Sharp Decay Research Articles

A MULTILINEAR EXTENSION INEQUALITY IN $\mathbb{R}^{n}$

Sharp decay estimates are provided in this paper for spherical averages of a certain multilinear extension operator on L2 (Sn−1) × … × L2 (Sn−1). 2000 Mathematics Subject Classification 42B10.

Polynomial decay and control of a 1− d hyperbolic–parabolic coupled system

In this paper we consider a linearized model for fluid–structure interaction in one space dimension. The domain where the system evolves consists in two parts in which the wave and heat equations evolve, respectively, with transmission conditions at the interface. First of all we develop a careful spectral asymptotic analysis on high frequencies for the underlying semigroup. It is shown that the semigroup governed by the system can be split into a parabolic and a hyperbolic projection. The dissipative mechanism of the system in the domain where the heat equation holds produces a slow decay of the hyperbolic component of solutions. According to this analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Next, we discuss the problem of null-controllability of the system when the control acts on the boundary of the domain where the heat equation holds. The key observability inequality of the dual system with observation on the heat component is derived though a new Ingham-type inequality, which in turn, thanks to our spectral analysis, is a consequence of a known observability inequality of the same system but with observation on the wave component.

A Sharp Decay Estimate for Positive Nonlinear Waves

We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers's equation with impulsive sources.

Three-body Coulomb breakup of 11Li in the complex scaling method

Coulomb breakup strengths of 11Li into a three-body 9Li+n+n system are studied in the complex scaling method. We decompose the transition strengths into the contributions from three-body resonances, two-body “10Li+n” and three-body “9Li+n+n” continuum states. In the calculated results, we cannot find the dipole resonances with a sharp decay width in 11Li. There is a low energy enhancement in the breakup strength, which is produced by both the two- and three-body continuum states. The enhancement given by the three-body continuum states is found to have a strong connection to the halo structure of 11Li. The calculated breakup strength distribution is compared with the experimental data from MSU, RIKEN and GSI.

A bilinear extension inequality in two dimensions

We provide sharp decay estimates for circular averages of a certain bilinear extension operator on L 2( S 1)×L 2( S 1) .

Control, observation and polynomial decay for a coupled heat-wave system

This Note is devoted to study the control, observation and polynomial decay of a linearized 1-d model for fluid–structure interaction, where a wave and a heat equation evolve in two bounded intervals, with natural transmission conditions at the point of interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. The controllability and observability of the system through the wave component are derived from sidewise energy estimate and Carleman inequalities. As for the control and observation through the heat component, we need to develop first a careful spectral high frequency analysis for the underlying semigroup, which yields a new Ingahm-type inequality. It is shown that the controllable/observable subspace for both cases are quite different. Also, we obtain a sharp polynomial decay rate for the energy of smooth solutions. To cite this article: X. Zhang, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Polynomial decay and control of a 1− d model for fluid–structure interaction

We consider a linearized and simplified 1− d model for fluid–structure interaction. The domain where the system evolves consists in two bounded intervals in which the wave and heat equations evolve respectively, with transmission conditions at the point of interface. First, we develop a careful spectral asymptotic analysis on high frequencies. Next, according to this spectral analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Finally, we prove the null-controllability of the system when the control acts on the boundary of the interval where the heat equation holds. The proof is based on a new Ingham-type inequality, which follows from the spectral analysis we develop and the null controllability result in Zuazua (in: J.L. Menaldi et al. (Eds.), Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 198–210) where the control acts on the wave component. To cite this article: X. Zhang, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

On L1 decay problem for the dissipative wave equation

AbstractWe study the decay estimates of solutions to the Cauchy problem for the dissipative wave equation in one, two, and three dimensions. The representation formulas of the solutions provide the sharp decay rates on L1 norms and also Lp norms. Copyright © 2003 John Wiley & Sons, Ltd.

Nonlinear Wave Equations in Infinite Waveguides

We present sharp decay rates as time tends to infinity for solutions to linear Kleinc-Gordon and wave equations in domains with infinite boundaries like infinite waveguides, as well as the global well-posedness and the asymptotics for small data for the solutions to the associated nonlinear initial-boundary value problems.

Sharp Lr asymptotics of the small solutions to the nonlinear Schrödinger equations

This paper studies the large time behavior of the small solution to the nonlinear Schrödinger equation with power type nonlinearity. If the power is large enough, then it is well known that the nonlinear solution asymptotically behaves like a linear solution as t→±∞ (see e.g. (J. Funct. Anal. 32 (1979) 1; J. Math. Pures Appl. 64 (1985) 363)). Our concern at the present work is to determine the sharp decay rate of the difference between the nonlinear solution and the linear solution in L r (2⩽r⩽∞) spaces.

