高阶Schrodinger方程的差分格式

高阶Ｓｃｈｒｏｄｉｎｇｅｒ方程的差分格式曾文平（泉州华侨大学应用数学系泉州３６２０ｉｆ）关键词：显式差分格式；稳定性分析；高阶Ｓｃｈｒｏｄｉｎｇｅｒ方程ＡＭＳ（１９９１）主题分类：６５Ｍ０６，６５Ｍ１２．高阶Ｓｃｈｒｓｄｉｎｇｅｒ方程在量子力学、非线...     

二维空间中具有积分型非线性Schrodinger方程组的初值问题

具有积分型非线性ｓｃｈｒｏｄｉｎｇｅｒ方程是在研究非线性Ｌａｎｇｍｕｉｒ波时考虑到离子惯性作用而导出的．本文讨论了二维空间中具有积分型非线性ｓｃｈｒｏｄｉｎｇｅｒ方程组的初值问题，用积分估计方法证明了整体解的存在唯一性．     

Schrodinger算子的极大耗散扩张

本文研究Ｓｃｈｒｏｄｉｎｇｅｒ算子的极大耗散扩张，引入广义不定度规空间，得到在自然边界空间中，Ｓｃｈｒｏｄｉｎｇｅｒ算子极大耗散扩张表示形式，为进一步研究非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程无穷维动力系统的长期混沌行为作准备。     

一类二维非线性Schrodinger方程的吸引子及其维数

本文研究了一类二维非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程解的有限维行为，我们得到了此方程存在吸引子，并得到了此吸引子维数的上界估计     

关于非线性Schrodinger方程混合问题解的破裂和质量集中

本文讨论了一类非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程混合问题解的破裂现象，ｂｌｏｗ－ｕｐ时刻的估计和质量集中定理，并推广和改进了［ｌ～３］中的结果．     

带弱阻尼的非线性Schrdinger方程谱逼近的大时间性态

本文讨论带弱阻尼的非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程周期初值问题当采用谱方法求解时近似解的大时间误差估计、近似吸引子ＡＮ的存在和它们的弱上半连续性ｄｗ（ＡＮ，Ａ）→０．     

一类高阶多维非线性伪双曲方程

本文研究了一类高阶多维非线性伪双曲方程，通过先验估计结合Ｓｏｂｏｌｅｖ嵌入定理及Ｌｃａｒｙ－Ｓｃｈａｕｄｅｒ不动点定现证明了初边值问题整体广义解的存在性和唯一性。     

具四维对称代数的变系数非线性Schrdinger方程的部分不变解

基于对称解的子群分类方法，得到了具四维对称代数的变系数非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程的部分不变解．     

一般非线性发展方程解的长时间行为

本文研究了一般形式的非线性发展方程Ｃａｕｃｈｙ问题的解关于时间的渐近性质．其结果覆盖并部分推广了导数非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程，Ｋｏｒｔｅｗｅｇ－ｄｅＶｒｉｅｓ方程和Ｂｅｎｊａｍｉｎ－Ｏｎｏ等方程的有关结果．     

具四维对称代数的变系数非线性Schrodinger方程的部分不变解

基于对称解的子解分类方法，得到了具四维对称代数的变系数非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程的部分不变解。     

高阶波动方程的时空估计与低能量散射

本文研究了高阶波动方程的低能量散射理论，基本工具是高阶线性波动方程解的时空估计．与经典的二阶波动方程解的时空估计证明不同，我们采用泛函分析的方法与待定指标技巧，首次给出了高阶线性波动方程的时空估计，藉此与非线性函数在齐次Ｓｏｂｏｌｅｖ空间中的估计，获得了高阶波动方程的低能量散射结论．与此同时，也得到了具临界增长的高阶波动方程的柯西问题在低能量条件下的整体存在唯一性．     

Gradient estimates for higher order elliptic equations on nonsmooth domains

We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical Wm,p, m=1,2,…, 1<p<∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.     

Uniform estimates for positive solutions of higher order quasilinear differential equations

For a higher order quasilinear differential equation, the existence of uniform estimates for positive solutions with common domain of definition is proved; these estimates depend on the estimates for the coefficients of the equation and do not depend on the coefficients themselves.     

SOME OSCILLATION THEOREMS OF HIGHER ORDER NON－HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH TRANSCENDENTAL MEROMORPIC COEFFICIENTS

ＳＯＭＥＯＳＣＩＬＬＡＴＩＯＮＴＨＥＯＲＥＭＳＯＦＨＩＧＨＥＲＯＲＤＥＲＮＯＮ－ＨＯＭＯＧＥＮＥＯＵＳＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷＩＴＨＴＲＡＮＳＣＥＮＤＥＮＴＡＬＭＥＲＯＭＯＲＰＩＣＣＯＥＦＦＩＣＩＥＮＴＳ￥ＣｈｅｎＺｏ...     

高次三角形有限元外推的探讨

探讨泊松方程高次三角形有限元外推公式.为此先推导离散格林函数的权模估计和有限元解的渐近不等式展开,然后给出公式的证明.     

The Decay Rates of Strong Solutions for Navier-Stokes Equations

In this paper, we deduce the estimates on decay rates of higher order derivatives about time variable and space variables for the strong solution to the Cauchy problem of the Navier-Stokes equations. The rate obtained is optimal in the sense that it coincides with that of solution to the heat equation.     

Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle

In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions. In order to prove the global estimates, we explore weighted Strichartz estimates for solutions of the wave equation when the Cauchy data and forcing term are compactly supported.

     

Discretization of elliptic control problems with time dependent parameters

We consider linear-quadratic problems of optimal control with an elliptic state equation and control constraints. For a discretization of the state equation by the method of Finite Differences and a piecewise approximation of the control we develop error estimates for the solution of the discrete problem and, based on the optimality conditions, we construct a new feasible control for which we derive error estimates of quadratic order.     

The cauchy problem for a shallow water type equation

We prove existence and uniqueness of local and global solutions of the periodic Cauchy problem for a higher order shallow water type equation under low regularity initial data. Using Fourier analysis we first prove local estimates in appropriate spaces and then use a contraction mapping argument and a conserved norm to get global existence.     

Uniqueness of the solution of the dirichlet problem for an n-dimensional pseudoparabolic equation

In an n-dimensional domain formed by the characteristic planes of an equation of order 2n with a double higher partial derivative, we derive sufficient conditions for the uniqueness of the solution of the Dirichlet problem. These conditions are written out in terms of the coefficients of the equation, and the argument is based on the method of a priori estimates.