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Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

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Acta Mathematica Sinica,English Series 2005年第5期21卷 1035-1044页

作者： shuang ping tao Shang Bin CUI Department of Mathematics Northwest Normal University Lanzhou 730070 P. R. China Department of Mathematics Zhongshan University Guangzhou 510275 P. R. China

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third ＋α the second partial dervative of u to x ,the second the third ＋β the third partial dervative of u to x ,the second the thire ＋γ the fifth partial dervative of u to x = 0,（x,t）∈R^*** first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo（x） ∈ H^s（R） with s ≥ 1/4, and a global solution exists if s ≥ 2.

关键词： Kawahara equation Initial value problem Solution Local existence Global existence

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

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Acta Mathematica Sinica,English Series 2005年第4期21卷 881-892页

作者： shuang ping tao Shang Bin CUI Department of Mathematics. Northwest Normal University Lanzhou 730070 P. R. China Department of Mathematics Zhongshan University Guangzhou 510275 P. R. China

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.

关键词： Kaup-Kupershmidt equation Initial value problem Solution Local existence Global existence

Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L^2 Initial Data

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Acta Mathematica Sinica,English Series 2006年第5期22卷 1457-1466页

作者： Shang Bin CUI Dong Gao DENG shuang ping tao Department of Mathematics Sun Yat-Sen University Guangzhou 510275 P. R. China Department of Mathematics Northwest Normal University Lanzhou 730070 P. R. China

In this paper we study solvability of the Cauchy problem of the Kawahara equation 偏导dtu ＋ au偏导dzu ＋ β偏导d^3xu ＋γ偏导d^5xu = 0 with L^2 initial data. By working on the Bourgain space X^r,s（R^2） associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H^r（R） and -1 〈 r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L^2（R）.

关键词： Kawahara equation Cauchy problem global solution

A Class of Oscillatory Singular Integrals with Hardy Kernels on Triebel-Lizorkin Spaces and Besov Spaces

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Journal of Mathematical Research and Exposition 2011年第3期31卷 509-520页

作者： Yao Ming NIU shuang ping tao Faculty of Mathematics Baotou teachers college Inner Mongolia 014030 P. R. China College of Mathematics and Information Science Northwest Normal University Gansu 730070 P. R. China

In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.

关键词： oscillatory singular integrals Triebel-Lizorkin spaces Besov spaces Hardy kernel.

Metal reflector-enhanced thermoacoustic imaging as a guidance for puncture biopsy

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Journal of Innovative Optical Health Sciences 2024年第6期17卷 83-96页

作者： shuang Du tao Qiang Zihui Chi Huabei Jiang School of Optoelectronic Engineering Chongqing University of Posts and TelecommunicationsChongqing 400065P.R.China Department of Medical Engineering University of South FloridaTampaFL 33620USA

Puncture biopsy is an important clinical technique to obtain diseased tissue for pathological diagnosis,where imaging guidance is *** this paper,we describe a metal reflector-enhanced microwave-induced thermoacoustic imaging(TAI)approach capable of guiding puncture biopsy for detection of breast cancer and joint *** experimentations simulating puncture guidance in breast cancer and knee gout models werefirst conducted using(CST STUDIO SUITE)(CST)software,and then ex-vivo experiments were performed followed by qualitative observations and semi-quantitative *** results of both the simulations and ex-vivo experiments showed that our reflector-enhanced TAI could image the puncture needle in high resolution with a large depth of>12 cm.

关键词： Microwave-induced thermoacoustic imaging puncture guidance reflector

High-precision analytical solutions for Earth coverage of constellation equipped with reentry glide vehicles

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Chinese Journal of Aeronautics 2024年第9期37卷 385-398页

作者： ping MA Weibo SUN Jie GAO tao CHAO Control and Simulation Center Harbin Institute of TechnologyHarbin 150001China Beijing Institute of Spacecraft System Engineering Beijing 100094China

To efficiently and accurately design satellite constellations equipped with Reentry Glide Vehicles(RGVs),new analytical solutions are developed for calculating their coverage ***,a new coverage model is established by approximating the Reentry Reachable Domain(RRD).However,the computation of real-time relative distances between satellites and targets,which is essential for coverage analysis based on this model,imposes a significant compu-tational *** address this challenge,a coverage analysis method based on two-dimensional map theory is *** method represents the coverage conditions of a target as a fixed area on a two-dimensional map and transforms the satellite trajectory into a series of parallel *** determining the intersection points between these lines and the area boundaries,the coverage ana-lytical solutions for a target point are *** this basis,coverage theorems are presented for rapid calculation of the constellation coverage performance for an *** results demon-strate the effectiveness and high precision of the proposed analytical solutions.

