National Science Foundation Grant No. MP 574-05783 A01     

Sp(2)-covariant quantization of the Freedman-Townsend model

A model antisymmetric rank two tensor field is quantized in both the Lagrangian and the Hamiltonian versions of the expanded BRST quantization. It is demonstrated that the Lagrangian quantization results in a unitary S-matrix.Study supported in part by the Russian Fund for Fundamental Research, Project No. 94-02-03234, and by the International Science Fund, Grant RI 1000.Tomsk Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 72–78, February, 1995.     

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.Research supported in part by NSF Grant No. MCS-76-05857Research supported in part by NSF Grant No. MCS-74-07313-A02     

Existence of black hole solutions for the Einstein-Yang/Mills equations

This paper provides a rigorous proof of the existence of an infinite number of black hole solutions to the Einstein-Yang/Mills equations with gauge groupSU(2), for any event horizon. It is also demonstrated that the ADM mass of each solutions is finite, and that the corresponding Einstein metric tends to the associated Schwarzschild metric at a rate 1/r ², asr tends to infinity.Research supported in part by the NSF, Contract No. DMS-89-05205Research supported in part by the DE, Contract No. De-FG 02-88 EF 25065     

Analysis of the properties of heat radiation of the axisymmetric semitransparent systems with permeable boundaries

Numerical analysis of hemispherical values of absorptive, transmitting, and reflective abilities of a flat layer and absorptivity of a sphere is presented in this paper depending on the optic thickness and refractive index. The work was financially supported by the Russian Foundation for Basic Research (Grant No. 08-08-00587) and the grant of the President of the Russian Federation for young scientists (Grant MK-601.2008-8).     

On the Cauchy problem of the relativistic Boltzmann equation

It is shown that the relativistic Boltzmann equation has a local solution through an initial distribution function, if the scattering cross section is bounded for high energies and if the initial distribution falls off exponentially with the energy.Supported by the National Aeronautics and Space Administration under Grant No. NGR 44-004-042.On leave from the University of Heidelberg.     

Stability of the nakedness of Weyl singularities

We investigate the stability of the nakedness of the Weyl singularities against matter perturbations by considering the effects of infalling test matter on the convergence of outgoing null rays. We show that the additional convergence induced by infalling test matter does not blow up sufficiently fast to reconverge diverging outgoing rays, at least in the equator, and the nakedness seems to be stable in this limited sense. Supported in part by the National Science Foundation, grant No. PHY 76-21454 and by NASA, grant No. NSG 7204.