Weak Solution of Generalized KdV Equation with High Order Perturbation Terms

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Estimates of suitable weak solutions to the Navier-Stokes equations in critical Morrey spac

Deﬁnition 1.2 The pair v and p is called a suitable weak solution to the Navier-Stokes equations in Q if
v ∈ L2,∞(Q) ∩ W21,0(Q),
p ∈ L 3 (Q); 2
(1.3)
the Navier-Stokes equations hold in Q in the sense of distributions; (1.4)
3
had to develop a new method based on the unique continuation theory for parabolic equations, see [3].
The aim of this paper is to contribute somehow to analysis of smoothness of suitable weak solutions under additional assumptions that certain functionals invariant with respect to natural scaling are bounded. We hope that our results can be regarded as a starting point for that analysis. Let us ftion 1.3 of regular points is due Ladyzhenskaya-Seregin [4]. In the most popular deﬁnition by Caﬀarelli-Kohn-Nirenberg, the H¨older space is replaced with the space of essentially bounded functions.

On the energy equality for weak solutions of the 3D Navier-Stokes equations

1. Introduction We consider the 3D incompressible Navier-Stokes equations (NSE) ∂t u − ν ∆u + (u · ∇)u + ∇p = f, ∇·u=0 (1) u(x, t) = 0 for x ∈ ∂ Ω,
t
Байду номын сангаас
(5) (u(t), ϕ(t)) +
0
{−(u, ∂s ϕ) + ν (∇u, ∇ϕ) + (u · ∇u, ϕ)} ds
t
= (u0 , ϕ(0)) +
0
(f, ϕ)ds,
for all t ∈ [0, T ) and ϕ as before. Solutions satisfying all the properties listed above are commonly called Leray-Hopf solutions. It is a famous open problem to show that a given Leray-Hopf solution to the Navier-Stokes system actually satisﬁes the energy equality
ENERGY EQUALITY FOR 3D NAVIER-STOKES
3
locally L2 -summable in space-time. Since p is given by a CalderonZygmund operator applied to u ⊗ u, Kukavica’s condition is implied by that of Lions and Ladyzhenskaya. In this present paper we prove (6) in a functional class with a better scaling than that of Shinbrot. Let us denote (8) H = {u ∈ L2 (Ω) : ∇ · u = 0, u · n|∂ Ω = 0}, and let P : L2 (Ω) → H be the L2 -orthogonal projection, referred to as the Helmholtz-Leray projector. Let A be the Stokes operator deﬁned by (9) Au = −P∆u. The Stokes operator is a self-adjoint positive sectorial operator with a compact inverse. We denote V s = D (As/2 ), s > 0, the domain of the fractional power of A (see the next section for details). Roughly, V s corresponds to the Sobolev space H s , and is in fact equal to H s under periodic boundary conditions. In particular one can give the following description of V = V 1 (see [2]): (10) V = {u ∈ H 1 (Ω) : ∇ · u = 0, u|∂ Ω = 0}. We prove the following result. Theorem 1.1. Every weak solution u(t) of (1) satisfying (5) on [0, T ) with u ∈ LH ∩ L3 ([0, T ); V 5/6 ) veriﬁes the energy equality (6). The dimensional Lr Ls -analogue of the functional class L3 ([0, T ); V 5/6 ) is L3 L9/2 , which exhibits the scaling (11) 2/s + 2/r = 10/9. It would be an interesting next step to try to prove the energy equality under this new scaling in the Lr Ls -class itself. Finally we note that under periodic boundary conditions the result of Theorem 1.1 follows from – although not explicitly stated in – the recent work by the authors on Onsager’s conjecture for the Euler equations [1]. So, the novelty of this present paper consists in the extension of Theorem 1.1 to the case of the Dirichlet boundary condition in a smooth domain. 2. Preliminaries First, let us introduce some notations and functional setting. Denote by (·, ·) and | · | the L2 (Ω)-inner product and the corresponding L2 (Ω)norm. Let H and V be as above (8), (10), and V ′ denote the dual of V . We endow V with the norm u = |∇u|. There exists an orthonormal

广义正则长波方程隐式差分格式

0.引言Pereg rine[1,2]首先提出广义正则长波(GRL W)方程u t +u x +β(u p)x -γu xx t =0(1.1)其中p 是正整数,β与γ是正的常数.正则长波(RLW)方程作为GRLW 方程的一个特例,它描述的波的运动与KdV 方程u t +u x +uu x -u xxx =0具有相同的逼近解,而且能较好的模拟KdV 方程的所有应用,因此引起了人们的重视.文[4,5,3,7]讨论了它的数值方法.文[6]对GRL W 方程进行了研究,提出了一个三层的格式.本文考虑以下GRL W 方程的Cauchy 问题u t +u x +β(u p )x -γu xx t =0(1.1)u|t =0=u 0(x)(1.2)其中p 为正整数,β>0,γ>0方程(1)满足以下守恒律,E(t)=u2L 2+γu x2L2=C onst (1.3)方程(1)有孤波解,u(x,t)=Asech 2p-1(k(x+x 0-σt))(1.4)其中A=(P+1)(σ-1)2β!"2p-1,k=p-12γσ-1σ#,p ≥2,σ,x 0为任意常数.1.差分格式与守恒律我们将应用下面的记号.对平面区域X l ,X r !"×0,!"T 作网格剖分,取空间方向步长h=(X r -X l )/J,时间方向步长为τ,x j =X l =jh ,t n =n τ,j=0,1,L,,J,N=T τ定义网格比r=τ/h,结点(x j ,t n )=(jn,n τ)处的数值解记作u n j ,即u n j =u(jh,n τ).(u nj )x=u nj +1-u n j h ,(u n j )x %=u nj -u nj -1h(u nj )t=un+1j τ,(u n j )t %=u n j-u n-1jτ(u nj )x$=u nj+1-u nj -12h ,u n+12=u n+1j2(u n,v n)=h j&u n j v nj ,vn2=(v n,v n)vn∞=sup jv nj ,,,.在本文中总假定为广义常数,即在不同的位置C 可以表示不同的数值.对于方程(1.1),(1.2)我们引入参数η(0≤η≤1),从而提出如下差分格式,1-η2(u n j+1+u n j-1)t +η(u nj )t+(u n+12j)x$-γ(u n+1j)xxt+p βp+1u jn+12()p-1u n+12j()x$!"(2.1)u 0j =u 0(X l +jh)(2.2)其中j=0,,L,J,n=1,L N.差分格式(2.1)对于(1.1)的数值模拟为:引理1差分格式(2.1)满足如下的离散守恒律,即E n=1-η2h Jj &(u n j+1u n j +u n j u nj-1)+ηun2+γu xn2=En-1=L =E 0(2.3)证明(2.1)式与u n+1+u n 作内积,得1-η2τh Jj&(u n+1j+1+u n +1j -1u n+1j -u n j+1u n j -u n j-1u nj )+ητ(un+12-u nx2)=0令E n=1-η2h Jj&(u n j+1u n j +u n j u n j-1)+ηun2+γu xn2则E n =E n-1=L E 0.W2.差分格式(2.1)的收敛性与稳定性下面我们将分析差分格式(2.1)的截断误差.考虑差分格式(2.1)的截断误差R n j ,记U n j =u(x J ,t n ),θn j =U n j -u nj 则R nj =1-η2(θn j+1+θn j-1)+η(θn j )t+(θn+12j)x$-γ(θn +1j)xxt+p βp+1(θn+12j)p-1(θn+12j)x$+(θn +12j)px$!"(3)在(x j ,t n )处,(3)式中的线性部分作T ay lor 展开,得R n j =1-η2(θn j+1+θn j-1)+η(θnj )t +(θn +12j)x$-γ(θn+1j)xxt =(u t +u x -γu x xt )|(x j ,t n )+O (τ2+h 2),记(3)式中的非线性部分为N=p βp+1(U n+12j )p -1(U n+12j )x$+(U n+12j )px$!",在(x j ,t n )处,N 作T ay lo r 展开并整理,得N=β(u p )x |(x j ,t n )+O(τ2+h 2),因此R n j =O(τ2+h 2)于是,我们就得到下面的引理:引理2假设u(x ,t)足够光滑,则差分格式(2.1)的局部截断误差为O(τ2+h 2)理3(离散的so blev 不等式)对于离散函数u 和任意给定的ε>0,存在仅依赖ε与n 常数n (ε,n ),使得un∞≤εu nx+C(ε,n)un引理4假设u 0∈H 10[X l ,X r ],则柯西问题(1.1),(1.2)的解满足如下估计:uL ≤C,u xL ≤C,uL ∞≤C证明由(3)式得广义正则长波方程隐式差分格式潘新田(潍坊学院山东潍坊261061)【摘要】首先提出了一个求解广义正则长波方程的新的守恒的隐式差分格式,引入了参数η(0≤η≤1),并对该格式作了稳定性与收敛性分析,格式的截断误差为O(τ2+h 2)数值结果表明,该方法具有较高的精度,是有效的,可靠的。

A Needed Narrowing Strategy

1
Introduction
Declarative programs are more abstract than equivalent imperative programs. Declarative languages replace pointers with algebraic data types, split complex computations into small, easily parameterizable units and avoid the manipulation of an explicit state through assignments. These features promise to ease some diﬃcult essential tasks of software development. For example, they simplify reasoning about programs (veriﬁcation of non-executable speciﬁcations), promote freedom of implementation (use of parallel architectures), and reduce both development time and maintenance eﬀorts (code is compact and easier to read and understand). All these advantages stem from various factors—the most important being the solid mathematical foundations of declarative computing paradigms. Currently, the ﬁeld of declarative programming is split into two main paradigms based on different mathematical formalisms: functional programming (lambda calculus) and logic programming (predicate logic). This situation has a negative impact on teaching, research and applications. Usually there are diﬀerent courses on functional programming and logic programming, and students do not perceive the similarities between them. In terms of research, each ﬁeld has its own community, conferences, and journals, and sometimes similar solutions are developed twice. Each ﬁeld also has its own application areas and some eﬀort has been devoted to show that one paradigm can cover applications of the other paradigm [71] instead of showing the advantages of declarative programming in various application ﬁelds. Each paradigm, of course, has its own advantages. Functional programming oﬀers nested expressions, eﬃcient evaluation by deterministic (often lazy) evaluation, and higher-order functions. Logic programming oﬀers existentially quantiﬁed variables, partial data structures, and built-in search. On the other hand, functional and logic languages have a common core and can be seen as diﬀerent facets of a single idea. Consequently, the interest in integrating functional and logic programming has grown over the last decade and resulted in various proposals of integrated functional logic languages that combine the advantages of both paradigms (see [31] for a survey). Functional logic languages extend both functional languages and logic languages. Functional languages are extended with facilities such as function inversion, partial data structures, and logic variables [65]. Logic languages are extended with nested expressions, a more eﬃcient operational behavior [30], and less need for impure control features such as the Prolog “cut. ” This paper concerns narrowing. Narrowing is a computation model of considerable importance both for declarative programming in general and for functional logic languages in particular. We explain why using an example. Example 1 Consider the following rules deﬁning the concatenation of lists (as an inﬁx operator ++) where we use the Prolog syntax for lists, i.e., [] denotes the empty list and [E |R] denotes a non-empty list consisting of a ﬁrst element E and a remaining list R: [] ++ L → L [E |R] ++ L → [E |R ++ L] In a functional language, this deﬁnition is used to concatenate two lists, e.g., [a, b] ++[c, d] evaluates to the list [a, b, c, d]. It is understood that (the value of) the arguments of ++ must be known in order to apply a rule. Narrowing extends the use of ++ without altering its deﬁnition in a remarkable way. Even if all or part of either argument of ++ is unknown, (i.e., is an uninstantiated variable), narrowing keeps computing. In principle, this is not diﬃcult—a value is assigned to the unknown parts of an argument—but the technical details that make this approach practical (sound, complete, and as eﬃcient as the best functional computation when the arguments are fully known) 2

Lie Symmetries and Solutions of KdV Equation

Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, I.R.Iran. e-mail: m nadjaﬁkhah@iust.ac.ir † e-mail: reza hejazi@iust.ac.ir
−u
∂ ∂ A2 ux utx + · · · − A5 utx = 0, ∂uxx ∂ux . . . ∂ ∂ − A2 + · · · − A7 = 0, ∂t ∂uxx ∂ ∂ − A4 + · · · + A2 utx = 0. ∂uxx ∂utx
Байду номын сангаас
After solving this system with respect to A1 , ..., A8 , we have A1 = c1 t + c2 , A2 = c1 x/3 + c3 t + c4 , A3 = −2c1 u/3 + c3 , A4 = −c3 ux − 5c1 ut /3, A5 = −c1 ux , A6 = −2c3 utx − 8c1 utt /3, A7 = −c3 uxx − 2c1 utx , A8 = −4c1 uxx /3, where c1 , c2 , c3 , c4 are arbitrary constants, thus v = (c1 t + c2 ) c1 ∂ 2 c1 ∂ 5 c1 ∂ ∂ + ( x + c3 t + c4 ) + (c3 − u) − (c3 u x + ut ) ∂t 3 ∂x 3 ∂u 3 ∂ut 8 c1 ∂ ∂ 4 c1 ∂ ∂ − (2c3 utx + utt ) − (c3 uxx + 2c1 utx ) − uxx . −c1 ux ∂ux 3 ∂utt ∂utx 3 ∂uxx

2Stochastic averaging of quasi-generalized Hamiltonian systems

Keywords: Stochastic averaging method Generalized Hamiltonian systems Gaussian white noises
ABSTRACT
A stochastic averaging method for generalized Hamiltonian systems (GHS) subject to light dampings and weak stochastic excitations is proposed. First, the GHS are briefly reviewed and classified into five classes, i.e., non-integrable GHS, completely integrable and non-resonant GHS, completely integrable and resonant GHS, partially integrable and non-resonant GHS and partially integrable and resonant GHS. Then, the averaged Itoˆ and FPK equations and the drift and diffusion coefficients for the five classes of quasi-GHS are derived. Finally, the stochastic averaging for a nine-dimensional quasi-partially integrable GHS is given to illustrate the application of the proposed procedure, and the results are confirmed by using those from Monte Carlo simulation.

