scholarly journals Asymptotic profiles for damped plate equations with rotational inertia terms

2020 ◽  
Vol 17 (03) ◽  
pp. 569-589
Author(s):  
Tomonori Fukushima ◽  
Ryo Ikehata ◽  
Hironori Michihisa
Keyword(s):  
Initial Data ◽  
High Frequency ◽  
Wave Equations ◽  
Splitting Method ◽  
Low Frequency ◽  
Rotational Inertia ◽  
Low Regularity ◽  
Frictional Damping ◽  
The Cauchy Problem ◽  
Plate Equations

We consider the Cauchy problem for plate equations with rotational inertia and frictional damping terms. We derive asymptotic profiles of the solution in [Formula: see text]-sense as [Formula: see text] in the case when the initial data have high and low regularity, respectively. Especially, in the low regularity case of the initial data one encounters the regularity-loss structure of the solutions, and the analysis is more delicate. We employ the so-called Fourier splitting method combined with the explicit formula of the solution (high-frequency estimates) and the method due to [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differential Equations 257 (2014) 2159–2177.] (low-frequency estimates). In this paper, we will introduce a new threshold [Formula: see text] on the regularity of the initial data that divides the property of the corresponding solution to our problem into two parts: one is wave-like, and the other is parabolic-like.

scholarly journals High-frequency homogenization for periodic media

2010 ◽  
Vol 466 (2120) ◽  
pp. 2341-2362 ◽  
Author(s):  
R. V. Craster ◽  
J. Kaplunov ◽  
A. V. Pichugin
Keyword(s):  
High Frequency ◽  
Wave Equations ◽  
Standing Waves ◽  
Low Frequency ◽  
Homogenization Theory ◽  
Periodic Media ◽  
Mode Spectrum ◽  
Bloch Mode ◽  
Long Wavelength Asymptotics ◽  
Model Equations

An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.

Non-uniform dependence on initial data for the generalized Degasperis–Procesi equation on the line

2016 ◽  
Vol 27 (03) ◽  
pp. 1650026
Author(s):  
Yanggeng Fu ◽  
Zanping Yu ◽  
Jianhe Shen
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
High Frequency ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Solution Map ◽  
High Frequency Part ◽  
Uniformly Continuous

In this paper, we show that the solution map of the generalized Degasperis–Procesi (gDP) equation is not uniformly continuous in Sobolev spaces [Formula: see text] for [Formula: see text]. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part. It also exploits the fact that the gDP equation conserves a quantity which is equivalent to the [Formula: see text] norm.

scholarly journals On Classical Global Solutions of Nonlinear Wave Equations with Large Data

2017 ◽  
Vol 2019 (19) ◽  
pp. 5859-5913 ◽  
Author(s):  
Shuang Miao ◽  
Long Pei ◽  
Pin Yu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Wave Equations ◽  
Global Solutions ◽  
Large Data ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Classical Solutions ◽  
Linear Wave ◽  
The Cauchy Problem

Abstract This article studies the Cauchy problem for systems of semi-linear wave equations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial energy. The choice of the large Cauchy initial data is inspired by Christodoulou's characteristic initial data in his work [2] on formation of black holes. The main innovation of the current work is that we discovered a relaxed energy ansatz which allows us to prove decay-in-time-estimate. Therefore, the new estimates can also be applied in studying the Cauchy problem for Einstein equations.

scholarly journals BILINEAR ESTIMATES AND APPLICATIONS TO NONLINEAR WAVE EQUATIONS

2002 ◽  
Vol 04 (02) ◽  
pp. 223-295 ◽  
Author(s):  
SERGIU KLAINERMAN ◽  
SIGMUND SELBERG
Keyword(s):  
Systematic Review ◽  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Large Data ◽  
Nonlinear Wave Equations ◽  
Well Posedness ◽  
Bilinear Estimates ◽  
The Cauchy Problem

We undertake a systematic review of results proved in [26, 27, 30-32] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates.

scholarly journals Compressive high-frequency waves riding on an Alfvén/ion-cyclotron wave in a multi-fluid plasma

2011 ◽  
Vol 77 (5) ◽  
pp. 693-707 ◽  
Author(s):  
DANIEL VERSCHAREN ◽  
ECKART MARSCH
Keyword(s):  
Acoustic Waves ◽  
High Frequency ◽  
Large Amplitude ◽  
Wave Equations ◽  
Low Frequency ◽  
Plasma Waves ◽  
Alpha Particles ◽  
Alfven Wave ◽  
Resonant Wave ◽  
High Frequency Waves

AbstractIn this paper, we study the weakly-compressive high-frequency plasma waves which are superposed on a large-amplitude Alfvén wave in a multi-fluid plasma consisting of protons, electrons, and alpha particles. For these waves, the plasma environment is inhomogenous due to the presence of the low-frequency Alfvén wave with a large amplitude, a situation that may apply to space plasmas such as the solar corona and solar wind. The dispersion relation of the plasma waves is determined from a linear stability analysis using a new eigenvalue method that is employed to solve the set of differential wave equations which describe the propagation of plasma waves along the direction of the constant component of the Alfvén wave magnetic field. This approach also allows one to consider weak compressive effects. In the presence of the background Alfvén wave, the dispersion branches obtained differ significantly from the situation of a uniform plasma. Due to compressibility, acoustic waves are excited and couplings between various modes occur, and even an instability of the compressive mode. In a kinetic treatment, these plasma waves would be natural candidates for Landau-resonant wave–particle interactions, and may thus via their damping lead to particle heating.