关键词： Constellation Analytical solutions Reentry glide vehicle Reentry reachable domain Two-dimensional map

Morrey空间上Marcinkiewicz积分与■交换子

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数学学报（中文版） 2019年第2期62卷 269-278页

作者：陶双平逯光辉西北师范大学数学与统计学院兰州730070

本文建立了Marcinkiewicz积分M与具离散系数的正则有界平均振荡空间■生成的交换子M_b在非齐性度量测度空间上的有界性.在控制函数λ满足∈-弱反双倍条件的假设下,当p∈(1,∞)时,证明了M_b在L^P(μ)上是有界的.另外,还得到了M_b在Morre... 详细信息

本文建立了Marcinkiewicz积分M与具离散系数的正则有界平均振荡空间■生成的交换子M_b在非齐性度量测度空间上的有界性.在控制函数λ满足∈-弱反双倍条件的假设下,当p∈(1,∞)时,证明了M_b在L^P(μ)上是有界的.另外,还得到了M_b在Morrey空间上的有界性.

关键词：非齐性度量测度空间 Marcinkiewicz积分交换子 RBMO(μ)空间 Morrey 空间

Effect of different bottom blowing elements on stirring characteristics of molten bath in converter

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Journal of Iron and Steel Research International 2024年第2期31卷 368-376页

作者： Ming Lv shuang-ping Chen Hong-min Guo Yi-jie Hao Xiang-dong Xing School of Metallurgical Engineering Xi’an University of Architecture and TechnologyXi’an710055ShaanxiChina Shaanxi Longmen Steel Company Hancheng715405ShaanxiChina

The bottom blowing element is the key equipment to ensure the bottom blowing effect of the *** types of bottom blowing elements,dispersive type(D1),double circular seam(D2)and straight cylinder type(D3),were built,and the effects of bottom blowing element type on molten bath flow,wall erosion and furnace bottom erosion were *** was found that when the bottom blowing elements of dispersive type(D1)and double circular seam(D2)were used,the dead zone area in the lower part of the molten bath was smaller,and the high-speed zone area was larger;therefore,the stirring effect on the bottom melt was *** the straight cylinder type(D3)bottom blowing element was used,the gas penetrated the molten bath at a faster rate to reach the surface of molten bath and failed to disperse in the bottom molten bath,and the wall shear stress near the nozzle outlet was *** argon was blown by three different bottom blowing elements,the area of the wall shear stress greater than 3 Pa was 4.8,5.6 and 8.7 cm2,respectively,within 0.2 m of the bottom blowing nozzle outlet.

关键词： Converter steelmaking Bottom blowing element Molten bath stirring Wall erosion

θ-型C-Z算子在加权变指数Morrey空间上的有界性

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数学学报（中文版） 2019年第3期62卷 503-514页

作者：杨沿奇陶双平西北师范大学数学与统计学院兰州730070

在满足一定的正则性假设条件下,建立了θ-型Calderón-Zygmund算子T_θ在一类变指数Lebesgue空间上的加权有界性.进一步得到了T_θ在加权变指数Herz空间和Herz-Morrey空间上的有界性.另外,还证明了相应的交换子[b,T_θ]在广义加权变指数... 详细信息

在满足一定的正则性假设条件下,建立了θ-型Calderón-Zygmund算子T_θ在一类变指数Lebesgue空间上的加权有界性.进一步得到了T_θ在加权变指数Herz空间和Herz-Morrey空间上的有界性.另外,还证明了相应的交换子[b,T_θ]在广义加权变指数Morrey空间上是有界的.

关键词： θ-型Calderón-Zygmund算子加权变指数Morrey空间加权变指数Herz-Morrey空间