Fractional order nonlinear systems with delay in iterativelearning control

Fractional order nonlinear systems with delay in iterative learning controlqLi Yan,Jiang Wei ⇑School of Mathematical Science,AnHui University,Hefei,Anhui 230601,PR Chinaa r t i c l e i n f o Keywords:Fractional order with delayGeneralized Gronwall inequality Iterative learning controla b s t r a c tIn this paper,we discuss a P-type iterative learning control (ILC)scheme for a class of frac-tional-order nonlinear systems with delay.By introducing the k -norm and using Gronwall inequality,the sufﬁcient condition for the robust convergence of the tracking errors is obtained.Based on this convergence,the P-type ILC updating laws can be determined.Ó2015Elsevier Inc.All rights reserved.1.IntroductionIterative learning control is suitable for repetitive movements of the controlled system.Its goal is to achieve full range of tracking tasks on ﬁnite interval.Iterative learning control (ILC)is one of the most active ﬁelds in control theories.ILC belongs to the intelligent control methodology,is an approach for improving the transient performance of systems that operate repetitively over a ﬁxed time interval.The combination of ILC and fractional calculus was ﬁrst propose in 2001.In the following ten years,many fractional-order ILC problems were presented aiming at enhancing the performance of ILC scheme for linear or nonlinear systems [2–6].In recent years,the application of ILC to the fractional-order system has become a new topic.The objective of ILC is to determine a control input iteratively,resulting in plant’s ability to track the given reference signal or the output trajectory over a ﬁxed time interval.Owing to its simplicity and effectiveness,ILC has been found to be a good alternative in many areas and applications,see recent surveys for detailed results [7–13].This paper is on the basis of [14].We discuss the tracking problem through the open-loop P-type iterative learning control and the closed-loop P-type iterative learning control.The sufﬁcient condition errors with respect to initial positioning error under P-type ILC is obtained by introducing the k -norm and using Gronwall inequality.The delay part of this paper,we reference [15–20].In Section 2some basic deﬁnitions of fractional calculus used in this paper are mentioned.Section 3the main results are shown.Finally,some conclusions are drawn in Section 4.Throughout this paper,the 2-norm for the n -dimensional vector w ¼ðw 1;w 2;...;w n Þis deﬁned as k w k ¼ðP n r ¼0w 2i Þ12,while the k -norm for a function is deﬁned as k Ák k ¼sup t 2½0;T f e Àk t j Ájg ,where k >0./10.1016/j.amc.2015.01.0140096-3003/Ó2015Elsevier Inc.All rights reserved.qThis research had been supported by National Natural Science Foundation of China (nos.11371027and 11471015);Doctoral Fund of Ministry of Education of China (no.20093401110001);Major Program of Educational Commission of Anhui Province of China (no.KJ2010ZD02);National Natural Science Tianyuan Specialty Foundation of China (11326115),Research Foundation for the Doctoral Program of Higher Education of China (20133401120013),Natural Science Program of Higher Colleges of Anhui Province (KJ2013A032).⇑Corresponding author.E-mail addresses:liyanahu@ (L.Yan),jiangwei@ (J.Wei).2.PreliminariesIn this section,some basic deﬁnitions and lemmas are introduced,which will be used in the following discussions.Deﬁnition 2.1.Riemann–Liouville’s fractional integral of order a >0for a function f :R +!R is deﬁned ast 0D Àat f ðt Þ¼1a Ztt 0ðt Às Þa À1f ðs Þds :ð2:1ÞDeﬁnition 2.2.The Caputo derivatives is deﬁned asC t 0D a t f ðt Þ¼t 0D a Àm tD mf ðt Þ;a 2½m À1;m Þ;ð2:2Þwhere m 2Z þ;D m is the classical m -order integral derivative.Deﬁnition 2.3.The deﬁnition of the two-parameter function of the Mittag–Lefﬂer type is described byE a ;b ðz Þ¼X1k ¼0z kða k þb Þ;a >0;b >0;z 2C :ð2:3ÞFor b ¼1we obtain the Mittag–Lefﬂer function of one-parameter:E a ðz Þ¼X1k ¼0z kC a ;a >0;z 2C :ð2:4ÞLemma 2.4.The fractional-order differentiation of the Mittag–Lefﬂer function ist 0D c t ½tb À1E a ;b ðk t a Þ ¼t b Àc À1E a ;b Àc ðk t a Þ;c <b :ð2:5ÞLemma 2.5.If the function f ðt ;x Þis continuous,then the initial value problemC t 0D at x ðt Þ¼f ðt ;x t Þ;0<a <1;x ðt 0Þ¼u :(ð2:6Þis equivalent to the following nonlinear Volterra integral equationx ðt Þ¼x ðt 0Þþ1C ða ÞZtt 0ðt Às Þa À1f ðs ;x s Þds :ð2:7Þand its solutions are continuous [12].Lemma 2.6.([14]Generalized Gronwall Inequality).Let u ðt Þbe a continuous function on t 2½0;T and let v ðt Às Þbe continuous and nonnegative on the triangle 06s 6T.Moreover,let w ðt Þbe a positive continuous and non-decreasing function on t 2½0;T .Ifu ðt Þ6w ðt ÞþZtv ðt Às Þu ðs Þds ;t 2½0;T ;ð2:8Þthenu ðt Þ6w ðt ÞeR tv ðt Às Þds;t 2½0;T :ð2:9Þ3.P-type iterative learning controlConsider the following SISO fractional-order nonlinear system with delayD a t x k ðt Þ¼f ðx k t ;u k ;t Þ;y k ðt Þ¼g ðx k t ;u k ;t Þ;(ð3:1Þwhere k is the number of iterations,k 2f 0;1;2;...g ;t 2½0;T ;a 2ð0;1Þ,L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–5525470<d 16j g u j ¼@g ðx k t ;u k ;t Þ@u k6d 2;0<d 36g x t ¼@g ðx k t ;u k ;t Þ@x t6d 4;k f ðx k t ;u k ;t ÞÀf ð x t k ; u k ;t Þk 6f 0j u k À u k j þf 0k x k t À x t k k ;and d 1;d 2;d 3;d 4and f 0are positive constants.x k t 2R nis the state variables in a time interval of length t,that isx k t ðh Þ¼x kðt þh Þ;ð3:2Þh 2½Às ;0 ;s >0;u k ðt Þ2R and y k ðt Þ2R are the control input and output,respectively.D a t denotes the Caputo derivative oforder a .The purpose of this paper is to ﬁnd an iterative control law to generate the control input u k ðt Þsuch that the system output y k ðt Þtracks the desired output trajectory y d ðt Þas accurately as possible when k goes to inﬁnity for all t 2½0;T .We discuss the tracking problem through the open-loop P-type iterative learning control and the closed-loop P-type iterative learning control,respectively.3.1.Open-loop P-type iterative learning controlFor the fractional-order nonlinear system with (3.1),we ﬁrst consider the following open-loop ILC updating law with initial state learning:x k þ1t 0¼x k t 0þLe k ðt 0Þ;u k þ1ðt Þ¼u k ðt Þþg 1e k ðt Þ;(ð3:3Þwhere e k ðt Þ¼y d ðt ÞÀy k ðt Þdenotes the tracking error.L and g 1are unknown parameters to be determined.In order to obtain our main result,the lemma is ﬁrst proved.Lemma 3.1.For fractional-order nonlinear system with delay (3.1)and a given reference y d ðt Þ,denote thatD u k ðt Þ¼u k þ1ðt ÞÀu k ðt Þ;D x k t ¼x k þ1t Àx k t ,ifmax 1Àg 1d 1ÀL d 3j j ;1Àg 1d 1ÀL d 4j j ;1Àg 1d 2ÀL d 3j j ;1Àg 1d 2ÀL d 4j j f g 6q 0<1;ð3:4Þwhere q 0is constant,then for all t 2½0;T ,and arbitrary input u 0ðt Þ,the open-loop-type ILC updating low (3.3)guarantees thatlim k !1j e k ðt 0Þj k ¼0:Proof.It follows from e k ðt Þ¼y d ðt ÞÀy k ðt Þand the mean value theorem thate k þ1ðt 0Þ¼e k ðt 0Þþy k ðt 0ÞÀy k þ1ðt 0Þ¼e k ðt 0Þþg ðx k t 0;u k ðt 0Þ;t 0ÞÀg ðx k þ1t 0;u k þ1ðt 0Þ;t 0Þ¼e k ðt 0ÞÀg x t ðn ÞD x k t 0Àg u ðn ÞD u kðt 0Þ;where n 2½x k t 0þr D x k t 0;u k ðt 0Þþr D u kðt 0Þ;t ,r 2ð0;1Þ.Then we havee k þ1ðt 0Þ¼e k ðt 0ÞÀLg x t ðn Þe k ðt 0ÞÀg 1g u ðn Þe k ðt 0Þ:ð3:5ÞTaking the k -norm,we obtainj e k þ1ðt 0Þj k 6j 1ÀLg x t ðn ÞÀg 1g u ðn Þjj e k ðt 0Þj k 6q 0j e k ðt 0Þj k :ð3:6ÞIf the condition (3.4)is satisﬁed,then q 0<1,we can obtainlim k !1j e k ðt 0Þj k ¼0:ð3:7ÞThe proof is complete.Then we give the main conclusion of this article.Theorem 3.2.For fractional-order nonlinear system (3.1)and a given reference y d ðt Þ,ifmax fj 1Àg 1d 1j ;j 1Àg 1d 2jg 6q 1<1;ð3:8Þwhere q 1is constant,then for all t 2½0;T ,and arbitrary input u 0ðt Þ,the open-loop-type ILC updating low (3.3)guarantees thatlim k !1y k ðt Þ¼y d ðt Þ:ð3:9Þ548L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–552Proof.It can be proved thate k þ1ðt Þ¼e k ðt Þþy k ðt ÞÀy k þ1ðt Þ¼e k ðt Þþg ðx k t ;u k ;t ÞÀg ðx k þ1t ;u k þ1;t Þ¼e k ðt ÞÀg x t ðn ÞD x k t Àg u ðn ÞD u kðt Þ;ð3:10Þwhere n ðt Þ2½x k t þr D x k t ;u k þr D u k;t ;r 2ð0;1Þ.Applying the 2-norm to Eq.(3.10)and following from (3.3),we havej e k þ1ðt Þj 6j 1Àg 1g u ðn Þjj e k ðt Þj þj g x t ðn Þjk D x k t k 6j 1Àg 1g u ðn Þjj e k ðt Þj þd 4k D x kt k :ð3:11ÞIn the next,we give an estimation for the upper bound of k D x k t k .It follows from Lemma 2.5and in accordance with the prop-erty of the fractional-order 0<a <1,we can prove that for any t 2½0;T .xk þ1ðt Þ¼xk þ1ðt 0Þþ1C ða ÞZtt 0ðt Às Þa À1f ðs ;x k þ1s;u k þ1ðs ÞÞds ;thusx kðt Þ¼x kðt 0Þþ1C ða ÞZtt 0ðt Às Þa À1f ðs ;x k s ;u kðs ÞÞds :k D x kðt Þk 6k D x kðt 0Þk þ1ða ÞZ t t 0ðt Às Þa À1½f ðs ;x k þ1s ;u k þ1ðs ÞÞÀf ðs ;x k s ;u kðs ÞÞ ds6k D x kðt 0Þk þfC a Z t t 0ðt Às Þa À1k D x k s k dsþfC a Z tt 0ðt Às Þa À1j D u k ðs Þj ds 6k D x k ðt 0Þk þf 0C ða ÞZt t 0ðt Às Þa À1k D x k s k ds þf 0C ða ÞZtt 0ðt Às Þa À1e k s ds j D u k j k :We deﬁnek D x k ðt þh Þk ¼sup h 2½Àr ;0k D x k ðt þh Þk :ð3:12Þk D x kðt þh Þk 6k D x kðt 0Þk þfC a Zt þht 0ðt þh Às Þa À1k D x k s k dsþfC a Zt þht 0ðt þh Às Þa À1e k s ds j D u k j k :It follows from [1],t 0D Àat f ðt Þ¼lim h !0nh ¼t h aX n r ¼0a r f ðt Àrh Þ¼1C ða ÞZ t t 0ðt Às Þa À1f ðs ;x s Þds ;if f ðt Þ>0,lim h !0nh ¼thaX n r ¼0a rf ðt Àrh Þ;increases,when t increases.Then1C ða ÞZtt 0ðt Às Þa À1f ðs ;x s Þds ;is an increasing function.k D x k t j6k D x kðt 0Þk þfC a Ztt 0ðt Às Þa À1k D x k s k dsþfC a Ztt 0ðt Às Þa À1e k s ds j D u k j k :On the other hand,it follows from Lemma 2.4and the deﬁnition of the Mittag–Lefﬂer function that for k >0,we havedt aE 1;1þa ðk t Þdt¼t a À1E 1;a ðk t Þ>0:Therefore,f 0C a Ztðt Às Þa À1e k s ds ¼f 0t a E 1;1þa ðk t Þ;is an increasing function.Settingv ðt Às Þ¼f 0C a ðt Às Þa À1;L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–552549w ðt Þ¼k D x kðt 0Þk þfa Ztt 0ðt Às Þa À1e k s ds j D u k j k ;and using Lemma 2.6,we havek D x k t k6ef 0T aC ða þ1Þk D x k ðt 0Þk þf 0e f 0T aC ða þ1ÞC ða ÞZtt 0ðt Às Þa À1e k s ds j D u k j k :Then fromZtt 0ðt Às Þa À1e k s ds <e k tk aC ða Þ;it yieldsk D x k t k 6c k D x kðt 0Þk þe k tf 0c ka j D u kj k ;ð3:13Þwhere c ¼e f 0T aC ða þ1Þis a constant.Now,substituting (3.13)into (3.11).we obtainj ek þ1ðt Þj 6j 1Àg 1g u ðn Þjj e kðt Þj þd 4c k D x kðt 0Þk þe k t f 0c d 4k aj D u k j k 6j 1Àg 1g u ðn Þjj e k ðt Þj þd 4c k D x k t 0Þk þe k tf 0c d 4k aj D u k j k :ð3:14ÞUsing (3.3)and multiplying both sides of the above inequality (3.14)by e Àk t and taking the k -norm,we havej ek þ1j k 6j 1Àg 1g u ðn Þjj e kj k þd 4cL j e kðt 0Þj k þf 0c g 1d 4k a j e k j k 6j 1Àg 1g u ðn Þj þf 0c g 1d 4kaj e k j k þd 4cL j e k ðt 0Þj k :By the assumption that max fj 1Àg 1d 1j ;j 1Àg 1d 2jg 6q 1<1,then there exists a sufﬁciently large k such thatj 1Àg 1g u ðn Þj þf 0cg 1d 4a<1:Therefore,from the inequality upon and Lemma 3.1,we have lim k !1j e k j k ¼0.It follows from that the equivalence of norms,we get that lim k !1j e k ðt Þj ¼0.which completes the proof.Next let us discuss the closed-loop P-type iterative control.3.2.Closed-loop P-type iterative controlThe above techniques that we used can be extended to the following closed-loop P-type ILC updating law:x k þ1t 0¼x k t 0þLe k ðt 0Þ;u k þ1ðt Þ¼u k ðt Þþg 2e k þ1ðt Þ;(ð3:15Þwhere L and g 2are unknown parameters to be determined.Lemma 3.3.For fractional-order nonlinear system with delay (3.1)and a given reference y d ðt Þ,denote thatD u k ðt Þ¼u k þ1ðt ÞÀu k ðt Þ;D x k t ¼x k þ1t Àx k t ,ifmax 1ÀL d 3g 21;1ÀL d 3g 22 ;1ÀL d 4g 21 ;1ÀL d 4g 226q 2<1;ð3:16Þwhere q 2is constant,then for all t 2½0;T ,and arbitrary input u 0ðt Þ,the closed-loop-type ILC updating low (3.15)guaranteesthatlim k !1j e k ðt 0Þj k ¼0:Proof.Similar to the proof of Lemma 3.1,we havee k þ1ðt 0Þ¼e k ðt 0ÞÀLg x ðf Þe k ðt 0ÞÀg 2g u ðf Þe k þ1ðt 0Þ:Thus,from the above equation,we gete k þ1ðt 0Þ¼1ÀLg x ðf Þ1þg 2g u ðf Þe kðt 0Þ:550L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–552Taking the k -norm,we can obtaine k þ1ðt 0Þ k¼1ÀLg x ðf Þ1þg 2g uðf Þj e k ðt 0Þj k 6q 2j e k ðt 0Þj k :ð3:17ÞBecause q 2<1,which yields lim k !1j e k ðt 0Þj k ¼0.The proof is complete.Theorem 3.4.For fractional-order nonlinear system with delay (3.1)and a given reference y d ðt Þ,if the assumptions in Lemma 3.5are met,and ifmax 11þg 2d 1;11þg 2d 2 6q 3<1;where q 3is constant,then for all t 2½0;T ,and arbitrary input u 0ðt Þ,the closed-loop P-type ILC updating low (3.15)guaran-tees thatlim k !1y k ðt Þ¼y d ðt Þ:Proof.With the same argument as Theorem 3.2and Lemma 3.3,it can be prove easily thatj ek þ1ðt Þj 61g 2u j e k ðt Þj þd 4g 2u k D x k t k :ð3:18ÞSubstituting (3.13)into (3.18),we obtainj ek þ1ðt Þj 611þg 2g u ðf Þ j e k ðt Þj þd 4c j 1þg 2g u ðf Þj k D x k ðt 0Þk þd 4f 0ce k t j 1þg 2g u ðf Þj k a j D u k j k 61g 2uj e k ðt Þj þd 4c g 2u k D x k t 0k þd 4f 0ce k t j 1þg 2g u ðf Þj k a j D u k j k ;where c ¼e f 0T aC a is a constant.j e k þ1j k 61j 1þg 2g u ðf Þjj e k j k þd 4cL j 1þg 2g u ðf Þj je k ðt 0Þj k þd 4f 0c g 2j 1þg 2g u ðf Þj k aj e k þ1j k :From the above equation,we getj e k þ1j k 61j 1þg 2g u ðf Þj Àd 4f 0c g 2k a j e k j k þd 4cL j 1þg 2g u ðf Þj Àd 4f 0c g 2kaj e k ðt 0Þj k :Similar to the Theorem 3.2,thus we omitted it.4.Numerical exampleIn this section,We have presented a numerical example to demonstrate our main conclusion.Consider the fractional-order nonlinear systemD 12t x kðt Þ¼3½x k t 2þ1u k ðt Þ;y k ðt Þ¼x k t þ910u k ðt Þ;(ð4:1ÞThe iterative learning control law is chosenx k þ1t 0¼x k t 0þ12e k ðt 0Þ;u k þ1ðt Þ¼u k ðt Þþe k ðt Þ;(ð4:2ÞIn this case,let the initial condition is x k ðt 0Þ¼À3,the initial control is u 0ðt Þ¼0and the reference is y d ðt Þ¼12t 2ð1Àt Þ.It can be easily proved that the condition of the Theorem 3.2is satisﬁed.Next,let the iterative learning control law be chosenx k þ1t 0¼x k t 0þ1e k ðt 0Þ;u k þ1ðt Þ¼u k ðt Þþe k þ1ðt Þ;(ð4:3ÞLet the initial control and the reference be the same.From numerical example,we can prove our main conclusion is satisﬁed.L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–552551552L.Yan,J.Wei/Applied Mathematics and Computation257(2015)546–5525.ConclusionThis paper based on the property of fractional order and the generalized Gronwall inequality.This paper mainly solves iterative learning control problems for a class of fractional with delay.Based on the robust convergence of the tracking errors, the P-type ILC updating laws can be determined.This is our main conclusion.Our future work will discuss the output delay. References[1]I.Podlubny,Fractional Differential Equations,Academic Press,San Diego,1999.[2]Yong Zhou,Basic Theory of Fractional Differential Equations,World Scientiﬁc,Singapore,2014.[3]J.X.Xu,Y.Tan,Linear and Nonlinear Iterative Learning Control,Spring-Verlag,Berlin,2003.[4]zarevic´,PD a-type iterative learning control for fractional LTI system,in:Proceedings of the16th International Congress of Chemical andProceedings Engineering,Praha,Czech Republic,2004,pp.22–26.[5]Y.Li,Y.Q.Chen,H.-S.Ahh,Fractional order iterative learning control,in:Int.Joint Conference,2009,pp.3106–3110.[6]Li Jia,Tian Yang,Minsen Chiu,An integrated iterative learning control strategy with model identiﬁcation and dynamic R-parameter for batch processes,J.Process Control vol.23(no.9)(2013)1332–1341.[7]Y.Li,Y.Q.Chen,H.-S.Ahh,On the PD a-type iterative learning control for the fractional-order nonlinear systems,in:American Control Conference,2011,pp.4320–4325.[8]Yong-Hong Lan,Yong Zhou,D-type iterative learning control for fractional-order linear time-delay systems,Asian J.Control15(3)(2013)669–677.[9]Y.Li,Y.Q.Chen,H.-S.Ahn,Fractional-order iterative learning control for fractional-order linear systems,Asian J.Control13(1)(2011)1–10.[10]Fali Ma,Chuandong Li,Tingwen Huang,Iterative learning control design of nonlinear multiple time-delay systems,put.vol.218(2011)4333–4340.[11]Ming-Tzong Lin,Chung-Liang Yen,Meng-Shiun Tsai,Hong-Tzong Yau,Application of robust iterative learning algorithm in motion control system,Mechatronics vol.23(no.5)(2013)530–540.[12]Y.Ye,A.Tayebi,X.Liu,All-passﬁltering in iterative learning control,Automatica45(1)(2009)257–264.[13]Yong Zhou,Feng Jiao,J.Pecaric,On the Cauchy problem for fractional functional differential equations in banach spaces,Topol.Methods NonlinearAnal.42(2013)119–136.[14]n,Iterative learning control with initial state learning for fractional order nonlinear systems,Comput.Math.Appl.64(2012)3210–3216.[15]Yong Zhou,Xiao Hui,Lu Zhang,Cauchy problem for fractional evolution equations with Caputo derivative,Eur.Phys.J.Special Topics222(2013)1747–1764.[16]Yong Zhou,Lu Zhang,Xiao Hui,Existence of mild solutions for fractional evolution equations,J.Int.Equ.Appl.25(2013)557–585.[17]Jiang Wei,The controllability of fractional control systems with control delay,Comput.Math.Appl.64(10)(2012)3153–3159.[18]Jiang Wei,The constant variation formulae for singular fractional differential systems with delay,Comput.Math.Appl.59(3)(2010)1184–1190.[19]Zhang Zhixin,Jiang Wei,Some results of the degenerate fractional differential system with delay,Comput.Math.Appl.62(3)(2011)1264–1291.[20]S.J.Sadati,D.Baleanu,A.Ranjbar,R.Ghaderi,T.Abdeljawad(Maraaba),Mittag–Lefﬂer stability theorem for fractional nonlinear system with delay,Abstr.Appl.Anal.vol.2010(2010)7.。

REGULARITY OF WEAK SOLUTIONS OF THE COMPRESSIBLE ISENTROPIC NAVIER-STOKES EQUATIONS

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LMENS - 97 - 1
January 1997
Laboratoire de Mathematiques de l'Ecole Normale Superieure 45 rue d'Ulm 75230 PARIS Cedex 05
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变系数_2_1_维非线性薛定谔方程的孤子解