Stationary nonlinear interaction between high- and low-frequency waves and the generation of density cavities in a plasma

1977 ◽  
Vol 17 (2) ◽  
pp. 185-199 ◽  
Author(s):  
K. Baumgärtel ◽  
K. Sauer
Keyword(s):  
Nonlinear Interaction ◽  
High Frequency ◽  
Wave Equations ◽  
Normal Modes ◽  
Low Frequency ◽  
Cavity Depth ◽  
Ion Acoustic Wave ◽  
Frequency Wave ◽  
Coupled Wave Equations ◽  
Density Disturbance

The nonlinear interaction of a high-frequency wave (transverse or longitudinal) and an ion-acoustic wave in a homogeneous plasma is investigated. The waves are assumed to be simultaneously excited by two localized external sinusoidal disturbances with frequencies ω0 and ωS in the plasma. From the coupled wave equations a system of ordinary differential equations for the spatial change of the amplitudes of the interacting waves (including the primary frequencies ω0, ωS and the mixed frequencies ω0±ωS) and a zero-frequency density disturbance is derived. The spatial development of the high-frequency amplitudes is characterized by localized regions of high intensities, coupled with local depressions of the plasma density. The influence of the initial amplitudes and the frequencies ω0, ωS on the maximum of the primary high-frequency amplitude and the ‘cavity’ depth is shown. The numerical results are compared with conclusions from a simplified model of three resonantly interacting normal modes.

Simulations of high-resolution images of amorphous silicon films

1986 ◽  
Vol 44 ◽  
pp. 544-545
Author(s):  
G. Y. Fan ◽  
J. M. Cowley
Keyword(s):  
Electron Microscope ◽  
High Frequency ◽  
Fourier Transformation ◽  
Amorphous Materials ◽  
Low Frequency ◽  
Chromatic Aberration ◽  
Structure Information ◽  
Frequency Components ◽  
High Resolution Images ◽  
Spatial Frequency Spectrum

It is well known that the structure information on the specimen is not always faithfully transferred through the electron microscope. Firstly, the spatial frequency spectrum is modulated by the transfer function (TF) at the focal plane. Secondly, the spectrum suffers high frequency cut-off by the aperture (or effectively damping terms such as chromatic aberration). While these do not have essential effect on imaging crystal periodicity as long as the low order Bragg spots are inside the aperture, although the contrast may be reversed, they may change the appearance of images of amorphous materials completely. Because the spectrum of amorphous materials is continuous, modulation of it emphasizes some components while weakening others. Especially the cut-off of high frequency components, which contribute to amorphous image just as strongly as low frequency components can have a fundamental effect. This can be illustrated through computer simulation. Imaging of a whitenoise object with an electron microscope without TF limitation gives Fig. 1a, which is obtained by Fourier transformation of a constant amplitude combined with random phases generated by computer.

SEM image sharpness analysis

1996 ◽  
Vol 54 ◽  
pp. 142-143
Author(s):  
M. T. Postek ◽  
A. E. Vladar
Keyword(s):  
High Frequency ◽  
Low Frequency ◽  
Quantitative Information ◽  
Primary Electron ◽  
Image Sharpness ◽  
Critical Dimensions ◽  
Sem Image ◽  
Frequency Changes ◽  
Electron Microscopes ◽  
Scanning Electron

Fully automated or semi-automated scanning electron microscopes (SEM) are now commonly used in semiconductor production and other forms of manufacturing. The industry requires that an automated instrument must be routinely capable of 5 nm resolution (or better) at 1.0 kV accelerating voltage for the measurement of nominal 0.25-0.35 micrometer semiconductor critical dimensions. Testing and proving that the instrument is performing at this level on a day-by-day basis is an industry need and concern which has been the object of a study at NIST and the fundamentals and results are discussed in this paper.In scanning electron microscopy, two of the most important instrument parameters are the size and shape of the primary electron beam and any image taken in a scanning electron microscope is the result of the sample and electron probe interaction. The low frequency changes in the video signal, collected from the sample, contains information about the larger features and the high frequency changes carry information of finer details. The sharper the image, the larger the number of high frequency components making up that image. Fast Fourier Transform (FFT) analysis of an SEM image can be employed to provide qualitiative and ultimately quantitative information regarding the SEM image quality.

Real Ear Sound Pressure Levels Developed by Three Portable Stereo System Earphones

1992 ◽  
Vol 1 (4) ◽  
pp. 52-55 ◽  
Author(s):  
Gail L. MacLean ◽  
Andrew Stuart ◽  
Robert Stenstrom
Keyword(s):  
High Frequency ◽  
Sound Pressure ◽  
Low Frequency ◽  
Test System ◽  
Output Frequency ◽  
Frequency Effects ◽  
Adult Men ◽  
Frequency Responses ◽  
Significant Difference ◽  
Sound Pressure Levels

Differences in real ear sound pressure levels (SPLs) with three portable stereo system (PSS) earphones (supraaural [Sony Model MDR-44], semiaural [Sony Model MDR-A15L], and insert [Sony Model MDR-E225]) were investigated. Twelve adult men served as subjects. Frequency response, high frequency average (HFA) output, peak output, peak output frequency, and overall RMS output for each PSS earphone were obtained with a probe tube microphone system (Fonix 6500 Hearing Aid Test System). Results indicated a significant difference in mean RMS outputs with nonsignificant differences in mean HFA outputs, peak outputs, and peak output frequencies among PSS earphones. Differences in mean overall RMS outputs were attributed to differences in low-frequency effects that were observed among the frequency responses of the three PSS earphones. It is suggested that one cannot assume equivalent real ear SPLs, with equivalent inputs, among different styles of PSS earphones.

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