Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

具阻尼项的非线性波动方程的稳定集与不稳定集

本文利用势井理论讨论一类非线性波动方程的初边值问题 .我们构造其稳定集 W和不稳定集 V,证明了当初值属于 W时 ,对 β∈ R整体弱解存在并且利用乘子法得到当 β>0解的指数衰减估计 ;当初值属于 V时 ,而 β<0时 ,解将爆破     

Adaptive nonparametric estimation of a multivariate regression function

We consider the kernel estimation of a multivariate regression function at a point. Theoretical choices of the bandwidth are possible for attaining minimum mean squared error or for local scaling, in the sense of asymptotic distribution. However, these choices are not available in practice. We follow the approach of Krieger and Pickands (Ann. Statist.9 (1981) 1066–1078) and Abramson (J. Multivariate Anal.12 (1982), 562–567) in constructing adaptive estimates after demonstrating the weak convergence of some error process. As consequences, efficient data-driven consistent estimation is feasible, and data-driven local scaling is also feasible. In the latter instance, nearest-neighbor-type estimates and variance-stabilizing estimates are obtained as special cases.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR GENERALIZED QUANTUM MHD EQUATION

We prove the existence of a weak solution for a generalized quantum MHD equation in a 2-dimensional periodic box for large initial data. The existence of a global weak solution is established through a three-level approximation,energy estimates, and weak convergence for the adiabatic exponent γ 1.     

一类非线性差分方程的持久性与全局吸引性

本文研究了差分方程ｘ（ｎ＋１）＝ｘｐｎｆ（ｘｎ，ｘ（ｎ－ｋ１），…，ｘ（ｎ－ｋｒ））的持久性与全局吸引性，解决了文［３］提出的两个研究课题，所得结果包含了文［３］的相应结果．     

高精度光电数据采集系统的实验研究

针对光电弱信号检测,采用Σ-ΔAD转换器AD7706和AVR单片机ATmega8515设计了适用于低频量测量的高精度数据采集器,采用Σ-ΔAD转换器ADC16471、AVR单片机ATmega8515、C51单片机C8051F020和以太网控制器RTL8019AS设计了适用于高精度线阵CCD的数据采集器,并分别对两套数据采集器进行了实验研究与性能分析。     

非线性自回归序列L1估计的渐近分布

由于时间序列数据中经常出现的厚尾特征使得通常的估计方法不再具有渐近的正态分布,在误差项二阶矩有限的条件下考虑了非线性自回归序列的L_1估计.采用局部线性近似的方法得到了具有凸样本路径的随机过程,在此基础上利用凸样本路径随机过程弱收敛的性质证明了非线性自回归序列L_1估计的渐近正态性及无偏性.     

一种超混沌映射的单双参量自适应同步

利用参量自适应法实现了超混沌系统Kawakami映射的同步.通过严密的数学推导,给出了Kawakami映射在单参量自适应下实现同步的充分条件,并在双参量自适应控制下系统同样实现了同步.数值模拟结果显示,该映射输出具有类似于白噪声的宽谱特性,因此比较适合用于保密通信.     

双波长分光光度法测定有机弱酸弱碱的解离常数

本文提出了一种测定有机一元弱酸弱碱解离常数的新方法———双波长分光光度法。该方法是在选定的一对工作波长下测定相同浓度不同pH值的有机一元弱酸弱碱水溶液的吸光度,利用所推导的公式作图求得pKa(或pKb)。利用本方法测定了甲基橙、溴甲酚绿、苯酚和苯胺等多种常见的有机一元弱酸弱碱的解离常数,结果十分满意。与传统的单波长分光光度法相比,双波长法测定结果的精密度和准确度更高。     

基于热成像仪从能量视角辨观化学反应*

“能量”是学生认识化学变化的重要维度,本实验借助手持红外热成像仪“看到”化学反应中的能量变化。在实验中通过拍摄等量的不同酸溶液与氢氧化钠溶液混合反应时的热成像图,从“温度”高低间接反映出热量变化,从宏观的能量变化建立模型,推理微观的成键、断键等过程,探究酸碱中和反应过程的放热以及弱酸电离过程的吸热,并培养学生的证据推理能力和科学探究意识。