第30卷第5期吉首大学学报(自然科学版)Vol.30No.5 2009年9月Journ al of Jishou University(Natural Science Edition)Sept.2009文章编号:1007-2985(2009)05-0056-04变系数(2+1)维非线性薛定谔方程的孤子解*徐四六,陈顺芳(咸宁学院电子与信息工程学院,湖北咸宁437005)摘要:利用F-展开法求解了一个变系数的(2+1)维非线性薛定谔方程,从而得到一个精确的明、暗孤子的解析解,这些孤子构成了一类空间孤子簇.关键词:空间孤子;F-展开法;非线性薛定谔方程中图分类号:O43文献标识码:A自然界普遍存在非线性物理现象,均可采用NLSE(非线性薛定谔方程)描述.近来,非线性薛定谔方程已经成为研究的热点,究其原因是因为它是自然界普遍存在的非线性物理现象的重要方程,是非常重要的一类非线性模型.它描述自然界许多物理现象,如玻色-爱因斯坦凝聚、非线性光学中的光脉冲传输、等离子物理和流体力学等都可以归于这一类模型.由于非线性偏微分方程的精确解可以解释自然界存在众多的非线性物理现象.因此,非线性偏微分方程再次引起人们的高度关注和研究热情.传统精确求解非线性方程的方法有:Zakhar ov和Shabat[1]的逆散射法(IST);Painleve展开法、H ir ota双线性法、Backlund[2]变换等.近年来,随着理论研究和实验探究的发展,一些处理非线性问题的数学方法与技巧得到飞速发展,尤其值得关注的是非线性偏微分方程解析解的解法研究领域中出现的各种代数方法,如齐次平衡原理[3-4]、扩展双曲函数法、F-展开技术[5-6]等.由于逆散射方法在一维情况下可以得到稳定的孤子解,但在高维情况(包含二维、三维)下,该方法已不再适用.因此,笔者将利用改进的齐次平衡原理[7]和F-展开技术[6-7],获得变系数含损耗或增益的(2+1)维非线性薛定谔方程(NLSE)的精确孤子解.1变系数的(2+1)维非线性薛定谔方程模型及解法在直角坐标系中,描述克尔介质中二维各向异性的空间孤子的传输特性,可以用一般的变系数(2+1)维非线性薛定谔模型来描述,其归一化形式为[7-8]:i 9u9t+12B1(t)92u9x2+12B2(t)92u9y2+C(t)|u|2u=i g(t)u,(1)其中:u(z,x,y)是光弹包络或物质波波包;t是归一化传输时间;x,y是归一化的两维空间坐标;B1(t), B2(t)的物理意义决定于具体的物理模型,在空间孤子模型中,B1,2(t)代表衍射系数[9];C(t)是Kerr(克尔)非线性系数,其值可以取正或负的值,当C(t)>0时,其对应于聚焦非线性介质,相反对应于散焦非线性介质;g(z)是增益或损耗系数.对于模型(1),当B1=1,B2=C=-1,g=0时,方程简化为Kerr(克尔)介质中标准的二维非线性薛*收稿日期:2009-07-20作者简介:徐四六(1968-),男,湖北咸宁人,湖北咸宁学院物理系讲师,华中科技大学博士生,主要从事光学空间孤子的理论和数值模拟研究.定谔方程[10-11].特别地,当t 和z 变化时,使B 1=0,B 2=1,方程(1)化简为(1+1)推广的分布系数的非线性薛定谔方程,此方程控制单模光纤中光孤子的传播规律[10,12].精确的啁啾自相似的孤子解已经获得,并且孤子解的稳定性得到了证实[13].3种类型的新奇自相似孤子解已经获得[14].进而当B 1=B 2时,方程(1)描述(2+1)维时空孤子(光弹)在层状的克尔非线性分布系数介质中的传输规律.同时,方程(1)体现了另一种描述波色-爱因斯坦凝聚的中的物质波的非线性G -P 方程.最近的研究表明,在周期性的光子晶格控制下,能够得到稳定的物质波.为了寻求方程(1)式的解,笔者定义复场包络为u(t,x ,y )=A(t,x ,y )e i B(t,x,y),(2)其中A(t,x ,y ),B(t,x ,y )分别为场的大小和相位,它们是时间空间坐标的实函数.将(2)式代入(1)式,让实部与虚部分别为零,则可以得到以下2个耦合方程:9A 9t +B 1(29A 9x 9B 9x 2+A 92B 9x 2)+B 2(29A 9y 9B 9y +B 2A 92B 9y 2)=g(t)A ,(3)-A 9B 9t +12B 1[92A 9x 2-A(9B 9x )2]+12B 2[92A 9y 2-A(9B 9y )2]+C (t)A 3=0.(4)根据推广的F -展开方法和齐次平衡原理[3-5],则A (t,x ,y,z )=f 1(t)+f 2(t)F(H )+f 3(t)F -1(H ),(6)H =k(t)x +l(t)y +X (t),(7)B(t,x ,y )=a(t)(x 2+y 2)+b(t)(x +y )+e(t),(8)其中:f 1,f 2,f 3,k,l,a,b,e 是待确定的系数;F(H )是Jacobi 椭圆函数,满足方程:(d F d H)2=q 0+q 2F 2+q 4F 4.2 变系数的(2+1)维非线性薛定谔方程的孤子解把(6)至(8)式代入(3)至(4)式,令x s F n ,y s F n (s =0,1,2;n =0,1,2,3)和q 0+q 2F 2+q 4F 4的系数都为0,其中:s =0,2;n =0,1,2,3.可以得到如下2种情况的周期性的行波解:(1)若B 1,2(t),C (t)和g(t)是传输时间的任意函数,则选取f 1=f 3=a =0,b(t)=b 0,k(t)=k 0,l(t)=l 0,可以得到:f 2=f 20exp [Q t 0g(S )d S ],(9)X (t)=-k 0b 0Q t 0B 1(S )d t -p 0l 0Q t 0B 2(S )d S +X 0,(10)e(t)=(12q 2k 20-b 20)Q t 0B 1(S )d S +12(q 2l 20-b 20)Qt 0B 2(S )d S +e 0,(11)C =-B 1k 20q 4+B 2l 20q 4f 220ex p [-2Q t0g (S )d S ].(12)其中带下标的/00表示变系数的初始值.应当引起注意的是,(9)至(11)式决定了一种新的类型的空间孤子参数,特别地,衍射参数B 1(t),B 2(t)紧密关联着非线性系数C (t)和增益系数g(t).将(9)至(11)式代入(2)式及(6)至(8)式,可以得到:H =k 0x +l 0y +p 0z +w (t),(13)u(t,x ,y )=f 20ex p [Q t 0g(S )d S ]F(H )e i [b 0(x+y)+e],(14)其中q 参数见文献[4-5]中的m 值,对于给定的m 值,q 0,q 2,q 4的值能够确定.特别地,当m y 1时,以上Jacobi 椭圆函数退化成双曲函数,即sn (H )y tan h(H ),cn (H )y sec h(H ),dn (H )y sec h(H ).所以周期性Jacobi 椭圆函数行波解可以变成如下的空间孤子解:u 1(t,x ,y ,z )=f 20exp [Q t 0g (S )d S ]tan h(H 1)exp {i [b 1-(k 20+12)Q t0B 1(S )d S -(l 20+12b 20)Q t 0B 2(S )d S ]+e 0},(15)57第5期徐四六,等:变系数(2+1)维非线性薛定谔方程的孤子解u 2(t,x ,y )=f 20ex p [Q t 0g(S )d S ]sec h(H 1)exp {i b 1+i [12k 20-b 20]Q t 0B 1(S )d S +i [(-12(l 20-b 20)]Q t 0B 2(S )d S ]+e 0},(16)此时可知衍射系数B 1,2(t),非线性系数C (t)和以及增益系数g(t)是相互关联的,当B 1(k 20+l 20)q 4+B 2p 20q 4f 220C<0时,空间孤子解(15),(16)式可以获得.(2)若B 1,2(t),C (t)和g(t)是传输时间的任意函数,则选取f 1=f 3=0,b(t)=b 0a(t),k (t)=k 0a(t),l(t)=l 0a(t),可以得到:f 2=f 20a exp [Q t 0g(S )d S ],(17)-12a 2d a d t =B 1=B 2,(18)X (t)=12(k 0+l 0)b 0a +X 0,(19)e(t)=-14q 2(k 20+l 20)a -34b 20a +e 0,(20)C =-q 4(k 20+l 20)B 1f 220a ex p [-2Qt 0g(S )d S ].(21)将(17)至(21)式代入(2)式及(6)至(8)式,则u(t,x ,y )=f 20a ex p [Q t 0C (S )d S ]F (H )e i[a(x 2+y 2)+b 0a(x+y )+e].(22)类似上述情况1,当m y 1时,周期性Jacobi 椭圆函数行波解可变成如下的空间孤子解:u 31(t,x ,y ,z )=f 20a ex p [Q t 0g(S )d S ]tan h(H 2)ex p {i [K +12(k 20+l 20)a -14b 20a+e 0]},(23)u 32(t,x ,y ,z )=f 20ex p [Q t 0g(S )d S ]sec h(H 2)ex p {i [K -14(k 20+l 20)a -14b 20a+e 0]}.(24)其中:H 2=a(x 2+y 2)+(k 0+l 0)b 0a+12(k 0+l 0)b 0a+X 0;K =a(x 2+y 2)+b 0a(x +y ).根据以上的解析解可以得到,存在着2种类型的空间孤子,并且得到的孤子解可以简化为(1+1)维的孤子解[15].特别值得注意的是,q 4(k 20+l 20)B 1f 220a C<0时,可以获得啁啾性的周期性Jaco bi 椭圆函数行波解和空间孤子解.3 结语利用推广的平衡原理和F-展开法研究了变系数含损耗或增益的(2+1)维非线性薛定谔方程,得到了一类精确的Jacobi 椭圆函数解.在极限情况下,这些周期性Jacobi 椭圆函数可以过渡到双曲函数,从而得到精确的时空孤子解.这些解受衍射或色散系数、非线性系数、损耗或增益函数之间的关系的限制.该求解方法也可推广到其他非线性数学物理方程的求解.参考文献:[1] ZA L T H AR OV V E,SH ABA T A B.Exact T heo ry of T wo -Dimensional Self -M odulatio n of W aves in N onlinear M edia[J].Sov.P hy s.JET P.,1972,34:62.[2] M A T V EEV V B,SA L LE M A.Darboux,T r ansfo rmatio ns and So litons [M ].Ber lin:Spr inger ,1991:125-128.[3] F AN E G,ZH AN G H Q.T wo T ypes of T raveling W ave So lutio ns to Bur ger s -KdV Equations [J].Phy s.Lett.A,1998,246:403-406.[4] FA N E G.Ex tended T anh -Function M etho d and I ts A pplicatio ns to No nlinear Equatio ns [J].P hy s.Lett.A,2000,265:353-357.[5] ZH OU Y B,WA N G M L ,WA N G Y M.P erio dic Wav e So lutions to a Coupled K dV Equations w ith V ariable Coeff-i 58吉首大学学报(自然科学版)第30卷cients [J].P hys.L ett.A,2003,308:31-36.[6] W AN G D S,ZH A NG H Q.Chaos,Solitons and Fr act als,F urther Impro ved F -Expansion M ethod and New Exact Solu -tio ns of K ono pelchenko -Dubr ovsky Equation,2005,25:601-610.[7] ZH ON G W P,XIE R H ,BELIC 'M ,et al.Exact Spatial Soliton Solutions o f the T w o -Dimensional G ener alized N onlinea rSchr Êdinger [J].Phys.Rev.A,2008,78:023821(1-6).[8] K RU G LO V V I,P EACOCK A C,H A RV EY J D OP T.Ex press,Self -Similar P ro pag atio n of H igh -P ow er Parabo lic Pu-lses in O pt ical F iber A mplifier s,2004,12:6198.[9] M A L OM ED B A ,M IH A L ACH E D ,WI SE F ,et al.Spatiotempor al O ptical So lito ns [J].J.O pt.B,2005(7):53.[10] ABDU L LA EV F K,BA IZA K OV B B,SAL ERN O M.Stable T wo -Dimensional Dispersion -M anag ed Solito n [J].Phy s.R ev.E,2003,68:066605.[11] PO NO M A RENK S A ,AG RA WA L G P.D o Solitonlike Self -Similar Wav es Ex ist in N onlinear O pt ical M edia [J].Phys.R ev.L ett.,2006,97:013901.[12] D AL F OV O F ,G IORG IN I S,G U IL LU M AS M ,et al.Co llect ive and Sing le -P article Excitat ions of a T r apped Bose Gas[J].Phys.R ev.A ,1997,56:3840.[13] EI G A ,G A M M A L A ,KA M CH AT N OV A M ,et al.Sig nat ur es fo r G enera lized M acr oscopic Superposit ions [J].Phys.R ev.L ett.,2006,97:180405.[14] CH EN S H ,Y I L.Chir ped Self -Similar Solutio ns of a Generalized No nlinear Schr Êding er Equat ion M o del [J].Phy s.R ev.E,2005,71:016606.[15] CH EN S,SHI D F,Y I L.T iming Jitter o f F emtosecond Solitons in Sing le -M ode O ptical Fiber s:A Pertur bation M odel[J].Phys.R ev.E,2004,69:046602.Soliton Solutions to (2+1)-Dimensional Nonlinear Schr ÊdingerEquation with Distributed CoefficientsXU S-i liu,CH EN Shun -fang(Scho ol of Electro nics and Info rmation Eng ineering ,Xianning Co lleg e,X ianning 437005,H ubei China)Abstract:F -ex pansion techniquel is used to construct solutions to (2+1)-dimensional Schr Êdinger equa -tion w ith distributed coefficients,and thus exact solutions to bright and dark so litons are g otten.T hose solitons co nstitute a class of so lito n cluster.Key words:spatial solitons;F -ex pansion technique;nonlinear Schr Êding er equation(责任编辑陈炳权)59第5期徐四六,等:变系数(2+1)维非线性薛定谔方程的孤子解。

