完全非线性偏微分方程解的Gevrey微局部正则性

本文中，我们首先简要回顾了Gevrey类中的仿微分运算，然后考察了相关的完全非线性偏微分方程的象征的一些性质。作为应用，我们得到解在椭圆点附近的Gevrey微局部正则性。     

关于一个拟线性椭圆边值问题的研究

本文研究一个主部为N维P-Laplace微分算子的非线性偏微分方程的边值问题.方程中的非线性项关于其变量可以具有奇性和间断性．这类方程的一种典型情形是大家熟知的具负指数的Emden-Fowler方程．本文利用摄动技巧，Schauder不动点定理，经对解的性质的精细分析．获得了解的存在性和唯一性结果．     

Pohozaev恒等式及其在非线性椭圆型方程中的应用

在非线性椭圆型偏微分方程的研究中,Pohozaev恒等式在研究非平凡解的存在性和非存在性时起着十分重要的作用.本文旨在介绍Pohozaev恒等式及其在非线性椭圆型问题研究中的应用.首先介绍有界区域和无界区域上几种典型的Pohozaev恒等式,并得到几类非线性椭圆型方程存在解的必要条件,进而得到对应的方程非平凡解的非存在性和存在性结果.其次将介绍非线性椭圆型方程的局部Pohozaev恒等式,由此证明非线性椭圆型微分方程近似解序列的紧性,并得到几类典型非线性椭圆型方程的无穷多解存在性.最后利用非线性椭圆型方程的局部Pohozaev恒等式来研究其波峰解,得到波峰解的局部唯一性,并由此判断波峰解的对称性等特征.     

关于一阶拟线性方程的整体解

对于非线性偏微分方程,通常局部可解性比较容易得到,而整体解问题则复杂得多.近年来,关于非线性偏微分方程的整体可解性已得到很多研究结果.比如[1]、[2]中讨论了非线性波动方程的整体可解性.[3]中讨论了某种双曲型方程组的整体可解性.[4]中讨论了某种非线性椭圆型方程的整体不可解性.也有大量工作讨论整体广义解的存在性.这些结果都是关于微分方程的初值问题或边值问题的整体可解性.但是如果我们期望得到全空间的整体解,那么如本文所得到的结果那样,微分方程本身是否存在这种整体解就是一个很值得研究的问题. 我们称方程的在全空间具有直到方程阶数的连续导数的解为全正则解.     

三维拟线性热弹性力学方程区域内部解的奇性传播规律

本文利用频率分析对角化的方法,研究了三维拟线性热弹性力学方程区域内部解的奇性传播规律.首先从微局部观点出发,利用仿微分算子和拟微分算子将方程仿线性化和对角化.然后,利用穿梭法和经典的双曲方程和抛物方程理论,证明了区域内部解的奇性传播也是沿耦合方程组的双曲算子的零次特征带传播,并且当初值的奇性沿方程组的双曲算子的前向光锥传播时,时间t也具有很好的正则性.     

三维拟线性热弹性力学方程区域内部解的奇性传播规律

本文利用频率分析对角化的方法,研究了三维拟线性热弹性力学方程区域内部解的奇性传播规律. 首先从微局部观点出发,利用仿微分算子和拟微分算子将方程仿线性化和对角化.然后,利用穿梭法和经典的双曲方程和抛物方程理论,证明了区域内部解的奇性传播也是沿耦合方程组的双曲算子的零次特征带传播,并且当初值的奇性沿方程组的双曲算子的前向光锥传播时,时间t也具有很好的正则性.     

仿Fourier积分算子在非线性奇性传播中的应用

本文应用文献[1,2]中建立的仿Fourier积分算子概念以及仿微分算子的EropoB定理,对主型的仿微分算子进行微局部化简。并由此建立它们的奇性传播定理,然后用此定理讨论非线性方程解的低频奇性传播。     

高阶脉冲中立型偏泛函微分方程的强迫振动性

研究一类具高阶Laplace算子的高阶脉冲非线性中立型偏泛函微分方程的强迫振动性,利用Green公式和微分不等式方法将所讨论的脉冲中立型偏微分方程转化为脉冲中立型微分不等式的问题,获得了这类方程在三类不同边值条件下所有解强迫振动的若干充分条件.     

一类锥Sobolev空间的Littlewood-Paley分解及其应用

本文结合仿微分算子理论研究了一类锥Sobolev空间上的Littlewood—Paley分解,讨论了该分解在非线性偏微分方程上的应用.     

一类非线性偏微分方程解在定重特征点的正则性传播

本文讨论并证明了满足一定条件的一类非线性偏微分方程的解为上述方程的线性化算子的定常重特征点,则u在(x_o,ε~0)的微局部正则性沿过这一点的零次特征是不变的。     

Regularity, Local and Microlocal Analysis in Theories of Generalized Functions

We introduce a general context involving a presheaf and a subpresheaf ℬ of  . We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the ℬ-local analysis of sections of  . But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a “frequential microlocal analysis” and into a “microlocal asymptotic analysis”. The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis is a new spectral study of singularities. It can inherit from the algebraic structure of ℬ some good properties with respect to nonlinear operations.      