Comment to ``Remarks on modification of Helgason's support theorem. II'' by T. Takiguchi

We slightly extend a uniqueness theorem for generalized Radon transforms $f \mapsto Rf$ proved in [B2], by giving sharp decay conditions on the function $f$ at infinity.

Diffusion and thermal diffusion of semidilute to concentrated solutions of polystyrene in toluene in the vicinity of the glass transition.

The approaching glass transition in polystyrene/toluene solutions leads to a sharp decay of both the collective diffusion coefficient D and the thermal diffusion coefficient D(T) at concentrations above 0.2 g/cm(3). The Soret coefficient S(T) = D(T)/D follows power-law scaling from semidilute to concentrated and is not influenced by the slowing down of the dynamics associated with the glass transition. Both D and D(T) are governed by the same friction coefficient. The scaling behavior of S(T) with concentration on approach of the glass transition is compared to the divergence of S(T) near a consolute critical point.

Deterministic non-linear source processes of volcanic tremor signals accompanying the 1996 Vatnajökull eruption, central Iceland

SUMMARY Observations and theoretical considerations have cast doubt on the suggestion that volcanic tremor source processes may be modelled by a linear oscillator that is set into resonance by a sustained disturbance. Volcanic tremor signals that accompanied the 1996 Vatnajokull subglacial eruption, central Iceland, have been analysed using methods from the discipline of non-linear dynamics in order to investigate the possibility that they originated from a non- linear source. The volcano-seismic phenomena associated with the eruption were recorded by a permanent network equipped with broad-band seismometers (HOTSPOT) using a sampling rate of 20 samples s −1 . The eruption was preceded by increased seismic activity for a period of 2 days, which also included a large earthquake with a moment magnitude of 5.6. The tremor during the first 2 days of the eruption has a high signal-to-noise ratio at the nearest station to the eruption site and starts as a continuous signal, later evolving to low-amplitude background tremor interrupted by high-amplitude, cigar-shaped bursts having an average duration of 250 s. The phase space, which describes the evolution of the behaviour of a non-linear system, was reconstructed from the original tremor seismograms using the delay embedding theorem suggested by Takens. The delay time used for the reconstruction was selected after examining the autocorrelation function, which showed a first zero crossing at a timelag of 4 samples and the average mutual information that showed no minimum, indicating that the tremor process may have been undersampled. Based also on phase space portraits for different delay times, a delay time of one sample interval (0.05 s) was used. The sufficient embedding dimension for phase space reconstruction was selected by applying the false nearest-neighbours method, which revealed complete unfolding of the tremor attractor at dimensions 7-8, implying upper bounds of its fractal dimension in the range 3.5-4.0. The phase space prediction errors of different segments of the tremor time-series were compared in order to check whether the attractor dynamics change substantially with time. It was found that for continuous tremor there was almost no dynamic variation, in contrast to the background tremor and the superposed bursts that gave a maximum prediction error when the former was used to predict the latter. This difference in dynamics also had an effect on their spectra: the amplitude spectrum of a burst or continuous tremor has a much sharper decay at high frequencies than that of the background tremor. A possible physical mechanism that may explain these observed characteristics involves turbulent slug flow of magma in a narrow cylindrical conduit, generating the different dynamic regimes as the Reynolds number varies.

Hole Transport in Self-Organized Oligosilane Thin Films with Highly Ordered Hopping Sites

ABSTRACTSelf-organized oligosilane thin films possess molecular orientation normal to substrates with multilayered structure. This unique order of σ-conjugated molecules results in good hole transport properties. In the present work, carrier transport properties at low temperature are studied for 1,10-diethyldecamethylsilane polycrystalline films. Even at a temperature as low as 173 K, a time-of-flight transient photocurrent waveform showed a clear plateau and a sharp decay, whose shape is similar to that at room temperature. Their hole mobility followed Arrhe-nius type temperature dependence with a small activation energy of 0.09 eV. The hole mobility of 6.3×10-5cm-2/Vs at 193 K was more than 2 orders of magnitude higher than that of typical polysilanes, which inevitably contain disordered structures hindering smooth carrier transport.