Stability of peakons for the Degasperis-Procesi equation

a rXiv:0712.27v1[mat h.A P]12D ec27Stability of peakons for the Degasperis-Procesi equation Zhiwu Lin ∗and Yue Liu †Abstract The Degasperis-Procesi equation can be derived as a member of a one-parameter family of asymptotic shallow water approximations to the Eu-ler equations with the same asymptotic accuracy as that of the Camassa-Holm equation.In this paper,we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line.By con-structing a Liapunov function,we prove that the shapes of these peakon solitons are stable under small perturbations.Keywords:Stabiltiy;Degasperis-Procesi equation;Peakons Mathematics Subject Classiﬁcation (2000):35G35,35Q51,35G25,35L051Introduction The Degasperis-Procesi (DP)equation y t +y x u +3yu x =0,x ∈R ,t >0,(1.1)with y =u −u xx ,was originally derived by Degasperis-Procesi [14]using the method of asymptotic integrability up to third order as one of three equations in the family of third order dispersive PDE conservation laws of the form u t −α2u xxt +γu xxx +c 0u x =(c 1u 2+c 2u 2x +c 3uu xx )x .(1.2)The other two integrable equations in the family,after rescaling and applying a Galilean transformation,are the Korteweg-de Vries (KdV)equationu t +u xxx +uu x =0and the Camassa-Holm (CH)shallow water equation [2,15,18,22],y t +y x u +2yu x =0,y =u −u xx .(1.3)These three cases exhaust in the completely integrable candidates for(1.2)by Painlev´e analysis.Degasperis,Holm and Hone[13]showed the formal integra-bility of the DP equation as Hamiltonian systems by constructing a Lax pair and a bi-Hamiltonian structure.The Camassa-Holm equation wasﬁrst derived by Fokas and Fuchassteiner [18]as a bi-Hamiltonian system,and then as a model for shallow water waves by Camassa and Holm[2].The DP equation is also in dimensionless space-time variables(x,t)an approximation to the incompressible Euler equations for shallow water under the Kodama transformation[20,21]and its asymptotic accuracy is the same as that of the Camassa-Holm(CH)shallow water equation, where u(t,x)is considered as theﬂuid velocity at time t in the spatial x-direction with momentum density y.Recently,Liu and Yin[24]proved that theﬁrst blow-up inﬁnite time to equation(1.1)must occur as wave breaking and shock waves possibly appear afterwards.It is shown in[24]that the lifespan of solutions of the DP equation (1.1)is not aﬀected by the smoothness and size of the initial proﬁles,but aﬀected by the shape of the initial proﬁles(for the CH equation,see[1,8]).This can be viewed as a signiﬁcant diﬀerence between the DP equation(or the CH equation )and the KdV.It is also noted that the KdV equation,unlike the CH equation or DP equation,does not have wave breaking phenomena[30].Under wave breaking we understand that development of singularities inﬁnite time by which the wave remains bounded but its slope becomes unbounded[31].It is well known that the KdV equation is an integrable Hamiltonian equation that possesses smooth solitons as traveling waves.In the KdV equation,the leading order asymptotic balance that conﬁnes the traveling wave solitons occurs between nonlinear steepening and linear dispersion.However,the nonlinear dispersion and nonlocal balance in the CH equation and the DP equation,even in the absence of linear dispersion,can still produce a conﬁned solitary traveling wavesu(t,x)=cϕ(x−ct)(1.4) traveling at constant speed c>0,whereϕ(x)=e−|x|.Because of their shape (they are smooth except for a peak at their crest),these solutions are called the peakons[2,13].Peakons of both equations are true solitons that interact via elastic collisions under the CH dynamics,or the DP dynamics,respectively.The peakons of the CH equation are orbitally stable[12].For waves that approximate the peakons in a special way,a stability result was proved by a variation method [11].Note that we can rewrite the DP equation asu t−u txx+4uu x=3u x u xx+uu xxx,t>0,x∈R.(1.5) The peaked solitons are not classical solutions of(1.5).They satisfy the Degasperis-Procesi equation in the conservation law formu t+∂x 12ϕ∗ 3where∗stands for convolution with respect to the spatial variable x∈R.This is the exact meaning in which the peakons are solutions.Recently,Lundmark and Szmigielski[26]presented an inverse scattering ap-proach for computing n-peakon solutions to equation(1.5).Holm and Staley[20] studied stability of solitons and peakons numerically to equation(1.5).Anal-ogous to the case of Camassa-Holm equation[6],Henry[19]and Mustafa[29] showed that smooth solutions to equation(1.5)have inﬁnite speed of propaga-tion.The following are three useful conservation laws of the Degasperis-Procesi equation.E1(u)= R y dx,E2(u)= R yv dx,E3(u)= R u3dx,where y=(1−∂2x)u and v=(4−∂2x)−1u,while the corresponding three useful conservation laws of the Camassa-Holm equation are the following:F1(u)= R y dx,F2(u)= R(u2+u2x)dx,F3(u)= R(u3+uu2x)dx.(1.7)The stability of solitary waves is one of the fundamental qualitative proper-ties of the solutions of nonlinear wave equations.Numerical simulations[13,25] suggest that the sizes and velocities of the peakons do not change as a result of collision so these patterns are expected to be stable.Furthermore,it is observed that the shape of the peakons remains approximately the same as time evolves. As far as we know,the case of stability of the peakons for the Camassa-Holm equation is well understood by now[11,12],while the Degasperis-Procesi equa-tion case is the subject of this paper.The goal of this paper is to establish a stability result of peaked solitons for equation(1.5).It is found that the corresponding conservation laws of the Degasperis-Procesi equation are much weaker than those of the Camassa-Holm equation. In particular,one can see that the conservation law E2(u)for the DP equation.In fact,by the Fourier transform,we haveis equivalent to u 2L2E2(u)= R yvdx= R1+ξ2Let us denoteE2(u)= u 2X.The following stability theorem is the principal result of the present paper. Theorem1(Stability)Let cϕbe the peaked soliton deﬁned in(1.4)traveling with speed c>0.Then cϕis orbitally stable in the following sense.If u0∈H sfor some s>3/2,y0=u0−∂2x u0is a nonnegative Radon measure ofﬁnite total mass,andu0−cϕ X<cε,|E3(u0)−E3(cϕ)|<c3ε,0<ε<16 ≤c√ε.(1.10)Remark1The state of aﬀairs about these maxima/minima implied by the pre-vious theorem is a consequence of the assumption on y0,as shown in Lemma 3.1.For an initial proﬁle u0∈H s,s>3/2,there exists a local solution u∈C([0,T),H s)of(1.5)with initial data u(0)=u0[32].Under the assump-tion y0=u0−∂2x u0≥0in Theorem1,the existence is global in time[24],that is T=+∞.For peakons cϕwith c>0,we have 1−∂2x (cϕ)=2cδ(hereδis the Dirac distribution).Hence the assumption on y0that it is a nonnegative measure is quite natural for a small perturbation of the peakons.Existence of global weak solution in H1of the DP equation is also proved in[16].Note that peakons cϕare not strong solutions,sinceϕ∈H s,only for s<3/2.The above theorem of orbital stability states that any solution starting close to peakons cϕremains close to some translate of cϕin the norm X,at any later time.More information about this stability is contained in(1.9)and (1.10).Notice that for peakons cϕ,the function v cϕis single-humped with the height1There are two standard methods to study stability issues of dispersive waveequations.One is the variational approach which constructs the solitary waves as energy minimizers under appropriate constraints,and the stability automat-ically follows.However,without uniqueness of the minimizer,one can onlyobtain the stability of the set of minima.The variational approach is used in [11]for the CH equation.It is shown in[11]that the each peakon cϕis theunique minimum(ground state)of constrained energy,from which its orbital stability is proved for initial data u0∈H3with y0=(1−∂2x)u0≥0.Their proof strongly relies on the fact that the conserved energy F2in(1.7)of the CHequation is the H1−norm of the solution.However,for the DP equation the energy E2in(1.8)is only the L2norm of the solution.Consequently,it is morediﬃcult to use such a variational approach for the DP equation.Another approach to study stability is to linearize the equation around the solitary waves,and it is commonly believed that nonlinear stability is governed by the linearized equation.However,for the CH and DP equations,the non-linearity plays the dominant role rather than being a higher-order correction to linear terms.Thus it is unclear how one can get nonlinear stability of peakons by studying the linearized problem.Morover,the peaked solitons cϕare not diﬀerentiable,which makes it diﬃcult to analyze the spectrum of the linearized operator around cϕ.To establish the stability result for the DP equation,we extend the approach in[12]for the CH equation.The idea in[12]is to directly use the energy F2as the Liapunov functional.By expanding F2in(1.7)around the peakon cϕ,the error term is in the form of the diﬀerence of the maxima of cϕand the perturbed solution u.To estimate this diﬀerence,they establish two integral relations g2=F2(u)−2(max u)2and ug2=F3(u)−4M3,M=max u(x)3and the error estimate|M−maxϕ|then follows from the structure of the above polynomial inequality.To extend the above approach to nonlinear stability of the DP peakons,we have to overcome several diﬃculties.By expanding the energy E2(u)around the peakon cϕ,the error term turns out to be max v cϕ−max v u,with v u= (4−∂2x)−1u.We can derive the following two integral relations for M1=max v u, E2(u)and E3(u)byg2=E2(u)−12M21and hg2=E3(u)−144M31with some functions g and h related to v u.To get the required polynomial inequality from the above two identities,we need to show h≤18max v u.How-ever,since h is of the form−∂2x v u±6∂x v u+16v u,generally it can not be5bounded by v u.This new diﬃculty is due to the more complicated nonlinearstructure and weaker conservation laws of the DP equation.To overcome it,we introduce a new idea.By constructing g and h piecewise according to mono-tonicity of the function v u,we then establish two new integral identities(3.7)and(3.9)for E2,E3and all local maxima and minima of v u.The crucial es-timate h≤18max v u can now be shown by using this monotonicity structureand properties of the DP solutions.This results in inequality(3.13)related to E2,E3and all local maxima and minima of v u.By analyzing the structure ofequality(3.13),we can obtain not only the error estimate|M1−max v cϕ|but more precise stability information from(1.10).We note that the same approach can also be used for the CH equation to gain more stability information(seeRemark2).Although the DP equation is similar to the CH equation in several aspects, we would like to point out that these two equations are truly diﬀerent.One of the novel features of the DP equation is it has not only peaked solitons[13], u(t,x)=ce−|x−ct|,c>0but also shock peakons[4,25]of the form1u(t,x)=−for t≥T.k+(t−T)On the other hand,the isospectral problem in the Lax pair for equation(1.5)is the third-order equationψx−ψxxx−λyψ=0cf.[13],while the isospectral problem for the Camassa-Holm equation is thesecond order equationψxx−1(in both cases y=u−u xx)cf.[2].Another indication of the fact that there is no simple transformation of equation(1.5)into the Camassa-Holm equation is the entirely diﬀerent form of conservation laws for these two equations[2,13]. Furthermore,the Camassa-Holm equation is a re-expression of geodesicﬂow on the diﬀeomorphism group[5,10]and on the Bott-Virasoro group[9,28],while no such geometric derivation of the Degasperis-Procesi equation is available.The remainder of the paper is organized as follows.In Section2,we recall the local well-posedness of the Cauchy problem of equation(1.5),the precise blow-up scenario of strong solutions,and several useful results which are crucial in the proof of stability theorem for equation(1.5)from[32,33].Section3is devoted to the proof of the stability result(Theorem1).Notation.As above and henceforth,we denote by∗convolution with respect to the spatial variable x∈R.We use · L p to denote the norm in the Lebesgue space L p(R)(1≤p≤∞),and · H s,s≥0for the norm in the Sobolev spaces H s(R).2PreliminariesIn the present section,we discuss the issue of well-posedness.The local exis-tence theory of the initial-value problem is necessary for our study of nonlinear stability.We brieﬂy collect the needed results from[24,32,33].Denote p(x):=12u2 =0,t>0,x∈R.(2.1) The local well-posedness of the Cauchy problem of equation(1.5)with initial data u0∈H s(R),s>32,there exist a maximal T=T(u0)>0and a unique solution u to equation(1.5)(or equation(2.1)),such thatu=u(·,u0)∈C([0,T);H s(R))∩C1([0,T);H s−1(R)). Moreover,the solution depends continuously on the initial data,i.e.the mapping u0→u(·,u0):H s(R)→C([0,T);H s(R))∩C1([0,T);H s−1(R))is continuous and the maximal time of existence T>0can be chosen to be independent of s.The following two lemmas show that the only way that a classical solution to(1.5)may fail to exist for all time is that the wave may break. Lemma2.2[32]Given u0∈H s(R),s>3Lemma2.3[24]Assume u0∈H s(R),s>32.Assume there exists x0∈R such thaty0(x)=u0(x)−u0,xx(x)≥0if x≤x0,y0(x)=u0(x)−u0,xx(x)≤0if x≥x0,and y0changes sign.Then,the corresponding solution to equation(1.5)blows up in aﬁnite time.Lemma2.7[24]Assume u0∈H s(R),s>3The following lemma is a special case of Lemma 2.7.Lemma2.8[24]Assumeu0∈H s (R ),s >32x −∞e ηy (t,η)dη+e x 2 x −∞e ηy (t,η)dη+e x2(k 1∓1)e −x x−∞e ηydη+14 ∞−∞e −2|x −ξ|w (t,ξ)dξ=14e 2x ∞xe −2ξw (t,ξ)dξand ∂x (4−∂2x )−1w =−12e 2x ∞xe −2ξw (t,ξ)dξ.Combining above two identities,we get(k 2±∂x )(4−∂2x )−1w =14(k 2±2)e 2x ∞xe −2ξw (t,ξ)dξ≥0. 93Proof of stabilityIn this primary section of the paper,we prove the stability theorem(Theorem 1)stated in the introduction.Note that the assumptions on the initial proﬁles guarantee the existence of an unique global solution of equation(1.5).The stability theorem provides a quantitative estimate of how closely the wave must approximate the peakon initially in order to be close enough to some translate of the peakon at any later time.That translate must be located at a point where the wave is tallest.The proof of Theorem1is based on a series of lemmas including some in the previous section.We take the wave speed c=1and the case of general c follows by scaling the estimates.Note thatϕ(x)=e−|x|∈H1(R)has the peak at x=0andE3(ϕ)= ∞−∞e−3|x|dx=24e−2|x|∗u.Thenvϕ(x)=13e−|x|−16.(3.3)Noteϕ−∂2xϕ=2δ.Here,δdenotes the Dirac distribution.For simplicity, we abuse notation by writing integrals instead of the H−1/H1duality pairing. Hence we haveE2(ϕ)= ϕ 2X= R(1−∂2x)ϕ(4−∂2x)−1ϕdx=2 Rδ(x)(4−∂2x)−1ϕ(x)dx=2vϕ(0)=1where use has been made of integration by parts and the fact that E2(ϕ)= ϕ 2X=2vϕ(0).This completes the proof of the lemma.In the next two lemmas,we establish two formulas related the critical values of v u to the two invariants E2(u)and E3(u).Consider a function0=u∈H s,s>3/2and u≥0.Then0<v u= 4−∂2x −1u∈H s+2⊂C2.Since v u is positive and decays at inﬁnity,it must have n points{ξi}n i=1with local maximal values and n−1points{ηi}n−1i=1with local minimal values for some integer n≥1.We arrange these critical points in their order by−∞<ξ1<η1<ξ2<η2<...<ξi−1<ηi−1<ξi<ηi<...<ηn−1<ξn<+∞. Letv u(ξi)=M i,1≤i≤n and v u(ηi)=m i,1≤i≤n−1.(3.5) Here,we assume n<+∞,that is,there are aﬁnite number of minima and maxima of v u.In the case when there are inﬁnitely many maxima and minima, the proofs below can be modiﬁed simply by changing theﬁnite sums to inﬁnite sums.Lemma3.2Let0=u∈H s,s>3/2and u≥0.By the above notations, deﬁne the function g by1≤i≤n.(3.6) g(x)= 2v u+∂2x v u−3∂x v u,ηi−1<x<ξi,2v u+∂2x v u+3∂x v u,ξi<x<ηi,withη0=−∞andηn=+∞.Then we haveR g2(x)dx=E2(u)−12 n i=1M2i−n−1 i=1m2i .(3.7)Proof.To simplify notations,we use v for v u below.Then u=4v−∂2x v.First, we note thatE2(u)= R (1−∂2x)u vdx= R(uv+∂x u∂x v)dx= 4v−∂2x v v+ 4−∂2x ∂x v ∂x v dx= 4v2+5(∂x v)2+ ∂2x v 2 dx.To show(3.7),we evaluate the integral of g2on each interval[ηi−1,ηi],1≤i≤n. We haveηiηi−1g2(x)dx= ξiηi−1(2v+∂xx v−3∂x v)2dx+ ηiξi(2v+∂xx v+3∂x v)2dx =I+II.11To estimate theﬁrst term,by integration by parts,we haveI= ξiηi−1 4v2+(∂xx v)2+9(∂x v)2+4v∂xx v−12v∂x v−6∂xx v∂x v dx= ξiηi−1 4v2+(∂xx v)2+5(∂x v)2 dx−6 v(ξi)2−v(ηi−1)2 ,where use has been made of the fact that∂x v(ξi)=∂x v(ηi−1)=0.Similarly,II= ηiξi 4v2+ ∂2x v 2+5(∂x v)2 dx+6 v(ηi)2−v(ξi)2 .Soηiηi−1g2(x)dx= ηiηi−1 4v2+ ∂2x v 2+5(∂x v)2 dx−12v(ξi)2+6v(ηi−1)2+6v(ηi)2andR g2(x)dx= R 4v2+ ∂2x v 2+5(∂x v)2 dx−n i=1 12v(ξi)2−6v(ηi−1)2−6v(ηi)2 =E2(u)−12 n i=1M2i−n−1 i=1m2i ,where use has been made of the fact that v(η0)=v(ηn)=0and the notationsin(3.5).Lemma3.3With the same assumptions and notations in Lemma3.2.Deﬁnethe function h byh(x)= −∂2x v u−6∂x v u+16v u,ηi−1<x<ξi,−∂2x v u+6∂x v u+16v u,ξi<x<ηi,1≤i≤n.(3.8)withη0=−∞andηn=+∞.Then we haveR h(x)g2(x)dx=E3(u)−144 n i=1M3i−n−1 i=1m3i .(3.9) Proof.We still use v for v u.First,note thatE3(u)= R(4v−∂xx v)3dx= R −(∂xx v)3+12(∂xx v)2v−48v2∂xx v+64v3 dx.To show(3.9),we evaluate the integral of h(x)g2(x)on each interval[ηi−1,ηi],1≤i≤n.We haveηiηi−1h(x)g2(x)dx= ξiηi−1(−∂xx v−6∂x v+16v)(2v+∂xx v−3∂x v)2dx+ ηiξi(−∂xx v+6∂x v+16v)(2v+∂xx v+3∂x v)2dx=I+II.12It is found that theﬁrst termI= ξiηi−1{−(∂xx v)3+12(∂xx v)2v+27∂xx v(∂x v)2−108v∂xx v∂x v+60v2∂xx v −54(∂x v)3+216(∂x v)2v−216v2∂x v+64v3}dx= ξiηi−1{−(∂xx v)3+12(∂xx v)2v+54(∂x v)3+60v2∂xx v−54(∂x v)3−108v2∂xx v+64v3}dx−72 v(ξi)3−v(ηi−1)3= ξiηi−1 −(∂xx v)3+12(∂xx v)2v−48v2∂xx v+64v3 dx−72 v(ξi)3−v(ηi−1)3 ,where use has been made of the following integral identities due to integrationby parts and∂x v(ξi)=∂x v(ηi−1)=0,ξiηi−1∂xx v(∂x v)2dx=1(∂x v)2 dx=−12v2 dx=−12∂x v3 dx=13Without changing the integral identities(3.7)and(3.9),we can rearrange M i and m i in the order:M1≥M2···≥M n≥0,m1≥···≥m n−1≥0.Moreover,since each local minimum is less than the neighboring local maximum, we have M i≥m i−1(2≤i≤n).The following two elementary inequalities are needed in the later proofs.Lemma3.4For any n≥2,assume{M i}n i=1and{m i}n−1i=1are2n−1numbers satisfyM1≥M2···≥M n≥0,m1≥···≥m n−1≥0and M i≥m i−1(2≤i≤n).Then(i)ni=2 M3i−m3i−1 ≤32≥ M31+n 2 M3i−m3i−1 1M1 M2i−m2i−121=−2and B n= M31+n 2 M3i−m3i−1 1if M1≥M2≥m1≥0.We haveM21+M22−m21 3− M31+M32−m31 2=3M41M22−3M41m21−2M31M32+2M31m31+3M21M42−6M21M22m21+3M21m41−3M42m21+2M32m31+3M22m41−2m61=(M2−m1)(M1−m1){3M1M32+3M31M2−2M1m31+3M31m1−2M2m31+3M32m1−2m41−2M21M22+M21m21+M22m21−2M1M2m21+M1M22m1+M21M2m1},which is obviously nonnegative by the assumption M1≥M2≥m1≥0.Assume the inequality A n≥B n is true for n≤k(k≥2).Our goal is to deduce A k+1≥B k+1.Since A k≥M1≥M k+1≥m k,we haveA6k+1= M21+k+1 2 M2i−m2i−1 3= A2k+M2k+1−m2k 3≥ A3k+M3k+1−m3k 2(by the n=2inequality)≥ B3k+M3k+1−m3k 2(since A k≥B k by the induction assumption)=B6k+1.Thus A k+1≥B k+1and A n≥B n is true for any n≥2.The following lemma is crucial in the proof of stability of the peakons. Lemma3.5Assume u0∈H s,s>3/2and y0≥0.Let M1=v u(t,ξ1)= max x∈R{v(t,x)}Then for t≥0,E3(u)−144B3n≤18M1 E2(u)−12A2n (3.13) where u is the global solution of equation(1.5)with initial value u0,v u=(4−∂2x)−1u,and A n and B n are deﬁned in(3.12).Proof.First,by Lemma2.8the global solution u of equation(1.5)satisﬁes u(t,x)≥0and y(t,x)=u−∂2x u≥0for all(t,x)∈R+×R.We now claim that h≤18v for(t,x)∈R+×R.To see this,we rewrite the expression of h ash(x)= − ∂2x+3∂x+12 v−3∂x v+18v,ηi−1<x<ξi,− ∂2x−3∂x+2 v+3∂x v+18v,ξi<x<ηi,1≤i≤n. Ifηi−1<x<ξi,1≤i≤n,then v x>0.On the other hand,it follows from Lemma2.10that− ∂2x+3∂x+2 v=−(2+∂x)(4−∂2x)−1(1+∂x)u≤0.Hence− ∂2x+3∂x+2 v−3∂x v+18v≤18v.(3.14)15A similar argument also shows that forξi<x<ηi,1≤i≤n,∂x v<0and− ∂2x−3∂x+2 v+3∂x v+18v(3.15)=−(2−∂x)(4−∂2x)−1(1−∂x)u+3∂x v+18v≤18v.The combination of(3.14)and(3.15)yieldsh≤18v≤18max v=18M1,∀(t,x)∈R+×R.By the notation in(3.12),the integral identities(3.7)and(3.9)becomeR g2(x)dx=E2(u)−12A2nandRh(x)g2(x)dx=E3(u)−144B3n.Note that when n=1,A1=B1=M1.Relating the above integrals,we getE3(u)−144B3n≤18M1(E2(u)−12A2n).Lemma3.6Assume u∈H s,s>3/2and y≥0.LetM1=v u(t,ξ1)=maxx∈R{v(t,x)}andA n=M21+ni=2 M2i−m2i−1where M i and m i−1(2≤i≤n)are local maxima and minima of v u.If|E2(u)−E2(ϕ)|≤δand|E3(u)−E3(ϕ)|≤δwith0<δ<1,then(i)M1−1δ,(3.16) recalling that vϕ(0)=16 <√3√Proof.To obtain (i),we ﬁrst claim thatM 31−172E 3(u )≤0.(3.19)In the case when M 1isthe onlylocal maximum of v u ,we have n =1,A 1=B 1=M 1and (3.19)follows directly from (3.13).When n ≥2,in view of (3.13)and inequality (3.10)in Lemma 3.4(i),there appears the relationM 31−172E 3(u )≤144 n i =2 M 3i −m 3i −1 −34E 2(u )y +13and E 3(ϕ)=212y +16 2 y +14(E 2(u )−E 2(ϕ))M 1−16 2≤324(E 3(u )−E 3(ϕ)).(3.22)On the other hand,observing E 2(u )−12A 2n ≥0,we have0<M 1≤A n ≤ (1/3+δ)/12<16 ≤ 4|E 2(u )−E 2(ϕ)|+1δ.We now prove claim (ii).When n =1,A 1=M 1and it is reduced to (i).When n ≥2,it is thereby inferred from (3.13)thatA 3n −172E 3(u )≤0,(3.24)due to 0≤M 1≤A n and 0≤B n ≤A n by Lemma 3.4(ii).In consequence,(3.17)follows from the same argument as in part (i).17(iii)can be obtained from (i)and (ii).In fact,combining (i)and (ii),we have 2√A n +M 1,which implies (3.18)by using (3.23).ProofofTheorem1.Letu∈C ([0,∞),H s ),s >3/2be the solution of (1.5)with initial data u (0)=u 0.Since E 2and E 3are both conserved by the evolution equation (1.5),we haveE 2(u (t,·))=E 2(u 0)and E 3(u (t,·)=E 3(u 0),∀t ≥0.(3.25)Since u 0−ϕ X <ε,we obtain|E 2(u 0)−E 2(ϕ)|=|( u 0 X − ϕ X )( u 0 X + ϕ X )|≤ε(2 ϕ X +ε)=ε 23+ε <2ε,under the assumption ε<16≤√2ε+4√E 2(u )/12.For peakons cϕ,we havemax v cϕ=6c .So among all waves of a ﬁxed energy E 2,the peakon is tallest in terms of v u .(2)In our proof,we use inequality (3.22)to get estimates (3.16)and (3.18)more directly,compared with the argument in [12]by analyzing the root struc-ture of the polynomial P (y ).Moreover,it implies that the peakons are energy minimizers with a ﬁxed invariant E 3,which explains their stability.Indeed,if E 3(u )=E 3(ϕ),it follows from (3.22)that E 2(u )≥E 2(ϕ).The same re-mark also applies to the CH equation and shows that the CH-peakons are energy minima with ﬁxed F 3.(3)Compared with [12],our construction of the integral relations (3.7)and (3.9)is more delicate.It not only is required in our current case,but also 18provides us more information about stability via(1.10).For the CH equation, even if the orbital stability is proved by a simpler construction[12],our approach can also give the additional stability information.More speciﬁcally,for the CH equation(1.3)with y0≥0,by reﬁning the integrals of[12,Lemma2]to each monotonic interval of u,one can obtain4F3(u)−[10]A.Constantin and B.Kolev,Geodesicﬂow on the diﬀeomorphism group ofthe circle,Comment.Math.Helv.,78(2003),787–804.[11]A.Constantin and L.Molinet,Obtital stability of solitary waves for a shallowwater equation,Physica D,157(2001),75–89.[12]A.Constantin and W.A.Strauss,Stability of peakons,Comm.Pure Appl.Math.,53(2000),603–610.[13]A.Degasperis,D.D.Holm,and A.N.W.Hone,A New Integral Equationwith Peakon Solutions,Theoretical and Mathematical Physics,133(2002),1463–1474.[14]A.Degasperis and M.Procesi,Asymptotic integrability,in Symmetry andPerturbation Theory,edited by A.Degasperis and G.Gaeta,World Scientiﬁc (1999),23–37.[15]H.R.Dullin,G.A.Gottwald,and D.D.Holm,An integrable shallow waterequation with linear and nonlinear dispersion,Phys.Rev.Letters,87(2001), 4501–4504.[16]J.Escher,Y.Liu,and Z.Yin,Global weak solutions and blow-up structurefor the Degasperis-Procesi equation,J.Funct.Anal.,241(2006),457–485. 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Exact Solutions of the Generalized Equal Width Wave Equation

+∞ −∞
(u )2 dξ = k2 , as ξ −→ +∞ ,
(13)
By substituting (14) into (13) and using the relation (11), we obtain the analytical expression of the following important square integral,
2
Derivation of the Exact Solutions
We concentrate on ﬁnding an exact solitary wave solution of the form u(x, t) = u(x − x0 − Ct) . (3)
This corresponds to a traveling-wave propagating with steady celerity C . We are interested in solutions depending only on the moving coordinate ξ = x − x0 − Ct as, u(x, t) = u(x − x0 − Ct) ≡ u(ξ ) . (4) Substituting into (2), the function u(ξ ) satisﬁes a third order nonlinear ordinary diﬀerential equation (ODE), −Cu + aup u − δu + µCu =0, (5)
Abstract. The equal width wave (EW) equation is a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. The EW-Burgers equation models the propagation of nonlinear and dispersive waves with certain dissipative eﬀects. In this work, we derive exact solitary wave solutions for the general form of the EW equation and the generalized EW-Burgers equation with nonlinear terms of any order. We also derive analytical expressions of three invariants of motion for solitary wave solutions of the generalized EW equation.

