Convergence of relaxation schemes to the equations of elastodynamics

We study the effect of approximation matrices to semi-discrete relaxation schemes for the equations of one-dimensional elastodynamics. We consider a semi-discrete relaxation scheme and establish convergence using the theory of compensated compactness. Then we study the convergence of an associated relaxation-diffusion system, inspired by the scheme. Numerical comparisons of fully-discrete schemes are carried out.

     

非线性各向异性椭圆方程的均匀化

通过结合各向异性Sobolev空间与经典的补偿紧性技巧,得到了一类非线性各向异性椭圆方程的均匀化结果.     

Relaxation limit for hyperbolic systems in chromatography

This paper is concerned with a nonlinear system which arises in chromatography. The global existence of solutions in space for a Cauchy problem with initial data is obtained for any fixed 0$">, and the convergence of to its equilibrium state , governed by a limit system is proved for the case by using the compensated compactness coupled with the framework of Tzavaras (1999).

     

Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order

In this paper we study the homogenization of degenerate quasilinear parabolic equations: where a(t, y, a, λ) is periodic in (t, y).     

AN APPLICATION OF COMPENSATED COMPACTNESS ON QUASILINEAR HYPERBOLIC SYSTEMS

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IRROTATIONAL APPROXIMATION TO STEADY SUPERSONIC FLOW IN TWO SPACE VARIABLES

On the assumption that the total variation of the initial data is sufficiently small, we can use the stability results of Dafermos to get the L2 estimate of the difference between the solutions to the isentropic steady Euler system and the potential flow equations with the same initial data.     

Non-periodic explicit homogenization and reduction of dimension: the linear case

The aim of this paper is to give explicit limit expressions,for diffusion equations involving a small parameter , describingboth nonperiodic homogenization and reduction of dimension.In other words, we give the limit behaviour, when tends tozero, of the diffusion equation in a thin domain, with thicknessof order , when the coefficients of the equation also dependon and may present rapid, nonperiodic oscillations, providedthey satisfy a suitable compensated compactness condition. Weconsider two kinds of reduction of dimension: the case of thinplates (3D 2D) and the case of thin cylinders (3D 1D). Inparticular, we give the limit diffusion equation for laminatedplates. This is completely explicit and requires no specialassumption, except stratification. In the case of thin cylinders,the formulae are less explicit, but we also indicate some simpleapplications.     

On two-scale convergence and related sequential compactness topics

A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L ²(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.     

Stochastic scalar conservation laws

We introduce a notion of stochastic entropic solution à la Kruzkov, but with Ito's calculus replacing deterministic calculus. This results in a rich family of stochastic inequalities defining what we mean by a solution. A uniqueness theory is then developed following a stochastic generalization of L¹ contraction estimate. An existence theory is also developed by adapting compensated compactness arguments to stochastic setting. We use approximating models of vanishing viscosity solution type for the construction. While the uniqueness result applies to any spatial dimensions, the existence result, in the absence of special structural assumptions, is restricted to one spatial dimension only.     

高维Weierstrass型映射的振幅

本文研究了一类d维Weierstrass型映射,利用概周期函数的性质,得到了该类函数关于振幅的估计.     

一类非线性双曲偶合方程组广义解的存在性

本文考虑了具有形式ｎ的弱偶合双曲守恒方程组广义解的存在性问题．极大值原理导出了粘性解Ｌ∞模先验估计，关于单个守恒律简化的补偿列紧方法给出了粘性解的收敛性，即广义解的存在性．     

An Extension Landau-Lifshitz Model in Studying Soft Ferromagnetic Films

In this paper, we propose a model in studying soft ferromagnetic films, which is readily accessible experimentally. By using penalty approximation and compensated compactness, we prove that the dynamical equation in thin film has a local weak solution. Moreover, the corresponding linear equation is also dealt with in great detail.     

半导体与超晶格的数学模型及其分析Ⅱ:分析

本文先介绍等熵可压缩Euler方程的相关补偿列紧框架.然后,我们综述基于补偿列紧方法的关于半导体流体动力模型的诸如整体弱解,松弛极限和拟中性-松弛极限的一些新近数学结果.     

外汇期权的多维跳-扩散模型

本文建立了外汇期权的多维跳-扩散模型,在此模型下将外汇欧式未定权益的定价问题归结为一类倒向随机微分方程的求解问题,证明了这类倒向随机微分方程适应解的存在唯一性问题,并给出了一个关于外汇欧式未定权益的定价公式.     

Existence of global solutions to isentropic gas dynamics equations with a source term

In this paper we prove existence of isentropic gas dynamic equations with a source term (1.2). To this end we construct a sequence of regular hyperbolic systems (1.1) to approximate the inhomogeneous system of isentropic gas dynamics (1.2). First,for each fixed approximation parameter δ and very general condition on P (ρ),we establish the existence of entropy solutions for the Cauchy problem (1.1) with bounded initial date (1.4). Second,letting=o(δ),we obtain a complete proof of the H-1loc compactness of weak entropy pairs of system (1.2) in the form η(ρ,u) =ρH(ρ,u) given in Chen-LeFloch (2003). Finally,for the conditions of P(ρ) given in Chen-LeFloch (2003),applied to the results in Theorems 1 and 2,we obtain the global existence of entropy solutions for the Cauchy problem (1.2) with bounded initial date (1.4).     

Relaxation Limit for Aw-Rascle System

We study the relaxation limit for the Aw-Rascle system of traffic flow. For this we apply the theory of invariant regions and the compensated compactness method to get global existence of Cauchy problem for a particular Aw-Rascle system with source, where the source is the relaxation term, and we show the convergence of this solutions to the equilibrium state.     

COMPENSATED COMPACTNESS AND THE STRATIFIED LIE GROUP

In this paper we prove that the Jacobian J(F) of a map F( f1, , fl) from Ginto Rl maps the product of Lebesgue space Lp1 × × Lpl into local Hardy space hγ(G),where Q/(Q+1)<γ≤ 1, and Q is the homogeneous dimension of the stratified Lie group G .     

共鸣非线性守恒律组的广义解

利用粘性方法和补偿列紧理论 ,得到了 2× 2非严格双曲守恒律组初值问题广义解的存在性     

Characterization of Clifford-valued Hardy spaces and compensated compactness

In this paper, the general Clifford -valued Hardy spaces and conjugate Hardy spaces are characterized. In particular, each function in -valued Hardy space can be determined by half of its function components through Riesz transform, and the explicit determining formulas are given. The products of two functions in the Hardy space give six kinds of compensated quantities, which correspond to six paracommutators, and their boundedness, compactness and Schatten-von Neumann properties are given.

     

一类具有非齐次项对称双曲系统整体弱解的存在性

研究一类具有非齐次项的对称双曲系统Cauchy问题,用补偿紧性方法和最大值原理,得到其有界弱解的整体存在性结果.这一结果推广文献[7]中对称双曲系统到非齐次对称双曲系统.这样一类问题产生于弹性理论、磁流体动力学和石油再开采领域中,具有一定理论和应用价值.