Hole Transport In Self-Organized Oligosilane Thin Films With Highly Ordered Hopping Sites

ABSTRACTSelf-organized oligosilane thin films possess molecular orientation normal to substrates with multilayered structure. This unique order of -conjugated molecules results in good hole transport properties. In the present work, carrier transport properties at low temperature are studied for 1,10-diethyldecamethylsilane polycrystalline films. Even at a temperature as low as 173 K, a time-of-flight transient photocurrent waveform showed a clear plateau and a sharp decay, whose shape is similar to that at room temperature. Their hole mobility followed Arrhenius type temperature dependence with a small activation energy of 0.09 eV. The hole mobility of 6.3×10-5cm2/Vs at 193 K was more than 2 orders of magnitude higher than that of typical polysilanes, which inevitably contain disordered structures hindering smooth carrier transport.

Large time behavior of solutions to the Cauchy problem for one-dimensional thermoelastic system with dissipation

In this paper we investigate the large time behavior of solutions to the Cauchy problem on R for a one-dimensional thermoelastic system with dissipation. When the initial data is suitably small, (S. Zheng, Chin. Ann. Math. 8B (1987), 142-155)established the global existence and the decay properties of the solution. Our aim is to improve the results and to obtain the sharper decay properties, which seems to be optimal. The proof is given by the energy method and the Green function method.

The nuclear spin–spin coupling constant in He2

The scalar nuclear spin–spin coupling constant was calculated for the helium dimer using full configuration interaction (FCI) and EOM-CCSD methods. The Fermi-contact was found to have nonnegligible value of 1.3 Hz at R=5.6 a.u., while the other contributions are zero. This suggests that the nuclear spin–spin coupling constants transmitted through van der Waals interactions are a much more common phenomenon than previously thought. All contributions exhibit sharp exponential decay with the internuclear distance.

Improvement of luminescence efficiency and photostability in polymer thin films

The photodegradation mechanism and luminescence efficiency of a series of thin polymer films prepared under a variety of conditions was studied. The conjugated polymer, poly( m-phenylene-co-2,5-dioctoxy- p-phenylenevinylene) (PmPV), is shown by infra-red spectroscopy to degrade via the chain scission of the carbon double bond along the polymer backbone. This causes a reduction in conjugation length and a blue shift in its absorption and photoluminescence (PL) spectra. To reduce photodegradation effects, films were prepared using argon (Ar) gas and were investigated in air and an oxygen free environment. The initial PL intensity increased by over 70% for Ar treated films. The PL decay in air was bi-exponential in nature, with a sharp initial decay linked to atmospheric oxygen, and a longer second decay linked to oxygen embedded in the film. The increase in both PL efficiency and degradation lifetime, coupled with device encapsulation, should significantly improve the performance of electroluminescent devices.

Investigation of efficiency and photostability in polymer films

The photodegradation mechanism and luminescence efficiency of a series of thin polymer films prepared under a variety of conditions was studied. The conjugated polymer, poly( m-phenylene-co-2,5-dioctoxy- p-phenylenevinylene), is shown by infrared spectroscopy to degrade via the chain scission of the carbon double bond along the polymer backbone. To reduce photodegradation effects films were prepared using argon (Ar) gas and were investigated in air and an oxygen free environment. The initial photolumiscence (PL) efficiency increased by 70% for Ar treated films. The PL decay in air was bi-exponential in nature, with a sharp initial decay linked to atmospheric oxygen and a longer second decay linked to oxygen embedded in the film. This novel preparation technique coupled with device encapsulation should significantly improve both device efficiency and lifetime.

Overall Evolution of Jetted Gamma‐Ray Burst Ejecta

Whether gamma-ray bursts are highly beamed or not is a very difficult but important problem that we are confronted with. Some theorists suggest that beaming effect usually leads to a sharp break in the afterglow light curve during the ultrarelativistic phase, with the breaking point determined by γ = 1/θ0, where γ is the Lorentz factor of the blast wave and θ0 is the initial half-opening angle of the ejecta, but numerical studies tend to reject the suggestion. We note that previous studies are uniformly based on dynamics that is not proper for nonrelativistic blast waves. Here we investigate the problem in more detail, paying special attention to the transition from the ultrarelativistic phase to the nonrelativistic phase. Due to some crucial refinements in the dynamics, we can follow the overall evolution of a realistic jet until its velocity is as small as βc ~ 10-3c. We find no obvious break in the optical light curve during the relativistic phase itself. However, an obvious break does appear at the transition from the relativistic phase to the Newtonian phase if the physical parameters involved are properly assumed. Generally speaking, the Newtonian phase is characterized by a sharp decay of optical afterglows, with the power-law timing index α ~ 1.8-2.1. This is due to the quick lateral expansion at this stage. The quick decay of optical afterglows from GRB 970228, 980326, and 980519 and the breaks in the optical light curves of GRB 990123 and 990510 may indicate the presence of highly collimated gamma-ray burst ejecta.