CHAPTER I

CHAPTER IGLOBAL WEAK FORMS,WEIGHTED RESIDUALS,FINITE ELEMENTS,BOUNDARY ELEMENTS,&LOCAL WEAK FORMS1.1IntroductionDifferential equations(“ordinary differential equations”in one dimensional, and“partial differential equations”in two or three dimensional physical do-mains)and boundary conditions are abstract,and often highly approximate, characterizations of physical processes in engineering.In physical processes that evolve with time,derivatives with respect to time,and appropriate initial conditions,may also appear in these equations.Exact mathematical solutions are often possible only for problems deﬁned in the simplest of geometrical domains,and only mostly for linear problems[i.e.,for differential equations and boundary/initial conditions wherein the unknown variable(s)and its(their) derivatives appear only in linear combinations.]However,most engineering processes occur mostly in complex geometrical domains,and most often,in-volve nonlinearities of the unknown variable(s)and its(their)derivatives.To solve such practical problems,to aid in engineering analysis,design,and syn-thesis,(digital)computer-based modeling&simulation techniques are neces-sary.We label this as“Computer-Modeling&Simulation-Based Engineering”. The aim is to use the power of the modern digital computer,high-speed sci-entiﬁc computing,visualization techniques,virtual reality,and the attendant information technologies,to achieve a near-real-time simulation of complex en-gineering phenomena,in order to facilitate integrated product&process design for real-life engineering systems.These include aircraft,spacecraft,missiles, automobiles,ships,power plants,bridges,biological systems,micro-electronic devices,computer chips,micro-electro-mechanical systems,systems involving heat and mass transfer,compressors,turbines,systems involvingﬂuidﬂow in closed and open domains,atmospheric polution,sustainable ecological tech-2Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms nologies,nano,micro,meso and macro mechanics of engineered materials,etc. In as much as there is a great deal of commonality in the differential equations governing these processes,with appropriately deﬁned unknown variable(s)gov-erning each process,one can envision a common“tool kit”that will enable com-puter modeling and simulation of a diverse variety of engineering processes.Engineering computations have made a startling progress over the past three decades.The most successful methods among them are theﬁnite element method and the boundary element method.However,as compared to its sound theoretical foundation,theﬁnite element method suffers from drawbacks such as locking(shear,membrane,incompressibility,etc),and tedious meshing or re-meshing in problems of crack propagation,shear-band formation,large de-formations,etc.Due to these reasons,in recent years,meshless approaches in solving the boundary value problems have received a considerable attention as new ways to alleviate some of these drawbacks.Meshless methods,as alternative numerical approaches to eliminate the well-known drawbacks in theﬁnite element and boundary element methods, have attracted much attention in recent decades,due to theirﬂexibility,and, most importantly,due to their potential in negating the need for the human-labor intensive process of constructing geometric meshes in a domain.Such meshless methods are especially useful in those problems with discontinuities or moving boundaries.The main objective of the meshless methods is to get rid of,or at least alleviate the difﬁculty of,meshing and remeshing the entire structure,by only adding or deleting nodes in the entire structure,instead.Meshless meth-ods may also alleviate some other problems associated with theﬁnite element method,such as locking,element distortion,and others.The initial idea of meshless methods dates back to the smooth particle hy-drodynamics(SPH)method for modeling astrophysical phenomena(Gingold and Monaghan,1977).The research into meshless methods has become very active,only after the publication of the Diffuse Element Method by Nayroles, Touzot&Villon(1992).Several so-called meshless methods[Element Free Galerkin(EFG)]by Belytschko,Lu&Gu(1994);Reproducing Kernel Particle Method(RKPM)by Liu,Chen,Uras&Chang(1996);the Partition of Unity Finite Element Method(PUFEM)by Babuska and Melenk(1997);hp-cloud method by Duarte and Oden1996;Natural Element Method(NEM)by Suku-mar,Moran&Belytschko(1998);Meshless Galerkin methods using Radial Basis Functions(RBF)by Wendland(1999);have also been reported in litera-1.1:Introduction3 ture since then.A detailed bibliography on meshless methods is provided at the end of this monograph.The major differences in these meshless methods come only from the techniques used for interpolating the trial function.Even though no mesh is required in these methods for the interpolation of the trial and test functions for the solution variables,the use of shadow elements is inevitable in these methods,for the integration of the weak-form,or of the energy.Therefore, these methods are not truly meshless.Recently,two truly meshless methods,the meshless local boundary inte-gral equation(LBIE)method,and the meshless local Petrov-Galerkin(MLPG) method,have been developed in Zhu,Zhang,&Atluri(1998a,b),Atluri&Zhu (1998a,b)and Atluri,Kim&Cho(1999),for solving linear and non-linear boundary problems.Both these methods are truly meshless,as no domain/or boundary meshes are required in these two approaches,either for the purposes of interpolation of the trial and test functions for the solution variables,or for the purposes of integration of the weak-form.All pertinent integrals can be easily evaluated over over-lapping,regularly shaped,domains(in general,spheres in three-dimensional problems)and their boundaries.In fact,The LBIE approach can be treated simply as a special case of the MLPG approach(Atluri,Kim, &Cho,1999).Zhang and Yao(2001)present a new regular hybrid boundary node method by combining the advantages of both LBIE and the boundary node method.Remarkable successes of the MLPG method have been reported in solving the convection-diffusion problems[Lin&Atluri(2000)];fracture me-chanics problems[Kim&Atluri(2000),Ching&Batra(2001)];Navier-Stokes ﬂows[Lin&Atluri(2001)];Shear-deformable beams[Cho&Atluri(2001)]; and plate bending problems[Gu&Liu(2001),Long&Atluri(2002)],etc.In summary,the MLPG is a truly meshless method,which involves not only a meshless interpolation for the trial functions(such as MLS,PU,Shepard func-tion or RBF),but also a meshless integration of the weak-form(i.e.all integra-tions are always performed over regularly shaped sub-domains such as spheres, parallelopipeds,and ellipsoids in3-D).The aim of this monograph is an exposition of the MLPG method,in all its generality,and in all its variations.It is shown that some of the variations of the MLPG method not only eliminate the human-labor cost of meshing a domain altogether,but also involve less computational cost as compared to theﬁnite element and boundary element methods.4Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms1.2Global weak forms and the weighted residual method(WRM) To set the stage for a discourse on the MLPG method,we start off by discussing a variety of modeling strategies,using Global Weak Forms,through an example problem.It can be shown that all mathematical formulations governing a variety of linear boundary value problems in engineering may be written in the form: L mΨf(1.1) where L m is a linear differential operator of order m.L m can be an“ordinary differential operator”in a one-dimensional problem,or a“partial differential operator”in2and3-dimensional physical domains.In a general3-D domain, imagine a line,going through the domain,and connecting two different points on the boundary.Along this line,one can imagine an ordinary differential equa-tion of order“m”.Then,at each point on the boundary,we may have m2 number of boundary conditions,each involving a differential operator of order m1.For example,we consider the linear Poisson’s equation in2-dimmensional space,deﬁned by the Cartesian Coordinates x i.Consider the domainΩto be as sketched in Fig.1.1.The boundary ofΩ,denoted byΓ[or,sometimes, by∂Ω],is deﬁned by the curve A-B-C-D-E-F-A,and the curve GH.Along the segment AB,the boundary conditionφφis prescribed.Like wise,the boundary conditions along the various segments are considered to be:∂φ∂n q along BC;φφand∂φ∂n q along parts of CD;∂φ∂n0along the slit (or“crack”)D-E-F;φφand∂φ∂n q along parts of FA;and∂φ∂n0 along the“hole”GH.The boundary value problem deﬁned in Fig.1.1is a fairly realistic engineering problem.Hence,AB belongs to the essential boundaryΓu, and EF,ED and GH belong to theﬂux boundaryΓq.The Poisson equation can be can be written as:∇2φx p x xΩ(1.2)where x x i e i is the vector of position of a point insideΩ,p is a given source function,and the domainΩis enclosed byΓΓuΓq,with boundary condi-tions:φφonΓu(1.3a)∂φq q onΓq(1.3b)∂n1.2:Global weak forms and the weighted residual method(WRM)5Figure1.1:A realistic boundary value problem,∇2φx p x.The condition(1.3a)is often referred to as the Dirichlet boundary condition; and(1.3b)as the Neumann boundary condition.The Poisson equation arises in very many branches of engineering:the torsion of a solid rod of an arbitrary cross-section,undergoing an inﬁnitesimal twist;heat transfer in homogeneous media;seepageﬂow;ﬂuidﬁlm lubrication;eletro-staticﬁeld;magneto-static ﬁeld,etc.Let u,hereafter called the trial function,be an approximate solution to the problem(1.2).In as much as u may not be an exact solution,substituting u in the differential equation and the boundary conditions Eqs.(1.2-1.3),we obtain the following relations for the“error residuals”.Interior Error:R I x∇2u x p x0(1.4)Boundary Error:R B1u x u0atΓuR B2∂u∂nq atΓq(1.5)6Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Formswhere R I is the interior error;and R Bii12are the errors in the boundary conditions.The“best”approximation is the one that nulliﬁes the errors R I and R Biin some fashion.In general,the trial functions must satisfy several minimum re-quirements to make the process of error nulliﬁcation meaningful.The functions that satisfy such minimum requirements are called the admissible functions.For example,in Eq.(1.4),u must be2times differentiable and u i must be contin-uous(or u is C1continuous);if u is not C1continuous,u ii will be“inﬁnite”where u i is discontinuous.In that case,R I u is inﬁnite at these points;and the minimization of R I u no longer makes any sense.Consider a linear combination of N admissible functions u i(i12N), such that:uN∑i1a i u i x a i u i x(1.6)where a i(i12N)are undetermined coefﬁcients.We use the convention that summation is implied over a repeated index.Here,choosing an admissible u x[i.e.,C1continuous u x in Eq.(1.4)]over a global domainΩ,of arbitrary shape is a non-trivial proposition.Much of Chapter2in this monograph is devoted to this topic.However,ifΩis a simple geometry,such as a circle, square,etc.,u x can be in the form of trigonometric or other functions.Figure1.2:Schematics of the point collocation methodWe now discuss several methods,to reduce the errors R I and R Bito be as1.2:Global weak forms and the weighted residual method(WRM)7 small as possible,and in the limit,make them uniformly zero over the globaldomainΩ.1.2.1Point collocation methodWe assume that u is C1continuous all overΩin Eq.(1.4).We consider making the error vanish at certain points x j j12M,in the domainΩ.This leads to the following equations for the undetermined coefﬁcients.R I x j a i∇2u i x j p x j0j12M(1.7) andR Bix j0i12(1.8) Suppose that we use the conditions of vanishing of the two types of boundary residuals[Eq.(1.8)]to generate N M equations;then,by using M collocation points in the interior of the domainΩ,we can generate N equations for the N undetermined coefﬁcients a i.While a solution a j for such a system of equationswill make the errors R I and R Bigo to zero identically at aﬁnite number of points,the residuals R I and R Bimay oscillate in between these points and may indeed be quite ing more collocation points than M will result in more equations than the unknowns.In this case,it is not possible,in general,toﬁnd a set of coefﬁcients a i that will drive the error at the collocation points to zero.Suppose that point collocation of Eqs.(1.7)at an arbitrarily large number of points,along with enforcing(1.8)at an arbitrary number of points,lead,in general,to the over-determined system of equations:A i j a j p i(1.9) where i12K;j12N and K N.We seek an approximate solu-tion a j for a j,such thatA i j a j p iεi0(1.10) and a j minimize the square error,εiεi.Thus,∂∂a j εiεi∂∂a j A ip a pp i A iq a q p i0(1.11)8Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak FormsorA i j A iq a q A i j p i(1.12) orB jq a q p i(1.13) where i12K;j12K and q12N.The above method is an algebraic least-squares procedure to approximately solve an over-determined system of equations,provided that B jq is positive def-inite.For an underdetermined system of equations,however,only some of the unknowns can be expressed in terms of the other unknowns.It is noted that Eq.(1.11)also amounts to minimizing the summed point-wise square error. 1.2.2Weighted integral square errorAs discussed above,in the point collocation method,we can use more colloca-tion points than the number of unknown coefﬁcients in the trial function,using a least square error approach.In the limit,as the number of collocation points tends to∞,we may consider minimizing the integrated square error:εσΩ∇2u x p x2dΩβΓuu u2dΓγΓq∂u∂nq2dΓ(1.14)whereσ,β,γare weighting constants,which are also often referred to as “penalty-parameters”.By minimizing the aboveεwith respect to a i,one may obtain the desired system of N algebraic equations to determine the N unknownsa i.1.2.3Subdomain integral/average error methodAlternatively,we may consider making each of the averages of the error,over a set of subdomains,go to zero.For example,we may setΩk R i x dΩ0;∂ΩkR i x dΓ0(1.15)The number of subdomains is chosen such that we have a solvable set of equa-tions for a i,i12N[i.e.N,or more than N subdomains,to yield a1.2:Global weak forms and the weighted residual method(WRM)9 system of N equations,or of more than N equations].This is illustrated in Fig.1.3.Note that the subdomainsΩi i12k are continguous(i.e.,non-overlapping),and span the entire domainΩ.While most ofΩi may be of a regular[say triangular]shape,others may be approximately triangular.Figure1.3:Schematics of the subdomains1.2.4Finite volume methodThis is similar to the Subdomain Integral method,and has been popularized in the literature dealing with unsteady aerodynamics,and heat transfer.The prin-ciple of theﬁnite volume scheme is to integrate Eq.(1.2)on a control volume (i.e.subdomain)Ωk,which yields:∂Ωk u i n i dΓΩkpdΩ0(1.16)where∂Ωk is the boundary of the subdomainΩk,andΩkΩ;next,theﬂuxes need to be approximated on the boundary∂Ωk of the control volume.Hence, we shall approximate the integral of u i n i over each edge of the mesh.Note, once again,that the sub-domainsΩk are contiguous,as in Fig.1.3.10Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms Another variant of the Finite V olume Method may be derived,by rewriting Eq.(1.2)in a“mixed”form:φiφi;φi i p(1.17) wherein,φandφi are treated as independent variables.If u and u i are the trial functions forφandφi,respectively,we may write the counterparts of(1.16)as:∂Ωk un i dΓΩku i dΩ0(1.18)and∂Ωk u i n i dΓΩkpdΩ0(1.19)Instead of the“global trial function”,u a k u k x;and u i a k u ki x,one may assume a linear interpolation of u and u i over each element,as shown in Fig.1.3.Note again,all the subdomainsΩk are contiguous in Fig. 1.3,and the method involves the generation of a“mesh”,for both the purpose of interpola-tion,as well as for purpose of integration,as in Eqs.(1.18)and(1.19)]1.2.5The weighted residual method and global weak formsIf the function F x satisﬁes the conditionΩF x g x dΩ0(1.20) for all g x onΩ,then,F x0onΩ.Conversely,therefore,we canﬁnd the trial function u a i u i such that R I u0[where R I is deﬁned in Eq.(1.7)] by enforcing the weak form equationΩR I x v x dΩ0(1.21) for all possible functions v x onΩ.We denote v x as“test”functions.In general,the trial function u x is chosen to be a linear combination of a set of basis functions,u i(i12N),with undetermined coefﬁcients,a i,i.e. u a i u i x,xΩ.Likewise,the test function v x is chosen as v b j v j x, j12M;where v j are the basis functions and b j are the undetermined1.2:Global weak forms and the weighted residual method(WRM)11 coefﬁcients.The“best”approximation is obtained by determining that combi-nation of u i that satisﬁes the weak form equation(1.21)for the chosen test func-tions v j.Different choices of the basis functions for the trial function and the test functions will lead to different approximation methods.For example,it reduces to the Point Collocation Method when Dirac Delta functions,v xδx x j, are used for the test functions.It reduces to the subdomain integral method, when the following piecewise constant functions[or Heaviside step functions] are used as test functions:v x1xΩk[see Fig.1.3]0elsewhere(1.22)When the test function v x is taken to be the same as the error function R I, Eq.(1.21)reduces to the least square error method.Now,consider the weighted interior residual equation for the Poisson’s prob-lem,governed by Eq.(1.2):Ω∇2u x p x v x dΩ0(1.23)which requires u to be twice differentiable and u i to be continuous[or u is C1 continuous].If not,u ii will be∞where u i is discontinuous and(1.23)makes no sense.On the other hand,there is no continuity requirement on the test function v in Eq.(1.23).Thus,the requirements on u and v are“unsymmetric”, in order to be admissible in Eq.(1.23).Hence,we denote Eq.(1.23)as a global unsymmetric weak form(GUSWF).To reduce this high-order differentiability requirement on u,we can inte-grate Eq.(1.23)by parts,by using∇2u v u ii v u i v i u i v i,to obtain the following global symmetric weak formulation(GSWF),Γu i n i vdΓΩu i v i pv dΩ0(1.24)which has the same differentiability requirements on both the trial function u and the test function v in the interior of the domainΩ,namely,both u and v are required to be once differentiable,and both u and v are required to be C0continuous.If they are not C0continuous,u i and v i may tend to∞when u and v are discontinuous and Eq.(1.24)is no longer meaningful.Thus,the12Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms requirement on the test function is“increased”,while the requirement on the trial function is“decreased”.We may directly substitute the natural boundary condition Eq.(1.3b)into the symmetric weak form equation,i.e.Eq.(1.24),and noticing that u i n i∂u∂n q in Eq.(1.24),we obtainΓu u i n i vdΓΓqqvdΓΩu i v i pv dΩ0(1.25)When all the boundary conditions are natural,(or“Generalized Force”type), we may rewrite Eq.(1.25)asΓq qvdΓΩu i v i pv dΩ0(1.26)We can see that Eq.(1.26)is actually the combined weak form equation for both the differential equation(D.E.)and the boundary conditions(B.C.),if these B.C. are purely of the“natural”or“kinetic”,or“generalized-force”type.Let u be approximated using u i(i12N),i.e.u a i u i i12N(1.27)where a i are undermined coefﬁcients.Note that,in order to be admissible,u i need only to be C0continuous in the domainΩ,,but they need not obey any boundary conditions a priori,in as much as the boundary condition for u,if they are of the“natural”or“kinetic”type,i.e.,Eq.(1.3b),is already embedded in the weak form Eq.(1.26).Let v be any function that can be represented asv b j v j j12M(1.28)where b i are arbitrary constants.Note again,that in as much as the prescribed boundary conditions for u are purely natural as in Eq.(1.3b),the test func-tion v need not obey any boundary conditions,as seen from Eq.(1.26).Using Eqs.(1.27)and(1.28)in Eq.(1.26),we obtainΓq qb j v j dΓΩa i u i kb j v j k pb j v j dΩ0(1.29)Since Eq.(1.29)is true for any b j,we have the following linear system of equa-tions for a i.A i j a i f j(1.30)1.2:Global weak forms and the weighted residual method(WRM)13 where i12N;j12M,andA i jΩu i k v j k dΩ(1.31)f jΩpv j dΩΓqqv j dΓ(1.32)If M N[i.e.,the number of basis functions in the trial function u,is the same as the number of basis functions in the test function v],and u j are identically the same as v j j12N,we have a symmetric A i j.When v j u j,the method is generally known as the Galerkin symmetric weak form approach.When the test functions v j are in general different from u j,the method is known as the Petrov-Galerkin approach.In this case,even when M N and even when a symmetric weak form is used,the coefﬁcient matrix A i j is unsymmetric.When M N,the system of Eq.(1.30)is over-determined,and can be solved by the method of algebraic least squares.Also,when m n,and u j v j,the coefﬁcient matrix A i j in Eq.(1.30)is symmetric and positive deﬁnite,as long as“rigid body modes”[i.e.u j is a constant or linear function],are removed.Suppose now that the B.C.are purely kinematic[i.e.,ΓΓu],i.e.essen-tial boundary condition Eq.(1.3a)exists all over the boundary.If we examine the GSWF(1.24),we notice that the above“lower-order”or purely“kinematic”B.C.(1.3a)cannot be directly substituted into the GSWF(1.24).Unless we choose to add additional terms to Eq.(1.24)to enforce essential boundary con-dition(1.3a)in a weak form,we must choose the trial function u to satisfy these conditions(1.3a)identically,a priori.In this case,we are left with the GSWF (1.24)with boundary terms as indicated in Eq.(1.24).We can eliminate these boundary terms onΓin Eq.(1.24),if we choose the test functions such that v x0,xΓ.If,on the otherhand,in the case when the boundary conditions are entirely kinematic,i.e.,only on u,and if,correspondingly,v does not vanish at the boundary,we have the discrete weak form(similar to1.29)as:Γa i u i k n k b j v j dΓΩa i u i kb j v j k pb j v j dΩ0(1.33)leading to the discrete equation A i j a j f i,whereA i jΩu i k v j k dΩΓu i k n k v j dΓ(1.34a)and f jΩpv j dΩ(1.34b)14Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms It is seen that nothing can be said of the positive deﬁniteness of A i j in (1.34a),which does not assure the solvability of the problem.On the other-hand,if v j vanish at the boundary,A i j reduces to that in Eq.(1.31),which is positive deﬁnite(when rigid modes are suppressed).Hence a solution always exists for a i.Thus,if the given problem has prescribed boundary conditions that are purely of the kinematic type,i.e.,as in Eq.(1.3a)[i.e.,ΓΓu],we choose u to obey these conditions a priori,and correspondingly choose v such that v x0,xΓ.In this case,the GSWF reduces to:Ωu i v i pv dΩ0(1.35)when all the B.C.are purely kinematic.If the trial function u,especially in2and3-dimensional problems,cannot be chosen so as to satisfy the essential boundary conditions a priori,one can enforce them by:(i)choosing a boundary weighting function that may depend on the higher-order derivatives of the test function v,such that the kinematic boundary conditions are enforced by the weak-form:Γuu u v i n i dΓ0(1.36)However,when the bilinear form arising out ofΓu uv i n i dΓis added to the A i jin Eq.(1.34b),the positive deﬁniteness of the resulting coefﬁcient matrix A i j cannot be guaranteed;(ii)by choosing an independent Lagrange multiplierﬁeld at the boundary[this is also not desirable,since the solvability of the problem will involve certain stability conditions which must be satisﬁed a priori[Brezzi (1974)];(iii)by using boundary collocation for the essential B.C.at a desired number of points;(iv)by using the least-squares integral error condition for boundary error residuals;or(v)by using a“Penalty”approach.Methods(iii)to (v)are in general simpler than the others.More will be said of these essential boundary conditions,and their satisfaction,later in this book,in the context of meshless methods.In general,in the GSWF(1.24),one can directly substitute any higher-order B.C.(natural)into(1.24),and thus enforce them a posteriori;while all lower-order(essential)B.C.must be satisﬁed a priori,and the GSWF is then simpli-ﬁed by enforcing v to satisfy the homogeneous condition atΓu.1.2:Global weak forms and the weighted residual method(WRM)15Now,we use the formalism that the test function v is considered to be a “variation”of the trial function u,in the traditional“calculus of variations”sense.For,example,in general,if L2m is a self-adjoint operator,i.e.L2m u vdΩL m v L m u dΩB(1.37) where B are boundary integral terms,then aπexist so thatδπ0leads to L2m u p,whereπ12L m u2pu dΩB(1.38) In a elastic system with a conservative(dead-load)forces,πis the“Total Po-tential Energy”,andδπ0is the condition of stationarity of the total potential energy.It may simply be viewed as a mathematical equivalent of the symmetric weak form,which,however,is more direct.In general,the weak form approach is,however,more general than the above“energy approach”,in as much as a weak form can be stated for non-self-adjoint operators as well.The Galerkin Finite Element Method(GFEM)is based on the Global Symmetric Weak Form (GSWF),Eq.(1.24).To reduce the C0continuity conditions as demanded by the Symmetric Weak form(1.24),we can integrate Eq.(1.23)by parts twice,by using∇2u vu ii v u i v i uv i i uv ii,to obtain another global unsymmetric weak for-mulation(GUSWF)as follows,Γu i n i vdΓΓuv i n i dΓΩu∇2v pv dΩ0(1.39)which requires the test function v to be twice differentiable and v i to be contin-uous[or v is C1continuous].If not,v ii will be∞where v i is discontinuous and (1.39)makes no sense.On the other hand,there is no continuity requirement on the trial function u in Eq.(1.39).Thus,the requirements on u and v are also “unsymmetric”,in order to be admissible in Eq.(1.39).Hence,we also denote Eq.(1.39)as a global unsymmetric weak form(GUSWF2).In this case,we may directly substitute the natrural and essential bound-ary conditions Eqs.(1.3a)and(1.3b)into the global unsymmetric weak form equation,i.e.Eq.(1.39),we obtainΓu qvdΓΓqqvdΓΓuuv i n i dΓΓquv i n i dΓΩu∇2v pv dΩ0(1.40)16Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms1.3The Galerkinﬁnite element methodIn the previous section,the basis functions u j and v j for the trial function u and the test function v,were at once assumed as“global”functions,and each of it was nonzero over the entire problem-domain,Ω.Figure1.4:Use of a patch work of generic triangular elements to approximate the given geometry in Fig.1.1In a one-dimensional problem,assuming component trial and test functions u j and v j over the entire problem-domain is relatively simple.On the other hand,if the problem is deﬁned by a partial differential equation in2or3dimen-sions,and if the global domain is of an arbitrary shape,assuming the trial and test functions as simple polynomials over the entire global domain is inconve-nient at best.To overcome this curse of arbitrary shaped geometrical domains wherein the problem is deﬁned,one may resort to assuming trial and test func-tions over a generic simpler-shaped subdomain(element)of the given problem, and create a non-overlapping patchwork of the subdomains(elements)to geo-metrically approximate the given domain(see Fig.1.4).We can assume the trial and test functions over each subdomain(element)as“local basis”or“element basis”functions.。

