雙曲守恒律系統(tǒng)文獻(xiàn)翻譯

1、附件1：外文資料翻譯譯文第1章預(yù)備知識(shí)雙曲守恒律系統(tǒng)是應(yīng)用在出現(xiàn)在交通流，彈性理論，氣體動(dòng)力學(xué)，流體動(dòng)力學(xué)等等的各種各樣的物理現(xiàn)象的非常重要的數(shù)學(xué)模型。一般來(lái)說(shuō)，古典解非線性雙曲方程柯西問(wèn)題解的守恒定律僅僅適時(shí)局部存在于初始數(shù)據(jù)是微小和平滑的.這意味著震波在解決方案里相配的大量時(shí)間里出現(xiàn)。既然解是間斷的而且不滿足給定的傳統(tǒng)偏微分方程式，我們不得不去研究廣義的解決方法，或者是滿足分布意義的方程式的函數(shù).我們考慮到如下形式的擬線性系統(tǒng)ut+fux=0.x,trr+, (1.0.1)這里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是給定表示保守項(xiàng)的

2、適量函數(shù)，這些方程式通常被叫做守恒律.讓我們假設(shè)一下，u是(1.0.1)在初始數(shù)據(jù)ux,0=u0(x) . (1.0.2)下的傳統(tǒng)解。使c01成為c1消失在緊湊子集外的函數(shù)的一類。我們用乘以(1.0.1)并且使t0的部分，得到t0(ut+fux)dxdt+t=0u0dx=0. (1.0.3)定義1.0.1 有l(wèi)p，10是恒定的.我們首先應(yīng)該得到一個(gè)關(guān)于柯西問(wèn)題(1.0.4),(1.0.2)對(duì)于任何一個(gè)依據(jù)下列拋物方程的一般理論存在的的解的序列u：定理1.0.2 (1)對(duì)于任意存在的0, (1.0.4)的柯西問(wèn)題在有界可測(cè)原始數(shù)據(jù)(1.0.2)對(duì)于無(wú)限小的總有一個(gè)局部光滑解u(x,t)c(r0,

3、)，僅依賴于以原始數(shù)據(jù)u0(x)的l.(2)如果解u有一個(gè)推理的l估量|u,t|lm(,t)對(duì)于任意的t0,t,于是解在r0,t上存在.(3)解u滿足：lim|x|u=0 如果limxu0x=0.( 4)特別的，如果在(1.0.4)系統(tǒng)中的一個(gè)解以t+(gu)x=xx (1.0.5)形式存在，這里g(u)是在urn上連續(xù)函數(shù)，cr,c0,0,如果0xc00 (1.0.6) 這里c0是一個(gè)正的恒量，而且當(dāng)變量t趨向無(wú)窮大或者趨向于0時(shí)，cr,c0,趨向于0.證明.在(1)中的局部存在的結(jié)果能簡(jiǎn)單的通過(guò)把收縮映射原則應(yīng)用到解的積分表現(xiàn)得到，根據(jù)半線性拋物系統(tǒng)標(biāo)準(zhǔn)理論.每當(dāng)我們有一個(gè)先驗(yàn)的l局部解的

4、評(píng)估，明顯的本地變量一步一步擴(kuò)展到t，因?yàn)橹鸩阶兞恳罁?jù)l基準(zhǔn).取得局部解的過(guò)程清晰地表現(xiàn)在(3)中的解的行為.定理1.0.2的(1)-(3)證明的細(xì)節(jié)在lsu,sm看到.接下來(lái)是bereux和sainsaulieu未發(fā)表的證明(cf. lu9, pe)我們改寫(xiě)方程式(1.0.5)如下：vt+guvx+g(u)x=(vxx+vx2) (1.0.7)當(dāng)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能被格林函數(shù)gx-y,t=1texp(-x-y24t)描寫(xiě)：v=-gx-y,tv0ydy+0t-vx-g

5、u22-g2u4-guxgx-y,t-sdyds. (1.0.9)由于-gx-y,tdy=1，-|gx-y,t|dymt,(1.0.9)轉(zhuǎn)化為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)因此對(duì)于任意一個(gè)，t0,如果我們?cè)偌偃鐄是在關(guān)于參數(shù)的空間lp(1p)上一致連續(xù)，即存在子序列（仍被標(biāo)記）u如下ux,tux,t, 在lp上弱對(duì)應(yīng) (1.0.11)而且有子序列f(u)如下f（ux,t）lx,t， 弱

6、對(duì)應(yīng) (1.0.12)在習(xí)慣于fu成長(zhǎng)適當(dāng)成長(zhǎng)性.如果lx,t=f（ux,t），a.e.， (1.0.13)然后明顯的ux,t是(1.01)使在(1.0.4)的趨近于0的一個(gè)初始值(1.0.2)的一個(gè)弱解.我們?nèi)绾蔚玫饺踹B續(xù)(1.0.13)的關(guān)于粘度解的序列u的非線性通量函數(shù)f(u)？補(bǔ)償密實(shí)度原理就回答了這個(gè)問(wèn)題.為什么這個(gè)理論叫補(bǔ)償密實(shí)度？粗略的講，這個(gè)術(shù)語(yǔ)源自于下列結(jié)果：如果一個(gè)函數(shù)序列滿足x,tx,t (1.0.14)與下列之一()2+()3()2+()3或者()2-()3()2-()3 (1.0.15)當(dāng)趨近于0時(shí)弱相關(guān)，總之，x,t不緊密.然而，明顯的，任何一個(gè)在(1.0.15)中

