双曲守恒律系统文献翻译.doc

附件1：外文资料翻译译文第1章预备知识双曲守恒律系统是应用在出现在交通流，弹性理论，气体动力学，流体动力学等等的各种各样的物理现象的非常重要的数学模型。一般来说，古典解非线性双曲方程柯西问题解的守恒定律仅仅适时局部存在于初始数据是微小和平滑的.这意味着震波在解决方案里相配的大量时间里出现。既然解是间断的而且不满足给定的传统偏微分方程式，我们不得不去研究广义的解决方法，或者是满足分布意义的方程式的函数.我们考虑到如下形式的拟线性系统ut+fux=0.x,trr+, (1.0.1)这里u=(u1,u2,un)trn,n1是代表物理量密度的未知矢量向量，fu=(f1u,fn(u)t是给定表示保守项的适量函数，这些方程式通常被叫做守恒律.让我们假设一下，u是(1.0.1)在初始数据ux,0=u0(x) . (1.0.2)下的传统解。使c01成为c1消失在紧凑子集外的函数的一类。我们用乘以(1.0.1)并且使t0的部分，得到t0(ut+fux)dxdt+t=0u0dx=0. (1.0.3)定义1.0.1 有lp，10是恒定的.我们首先应该得到一个关于柯西问题(1.0.4),(1.0.2)对于任何一个依据下列抛物方程的一般理论存在的的解的序列u：定理1.0.2 (1)对于任意存在的0, (1.0.4)的柯西问题在有界可测原始数据(1.0.2)对于无限小的总有一个局部光滑解u(x,t)c(r0,)，仅依赖于以原始数据u0(x)的l.(2)如果解u有一个推理的l估量|u,t|lm(,t)对于任意的t0,t,于是解在r0,t上存在.(3)解u满足：lim|x|u=0 如果limxu0x=0.( 4)特别的，如果在(1.0.4)系统中的一个解以t+(gu)x=xx (1.0.5)形式存在，这里g(u)是在urn上连续函数，cr,c0,0,如果0xc00 (1.0.6) 这里c0是一个正的恒量，而且当变量t趋向无穷大或者趋向于0时，cr,c0,趋向于0.证明.在(1)中的局部存在的结果能简单的通过把收缩映射原则应用到解的积分表现得到，根据半线性抛物系统标准理论.每当我们有一个先验的l局部解的评估，明显的本地变量一步一步扩展到t，因为逐步变量依据l基准.取得局部解的过程清晰地表现在(3)中的解的行为.定理1.0.2的(1)-(3)证明的细节在lsu,sm看到.接下来是bereux和sainsaulieu未发表的证明(cf. lu9, pe)我们改写方程式(1.0.5)如下：vt+guvx+g(u)x=(vxx+vx2) (1.0.7)当v=logw.然后vt=vxx+(vx-gu2)2-g(u)x-g2(u)4. (1.0.8) 以初值v0x=log(0(x)(1.0.8)的解v能被格林函数gx-y,t=1texp(-x-y24t)描写：v=-gx-y,tv0ydy+0t-vx-gu22-g2u4-guxgx-y,t-sdyds. (1.0.9)由于-gx-y,tdy=1，-|gx-y,t|dymt,(1.0.9)转化为v-gx-y,tv0ydy+0t-g2u4-guxgx-y,t-sdyds =-gx-y,tv0ydy+0t-(g(u)gyx-y,t-s-g2u4gx-y,t-s)dydslogc0-mt-m1t1212-ct,c0,-. (1.0.10)因此对于任意一个，t0,如果我们再假如u是在关于参数的空间lp(1p)上一致连续，即存在子序列（仍被标记）u如下ux,tux,t, 在lp上弱对应 (1.0.11)而且有子序列f(u)如下f（ux,t）lx,t， 弱对应 (1.0.12)在习惯于fu成长适当成长性.如果lx,t=f（ux,t），a.e.， (1.0.13)然后明显的ux,t是(1.01)使在(1.0.4)的趋近于0的一个初始值(1.0.2)的一个弱解.我们如何得到弱连续(1.0.13)的关于粘度解的序列u的非线性通量函数f(u)？补偿密实度原理就回答了这个问题.为什么这个理论叫补偿密实度？粗略的讲，这个术语源自于下列结果：如果一个函数序列满足x,tx,t (1.0.14)与下列之一()2+()3()2+()3或者()2-()3()2-()3 (1.0.15)当趋近于0时弱相关，总之，x,t不紧密.然而，明显的，任何一个在(1.0.15)中的弱紧密度能补偿使其成为的紧密度.事实上，如果我们将其相加，得到()2()2 (1.0.16)当趋近于0时弱相关,与(1.0.14)结合意味着的紧密度.在这本书里，我们的目标是介绍一些补偿紧密度方法对标量守恒律的应用，和一些特殊的两到三个方程式系统.此外，一些具有松弛扰动参量的物理系统也被考虑进来。这本书的准备情况如下：在第2章我们介绍一些基本定理关于补偿紧密度原理.章节2.1是关于22行列式的弱连续定理,和来自于ta的证明.章节2.2是关于弱极限理论的表现的young式测量，我们用了lin的证明.章节2.3是关于缪拉紧密嵌入式定理，在这个部分我们介绍两种原理.定理2.32的证明是和在dcl1给出的证明是一样的，2.3.4的证明是从法国人缪拉的论文中摘抄的.有必要提出的是定理2.34是独立于本书，读者可以不用考虑细节的掠过它.我们把它收集在这是因为它被用于一些研究论文中(cf.cll, jpp). 在第3章，我们分别考虑在l和lp(1p3我们的证明来自于lpt的论文.在13的情况下，只用弱熵熵流对的四个部分，我们给出一个简短的证明，通过假定解总是来自空间而且小(cf. cl2). 在第9章，第6和7章的方法再次延伸去研究一维欧拉方程的两个特殊系统，都是在真空线 = 0非严格双曲的.为了平滑解，他们分别等同于多变气体动力学系统在绝热指数3 and = 的情况.我们的证明来自于lu2 和 lu8. 在第10章，我们考虑一般的可压缩液流一维欧拉方程.这个更一般的系统再次在真空线 = 0非严格双曲的.通过运用补偿紧密度学习这个系统，一个基本的困难时如何构造熵熵流对和得出关于这些熵的必要估计.由于构建严格双曲的松弛型熵熵流对的方法不再有效，在这一章，我们延伸了diperna关于非严格双曲系统的方法，我们介绍了一个松弛熵的特殊模壳，在其中级数术语是一个单一变量的函数. 这个必要的专业术语估计所得的奇异摄动微分方程理论二次顺序.证明是这个章节来自于lu6 在第11章，我们延伸了第十章给出的研究拓展在l空间弹性系统的方法.这个证明也来此lu6. 在12章，一些重要的关于lp(1p),弹性系统的弱解的结果被介绍，包括一个针对这个系统的通过lin粘度解决方法的紧密度框架，和一个shearer研究的物理粘度紧密度框架.shearer研究的绝热气体通过多孔介质的紧密度框架也被考虑了. 但是，为了避免棘手的数学公式，我们选择不去提供书中的这两个紧密度框架的证明.尽管他们十分重要，形成了16章三位双曲系统的松弛问题的基准. 从13章到16章，我们介绍一些关于松弛问题补偿紧密度的应用. 在13章，介绍一个松弛单一问题的总则. 在14章，考虑了硬性松弛的单一极限和一般2 2非线性守恒系统的显性扩散.这些包括弹性系统在l上的解决方案，等熵流体动力学在欧拉坐标和延伸的交通流量模型.lp(1p0, to gett0(ut+fux)dxdt+t=0u0dx=0. (1.0.3)definition 1.0.1 an lp 10 is a constant.we may first get a sequence of solutions uof the cauchy problem (1.0.4),(1.0.2) for any fixed by the following general theorem for parabolic equations:theorem 1.0.2 (1) for any fixed 0, the cauchy problem (1.0.4) with the bounded measurable initial data (1.0.2) always has a local smooth solution u(x,t)c(r0,) for a small time , which depends only on the l norm of the initial data u0(x).