The Homogenization Method Applied to the Computation of E¤ective Thermal Conductivities of

The Homogenization Method Applied to the Computation of E¤ective ThermalConductivities of Composite MaterialsJOHAN BYSTRÖMDepartment of Mathematics,LuleåUniversity of Technology,S-97187Luleå,SwedenAbstractTo solve boundary value problems of stationary heat distribution in com-posite materials,one has to solve partial di¤erential equations with rapidly oscillating coe¢cients.These equations are in general impossible to solve numerically by standard methods.Instead,one wants to…nd constant,ef-fective conductivities and solve the corresponding boundary value problem to get an approximate solution to the original problem.To compute these e¤ective parameters,we have developed a…nite element computer program with integrated homogenization routines.Contents1Introduction21.1Notations and conventions (4)2The Homogenization Method in R262.1The model problem (6)2.2Asymptotic expansion (7)3Numerical computation of the homogenized coe¢cients123.1Weak formulation of the cell problem (12)3.2Finite element solutions of the cell problem (13)4Existence and uniqueness of weak solutions for elliptic prob-lems18 5Numerical results245.1The computer program (24)5.1.1Input of data (25)5.1.2Construction and representation of the triangulation.265.1.3Computation of the element sti¤ness matrices a K n andelement loads b K n(k) (27)5.1.4Assembly of global sti¤ness matrix and load vectors..285.1.5Solution of the linear systems (28)5.1.6Presentation of the results (29)5.2A comparison of computed coe¢cients with various bounds..295.2.1Discussion (31)6Some concluding words321Chapter1IntroductionBecause of the highly increased use of…bre reinforced composite materials, it is necessary to have mathematical methods for the prediction of material properties.The actual distribution of di¤erent phases of the composite is more or less unknown,but a reasonable assumption is that the structure of the material is periodic.This assumption leads to di¤erential equations with periodically oscillating coe¢cients describing the spatial structure of the material.For instance,consider an elliptic boundary value problem modelling a stationary heat distribution in a region½R n without internal heat source¡@@xiÃa ij(x)@~u@x j!=0in;1·i;j·n(1.1) ~u=~u0on@;(1.2)where a ij(x),~u(x)and~u0(x)denotes the thermal conductivity tensor,the temperature…eld and the boundary temperature.In the above formulation, we have used the Einstein summation convention for repeated indices,and if nothing else is indicated,we will use it in the sequel.This problem can be reformulated asL(u)=¡@@x iÃa ij(x)@u@x j!=f(x)in;(1.3) u=0on@;(1.4)by…nding a function g(x)such that g(x)=~u0(x)on@:Then~u(x)=u(x)+g(x)and f(x)=@i (a ij(x)@gj).This problem is easily solved numerically2when the tensor a ij(x)is constant or slowly varying in x.But when a ij(x) oscillates rapidly in x,the numerical solution becomes cumbersome.This is because a very…ne subdivision of the region is required to obtain an accurate solution.Instead,one wants to…nd an approximate solution of the problem by replacing a ij(x)by a constant conductivity tensor q ij(x)and solve the corresponding problem by some standard method.This constant tensor is called the e¤ective conductivity tensor and may be found by an averaging procedure called the homogenization method.The main idea of homogenization is the following:Letbe a domain with boundary@.For a…xed real number">0,let A"be a linear partial di¤erential operator onand study the boundary value problemA"u"=f"(1.5) in:It is here assumed thatis divided into identical cells of a size of order ".By varying"we obtain a family of partial di¤erential operators{A"}and a corresponding family of solutions{u"}.Then,a multiple scales expansion of the type u"=u0+"u1+"2u2+:::is made.If certain conditions are ful…lled(see e.g.Persson et al.[15]),then as "!0,u"converges in a weak sense to u0,where u0is the unique solution to the problemA0u0=f,(1.6) where the coe¢cients of the homogenized operator A0are constant.31.1Notations and conventionsLet denote an open and bounded set in R n with boundary @.We say that has the Lipschitz property if @is locally Lipschitz,that is,if each point x of @has a neighborhood U x such that the intersection of U x and @is the graph of a Lipschitz continuous function.We de…ne some vector spaces of real valued functions on and their corresponding norms (1·p;q <1):C 10()=f u :u is smooth with support on a compact set K ½g ;L p ()=f u :Zj u j p dx <1g ;k u k L p ()=(Z j u j p dx )1;L 1()=f u :essup x 2j u (x )j <1g ;k u k L 1()=essup x 2j u (x )j ;H 1;p ()=f u 2L p ():@u @x i 2L p (),i =1;2;:::;n g ;k u k H 1;p ()=(k u k pL p ()+n X i =1°°°°°@u @x i °°°°°p L p ())1;H 1;p 0()=f u 2H 1;p ():u =0on @g ;k u k H 1;p 0()=k u k H 1;p ();H 1;p ()=R =f u ¡c :u 2H 1;p ()and c 2R g ;k u k H 1;p ()=R =inf c 2Rk u ¡c k H 1;p ():In H 1;p (),the derivatives @u @x i are taken in the distributional sense,that is,@u i is the (unique)L p -function that ful…lls R @u i ´dx =¡R u @´i dx for all test functions ´2C 10():In H 1;p 0();u =0on in the trace sense,that is,°u =0where °:H 1;p ()!L p (@)is the unique bounded linear operator which ful…lls °u =u j @for all u 2H 1;p ()that is continuous up to the boundary.The existence and uniqueness of the trace operator °is a special case of the trace theorem,given in Alt [1],p.190.4In the factor space H1;p()=R we identify functions u and v such that u¡v=c;c constant.For the case p=2we simply write H1()and H10()instead of H1;2() and H1;20();respectively.By solutions of the boundary value problem(1.3)-(1.4)we understand weak solutions,that is solutions taken in the distributional sense.This means that we will consider functions u2H10()satisfyinga(u;v)=(f;v)for all v2H10();(1.7) where the bilinear form a:H10()£H10()!R is given bya(u;v)=Za ij(x)@u@x j@v@x i dx;(1.8) and the linear functional(f;v)by(f;v)=Zfvdx:(1.9)However,in the following chapters,we will consider an even more general boundary condition,leading to another de…nition of the linear functional.In (1.8)and(1.9),we have the restrictions a ij(x)2L1()and f2L2():We will use h¢i to denote the arithmetic mean over a domain D,that ish f i=1m(D)Z D fdx;(1.10)where m(D)is the Lebesgue measure(area in R2)of D:Later,D will be the Y-cell which we will de…ne in the following chapter.We will write j Y j instead of m(Y):5Chapter2The Homogenization Methodin R22.1The model problemLetbe a region in R2with boundary@containing two di¤erent isotropicmaterials,material1and material2,each of them possessing a constantthermal conductivity.Assume that material1occupies the region1½and that material2occupies the region2=n1.Furthermore,assumethat the boundary@can be divided into two disjunct parts,¡1and¡2,that is@=¡1[¡2and¡1\¡2=;:Let a ij(x),u(x),f(x)and g(x)denote the thermal conductivity tensor,the temperature…eld,the internal heat source density and the heat‡ow,respectively,at the spatial point x:Finally, let n=(n i)denote the outer unit normal to@:Then the temperature distribution in the material is given by the elliptic boundary value problem¡@@xiÃa ij(x)@u@x j!=f(x)in;(2.1)u=0on¡1and a ij(x)@u@x jn i=g on¡2:(2.2)The above equation is satis…ed in each of the regions1and2,together with the imposed boundary conditions.To get a well-posed problem in;we assume the physically based conditions that the temperature and the normal components of the heat‡ows are continuous at the interfaces between1and 2.Let us from now on assume that material2is distributed in material1.6For materials with few interfaces,we can easily solve the problem.But for geometries with several interfaces,the conductivity tensor a ij(x)is rapidly oscillating with x and the problem becomes hard to solve numerically.A special case of this is a…bre composite,where material2are…bres(for instance glass)distributed in a continuous matrix of material1(for instance epoxy).The composite material is assumed to have a periodic structure,so that R2can be divided into identical unit cells Y.Speci…cally,we will use quadratic or hexagonal unit cells in the numerical treatment of the prob-lem.This assumption implies that the conductivity tensor a ij(x)must be periodic.To describe the Y-cell,we will use a local variable y de…ned as y=x=":Here,the parameter">0is a scale factor such that the conduc-tivity tensor for the body is de…ned as a"ij(x)=a ij(x=")for x2: The material is then described by a Y-periodic conductivity tensor a ij(y), de…ned on R2,where Y-periodicity means that a ij(y1)=a ij(y2)whenever y1 and y2have the same positions in the corresponding cells.In particular,we note that in a cell Y,Y-periodic functions take on the same boundary values twice,but with opposite outer normals n=(n i):Thus,for a Y-periodic vector…eld(f i(y));Z Y@f i@y i dy=I@Y f i n i ds=0(2.3)by Gauss’theorem.This is a crucial fact that will be frequently used in the sequel.Above,ds denotes the curve element of@:Further,we assume that a ij(y)belongs to L1()and that a ij(y)is positive de…nite,that isa ij(y)»i»j¸®»i»i(2.4) for some®>0:Now we let"vary and thereby obtain a class of boundary value problems¡@@xiÃa"ij(x)@u"@x j!=f"(x)in;(2.5)u"=0on¡1and a"ij(x)@u"@x jn i=g on¡2:(2.6)2.2Asymptotic expansionTo solve(2.5)with the boundary conditions(2.6),we will perform a multiple scales expansion(see e.g.Nayfeh[13])of the solution u"in(2.5).We assume7that u"and f"can be represented by asymptotic expansions of the form u"(x)=w0(x;y)+"w1(x;y)+"2w2(x;y)+:::f"(x)=f0(x;y)+"f1(x;y)+"2f2(x;y)+::: respectively,where w i,i=0;1;2;:::;are assumed to be Y-periodic in thevariable y;wherey i=x i":(2.7)We de…ne the operators A"byA"ª=¡@@x i(a"ij(x)@ª@x j):(2.8)By settingª(x)=©(x;y)we get@ª@x i =@©@x i+1"@©@y i(2.9)by the chain rule so thatA"ª=¡@@x i(a"ij(x)@ª@x j)=¡(@@x i+1"@@y i)(a ij(y)(@©@x j+1"@©@y j))==¡[a ij(y)@2©@x i@x j+a ij(y)1"@2©@x i@y j+1"@a ij(y)@y i@©@x j+1"a ij(y)@2©@y i@x j+ +1"2@a ij(y)@y i@©@y j+1"2a ij(y)@2©@y i@y j]:(2.10)We can therefore represent A"asA"ª=("¡2A0+"¡1A1+A2)©;(2.11)whereA0=¡@@y i(a ij(y)@@y j);(2.12)A1=¡@@y i(a ij(y)@@x j)¡a ij(y)@2@x i@y j;(2.13) A2=¡a ij(y)@2i j:(2.14) 8Equation(2.5)then becomesA"u"(x)=("¡2A0+"¡1A1+A2)(w0+"w1+"2w2+:::)= ="¡2A0w0+"¡1(A0w1+A1w0)+(A0w2+A1w1+A2w0)+:::==f0+"f1+"2f2+:::(2.15) Identi…cation of powers of"gives the three lowest order equationsA0w0=0;(2.16)A0w1+A1w0=0;(2.17)A0w2+A1w1+A2w0=f0:(2.18) To solve(2.16)-(2.18),we need the following lemma:Lemma1Consider the boundary value problemA0©=F in a unit Y-cell,(2.19) where©is Y-periodic and F2L2(Y):Then the following holds:1.There exists a weak Y-periodic solution©if and only if h F i=0:2.If there exists a weak Y-periodic solution©;then it is unique up to anadditive constant,that is,if we…nd one solution©0(y);every solution is of the form©(y)=©0(y)+c,where c is a constant.Proof:See Chapter4,page21.The operator A0contains only derivatives with respect to y;so x is just a parameter in the solution of(2.16).We note that one solution to(2.16)is w0(x;y)´0:Then,by Lemma1,w0(x;y)is a solution of(2.16)if and only if w0is a constant with respect to the variable y,that is,w0(x;y)=u(x)(2.20) for some su¢ciently di¤erentiable function u(x):By(2.17),(2.20)and(2.13), we getA0w1=¡A1w0=@a iji (y)@uj(x):(2.21)9Here again,x is just a parameter because of the construction of A0;so that the equation(2.21)may be regarded as a problem depending on the variable y only.Therefore it su¢ces to consider the cell problemA0Âk=¡@a ik@y i(y);Âk(y)Y-periodic.(2.22)Now,assume that a solutionÂk(y)of(2.22)is given.From Lemma1and the fact that A0does not include di¤erentiation with respect to x,we can conclude that every solution to(2.22)is of the formÂk1(x;y)=Âk(y)+~u k1(x);(2.23) since by(2.3)and the Y-periodicity of a ij(y)*@a ij@y i+=1j Y j Z Y@a ij@y i dy=1j Y jI@Y a ij n i ds=0:(2.24)By using linearity we get thatw1(x;y)=¡Âk(y)@u@x k(x)+u1(x)(2.25)is a solution to(2.21),where u1(x)=¡~u k1(x)@uk (x);and,as pointed outabove,every solution to(2.21)is of this form.Furthermore,by rewriting (2.18)as A0w2=f0¡A1w1¡A2w0and using Lemma1once more,we conclude that(2.18)possesses a Y-periodic solution if and only ifZ Y(f0¡A1w1¡A2w0)dy=0:(2.26)By inserting the expressions(2.20)and(2.25)for w0and w1,respectively, into(2.26)we get10Z Y(f0¡@i(a ijÂk)@2u j k+@a ij i@u1j¡¡a ij@Âk@yj@2u@x i@x k+a ij@2u@x i@x j)dy=0(2.27)Since the functionsÂk(y)and a ij(y)are Y-periodic and the functions u(x) and u1(x)are independent of y,the second and third term in(2.27)vanish by Gauss’theorem(2.3),and(2.27)is reduced toZ Y(f0¡a ij@Âk@y j@2u@x i@x k+a ik@2u@x i@x k)dy=0;(2.28) which can be rewritten as¡"Z y(a ik¡a ij@Âk j)dy#@2u i k=Z Y f0(x)dy=h f0i j Y j:(2.29) Dividing both sides with j Y j gives¡q ik@2u@x i@x k=h f0i;(2.30) which is the homogenized equation,where the homogenized coe¢cients q ik are given byq ik=*a ik¡a ij@Âk@y j+=1j Y j Z Y(a ik¡a ij@Âk@y j)dy:(2.31) The main steps described above can be summarized as the following homog-enization procedure:1.Identify and solve the cell problem(2.22).2.Insert the solution of the cell problem into(2.31)and compute thehomogenized coe¢cients.3.Impose appropriate boundary conditions and solve the homogenizedproblem(2.30).The solution of(2.30)represents an averaged solution of the original prob-lem(2.5)-(2.6).11Chapter3Numerical computation of the homogenized coe¢cients3.1W eak formulation of the cell problemThe homogenized coe¢cients are given by(2.31)q ik=*a ik¡a ij@Âk j+=1Z Y(a ik¡a ij@Âk j)dy:(3.1)In order to compute these coe¢cients,we need to solve the cell problem (2.22)@a ikA0Âk=¡(y);Âk(y)Y-periodic(3.2)iforÂk so that we can compute the integral in(3.1).To get a weak formulation of(3.2),we note that¡A0y k=¡"¡@i(a ij(y)@j)#y k=@a ij i±jk=@a ik i;(3.3) so that(3.2)can be writtenA0(Âk¡y k)=0;Âk(y)Y-periodic.(3.4) Now we introduce the space of all Y-periodic functions in H1(Y)asW(Y)=fª2H1(Y):ªis Y-periodic g:12Assume that we have a solution#k=Âk¡y k to(3.4)and#2W(Y).Then we take³2W(Y)and multiply it with A0#and integrate over Y to getZ Y(A0#)³dy=¡Z Y@@y i(a ij(y)@#@y j)³dy==Z Y a ij(y)@#@y j@³@y i dy¡I@Y a ij(y)@#@y j³n i ds(3.5) by Green’s theorem.We then use the fact that a ij(y)is Y-periodic together with Gauss’theorem(2.3)and getZ Y(A0#)³dy=Z Y a ij(y)@#j@³i dy:(3.6) If we de…ne a Y(#;³)througha Y(#;³)=Z Y a ij(y)@#@y j@³@y i dy;(3.7)we can write the weak formulation of(3.2)as:FindÂk2W(Y)such thata Y(Âk¡y k;ª)=0for allª2W(Y);(3.8) orFindÂk2W(Y)such thata Y(Âk;ª)=a Y(y k;ª)=L k Y(ª)for allª2W(Y);(3.9) whereL k Y(#)=Z Y a ij(y)@y k@y j@#@y i dy=Z Y a ik(y)@#@y i dy:(3.10) It should be noted that the solution of(3.8)or(3.9)is only unique up to an additive constant,as will be shown in Chapter4.3.2Finite element solutions of the cell prob-lemIn order to obtain a FEM formulation of(3.8)or(3.9),we now make a subdivision of the unit cell Y in N triangles K n and M nodes N i according13to Johnson[10].Let W h(Y)be a…nite-dimensional subspace of W(Y)of dimension P:We introduce…nite element base functions'j(y)2W h(Y);j= 1;2;:::;P;where P denotes the number of degrees of freedom.Generally,P is seldom equal to M;because in the case of Y-periodicity,nodes at opposite positions on the boundary of the Y-cell must correspond to the same degrees of freedom.To relate the nodes to the degrees of freedom,we let Q be a mapping from node N i to the corresponding degree of freedom l;that is l=Q(N i).We also choose the base functions to be piecewise linear and de…ne them by'j(N i)=±jl;where l=Q(N i):(3.11) We assume an approximate solution to(3.9)of the formÂk h(y)=PX j=1»k j'j(y);(3.12)where»k j are constants to be determined.LetÃ2W h(Y)so thatÃhas the unique representationÃ=PX i=1´i'i:(3.13)We can now formulate the discrete analogue to the problem(3.9):FindÂk h2W h(Y)such thata Y(Âk h;Ã)=a Y(y k;Ã)=L k Y(Ã)for allÃ2W h(Y):(3.14) Because(3.14)must be valid for everyÃ2W h(Y);(3.14)will be valid for Ã='i(y);j=1;2;::;P:This is mainly the idea behind the Galerkin method. WithÃ='i(y);(3.14)becomesPX j=1A ij»k j=b k i;i=1;2;:::;P;(3.15)or in matrix formA»k=b k;(3.16) whereA ij=a Y('j;'i)(3.17) andb k i=L k Y('i):(3.18)14In order to prove existence and uniqueness of the solutions of(3.16),we will now state some lemmas and theorems.For the matrix(A ij)the following holds:Lemma2The reduced matrix obtained from the sti¤ness matrix A by re-moving row m and column m for a…xed m is positive de…nite.Proof:From the representation(3.13),we havea Y(Ã;Ã)=a Y(PX i=1´i'i;P X j=1´j'j)=P X i=1P X j=1´i a Y('i;'j)´j=´¢A´;where the dot denotes the usual scalar product in R P:But we also have, recalling(2.4),that a ij is positive de…nite,and hence´¢A´=a Y(Ã;Ã)=Z Y a ij@Ã@y j@Ã@y i dy¸®Z Y jrÃj2dy¸0with equality if and only if rÃ=0;that is,ifÃ(y)=PX i=1´i'i(y)=C´,C´constant:By the de…nition of'i;this holds only if´1=´2=:::=´P=C´:Now…x´1=0:ThenPX i=1P X j=1´i a Y('i;'j)´j=P X i=2P X j=2´i a Y('i;'j)´j=0if and only if´2=´3=:::=´P=0:Hence the reduced matrix obtained by removing row1and column1is positive de…nite.The above procedure can be done on any row and column m so the lemma follows.We now need the following de…nition and lemma:15De…nition3A positive cone in a real linear space is a subset M with the property that if u2M;then®u2M for all real numbers®¸0:Lemma4(Poincaréinequality version1):Letbe an open,bounded and connected set in R n with the Lipschitz property and let M be a closed positive cone of H1(),not containing the constant functions=0:ThenZj u j2dx·C Z@u@x i@u@x i dx=C Zjr u j2dx(3.19) for all u2M and some constant C=C():Proof:See e.g.Alt[1],p.170-171.If we assume that#is not a nonzero constant function,we can state the following inequalities:Lemma5The following statements holds true for the bilinear form a Y(#;³) and the linear functional L k(#)if#is not a nonzero constant function:Y1.a Y(#;³)is coercive,that is,there is a constant¯>0such that,8#2W(Y):(3.20)a Y(#;#)¸¯k#k2W(Y)2.L k Y(#)is bounded,that is,there is a constant¤>0such that¯¯¯L k Y(#)¯¯¯·¤k#k W(Y);8#2W(Y):(3.21)Proof:1.First we note that W(Y)is a closed positive cone of H1(); then we use the Poincaréinequality together with(2.4)and geta Y(#;#)=Z Y a ij(y)@#@y j@#@y i dy¸Z Y®@#@y i@#@y i dy¸¸®1+C Z Y j#j2+jr#j2dy=¯k#k2W(Y);where¯=®:2.We use Schwartz’inequality on(3.10)and get¯¯¯L k Y(#¯¯¯=¯¯¯¯¯Z Y a ik@#i dy¯¯¯¯¯·k a ik k L1(Y)Z Y¯¯¯¯¯@#i¯¯¯¯¯dy··k a ik k L1(Y)°°°°°@#i°°°°°L2(Y)k1k L2(Y)·¤k#k W(Y);where¤=k a ik k L1(Y)j Y j:16Theorem 6There exists a unique solution »k 2R P to the problem (3.15)if we …x for instance »k 1=0:This means that there exists a unique solution Âkh 2W h (Y )to (3.14).Moreover,the following stability estimate holds:°°°Âk h °°°W (Y )·¤(3.22)Proof:Lemma 2says that A is positive de…nite if we for instance …x »k 1=0.Hence A is non-singular,which proves existence and uniqueness.Fixing one »k i =0for some number i means that the only possible constantfunction Âk h is Âk h =0:The stability estimate follows by choosing Ã=Âkh in (3.14)which gives,using (3.20)and (3.21),¯°°°Âk h °°°2W (Y )·a Y (Âk h ;Âkh )=L k Y (Âkh )·¤°°°Âk h °°°W (Y );(no summation over k ),from which (3.22)follows upon division by ¯°°°Âk h °°°W (Y )=0:We can now compute the approximate homogenized coe¢cients by insert (3.12)in (3.1)to getq ik =1Z Y (a ik ¡a ij P X l =1»k l@'lj )dy ==1ZYa ik dy ¡1P Xl =1»klZYa ij@'lj dy:(3.23)We now note thata Y ('l ;y i )=Z Ya mj @'l @y j @y i @y mdy =ZYa ij@'l@y jdy;(3.24)because@y i m=±im and henceq ik =1j Y jZYa ik dy ¡1j Y j P Xl =1a Y ('l ;y i )»k l :(3.25)If a ik is symmetric,(3.10)and (3.18)givesZYa ij @'l @y j dy =ZYa ji@'l @y jdy =L i Y ('l )=b il (3.26)so thatq ik =1ZYa ik dy ¡1P X l =1b i l »kl :(3.27)17Chapter4Existence and uniqueness of weak solutions for elliptic problemsIn this section,we will prove existence and uniqueness of weak solutions of the boundary value problems considered in the previous sections.Moreover, we will prove Lemma1and some properties of the homogenized coe¢cients. To prove this,we will need another version of the Poincaréinequality and the Lax-Milgram lemma:De…nition7Let X be a real Hilbert space.A mappinga(¢;¢):X£X!Ris called a bilinear form on X if1.a(x1+x2;y)=a(x1;y)+a(x2;y)for all x1;x2;y2X;2.a(®x;y)=®a(x;y)for all x;y2X and®2R;3.a(x;y1+y2)=a(x;y1)+a(x;y2)for all x;y1;y22X;4.a(x;¯y)=¯a(x;y)for all x;y2X and¯2R:De…nition8Let a(¢;¢)be a bilinear form on the real Hilbert space X:We say that a(¢;¢)is181.bounded if there exists a constant°>0such that j a(x;y)j·°k x kX k y k X for all x;y2X:2.symmetric if a(x;y)=a(y;x)for all x;y2X:3.coercive if there exists a constant®>0such that a(x;x)¸®k x k2X forall x2X:Lemma9(Poincaréinequality version2):Letbe an open and bounded set in R n which has the Lipschitz property.Thenk u k2H1()=R·C Z@u@x i@u@x i dx(4.1) for each u2H1()=R and some constant C=C():Proof:See e.g.Necas[14].Lemma10(Lax-Milgram):Let X be a real Hilbert space.If the bilinear form a(¢;¢):X£X!R is bounded and coercive and if,in addition,the linear functional f:X!R is bounded,then there exists a unique element x2X such that a(x;y)=(f;y)for all y2X:Proof:See e.g.Alt[1],p.118-119.We recall equations(2.1)and(2.2),that is,¡@iÃa ij(x)@uj!=f(x)in;(4.2)u=0on¡1and a ij(x)@u@x jn i=g on¡2;(4.3)where¡1[¡2=@and¡1\¡2=;:We assume that²a ij2L1();²a ij is positive de…nite,that is,a ij(y)»i»j¸®»i»i almost everywhere for some constant®>0and all»2R n;²f2L2()and g2L2(¡2):19We de…ne V=f v2H1():v j¡1=0g and note that H10()½V½H1():Furthermore,we de…ne the bilinear forma(u;v)=Za ij(x)@u@x j@v@x i dx(4.4) and the linear functionalL(v)=Zfvdx+Z¡2gvdS:(4.5)The weak formulation of(2.1)and(2.2)takes the following form: Find u2V such thata(u;v)=L(v)for all v2V(4.6) This follows by multiplying each side of(2.1)by a function v2V;integrating overand applying Gauss’theorem on the left hand side.We have the following theorem:Theorem11Letbe an open,bounded and connected set in R n with the Lipschitz property.Then1.If¡1has strictly positive surface measure,then there exists a uniquesolution u2V of equation(4.6).2.If¡1has surface measure zero,then there exists a solution u2V ofequation(4.6).This solution is unique up to an additive constant.Proof:(i):V is a closed subspace of H1()by the trace theorem(see e.g.Alt[1],p.190).In particular,V is a closed positive cone in H1()and the boundary condition implies that V contains no other constant functions than u´0:Thus Lemma4and the positiveness of a ij yield®1k v k2V·a(v;v)for some positive constant®1:Furthermore,the bilinear form is bounded, since,by Schwarz inequality,j a(u;v)j·k a ij k L1(°°°°°@u@x i°°°°°2L2)1(°°°°°@u@x j°°°°°2L2)1·C0k u k V k v k V20。