7、的弱緊密度能補(bǔ)償使其成為的緊密度.事實(shí)上，如果我們將其相加，得到()2()2 (1.0.16)當(dāng)趨近于0時(shí)弱相關(guān),與(1.0.14)結(jié)合意味著的緊密度.在這本書(shū)里，我們的目標(biāo)是介紹一些補(bǔ)償緊密度方法對(duì)標(biāo)量守恒律的應(yīng)用，和一些特殊的兩到三個(gè)方程式系統(tǒng).此外，一些具有松弛擾動(dòng)參量的物理系統(tǒng)也被考慮進(jìn)來(lái)。這本書(shū)的準(zhǔn)備情況如下：在第2章我們介紹一些基本定理關(guān)于補(bǔ)償緊密度原理.章節(jié)2.1是關(guān)于22行列式的弱連續(xù)定理,和來(lái)自于ta的證明.章節(jié)2.2是關(guān)于弱極限理論的表現(xiàn)的young式測(cè)量，我們用了lin的證明.章節(jié)2.3是關(guān)于繆拉緊密嵌入式定理，在這個(gè)部分我們介紹兩種原理.定理2.32的證明是和在dcl1

8、給出的證明是一樣的，2.3.4的證明是從法國(guó)人繆拉的論文中摘抄的.有必要提出的是定理2.34是獨(dú)立于本書(shū)，讀者可以不用考慮細(xì)節(jié)的掠過(guò)它.我們把它收集在這是因?yàn)樗挥糜谝恍┭芯空撐闹?cf.cll, jpp). 在第3章，我們分別考慮在l和lp(1p3我們的證明來(lái)自于lpt的論文.在13的情況下，只用弱熵熵流對(duì)的四個(gè)部分，我們給出一個(gè)簡(jiǎn)短的證明，通過(guò)假定解總是來(lái)自空間而且小(cf. cl2). 在第9章，第6和7章的方法再次延伸去研究一維歐拉方程的兩個(gè)特殊系統(tǒng)，都是在真空線 = 0非嚴(yán)格雙曲的.為了平滑解，他們分別等同于多變氣體動(dòng)力學(xué)系統(tǒng)在絕熱指數(shù)3 and = 的情況.我們的證明來(lái)自于lu2

9、和 lu8. 在第10章，我們考慮一般的可壓縮液流一維歐拉方程.這個(gè)更一般的系統(tǒng)再次在真空線 = 0非嚴(yán)格雙曲的.通過(guò)運(yùn)用補(bǔ)償緊密度學(xué)習(xí)這個(gè)系統(tǒng)，一個(gè)基本的困難時(shí)如何構(gòu)造熵熵流對(duì)和得出關(guān)于這些熵的必要估計(jì).由于構(gòu)建嚴(yán)格雙曲的松弛型熵熵流對(duì)的方法不再有效，在這一章，我們延伸了diperna關(guān)于非嚴(yán)格雙曲系統(tǒng)的方法，我們介紹了一個(gè)松弛熵的特殊模殼，在其中級(jí)數(shù)術(shù)語(yǔ)是一個(gè)單一變量的函數(shù). 這個(gè)必要的專業(yè)術(shù)語(yǔ)估計(jì)所得的奇異攝動(dòng)微分方程理論二次順序.證明是這個(gè)章節(jié)來(lái)自于lu6 在第11章，我們延伸了第十章給出的研究拓展在l空間彈性系統(tǒng)的方法.這個(gè)證明也來(lái)此lu6. 在12章，一些重要的關(guān)于lp(1p),彈

10、性系統(tǒng)的弱解的結(jié)果被介紹，包括一個(gè)針對(duì)這個(gè)系統(tǒng)的通過(guò)lin粘度解決方法的緊密度框架，和一個(gè)shearer研究的物理粘度緊密度框架.shearer研究的絕熱氣體通過(guò)多孔介質(zhì)的緊密度框架也被考慮了. 但是，為了避免棘手的數(shù)學(xué)公式，我們選擇不去提供書(shū)中的這兩個(gè)緊密度框架的證明.盡管他們十分重要，形成了16章三位雙曲系統(tǒng)的松弛問(wèn)題的基準(zhǔn). 從13章到16章，我們介紹一些關(guān)于松弛問(wèn)題補(bǔ)償緊密度的應(yīng)用. 在13章，介紹一個(gè)松弛單一問(wèn)題的總則. 在14章，考慮了硬性松弛的單一極限和一般2 2非線性守恒系統(tǒng)的顯性擴(kuò)散.這些包括彈性系統(tǒng)在l上的解決方案，等熵流體動(dòng)力學(xué)在歐拉坐標(biāo)和延伸的交通流量模型.lp(1p0

11、, 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 t

12、he 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 sati

13、sfies: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 z

14、ero.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

15、 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

16、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):

17、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 ar

18、e 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 c

19、learly 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

20、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

21、 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 applicatio

22、ns 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, w

23、e 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 a

24、bout 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

25、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

26、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 th

27、e 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 flo

28、w. 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 entr

29、opy-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 es

30、timates 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

31、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 applicati

32、on 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 import

33、ant 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