双曲守恒律系统文献翻译

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1、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,tR×R+, (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

2、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t>0的部分，得到t>0(ut+fux)dxdt+t=0u0dx=0. (1.0.3)定义1.0.1 有Lp，1<p,有界函数u(x,t)叫做在以原始数据u0为边界条件下Lp，(1.0.1)初值问题的一个弱解，在(1.0.3)适用于所有C01(R×R+). 非线性系统守恒理论的一个重要方面是这些方程解的存在疑问性.它正确的帮助解答在手边的已经建立的

3、自然现象的模型的问题，而且如果在问题是适定的.为了得到一个总体的弱解或者一个考虑到双曲守恒律的普遍的解，一个为了在(1.0.1)右手边增加一个微小抛物摄动限：ut+f(u)x=uxx (1.0.4)在这>0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(R×0,)，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·,

4、t|LM(,T)对于任意的t0,T,于是解在R×0,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0xc0>0 (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

5、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

6、. (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，t<有一个正的下界ct,c0,.在定理1.0.2中获得的解叫做粘性解.然后我们有了粘性解的序列u,>0,如果我们再假如u是在关于参数的空间Lp(1<p)上一致连续，即存在子序列（仍被标记）u如下ux,tux,t, 在Lp

7、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

8、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2×2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

9、.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(1<p<)初始数据下的标量方程的柯西问题.在这一部分，一个没用到Young式测量的简化的证明会给出. 在第4章的第一部分，我们介绍一些基本的二维方程系统的定义，比如严格双曲性,绝对非线性,线性退化.黎曼不等式，熵熵流对等等(cf. La2, La3, S

10、m). 在第二部分，一个来自CCS叫做不变区域理论的取得二维方程系统L粘度估计解的框架将被介绍. 在第5章，我们考虑一个特殊的对称二维方程系统(Ch3).这个系统与标量方程非常相似，因为一个特征域经常线性退化，尽管另外的域是绝对非线性的.这个系统有趣是因为绝对非线性特征域，在L区间获得的粘性密集解不再有有规律的情况.然而，沿着线性退化的特征域有更多的规律情况，像BV评价一定会添加到粘度解的序列的强紧密度. 在第6章，我们考虑一个带有二次变量的二维方程系统，这个系统是非严格按原点双曲的，一个特征域在u轴正半部分是线性退化的，当另外的域沿u轴负方向线性退化的时候，它的熵方程和绝热指数=2的多变气体

11、动力学系统一样.在学习这个偿还紧密度系统中主要的困难是熵熵流对在原点是异常的.通过一个经典富克斯方程的精确解我们详尽的得到这个系统的松弛型熵熵流对. 对从富克斯方程解的分析熵的大项估计的必要. (cf. Lu4). 在第7章，我们延伸第6章给出的方法到Le Roux系统，一个在原点不绝对双曲的系统，但是熵方程和=53的多变气体动力学系统一样，这个系统有趣在它是一个典型的庙宇形态系统，特征域都是直线.这章的证明来自LMR. 在8章，我们考虑最典型的二维双曲守恒系统，所谓的多变气体动力系统（或者叫-law） ，在>3我们的证明来自于LPT的论文.在1<3的情况下，只用弱熵熵流对的四个部

12、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

13、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(1<p<),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

14、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(1<p<)下一些物理模型的解决方法，没有了L有界估计，如等熵流体动力学在拉格朗日坐标，和不同国家的交通流量模型.所有的证明在Lu9找到. 在15章，一个应用于一般2 × 2双曲保持系统（不必要严格双曲）被介绍. 这个框架在非严格双曲系统的一个应用，所谓

15、的扩展交通流量系统也被得到. 在16章，一般3×3化学反应系统严格松弛和显性扩散单侧极限被考虑.纯松弛极限（没有粘度）化学反应的一个特例也介绍了.附件2：外文原文Chapter 1PreliminarySystems of hyperbolic conservation laws are very important mathematical models for a variety of physical phenomena that appear in traffic flow, theory of elasticity, gas dynamics, fluid dynamics

16、 and so on.In general, the classical solution of the Cauchy problem for nonlinear hyperbolic conservation laws exists only locally in time even if the initial data are small and smooth. This means that shock waves always appear in the solution for a suitable large time. Since the solution is discont

17、inuous and does not satisfy the given partial differential equationsin the classical sense, we have to study the generalized solutions,or functions which satisfy the equations in the sense of distributions.We consider the quasi-linear systems of the formut+fux=0.x,tR×R+.(1.0.1)whereu=(u1,u2,un)

18、TRn,n1is the unknown vector function standing for the density of physical quantities, fu=(f1u,fn(u)Tis a given vector function denoting the conservative term. These equations are commonly called conservation laws. Let us suppose for the moment, that u is a classical solution of (1.0.1) with the init

19、ial dataux,0=u0(x) . (1.0.2)Let C01be the class of C1function which vanishes outside of a compact subset. We multiply (1.0.1)by and integrate by parts over t>0, to gett>0(ut+fux)dxdt+t=0u0dx=0. (1.0.3)Definition 1.0.1 An Lp 1<p, bounded function u(x,t)is calleda weak solution of the initial