N Soliton Solutions to The Bogoyavlenskii-Schiff Equation and A Quest for The Soliton Solut

(5)
3 We have, in general, an exact solution τN whichN = 1 +
n=1 N Cn
ηi1 ···in exp(λi1 + · · · + λin ), ω j = − p3 j,
(6) (7) (8) (9)
equation (2) is transformed into the bilinear form
3 Dx (Dt + Dx )τ · τ = 0,
(4)
where the Hirota’s derivative D operating on f · g is deﬁned by
n Dz f (z ) · g (z ) ≡ (∂z1 − ∂z2 )n f (z1 )g (z2 ) |z1 =z2 =z .
Short title: Soliton Solutions to The BS Equation February 9, 2008
† fpc30017@bkc.ritsumei.ac.jp ‡ sph20063@bkc.ritsumei.ac.jp § sasa@sugar.tokai.jaeri.go.jp fukuyama@bkc.ritsumei.ac.jp
N Soliton Solutions to The Bogoyavlenskii-Schiﬀ Equation and A Quest for The Soliton Solution in (3 + 1) Dimensions
arXiv:solv-int/9801003v1 3 Jan 1998
Here Φ(u) has the same form as that in equation (1) with argument x. Using the potential u ≡ φx , this equation reads φxt + φxxxz + 4φx φxz + 2φxx φz = 0. (11)