(2) if the solution u has an a priori l estimate |u,t|lm(,t) for any t0,t, then the solution exists on r0,t.(3) the solution u satisfies:lim|x|u=0 if limxu0x=0. (4) particularly, if one of the equations in system (1.0.4) is in theformt+(gu)x=xx (1.0.5)where g(u) is a continuous function of urn thencr,c0,0, if 0xc00 (1.0.6),where c0is a positive constant and cr,c0, could tend to zero as the time t tends to infinity or tends to zero.proof. the local existence result in (1) can be easily obtained by applying the contraction mapping principle to an integral representation for a solution, following the standard theory of semilinear parabolic systems.whenever we have an a priori l estimate of the local solution, itis clear that the local time can be extended to tstep by step since the step time depends only on the l norm. the process to get the local solution clearly shows the behavior of the solution in (3). the details about the proofs of (1)-(3) in theorem 1.0.2 can be seen in lsu, sm. the following is the unpublished proof of (1.0.6) by bereux and sainsaulieu (cf. lu9, pe).we rewrite equation (1.0.5) as follows:vt+guvx+g(u)x=(vxx+vx2) (1.0.7)where v=logw. thenvt=vxx+(vx-gu2)2-g(u)x-g2(u)4. (1.0.8)the solution v of (1.0.8) with initial data v0x=log(0(x) can berepresented by a green function gx-y,t=1texp(-x-y24t):v=-gx-y,tv0ydy+0t-vx-gu22-g2u4-guxgx-y,t-sdyds. (1.0.9)it follows from (1.0.9) thatv-gx-y,tv0ydy+0t-g2u4-guxgx-y,t-sdyds =-gx-y,tv0ydy+0t-(g(u)gyx-y,t-s-g2u4gx-y,t-s)dydslogc0-mt-m1t1212-ct,c0,-. (1.0.10)thus has a positive lower bound ct,c0, for any fixed and t0, if we furthermore.suppose that u are uniformly bounded in lp(1p) space with respect to the parameter , then there exists a subsequence (still labelled) usuch thatux,tux,t, weakly in lp, (1.0.11)and also a subsequence f(u) such thatf（ux,t）lx,t weakly (1.0.12)under suitable growth conditions on f(u). iflx,t=f（ux,t），a.e.， (1.0.13)then clearly ux,t is a weak solution of system (1.0.1) with the initial data (1.0.2) by letting tend to zero in (1.0.4).how could we obtain the weak continuity (1.0.13) of the nonlinear flux function f(u) with respect to the sequence of viscosity solutions u? the theory of compensated compactness is just to answer thisquestion.why is this theory called compensated compactness? roughly speaking, this term comes from the following fact:if a sequence of functions satisfiesx,tx,t (1.0.14)with either()2+()3()2+()3或者()2-()3()2-()3 (1.0.15)weakly as tends to zero, in general, x,t is not compact. however,it is clear that any one weak compactness in (1.0.15) can compensate for another to make the compactness of . in fact, if we add them together, we get()2()2 (1.0.16)weakly as tends to zero, which combining with (1.0.14) implies the compactness of .in this book, our goal is to introduce some applications of the method of compensated compactness to the scalar conservation law as well as some special systems of two or three equations. moreover,applications to some physical systems with a relaxation perturbationparameter are also considered. the arrangement of this book is as follows:in chapter 2, we introduce some elemental theorems in the theory