A TWO-DIMENSIONAL GLIMM TYPE SCHEME ON CAUCHY PROBLEM OF TWO-DIMENSIONAL SCALAR CONSERVATION LAW

In this paper,we construct a new two-dimensional convergent scheme to solve Cauchy problem of following two-dimensional scalar conservation law{ tu ＋ xf（u） ＋ yg（u） = 0,u（x,y,0） = u0（x,y）.In which initial data can be unbounded.Although the existence and uniqueness of the weak entropy solution are obtained,little is known about how to investigate two-dimensional or higher dimensional conservation law by the schemes based on wave interaction of 2D Riemann solutions and their estimation.So we construct such scheme in our paper and get some new results.

Glimm方法对于燃烧模型含爆燃波及爆轰波解的收敛性

该文对于反应速率无穷的燃烧模型组的初值问题 ( u + qz) t + f( u) x =0 ,z( x,t) =z( x,0 ) , sup0≤τ≤ tu( x,τ)≤ 0 ,-∞ <x <∞ ,t≥ 0 ,0 , 其他 ,( 1 )( u( x,0 ) ,z( x,0 ) ) =( u0 ( x) ,z0 ( x) ) , -∞ <x <+∞ ,( 2 )证明了当初值 u0 ( x) ,z0 ( x)的变差有界时 ,对于含爆燃 ( DF)波及爆轰 ( DT)波解的 Glimm格式的收敛性 ,其中 f′( u) >- Λ,f′( 0 ) <0 ,f″( u) >0 ,f″( u)≥ δ>0 ( u≥ 0 ) ,常数 q>0 ,以及当 u0( x) >0时 z0 ( x)

STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem （initial value problem） in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

INITIAL-BOUNDARY PROBLEM FOR THE 1-D EULER-BOLTZMANN EQUATIONS IN RADIATION HYDRODYNAMICS

We study the initial-boundary value problem for the one dimensional Euler-Boltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions （U△t,d, I△t,d） and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.

带有阻尼项的三维欧拉方程组球对称解

研究了一类带有阻尼项的三维等熵可压缩欧拉方程组经典解局部存在性和弱解的整体存在性。一方面利用对称双曲型方程组解的存在性结论,将该方程组转化为对称双曲型方程组,得到了经典解的局部存在性;另一方面由新坐标的引入,该方程组转化为比较熟悉的p-系统,并通过Glimm格式证明了它的整体弱解的存在性。

EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ（T ） is assumed to have a positive lower bound. We first consider the Cauchy problem （the initial value problem）, that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an ad-ditional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave （weak or strong, determined by the wedge angle at the wedge vertex） when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.

关于具有初始激波流的均熵气体动力学方程组的解不产生真空态的一个定理

应用Ｇｌｉｍｍ方法证明具有初始弱激波流的均熵流，在任意有限时间内不会产生真空，且其比容ｖ（ｘ，ｔ）不会超出初始比容ｖ（ｘ。

EXISTENCE OF PERIODIC SOLUTIONS TO AN ISOTHERMAL RELATIVISTIC EULER SYSTEM

In this paper,we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space.First,we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant plane.Based on these properties,we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme,and prove that there exists an entropy solution V(x,t)which belongs to L^(∞)∩BV_(loc)(R×R_(+)).

THE GLOBAL LIPSCHITZ SOLUTION FOR A PEELING MODEL

This paper focusses on a peeling phenomenon governed by a nonlinear wave equation with a free boundary.Under the hypotheses that the total variation of the intial data and the boundary data are small,the global existence of a weak solution to the nonlinear problem(1.1)-(1.3)is proven by a modified Glimm scheme.The regularity of the peeling front is established,and the asymptotic behaviour of the obtained solution and the peeling front at infinity is also studied.

压差方程的一维活塞问题

用改进的Glimm格式的方法,研究压差方程的一维活塞问题:当活塞的运动速度是一个常数的扰动时,含有激波的弱解的存在性.对波的相互作用以及扰动波在主激波和活塞上的反射给出了精确的估计,在对主激波的强度不加限制的情况下证明激波解的整体存在性.