On the sensitivity of solutions of hyperbolic

The goal of this work is to determine appropriate domain and range of the map from the coefficients to the solutions of the wave equation for which its linearization or formal derivative is bounded and the properties of the coefficients on which the bound depends.Such information is indispensable in the study of the inverse (coefficient identification) problem vio smooth optimization methods. The main result of this paper is an explicit microlocal Sobolev estimate for the linearized forward map. In view of results of Rakesh [19] for the smooth coefficient case, the order of our regularity result is optimal. Our proof is based on the method of nonsmooth microlocal analysis, in particular various results on propagation of singularities, the method of progressing wave expansions, microlocal study of solutions of the transport equations, study of conormal properties of the fundamental solution, and a duality technique.     

Propagation of singularities in Cauchy problems for quasilinear thermoelastic systems in three space variables

By using microlocal analysis, the propagation of weak singularities in Cauchy problems for quasilinear thermoelastic systems in three space variables are investigated. First, paradifferential operators are employed to decouple the quasilinear thermoelastic systems. Second, by investigating the decoupled hyperbolic-parabolic systems and using the classical bootstrap argument, the property of finite propagation speeds of singularities in Cauchy problems for the quasilinear thermoelastic systems is obtained. Finally, it is shown that the microlocal weak singularities for Cauchy problems of the thermoelastic systems are propagated along the null bicharacteristics of the hyperbolic operators.     

Étude symbolique de la perte de régularité C∞ par morceaux dans l'interaction d'ondes semi-linéaires

We consider the interaction of two piecewise smooth waves for a first order semilinear system. We know that under geometric hypotheses the piecewise smooth regularity can be not conserved after the interaction. A symbolic study allows us to prove that this loss does not appear on principal singularities. We then show the microlocal mechanism of the creation and the propagation of a logarithmic singularity.     

关于Gevrey-Sobolev波前集的刻划

本文中,我们利用微局部能量方法给出了Gevrey-Sobolev波前集的刻划,这 种刻划可被用在Gevreg类的非线性微局部分析理论的研究中.     

A Characterization for the Gevrey-Sobolev Wave Front Set

In this note, we use the so-called microlocal energy method to give a characterization of the Gevrey-Sobolev wave front set , which will be useful in the study of non-linear microlocal analysis in Gevrey classes. Research supported by grants of the Natural Science Foundation of China, the State Education Committee and the Huacheng Foundation.     

Propagation des singularités Gevrey pour l’équation de Cauchy-Riemann dégénérée

In this paper we study the degenerate Cauchy-Riemann equation in Gevrey classes. We first prove the local solvability in Gevrey classes of functions and ultra-distributions. Using microlocal techniques with Fourier integral operators of infinite order and microlocal energy estimates, we prove a result of propagation of singularities along one dimensional bicharacteristics.      

Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients

We are concerned with analyzing hyperbolic equations with distributional coefficients. We focus on the case of coefficients with jump discontinuities considered earlier by Hurd and Sattinger in their proof of the breakdown of global distributional solutions. Within the framework of Colombeau generalized functions, however, Oberguggenberger showed the existence and uniqueness of a global solution. Within this framework we develop further a microlocal analysis to understand the propagation of singularities of such Colombeau solutions. To achieve this we introduce a refined notion of a wave-front set, extending Hörmander's definition for distributions. We show how the coefficient singularities modify the classical relation of the wave front set of the solution and the characteristic set of the operator, with a generalized notion of characteristic set.     

PROPAGATION OF SINGULARITES FOR SOLUTION OF SEMILINER WAVE EQUATIONS

In order better to research the singularities of the solutions $\[u \in H_{loc}^s(\Omega ),\Omega \subset {R^n},s > \frac{n}{2} + 1\]$ , for semilinear hyperbolic equations $\[u = f(u,Du)\]$, in this paper, a kind of weighted Sobolev space $\[({H^s})_{{P_\mu }}^\alpha \],\[\mu = 1,2,{p_1} = {D_i} - \left| {{D_x}} \right|,{P_2} = {D_i} + \left| {{D_x}} \right|\]$, closely related with the solutions of the equations, is presented. It is discussed that their products tacitly keep roughly $\[{H^{3x - n}}\]$ microlocal regularity on the characteristic directions for $\[{P_\mu }\]$ and invariance under nonlinear maps. Then it is obtained that roughly $\[{H^{3x - n}}\]$ propagation of singularities theorem is valid for $\[u = f(u)\]$.     

Pompeiu transforms on geodesic spheres in real analytic manifolds

We prove a support theorem for Pompeiu transforms integrating on geodesic spheres of fixed radiusr>0 on real analytic manifolds when the measures are real analytic and nowhere zero. To avoid pathologies, we assume thatr is less than the injectivity radius at the center of each sphere being integrated over. The proof of the main result is local and it involves the microlocal properties of the Pompeiu transform and a theorem of Hörmander, Kawai, and Kashiwara on microlocal singularities.