黎曼流形上薛定谔方程的Harnack估计

推导了薛定谔方程正解的一种新的整体梯度估计和Harnack不等式,推广了一些有关热方程的结论,并且得到了一个关于薛定谔算子的刘维尔定理.     

非线性抛物型方程的全离散有限元方法

§1.引言关于非线性抛物型方程有限元方法的研究已有许多工作.但所讨论的方程关于梯度是线性的,仅得到全离散有限元逼近的L_2误差估计及较弱意义下(两层平均)的H~1误差估计.本文讨论关于梯度亦是非线性的抛物型方程.得到了最佳L_2、H~1(较强意义下)、L.及时间导数的误差估计. 考虑下述抛物型方程的混合问题:     

一种非线性抛物方程在一般几何流下的梯度估计（英文）

本文通过Li-Yau梯度估计的方法和Jun Sun对热方程在一般几何流下梯度估计的研究,推导出一类重要的非线性抛物方程在一般几何流演化下的梯度估计,并得到了哈拿克不等式等一些结论.推广了Wang的结果.     

多孔介质方程在一般几何流演化下的梯度估计

本文研究了多孔介质方程在一般几何流下的梯度估计.通过Aroson和Bénilan对多孔介质方程的研究结果以及运用Li-Yau梯度估计的方法,获得了对多孔介质方程的正解对于Laplace算子以及drifing Laplace算子在一般几何流演化下的一些梯度估计,推广了Zhu Xiao-bao和Deng Yi-hua的结果.     

Riemann流形上改进的p-Laplace方程的梯度估计

本文研究Riemann流形上的改进的p-Laplace方程,运用截断函数的估计、Hessian比较定理和Laplace比较定理,得到该方程正解的梯度估计.并应用该结论,得到在Riemann流形上关于改进的p-Laplace方程正解的Harnack不等式和Liouville型定理.     

完备流形上一类含负指数项抛物型方程的梯度估计（英文）

设M为完备非紧的黎曼流形,本文在M×[0,+∞)上得到了如下一类含负指数项抛物型方程的正解的梯度估计:u=t=△u+cu~(-α),其中α0和c为两个固定常数.这一结果推广了杨云雁教授关于流形上含负指数项椭圆型方程梯度估计的结论[Acta Math.Sin.,Engl.Ser.,2010,26(6):1177-1182].同时这一结果也可以看作是对李嘉禹教授关于流形上含非线性项椭圆与抛物型方程梯度估计[J.Funct.Anal.,1991,100(2):153-201]的一个补充.     

p-Laplace Schrdinger热方程的椭圆型梯度估计

本文对p-Laplace Schr(o|¨)dinger热方程正连续弱解做了椭圆型梯度估计;作为应用,本文得到了一个关于p-Laplace算子的Liouville型结果;并证明了关于p-LaplaceSchr(o|¨)dinger热方程正连续弱解的Harnack不等式.     

光滑度量测度空间上的加权扩散方程的梯度估计

本文主要考虑了一类加权非线性扩散方程正解的梯度估计.在m-维Bakry-(E)mery Ricci曲率下有界的假设下,得到加权多孔介质方程(γ＞1)正解的Li-Yau型梯度估计,此外对于加权快速扩散方程(0＜γ＜1),证明了Hamilton型椭圆梯度估计,结论分别推广了Lu,Ni,Vázquez and Villani在文[1]和Zhu在文[2]中的结果.     

Laplace方程斜边值问题的梯度估计

该文介绍了Laplace方程斜边值问题解的梯度估计的两种证明方法:第一种证明重新整理文献[1]中的梯度估计;第二种证明采用不同于文献[1]的辅助函数得到估计.两种方法都充分利用函数在极大值点的性质,得到边界梯度估计和近边梯度估计,结合文献[2]中已有的梯度内估计,从而得到解的全局梯度估计.     

黎曼流形上一类非线性抛物方程的梯度估计（英文）

本文得到了黎曼流形上一类非线性抛物方程的局部Hamilton梯度估计.利用这个局部梯度估计,得到了一个Harnack型不等式.     

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.     

Dirichlet problem for the Stokes equation

The paper presents some coercive a priori estimates of the solution of the Dirichlet problem for the linear Stokes equation relating vorticity and the stream function of an axially symmetric flow of an incompressible fluid. This equation degenerates on the axis of symmetry. The method used to obtain the estimates is based on a differential substitution transforming the Stokes equation into the Laplace equation and on the subsequent transition from cylindrical to Cartesian coordinates in three-dimensional space.     

Compactness of solutions to the Yamabe problem. III

For a sequence of blow up solutions of the Yamabe equation on non-locally conformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates would imply the compactness of the set of solutions of the Yamabe equation on such manifolds.     

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．     

Strichartz Estimates for the Schrödinger Equation on Polygonal Domains

We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone.     

Spectrum structure and decay rate estimates on the Landau equation with Coulomb potential

In this paper we first deduce the estimates on the linearized Landau operator with Coulomb potential and then analyze its spectrum structure by using semigroup theory and linear operator perturbation theory. Based on these estimates, we give the precise time decay rate estimates on the semigroup generated by the linearized Landau operator so that the optimal time decay rates of the nonlinear Landau equation follow. In addition, we present a similar result for the non-angular cutoff Boltzmann equation with soft potentials.

     

Strichartz estimates for non-degenerate Schrödinger equations

We consider the Schrödinger equation with a non-degenerate metric on the Euclidean space. We study local in time Strichartz estimates for the Schrödinger equation without loss of derivatives including the endpoint case. In contrast to the Riemannian metric case, we need the additional assumptions for the well-posedness of our Schrödinger equation and for proving Strichartz estimates without loss.     

Uniform null controllability of the heat equation with rapidly oscillating periodic density

We consider the heat equation with fast oscillating periodic density, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform controllability result is the first in the multidimensional setting for the heat equation with oscillating density. Finally, we prove that the sequence of null controls converges to the optimal null control of the limit equation when the period tends to zero. To cite this article: L. Tebou, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

Endpoint Strichartz estimates for the Klein–Gordon equation in two space dimensions and some applications

We prove the endpoint Strichartz estimates for the Klein–Gordon equation in mixed norms on the polar coordinates in two space dimensions. As an application, similar endpoint estimates for the Schrödinger equation in two space dimensions are shown by using the non-relativistic limit. The existence of global solutions for the cubic nonlinear Klein–Gordon equation in two space dimensions for small data is also shown.