一维活塞问题激波解的整体存在性

本文用改进的Glimm格式的方法,研究一维活塞问题当活塞的运动速度是一个常数的扰动时含有激波的弱解的存在性.对波的相互作用以及扰动波在主激波和活塞上的反射作出了精确的估计,在对主激波的强度不加限制的情况下证明了激波解的整体存在性.

一维相对论Euler方程组活塞问题激波解的整体存在性

研究等熵流相对论Euler方程组的一维活塞问题,证明了当活塞速度是一个常数的扰动时其激波解的整体存在性.通过采用改进的Glimm格式构造问题的近似解,然后对基本波的相互作用作出精确的估计,最后构造Glimm泛函并证明其单调性.

磁流体力学方程组一维活塞问题激波解的整体存在性

研究磁流体力学方程组一维活塞问题,证明了当活塞速度是一个常数的扰动时,其激波解的整体存在性.通过改进的Glimm格式先构造问题的近似解,然后对基本波的相互作用作出精确的估计,最后构造Glimm泛函并证明其单调性.

带耗散项等温的p-方程组大初值解的整体存在性

研究带耗散项的等温情形的p-方程组大初值解的存在性.对于此类问题我们主要是应用改进的Glimm格式来证明解的存在性.C.M.Dafermous和L.Hsiao曾对一般的带耗散的守恒律做出了相关的研究,但主要适用于小初值的情形.该文中通过对于Riemann问题和波强度进行详细的分析,得到了在L∞∩BV(R)空间中弱解的整体存在性.

压差方程整体解的存在性

利用改进的Glimm格式讨论了压差方程整体解的存在性.通过研究压差方程Riemann不变量的性质来研究波的性质,证明了随着左状态的改变右状态集合组成的激波曲线满足Diperna条件,再利用Glimm格式构造近似解,在Riemann问题的解的变差由Riemann不变量控制,最后得到整体解的存在性.

一类守恒律方程组周期解的存在性

利用改进版本的Glimm格式得到一类源自于等熵多方气体动力学方程组的守恒律方程组周期解的存在性.这类方程组的周期解最早被Frid研究.文中通过对近似解更精细的估计,改进了Frid的结论,得到当周期初值U_0∈L∞∩BV_(loc)(R),方程组存在属于L∞∩BV_(loc)(R×R_+)的周期解U(x,t).首先构造近似解,并得到近似解的有界性和全变差的有界性,然后根据Glimm格式的框架得到周期解的存在性.

Error Bounds for Glimm Difference Approximations for Scalar Conservation Laws Without Convexity

In last century, D. Hoff and J. Smoller derived the error bounds for the Glimm difference approximations of the solutions to scalar conservation laws with convexity. Our work is to extend the corresponding result of them to the case without convexity.

Global Entropy Solutions of the Cauchy Problem for Nonhomogeneous Relativistic Euler System

We analyze the 2 × 2 nonhomogeneous relativistic Euler equations for perfect fluids in special relativity. We impose appropriate conditions on the lower order source terms and establish the existence of global entropy solutions of the Cauchy problem under these conditions.

Weak Solutions of the Isothermal Bipolar Hydrodynamic Model for Semiconductors with Large Data

This paper is devoted to weak solutions of Cauchy problem to the isothermal bipolar hydrodynamic model with large data. The model takes the bipolar Euler-Poisson form, with electric field and relaxation terms added to the momentum equations. Using Glimm scheme to the hyperbolic part and the standard theory to the ordinary differential equations, we first construct the approximation solutions, then from the facts that the total charge is quasi-conservation, we can obtain a uniform estimate of the total variation of the electric field, which allows to prove the L∞ estimate of densities and velocities, and the convergence of the scheme. Then we can prove the global existence of weal solution to Cauchy problem with large data.

Existence in the Large for Pressure-Gradient System

In this paper,the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data.To this end,some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions(see[Diperna,R.J.,Existence in the large for quasilinear hyperbolic conservation laws,Arch.Ration.Mech.Anal.,52(3),1973,244–257])are studied.Then they construct the approximate solution sequence through Glimm scheme.By establishing accurate local interaction estimates,they prove the boundedness of the approximate solution sequence and its total variation.