非线性Sobolev Galpern方程的近似惯性流形

近似惯性流形概念与耗散偏微分方程的长时间行为研究有关, 该文对非线性Sobolev Galpern方程构造了两个近似惯性流形. 证明了非平滑近似惯性流形Σ和平滑近似惯性流形Σ_0=P_mH对整体吸引子有相同的逼近阶数.     

加罚N-S方程的有限元非线性Galerkin方法

非线性Galerkin方法是对耗散型非线性发展方程的一种数值解法,其空间变量不象一般Galerkin方法那样在线性空间上离散,而是在非线性流形上离散,所得逼近解在时间变量增大时可以更快地逼近其精确解.精细的理论分析可见[1],[2]等,在有限元逼近基础上将此方法应用到Navier-Stokes方程上的工作可参见[3],[4],这些文章主要针对速度与压力同时求解的混合元情形做了讨论.本文在[4]的基础上对加罚Navier-Stokes方程的一种非线性Galerkin方法的半离散和全离散有限元逼近格式分别进行了误差估     

加罚N-S方程的有限元非线性Galerkin方法

非线性Galerkin方法是对耗散型非线性发展方程的一种数值解法,其空间变量不象一般Galerkin方法那样在线性空间上离散,而是在非线性流形上离散,所得逼近解在时间变量增大时可以更快地逼近其精确解.精细的理论分析可见[1],[2]等,在有限元逼近基础上将此方法应用到Navier-Stokes方程上的工作可参见[3],[4],这些文章主要针对速度与压力同时求解的混合元情形做了讨论.本文在[4]的基础上对加罚Navier-Stokes方程的一种非线性Galerkin方法的半离散和全离散有限元逼近格式分别进行了误差估     

N-S方程一般近似惯性流形构造和逼近

1．引言为了用有限维常微分方程来研究Navier七lobes（N七）方程的长时间动力学行为,Foias,Sell和Temaml］引入了耗散系统惯性流形的概念．但惯性流形存在的一个相当苛刻的条件一谱间隔条件一是包括N习方程在内的很多耗散系统无法满足的．因此,Foias,Manlea和TemamZ］随后又提出了近似惯性流形的概念．近似惯性流形也是一个光滑的Lipschitz流形,所有原方程的解在时间充分大时,将被吸引进入该流形的一个三邻域中．因其存在性不需要谱间隔条件来保证,从而可证明包括N－S方程在内的一大类耗散系统存在近似惯性流形．利用近似惯性…     

强阻尼非线性波方程的近似惯性流形

本文研究一类强阻尼非线性波方程．通过交换的方法，我们构造了方程的近似惯性流形序列，并得到了对于方程的整体吸引子的逼近估计．     

惯性算法的收敛性和稳定性

1．引言对于非线性发展方程，人们感兴趣的是解的渐近行为．当某一物理参数人很小时，非定常解趋向定常解，而当入充分大时，非定常解的渐近行为完全表现在一个吸引子的结构上，这个吸引子可能是具有分数维数的分形结构．在试图逼近这个吸引子的设想当中，惯性流形显示了它的巨大优越性［1－4］．一个系统的惯性流形是一个光滑的有限维流形，它以指数级速度逼近吸引子．在这个光滑的流形上，一个偏微系统可以用它的惯性形式即有限维常微系统来得到．然而在目前状况下，人们知道存在惯性流形的非线性发展方程为数不多．而绝大部分非线性发展…     

Cahn—Hilliard方程的近似惯性流形

通过压缩映象原理，本文对Ｃａｈｎ－Ｈｉｌｉａｒｄ方程构造了一簇近似惯性流形，这些近惯性流越来越逼近全局吸引子。     

弱阻尼KdV方程中长期动力学行为研究*

证明了周期边界条件下弱阻尼KdV方程存在近似惯性流形．该流形使吸引子被确切定义的有限维光滑流形逼近．并由此概念引出新的数值方法很好地用于研究动力系统长期行为．     

Stochastic inertial manifold of dissipative wave equations with time delays

利用修正的Lyapunov-Perron方法研究随机耗散时滞波方程不变流形的存在性,证明了当谱间隙条件成立和时滞适当小时,随机耗散时滞波方程存在随机惯性流形,并且谱间隙条件与确定型时滞耗散波方程的一致.     