of compensated compactness. section 2.1 is about the weak continuity theorems of 22 determinants, and the proofs come from ta. section2.2 is about the young measure representation theorems of weak limits and we use the proofs in lin. section 2.3 is about the murat compact embedding theorems. in this part, we introduce two theorems. the proof of theorem 2.3.2 is the same as that given in dcl1 and the proof of theorem 2.3.4 is copied from the french paper by murat mu.it is necessary to point out that theorem 2.3.4 is independent of this book and the readers could pass over it without considering the details.we collect it here because it was used in some research papers (cf.cll, jpp).in chapter 3, we consider the cauchy problem of the scalar equation with l and lp(1p 3, our proof is copied from the paper lpt. for the case of 1 3, using only four pairs ofweak entropy-entropy flux, we give a short proof by assuming that the solution is away from vacuum and small (cf. cl2).in chapter 9, the methods in chapters 6 and 7 are again extended to study two special systems of one-dimensional euler equations, which are nonstrictly hyperbolic on the vacuum line = 0. for smooth solutions, they are equivalent to the systems of polytropic gas dynamics with the adiabatic exponents 3 and = , respectively. our proofs in this chapter come from lu2 and lu8.in chapter 10, we consider the general euler equations of onedimensional,compressible fluid flow. this more general system is again nonstrictly hyperbolic on the vacuum line = 0. to study this system by using the compensated compactness, one basic difficulty is how to construct entropy-entropy flux pairs and obtain the necessary estimates on these entropies. since the method to construct entropy-entropy flux pairs of lax type (cf. la1) to strictly hyperbolic systems does not work here, in this chapter we extend dipernas method to nonstrictly hyperbolic systems. we introduce a special form of lax entropy, in which the progression terms are functions of a single variable. the necessary estimates for the major terms are obtained by the singular perturbation theory of the ordinary differential equations of second order. the proof in this chapter comes from lu6.in chapter 11, we extend the method given in chapter 10 to study some extended systems of elasticity in l space. the proof is alsofrom lu6.in chapter 12, some important results about lp(1p), weak solutions for the system of elasticity are introduced, which include a compactness framework of artificial viscosity solutions to this system by lin lin and a compactness framework of physical viscosity by shearer sh. an application of the latter compactness framework by shearer on the system of adiabatic gas flow through porous media is also considered (cf. lk1). however, to avoid knotty mathematical formulas, we choose not to provide the proofs of these two compactness frameworks in this book, although they are very important and form a basis on relaxation problems of hyperbolic systems of three equations in chapter 16.from chapter 13 to chapter 16, we introduce some applications of the compensated compactness on the relaxation problems.in chapter 13, a general description of the relaxation singular problemis introduced.in chapter 14, singular limits of stiff relaxation and dominant diffusionfo