20、-value problem (1.0.1) with Lpbounded initial data u0, provided that (1.0.3) holds for all C01R×R+ An important aspect of the theory of nonlinear system of conservation laws is the question of existence of solutions to these equations. Ithelps to answer the question if the modelling of the natu

21、ral phenomena at hand has been done correctly, and if the problem is well posed. To get a global weak solution or a generalized solution for given hyperbolic conservation laws, a standard method is to add a smallparabolic perturbation term to the right-hand side of (1.0.1):ut+f(u)x=uxx (1.0.4)where

22、>0 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 l

23、ocal smooth solution u(x,t)C(R×0,) 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 R×0,T.(3) The solution u satisfies:lim|x|u=0 if limxu0x=0. (4) Parti

24、cularly, 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 0xc0>0 (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

25、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

26、 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

27、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-

28、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 t<. The solution obtained in Theorem 1.0.2 is called viscosity

29、solution. After we have the sequence of viscosity solutions u, >0, if we furthermore.suppose that u are uniformly bounded in Lp(1<p) 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

30、 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 functi

31、on 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 eit

32、her()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, whi

33、ch 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 rel

34、axation 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 2×2 determinants, and the proofs come from Ta. Section2

35、.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

36、 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

37、of the scalar equation with L and Lp(1<p<) initial data, respectively. In this part, a simplified proof (cf. CL1, Lu1) without using the Young measure isgiven.In the first part of Chapter 4, we introduce some basic definitions of systems of two equations, such as the strict hyperbolicity, genu

38、ine nonlinearity, linear degeneration, Riemann invariants, entropy-entropy flux pair and so on (cf. La2, La3, Sm).In the second part, a framework to obtain L estimates of viscositysolutions for systems of two equations, called the Invariant RegionTheory from CCS, is introduced.In Chapter 5, we consi

39、der a special symmetric system of two equations(Ch3). This system is very similar to the scalar equation because one characteristic field is always linearly degenerate, although the other field is genuinely nonlinear. This system is of interest because along thegenuinely nonlinear characteristic fie

40、ld, the compactness of viscosity solutions is obtained in Lspace without any more regular condition. However, along the linearly degenerate characteristic field, some more regular conditions, such as BV estimates, must be added to ensure the strong compactness of the sequence of viscosity solutions.

41、In Chapter 6, we consider a system of two equations with quadratic flux. This system is nonstrictly hyperbolic at the original point, one characteristic field is linearly degenerate on the positive half axis of u, while the other field is linearly degenerate on the negative half axis ofu. Its entrop

42、y equation is the same as that of the system of polytropic gas dynamics with the adiabatic exponent = 2. The main difficulty in studying this system by the compensated compactness is that the entropy-entropy flux pairs are singular at the original point. Througha careful construction of exact soluti

43、ons of the classical Fuchsia equation, we obtain the explicit entropy-entropy flux pair of Lax type for this system. Then the necessary estimates for the major terms of these entropies follow from the analysis of solutions of the Fuchsia equation(cf. Lu4).In Chapter 7, we extend the method given in

44、Chapter 6 to the Le Roux system, which is also nonstrictly hyperbolic at the original point, but the entropy equation is the same as that of = 5/3 for the polytropic gas. This system is of interest because it is a typical system of Temple type, whose characteristic fields are both straight lines. Th

45、e proof in this chapter is from LMR.In Chapter 8, we consider the most typical hyperbolic conservation system of two equations, the so-called system of the polytropic gas dynamics (or -law). For the case of > 3, our proof is copied from the paper LPT. For the case of 1 < 3, using only four pai

46、rs 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

47、 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 f

48、low. 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 en

49、tropy-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

50、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 i

51、s alsofrom Lu6.In Chapter 12, some important results about Lp(1<p<), 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

52、 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 v

53、ery 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 problem

54、is introduced.In Chapter 14, singular limits of stiff relaxation and dominant diffusionfor general 2 × 2 nonlinear systems of conservation laws are considered. These include the L solutions of the system of elasticity, the system of isentropic fluid dynamics in Eulerian coordinates and theexten

55、ded models of traffic flows; the Lp(1<p<), solutions for some physical models, without L bounded estimates, such as the system of isentropic fluid dynamics in Lagrangian coordinates, and the models of traffic flows in different states. All proofs in this chapter can be found in Lu9.In Chapter

56、15, a framework for the singular limits of stiff relaxation for general 2×2 hyperbolic conservation systems (not necessary strictly hyperbolic) is introduced (cf. CLL, Lu10).An application of this framework on a nonstrictly hyperbolic system,the so-called system of extended traffic flow, is als

57、o obtained. Theproof is from Lu10.In Chapter 16, singular limits of stiff relaxation and dominant diffusion for the general 3×3 systemof chemical reaction are considered (cf.Lu12). The pure relaxation limit (without viscosity) for a special case of this chemical reaction system is also introduced (cf. LK1, LK2, Tz).11