Davey-Stewartson方程组的近似惯性流形

本文研究了二维Davey--Stewartson方程组,证明了解的时间解析性和Gevrey类正则性,构造了指数式的近似惯性流形.本文结果表明,如果一个方程的线性主部算子能生成一个解析半群,那么其平坦近似惯性流形(Galerkin近似)和非平坦近似惯性流形具有相同的逼近精度,都可以是指数式的.     

Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses

We present a non-periodic averaging principle for measure functional differential equations and, using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales (see Federson et al., 2012 [8]), we obtain a non-periodic averaging result for functional dynamic equations on time scales. Moreover, using the relation between measure functional differential equations and impulsive measure functional differential equations, we get a non-periodic averaging theorem for these equations. Also, it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales (see Federson et al., 2013 [9]). Therefore, applying this correspondence to our averaging principle, we obtain a non-periodic averaging theorem for impulsive functional dynamic equations on time scales.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

Characterizations of linear Volterra integral equations with nonnegative kernels

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.     

一类非牛顿渗流方程爆破界的估计

本文利用拟线性常微分方程解的非存在性定理得到了一类拟线性反应扩散方程(非牛顿渗流方程)爆破界的估计,从而推广了半线性反应扩散方程(牛顿渗流方程)相应结果.     

Integrating factors, adjoint equations and Lagrangians

Integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether's theorem is applied to the Maxwell equations.     

层状二维流动的基本方程式

在很多海洋、大气等二维流动问题中所用的动力学方程往往沿用推广后的河流水力学方程或"纳维-斯托克斯方程"其中把湍流阻力项写成这样的方程式和湍流阻力项用到实际问题上去,无疑是存在着极大的局限性,而将导致矛盾百出.本文则从雷诺方程出发,把所有的物理量沿深度加以平均,求出平均以后的物理量所满足的运动方程,连续方程和扩散方程.     

Existence of solutions and Gevrey class regularity for Leray-alpha equations

In this paper we consider some equations similar to Navier-Stokes equations, the three-dimensional Leray-alpha equations with space periodic boundary conditions. We establish the regularity of the equations by using the classical Faedo-Galerkin method. Our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the Leray-alpha equations, furthermore, the strong solutions are analytic in time with values in the Gevrey class of functions (for the space variable). The relations between the Leray-alpha equations and the Navier-Stokes equations are also considered.     

Derivation of equations of seismic and acoustic wave propagation and equations of filtration via homogenization of periodic structures

A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations involve nonsmooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification, under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size of the pores tends to zero, while the porous body is geometrically periodic. As the results for different ratios between physical parameters, we derive Biot’s equations of poroelasticity, a system consisting of nonisotropic Lamé’s equations for the solid component and acoustic equations for the liquid component, nonisotropic Lamé’s equations or equations of viscoelasticity for one-velocity continuum, decoupled system consisting of Darcy’s system of filtration or acoustic equations for the liquid component (first approximation) and nonisotropic Lamé’s equations for the solid component (second approximation), a system consisting of nonisotropic Stokes equations for the liquid component and acoustic equations for the solid component, nonisotropic Stokes equations for one-velocity continuum, or, finally a different type of acoustic equations for one- or two-velocity continuum. The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures.     

An examination of the homentropic Euler equations with averaged characteristics

This paper examines the properties of the homentropic Euler equations when the characteristics of the equations have been spatially averaged. The new equations are referred to as the characteristically averaged homentropic Euler (CAHE) equations. An existence and uniqueness proof for the modified equations is given. The speed of shocks for the CAHE equations are determined. The Riemann problem is examined and a general form of the solutions is presented. Finally, numerically simulations on the homentropic Euler and CAHE equations are conducted and the behaviors of the two sets of equations are compared.     

Dichotomies for generalized ordinary differential equations and applications

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.