中英文力学对准

一般力学类：分析力学 analytical mechanics拉格朗日乘子 Lagrange multiplier拉格朗日[量] Lagrangian拉格朗日括号 Lagrange bracket循环坐标 cyclic coordinate循环积分 cyclic integral哈密顿[量] Hamiltonian哈密顿函数 Hamiltonian function正则方程 canonical equation正则摄动 canonical perturbation正则变换 canonical transformation正则变量 canonical variable哈密顿原理 Hamilton principle作用量积分 action integral哈密顿-雅可比方程 Hamilton-Jacobi equation作用--角度变量 action-angle variables阿佩尔方程 Appell equation劳斯方程 Routh equation拉格朗日函数 Lagrangian function诺特定理 Noether theorem泊松括号 poisson bracket边界积分法 boundary integral method并矢 dyad运动稳定性 stability of motion轨道稳定性 orbital stability李雅普诺夫函数 Lyapunov function渐近稳定性 asymptotic stability结构稳定性 structural stability久期不稳定性 secular instability弗洛凯定理 Floquet theorem倾覆力矩 capsizing moment自由振动 free vibration固有振动 natural vibration暂态 transient state环境振动 ambient vibration反共振 anti-resonance衰减 attenuation库仑阻尼 Coulomb damping同相分量 in-phase component非同相分量 out-of -phase component超调量 overshoot 参量[激励]振动 parametric vibration模糊振动 fuzzy vibration临界转速 critical speed of rotation阻尼器 damper半峰宽度 half-peak width集总参量系统 lumped parameter system 相平面法 phase plane method相轨迹 phase trajectory等倾线法 isocline method跳跃现象 jump phenomenon负阻尼 negative damping达芬方程 Duffing equation希尔方程 Hill equationKBM方法 KBM method, Krylov-Bogoliu- bov-Mitropol'skii method马蒂厄方程 Mathieu equation平均法 averaging method组合音调 combination tone解谐 detuning耗散函数 dissipative function硬激励 hard excitation硬弹簧 hard spring, hardening spring谐波平衡法harmonic balance method久期项 secular term自激振动 self-excited vibration分界线 separatrix亚谐波 subharmonic软弹簧 soft spring ,softening spring软激励 soft excitation邓克利公式 Dunkerley formula瑞利定理 Rayleigh theorem分布参量系统 distributed parameter system优势频率 dominant frequency模态分析 modal analysis固有模态natural mode of vibration同步 synchronization超谐波 ultraharmonic范德波尔方程 van der pol equation频谱 frequency spectrum基频 fundamental frequencyWKB方法 WKB methodWKB方法Wentzel-Kramers-Brillouin method缓冲器 buffer风激振动 aeolian vibration嗡鸣 buzz倒谱cepstrum颤动 chatter蛇行 hunting阻抗匹配 impedance matching机械导纳 mechanical admittance机械效率 mechanical efficiency机械阻抗 mechanical impedance随机振动 stochastic vibration, random vibration隔振 vibration isolation减振 vibration reduction应力过冲 stress overshoot喘振surge摆振shimmy起伏运动 phugoid motion起伏振荡 phugoid oscillation驰振 galloping陀螺动力学 gyrodynamics陀螺摆 gyropendulum陀螺平台 gyroplatform陀螺力矩 gyroscoopic torque陀螺稳定器 gyrostabilizer陀螺体 gyrostat惯性导航 inertial guidance 姿态角 attitude angle方位角 azimuthal angle舒勒周期 Schuler period机器人动力学 robot dynamics多体系统 multibody system多刚体系统 multi-rigid-body system机动性 maneuverability凯恩方法Kane method转子[系统]动力学 rotor dynamics转子[一支承一基础]系统 rotor-support- foundation system静平衡 static balancing动平衡 dynamic balancing静不平衡 static unbalance动不平衡 dynamic unbalance现场平衡 field balancing不平衡 unbalance不平衡量 unbalance互耦力 cross force挠性转子 flexible rotor分频进动 fractional frequency precession半频进动half frequency precession油膜振荡 oil whip转子临界转速 rotor critical speed自动定心 self-alignment亚临界转速 subcritical speed涡动 whirl固体力学类：弹性力学 elasticity弹性理论 theory of elasticity均匀应力状态 homogeneous state of stress 应力不变量 stress invariant应变不变量 strain invariant应变椭球 strain ellipsoid均匀应变状态 homogeneous state of strain应变协调方程 equation of strain compatibility拉梅常量 Lame constants各向同性弹性 isotropic elasticity旋转圆盘 rotating circular disk 楔wedge开尔文问题 Kelvin problem布西内斯克问题 Boussinesq problem艾里应力函数 Airy stress function克罗索夫--穆斯赫利什维利法 Kolosoff- Muskhelishvili method基尔霍夫假设 Kirchhoff hypothesis板 Plate矩形板 Rectangular plate圆板 Circular plate环板 Annular plate波纹板 Corrugated plate加劲板 Stiffened plate,reinforcedPlate中厚板 Plate of moderate thickness弯[曲]应力函数 Stress function of bending 壳Shell扁壳 Shallow shell旋转壳 Revolutionary shell球壳 Spherical shell[圆]柱壳 Cylindrical shell锥壳Conical shell环壳 Toroidal shell封闭壳 Closed shell波纹壳 Corrugated shell扭[转]应力函数 Stress function of torsion 翘曲函数 Warping function半逆解法 semi-inverse method瑞利--里茨法 Rayleigh-Ritz method松弛法 Relaxation method莱维法 Levy method松弛 Relaxation量纲分析 Dimensional analysis自相似[性] self-similarity影响面 Influence surface接触应力 Contact stress赫兹理论 Hertz theory协调接触 Conforming contact滑动接触 Sliding contact滚动接触 Rolling contact压入 Indentation各向异性弹性 Anisotropic elasticity颗粒材料 Granular material散体力学 Mechanics of granular media热弹性 Thermoelasticity超弹性 Hyperelasticity粘弹性 Viscoelasticity对应原理 Correspondence principle褶皱Wrinkle塑性全量理论 Total theory of plasticity滑动 Sliding微滑Microslip粗糙度 Roughness非线性弹性 Nonlinear elasticity大挠度 Large deflection突弹跳变 snap-through有限变形 Finite deformation 格林应变 Green strain阿尔曼西应变 Almansi strain弹性动力学 Dynamic elasticity运动方程 Equation of motion准静态的Quasi-static气动弹性 Aeroelasticity水弹性 Hydroelasticity颤振Flutter弹性波Elastic wave简单波Simple wave柱面波 Cylindrical wave水平剪切波 Horizontal shear wave竖直剪切波Vertical shear wave体波 body wave无旋波 Irrotational wave畸变波 Distortion wave膨胀波 Dilatation wave瑞利波 Rayleigh wave等容波 Equivoluminal wave勒夫波Love wave界面波 Interfacial wave边缘效应 edge effect塑性力学 Plasticity可成形性 Formability金属成形 Metal forming耐撞性 Crashworthiness结构抗撞毁性 Structural crashworthiness 拉拔Drawing破坏机构 Collapse mechanism回弹 Springback挤压 Extrusion冲压 Stamping穿透Perforation层裂Spalling塑性理论 Theory of plasticity安定[性]理论 Shake-down theory运动安定定理 kinematic shake-down theorem静力安定定理 Static shake-down theorem 率相关理论 rate dependent theorem载荷因子load factor加载准则 Loading criterion加载函数 Loading function加载面 Loading surface塑性加载 Plastic loading塑性加载波 Plastic loading wave简单加载 Simple loading比例加载 Proportional loading卸载 Unloading卸载波 Unloading wave冲击载荷 Impulsive load阶跃载荷step load脉冲载荷 pulse load极限载荷 limit load中性变载 nentral loading拉抻失稳 instability in tension加速度波 acceleration wave本构方程 constitutive equation完全解 complete solution名义应力 nominal stress过应力 over-stress真应力 true stress等效应力 equivalent stress流动应力 flow stress应力间断 stress discontinuity应力空间 stress space主应力空间 principal stress space静水应力状态hydrostatic state of stress对数应变 logarithmic strain工程应变 engineering strain等效应变 equivalent strain应变局部化 strain localization应变率 strain rate应变率敏感性 strain rate sensitivity应变空间 strain space有限应变 finite strain塑性应变增量 plastic strain increment 累积塑性应变 accumulated plastic strain 永久变形 permanent deformation内变量 internal variable应变软化 strain-softening理想刚塑性材料 rigid-perfectly plastic Material刚塑性材料 rigid-plastic material理想塑性材料 perfectl plastic material 材料稳定性stability of material应变偏张量deviatoric tensor of strain应力偏张量deviatori tensor of stress 应变球张量spherical tensor of strain应力球张量spherical tensor of stress路径相关性 path-dependency线性强化 linear strain-hardening应变强化 strain-hardening随动强化 kinematic hardening各向同性强化 isotropic hardening强化模量 strain-hardening modulus幂强化 power hardening塑性极限弯矩 plastic limit bending Moment塑性极限扭矩 plastic limit torque弹塑性弯曲 elastic-plastic bending弹塑性交界面 elastic-plastic interface弹塑性扭转 elastic-plastic torsion粘塑性 Viscoplasticity非弹性 Inelasticity理想弹塑性材料 elastic-perfectly plastic Material极限分析 limit analysis极限设计 limit design极限面limit surface上限定理 upper bound theorem上屈服点upper yield point下限定理 lower bound theorem下屈服点 lower yield point界限定理 bound theorem初始屈服面initial yield surface后继屈服面 subsequent yield surface屈服面[的]外凸性 convexity of yield surface截面形状因子 shape factor of cross-section 沙堆比拟 sand heap analogy屈服Yield屈服条件 yield condition屈服准则 yield criterion屈服函数 yield function屈服面 yield surface塑性势 plastic potential能量吸收装置 energy absorbing device能量耗散率 energy absorbing device塑性动力学 dynamic plasticity塑性动力屈曲 dynamic plastic buckling塑性动力响应 dynamic plastic response塑性波 plastic wave运动容许场 kinematically admissible Field静力容许场 statically admissibleField流动法则 flow rule速度间断 velocity discontinuity滑移线 slip-lines滑移线场 slip-lines field移行塑性铰 travelling plastic hinge塑性增量理论 incremental theory ofPlasticity米泽斯屈服准则 Mises yield criterion普朗特--罗伊斯关系 prandtl- Reuss relation特雷斯卡屈服准则 Tresca yield criterion洛德应力参数 Lode stress parameter莱维--米泽斯关系 Levy-Mises relation亨基应力方程 Hencky stress equation赫艾--韦斯特加德应力空间Haigh-Westergaard stress space洛德应变参数 Lode strain parameter德鲁克公设 Drucker postulate盖林格速度方程Geiringer velocity Equation结构力学 structural mechanics结构分析 structural analysis结构动力学 structural dynamics拱 Arch三铰拱 three-hinged arch抛物线拱 parabolic arch圆拱 circular arch穹顶Dome空间结构 space structure空间桁架 space truss雪载[荷] snow load风载[荷] wind load土压力 earth pressure地震载荷 earthquake loading弹簧支座 spring support支座位移 support displacement支座沉降 support settlement超静定次数 degree of indeterminacy机动分析 kinematic analysis 结点法 method of joints截面法 method of sections结点力 joint forces共轭位移 conjugate displacement影响线 influence line三弯矩方程 three-moment equation单位虚力 unit virtual force刚度系数 stiffness coefficient柔度系数 flexibility coefficient力矩分配 moment distribution力矩分配法moment distribution method力矩再分配 moment redistribution分配系数 distribution factor矩阵位移法matri displacement method单元刚度矩阵 element stiffness matrix单元应变矩阵 element strain matrix总体坐标 global coordinates贝蒂定理 Betti theorem高斯--若尔当消去法 Gauss-Jordan elimination Method屈曲模态 buckling mode复合材料力学 mechanics of composites 复合材料composite material纤维复合材料 fibrous composite单向复合材料 unidirectional composite泡沫复合材料foamed composite颗粒复合材料 particulate composite层板Laminate夹层板 sandwich panel正交层板 cross-ply laminate斜交层板 angle-ply laminate层片Ply多胞固体 cellular solid膨胀 Expansion压实Debulk劣化 Degradation脱层 Delamination脱粘 Debond纤维应力 fiber stress层应力 ply stress层应变ply strain层间应力 interlaminar stress比强度 specific strength强度折减系数 strength reduction factor强度应力比 strength -stress ratio横向剪切模量 transverse shear modulus 横观各向同性 transverse isotropy正交各向异 Orthotropy剪滞分析 shear lag analysis短纤维 chopped fiber长纤维 continuous fiber纤维方向 fiber direction纤维断裂 fiber break纤维拔脱 fiber pull-out纤维增强 fiber reinforcement致密化 Densification最小重量设计 optimum weight design网格分析法 netting analysis混合律 rule of mixture失效准则 failure criterion蔡--吴失效准则 Tsai-W u failure criterion 达格代尔模型 Dugdale model断裂力学 fracture mechanics概率断裂力学 probabilistic fracture Mechanics格里菲思理论 Griffith theory线弹性断裂力学 linear elastic fracturemechanics, LEFM弹塑性断裂力学 elastic-plastic fracture mecha-nics, EPFM断裂 Fracture脆性断裂 brittle fracture解理断裂 cleavage fracture蠕变断裂 creep fracture延性断裂 ductile fracture晶间断裂 inter-granular fracture准解理断裂 quasi-cleavage fracture穿晶断裂 trans-granular fracture裂纹Crack裂缝Flaw缺陷Defect割缝Slit微裂纹Microcrack折裂Kink椭圆裂纹 elliptical crack深埋裂纹 embedded crack[钱]币状裂纹 penny-shape crack预制裂纹 Precrack 短裂纹 short crack表面裂纹 surface crack裂纹钝化 crack blunting裂纹分叉 crack branching裂纹闭合 crack closure裂纹前缘 crack front裂纹嘴 crack mouth裂纹张开角crack opening angle,COA裂纹张开位移 crack opening displacement, COD裂纹阻力 crack resistance裂纹面 crack surface裂纹尖端 crack tip裂尖张角 crack tip opening angle,CTOA裂尖张开位移 crack tip openingdisplacement, CTOD裂尖奇异场crack tip singularity Field裂纹扩展速率 crack growth rate稳定裂纹扩展 stable crack growth定常裂纹扩展 steady crack growth亚临界裂纹扩展 subcritical crack growth 裂纹[扩展]减速 crack retardation止裂crack arrest止裂韧度 arrest toughness断裂类型 fracture mode滑开型 sliding mode张开型 opening mode撕开型 tearing mode复合型 mixed mode撕裂 Tearing撕裂模量 tearing modulus断裂准则 fracture criterionJ积分 J-integralJ阻力曲线 J-resistance curve断裂韧度 fracture toughness应力强度因子 stress intensity factorHRR场 Hutchinson-Rice-Rosengren Field守恒积分 conservation integral有效应力张量 effective stress tensor应变能密度strain energy density能量释放率 energy release rate内聚区 cohesive zone塑性区 plastic zone张拉区 stretched zone热影响区heat affected zone, HAZ延脆转变温度 brittle-ductile transitiontemperature剪切带shear band剪切唇shear lip无损检测 non-destructive inspection双边缺口试件double edge notchedspecimen, DEN specimen单边缺口试件 single edge notchedspecimen, SEN specimen三点弯曲试件 three point bendingspecimen, TPB specimen中心裂纹拉伸试件 center cracked tension specimen, CCT specimen中心裂纹板试件 center cracked panelspecimen, CCP specimen紧凑拉伸试件 compact tension specimen, CT specimen大范围屈服large scale yielding小范围攻屈服 small scale yielding韦布尔分布 Weibull distribution帕里斯公式 paris formula空穴化 Cavitation应力腐蚀 stress corrosion概率风险判定 probabilistic riskassessment, PRA损伤力学 damage mechanics损伤Damage连续介质损伤力学 continuum damage mechanics细观损伤力学 microscopic damage mechanics累积损伤 accumulated damage脆性损伤 brittle damage延性损伤 ductile damage宏观损伤 macroscopic damage细观损伤 microscopic damage微观损伤 microscopic damage损伤准则 damage criterion损伤演化方程 damage evolution equation 损伤软化 damage softening损伤强化 damage strengthening 损伤张量 damage tensor损伤阈值 damage threshold损伤变量 damage variable损伤矢量 damage vector损伤区 damage zone疲劳Fatigue低周疲劳 low cycle fatigue应力疲劳 stress fatigue随机疲劳 random fatigue蠕变疲劳 creep fatigue腐蚀疲劳 corrosion fatigue疲劳损伤 fatigue damage疲劳失效 fatigue failure疲劳断裂 fatigue fracture疲劳裂纹 fatigue crack疲劳寿命 fatigue life疲劳破坏 fatigue rupture疲劳强度 fatigue strength疲劳辉纹 fatigue striations疲劳阈值 fatigue threshold交变载荷 alternating load交变应力 alternating stress应力幅值 stress amplitude应变疲劳 strain fatigue应力循环 stress cycle应力比 stress ratio安全寿命 safe life过载效应 overloading effect循环硬化 cyclic hardening循环软化 cyclic softening环境效应 environmental effect裂纹片crack gage裂纹扩展 crack growth, crack Propagation裂纹萌生 crack initiation循环比 cycle ratio实验应力分析 experimental stressAnalysis工作[应变]片 active[strain] gage基底材料 backing material应力计stress gage零[点]飘移zero shift, zero drift应变测量 strain measurement应变计strain gage应变指示器 strain indicator应变花 strain rosette应变灵敏度 strain sensitivity机械式应变仪 mechanical strain gage 直角应变花 rectangular rosette引伸仪 Extensometer应变遥测 telemetering of strain横向灵敏系数 transverse gage factor 横向灵敏度 transverse sensitivity焊接式应变计 weldable strain gage 平衡电桥 balanced bridge粘贴式应变计 bonded strain gage粘贴箔式应变计bonded foiled gage粘贴丝式应变计 bonded wire gage 桥路平衡 bridge balancing电容应变计 capacitance strain gage 补偿片 compensation technique补偿技术 compensation technique基准电桥 reference bridge电阻应变计 resistance strain gage温度自补偿应变计 self-temperature compensating gage半导体应变计 semiconductor strain Gage集流器slip ring应变放大镜 strain amplifier疲劳寿命计 fatigue life gage电感应变计 inductance [strain] gage 光[测]力学 Photomechanics光弹性 Photoelasticity光塑性 Photoplasticity杨氏条纹 Young fringe双折射效应 birefrigent effect等位移线 contour of equalDisplacement暗条纹 dark fringe条纹倍增 fringe multiplication干涉条纹 interference fringe等差线 Isochromatic等倾线 Isoclinic等和线 isopachic应力光学定律 stress- optic law主应力迹线 Isostatic亮条纹 light fringe 光程差optical path difference热光弹性 photo-thermo -elasticity光弹性贴片法 photoelastic coating Method光弹性夹片法 photoelastic sandwich Method动态光弹性 dynamic photo-elasticity空间滤波 spatial filtering空间频率 spatial frequency起偏镜 Polarizer反射式光弹性仪 reflection polariscope残余双折射效应 residual birefringent Effect应变条纹值 strain fringe value应变光学灵敏度 strain-optic sensitivity 应力冻结效应 stress freezing effect应力条纹值 stress fringe value应力光图 stress-optic pattern暂时双折射效应 temporary birefringent Effect脉冲全息法 pulsed holography透射式光弹性仪 transmission polariscope 实时全息干涉法 real-time holographicinterfero - metry网格法 grid method全息光弹性法 holo-photoelasticity全息图Hologram全息照相 Holograph全息干涉法 holographic interferometry 全息云纹法 holographic moire technique 全息术 Holography全场分析法 whole-field analysis散斑干涉法 speckle interferometry散斑Speckle错位散斑干涉法 speckle-shearinginterferometry, shearography散斑图Specklegram白光散斑法white-light speckle method云纹干涉法 moire interferometry[叠栅]云纹 moire fringe[叠栅]云纹法 moire method云纹图 moire pattern离面云纹法 off-plane moire method参考栅 reference grating试件栅 specimen grating分析栅 analyzer grating面内云纹法 in-plane moire method脆性涂层法 brittle-coating method条带法 strip coating method坐标变换 transformation ofCoordinates计算结构力学 computational structuralmecha-nics加权残量法weighted residual method有限差分法 finite difference method有限[单]元法 finite element method配点法 point collocation里茨法 Ritz method广义变分原理 generalized variational Principle最小二乘法 least square method胡[海昌]一鹫津原理 Hu-Washizu principle 赫林格-赖斯纳原理 Hellinger-Reissner Principle修正变分原理 modified variational Principle约束变分原理 constrained variational Principle混合法 mixed method杂交法 hybrid method边界解法boundary solution method有限条法 finite strip method半解析法 semi-analytical method协调元 conforming element非协调元 non-conforming element混合元 mixed element杂交元 hybrid element边界元 boundary element强迫边界条件 forced boundary condition 自然边界条件 natural boundary condition 离散化 Discretization离散系统 discrete system连续问题 continuous problem广义位移 generalized displacement广义载荷 generalized load广义应变 generalized strain广义应力 generalized stress界面变量 interface variable 节点 node, nodal point[单]元 Element角节点 corner node边节点 mid-side node内节点 internal node无节点变量 nodeless variable杆元 bar element桁架杆元 truss element梁元 beam element二维元 two-dimensional element一维元 one-dimensional element三维元 three-dimensional element轴对称元 axisymmetric element板元 plate element壳元 shell element厚板元 thick plate element三角形元 triangular element四边形元 quadrilateral element四面体元 tetrahedral element曲线元 curved element二次元 quadratic element线性元 linear element三次元 cubic element四次元 quartic element等参[数]元 isoparametric element超参数元 super-parametric element亚参数元 sub-parametric element节点数可变元 variable-number-node element拉格朗日元 Lagrange element拉格朗日族 Lagrange family巧凑边点元 serendipity element巧凑边点族 serendipity family无限元 infinite element单元分析 element analysis单元特性 element characteristics刚度矩阵 stiffness matrix几何矩阵 geometric matrix等效节点力 equivalent nodal force节点位移 nodal displacement节点载荷 nodal load位移矢量 displacement vector载荷矢量 load vector质量矩阵 mass matrix集总质量矩阵 lumped mass matrix相容质量矩阵 consistent mass matrix阻尼矩阵 damping matrix瑞利阻尼 Rayleigh damping刚度矩阵的组集 assembly of stiffnessMatrices载荷矢量的组集 consistent mass matrix质量矩阵的组集 assembly of mass matrices 单元的组集 assembly of elements局部坐标系 local coordinate system局部坐标 local coordinate面积坐标 area coordinates体积坐标 volume coordinates曲线坐标 curvilinear coordinates静凝聚 static condensation合同变换 contragradient transformation形状函数 shape function试探函数 trial function检验函数test function权函数 weight function样条函数 spline function代用函数 substitute function降阶积分 reduced integration零能模式 zero-energy modeP收敛 p-convergenceH收敛 h-convergence掺混插值 blended interpolation等参数映射 isoparametric mapping双线性插值 bilinear interpolation小块检验 patch test非协调模式 incompatible mode 节点号 node number单元号 element number带宽 band width带状矩阵 banded matrix变带状矩阵 profile matrix带宽最小化minimization of band width波前法 frontal method子空间迭代法 subspace iteration method 行列式搜索法determinant search method逐步法 step-by-step method纽马克法Newmark威尔逊法 Wilson拟牛顿法 quasi-Newton method牛顿-拉弗森法 Newton-Raphson method 增量法 incremental method初应变 initial strain初应力 initial stress切线刚度矩阵 tangent stiffness matrix割线刚度矩阵 secant stiffness matrix模态叠加法mode superposition method平衡迭代 equilibrium iteration子结构 Substructure子结构法 substructure technique超单元 super-element网格生成 mesh generation结构分析程序 structural analysis program 前处理 pre-processing后处理 post-processing网格细化 mesh refinement应力光顺 stress smoothing组合结构 composite structure流体动力学类：流体动力学 fluid dynamics连续介质力学 mechanics of continuous media介质medium流体质点 fluid particle无粘性流体 nonviscous fluid, inviscid fluid连续介质假设 continuous medium hypothesis流体运动学 fluid kinematics水静力学 hydrostatics 液体静力学 hydrostatics支配方程 governing equation伯努利方程 Bernoulli equation伯努利定理 Bernonlli theorem毕奥-萨伐尔定律 Biot-Savart law欧拉方程Euler equation亥姆霍兹定理 Helmholtz theorem开尔文定理 Kelvin theorem涡片 vortex sheet库塔-茹可夫斯基条件 Kutta-Zhoukowskicondition布拉休斯解 Blasius solution达朗贝尔佯廖 d'Alembert paradox 雷诺数 Reynolds number施特鲁哈尔数 Strouhal number随体导数 material derivative不可压缩流体 incompressible fluid 质量守恒 conservation of mass动量守恒 conservation of momentum 能量守恒 conservation of energy动量方程 momentum equation能量方程 energy equation控制体积 control volume液体静压 hydrostatic pressure涡量拟能 enstrophy压差 differential pressure流[动] flow流线stream line流面 stream surface流管stream tube迹线path, path line流场 flow field流态 flow regime流动参量 flow parameter流量 flow rate, flow discharge涡旋 vortex涡量 vorticity涡丝 vortex filament涡线 vortex line涡面 vortex surface涡层 vortex layer涡环 vortex ring涡对 vortex pair涡管 vortex tube涡街 vortex street卡门涡街 Karman vortex street马蹄涡 horseshoe vortex对流涡胞 convective cell卷筒涡胞 roll cell涡 eddy涡粘性 eddy viscosity环流 circulation环量 circulation速度环量 velocity circulation 偶极子 doublet, dipole驻点 stagnation point总压[力] total pressure总压头 total head静压头 static head总焓 total enthalpy能量输运 energy transport速度剖面 velocity profile库埃特流 Couette flow单相流 single phase flow单组份流 single-component flow均匀流 uniform flow非均匀流 nonuniform flow二维流 two-dimensional flow三维流 three-dimensional flow准定常流 quasi-steady flow非定常流unsteady flow, non-steady flow 暂态流transient flow周期流 periodic flow振荡流 oscillatory flow分层流 stratified flow无旋流 irrotational flow有旋流 rotational flow轴对称流 axisymmetric flow不可压缩性 incompressibility不可压缩流[动] incompressible flow 浮体 floating body定倾中心metacenter阻力 drag, resistance减阻 drag reduction表面力 surface force表面张力 surface tension毛细[管]作用 capillarity来流 incoming flow自由流 free stream自由流线 free stream line外流 external flow进口 entrance, inlet出口exit, outlet扰动 disturbance, perturbation分布 distribution传播 propagation色散 dispersion弥散 dispersion附加质量added mass ,associated mass收缩 contraction镜象法 image method无量纲参数 dimensionless parameter几何相似 geometric similarity运动相似 kinematic similarity动力相似[性] dynamic similarity平面流 plane flow势 potential势流 potential flow速度势 velocity potential复势 complex potential复速度 complex velocity流函数 stream function源source汇sink速度[水]头 velocity head拐角流 corner flow空泡流cavity flow超空泡 supercavity超空泡流 supercavity flow空气动力学 aerodynamics低速空气动力学 low-speed aerodynamics 高速空气动力学 high-speed aerodynamics 气动热力学 aerothermodynamics亚声速流[动] subsonic flow跨声速流[动] transonic flow超声速流[动] supersonic flow锥形流 conical flow楔流wedge flow叶栅流 cascade flow非平衡流[动] non-equilibrium flow细长体 slender body细长度 slenderness钝头体 bluff body钝体 blunt body翼型 airfoil翼弦 chord薄翼理论 thin-airfoil theory构型 configuration后缘 trailing edge迎角 angle of attack失速stall脱体激波detached shock wave 波阻wave drag诱导阻力 induced drag诱导速度 induced velocity临界雷诺数critical Reynolds number前缘涡 leading edge vortex附着涡 bound vortex约束涡 confined vortex气动中心 aerodynamic center气动力 aerodynamic force气动噪声 aerodynamic noise气动加热 aerodynamic heating离解 dissociation地面效应 ground effect气体动力学 gas dynamics稀疏波 rarefaction wave热状态方程thermal equation of state喷管Nozzle普朗特-迈耶流 Prandtl-Meyer flow瑞利流 Rayleigh flow可压缩流[动] compressible flow可压缩流体 compressible fluid绝热流 adiabatic flow非绝热流 diabatic flow未扰动流 undisturbed flow等熵流 isentropic flow匀熵流 homoentropic flow兰金-于戈尼奥条件 Rankine-Hugoniot condition状态方程 equation of state量热状态方程 caloric equation of state完全气体 perfect gas拉瓦尔喷管 Laval nozzle马赫角 Mach angle马赫锥 Mach cone马赫线Mach line马赫数Mach number马赫波Mach wave当地马赫数 local Mach number冲击波 shock wave激波 shock wave正激波normal shock wave斜激波oblique shock wave头波 bow wave附体激波 attached shock wave激波阵面 shock front激波层 shock layer压缩波 compression wave反射 reflection折射 refraction散射scattering衍射 diffraction绕射 diffraction出口压力 exit pressure超压[强] over pressure反压 back pressure爆炸 explosion爆轰 detonation缓燃 deflagration水动力学 hydrodynamics液体动力学 hydrodynamics泰勒不稳定性 Taylor instability 盖斯特纳波 Gerstner wave斯托克斯波 Stokes wave瑞利数 Rayleigh number自由面 free surface波速 wave speed, wave velocity 波高 wave height波列wave train波群 wave group波能wave energy表面波 surface wave表面张力波 capillary wave规则波 regular wave不规则波 irregular wave浅水波 shallow water wave深水波deep water wave重力波 gravity wave椭圆余弦波 cnoidal wave潮波tidal wave涌波surge wave破碎波 breaking wave船波ship wave非线性波 nonlinear wave孤立子 soliton水动[力]噪声 hydrodynamic noise 水击 water hammer空化 cavitation空化数 cavitation number 空蚀 cavitation damage超空化流 supercavitating flow水翼 hydrofoil水力学 hydraulics洪水波 flood wave涟漪ripple消能 energy dissipation海洋水动力学 marine hydrodynamics谢齐公式 Chezy formula欧拉数 Euler number弗劳德数 Froude number水力半径 hydraulic radius水力坡度 hvdraulic slope高度水头 elevating head水头损失 head loss水位 water level水跃 hydraulic jump含水层 aquifer排水 drainage排放量 discharge壅水曲线back water curve压[强水]头 pressure head过水断面 flow cross-section明槽流open channel flow孔流 orifice flow无压流 free surface flow有压流 pressure flow缓流 subcritical flow急流 supercritical flow渐变流gradually varied flow急变流 rapidly varied flow临界流 critical flow异重流density current, gravity flow堰流weir flow掺气流 aerated flow含沙流 sediment-laden stream降水曲线 dropdown curve沉积物 sediment, deposit沉[降堆]积 sedimentation, deposition沉降速度 settling velocity流动稳定性 flow stability不稳定性 instability奥尔-索末菲方程 Orr-Sommerfeld equation 涡量方程 vorticity equation泊肃叶流 Poiseuille flow奥辛流 Oseen flow剪切流 shear flow粘性流[动] viscous flow层流 laminar flow分离流 separated flow二次流 secondary flow近场流near field flow远场流 far field flow滞止流 stagnation flow尾流 wake [flow]回流 back flow反流 reverse flow射流 jet自由射流 free jet管流pipe flow, tube flow内流 internal flow拟序结构 coherent structure 猝发过程 bursting process表观粘度 apparent viscosity 运动粘性 kinematic viscosity 动力粘性 dynamic viscosity 泊 poise厘泊 centipoise厘沱 centistoke剪切层 shear layer次层 sublayer流动分离 flow separation层流分离 laminar separation 湍流分离 turbulent separation 分离点 separation point附着点 attachment point再附 reattachment再层流化 relaminarization起动涡starting vortex驻涡 standing vortex涡旋破碎 vortex breakdown 涡旋脱落 vortex shedding压[力]降 pressure drop压差阻力 pressure drag压力能 pressure energy型阻 profile drag滑移速度 slip velocity无滑移条件 non-slip condition 壁剪应力 skin friction, frictional drag壁剪切速度 friction velocity磨擦损失 friction loss磨擦因子 friction factor耗散 dissipation滞后lag相似性解 similar solution局域相似 local similarity气体润滑 gas lubrication液体动力润滑 hydrodynamic lubrication 浆体 slurry泰勒数 Taylor number纳维-斯托克斯方程 Navier-Stokes equation 牛顿流体 Newtonian fluid边界层理论boundary later theory边界层方程boundary layer equation边界层 boundary layer附面层 boundary layer层流边界层laminar boundary layer湍流边界层turbulent boundary layer温度边界层thermal boundary layer边界层转捩boundary layer transition边界层分离boundary layer separation边界层厚度boundary layer thickness位移厚度 displacement thickness动量厚度 momentum thickness能量厚度 energy thickness焓厚度 enthalpy thickness注入 injection吸出suction泰勒涡 Taylor vortex速度亏损律 velocity defect law形状因子 shape factor测速法 anemometry粘度测定法 visco[si] metry流动显示 flow visualization油烟显示 oil smoke visualization孔板流量计 orifice meter频率响应 frequency response油膜显示oil film visualization阴影法 shadow method纹影法 schlieren method烟丝法smoke wire method丝线法 tuft method。

ultra- weak solution意思

ultra- weak solution意思
Ultra-weak solution（超弱解）是一个在数学和物理学领域中使用的术语，通常用于描述偏微分方程的解。

它指的是某些偏微分方程的解在数学上的弱性质的一种极端情况。

在偏微分方程理论中，通常将解分为强解和弱解。

强解是指满足偏微分方程的各种要求，如连续性、可微性等。

而弱解则是对偏微分方程更宽泛的一种解释，通常是在某种广义意义上满足方程，但在数学严格性方面可能较强解要弱。

Ultra-weak solution则更进一步，描述了偏微分方程的解在弱性质方面的极端情况。

它可能指的是某些偏微分方程的解在数学上非常弱，可能甚至不满足传统意义上的弱解的要求。

这些解可能在分布理论（Distribution Theory）或广义函数（Generalized Function）的框架下被定义和处理。

Uniqueness of weak solutions

F. Otto
Institut fur Angewandte Mathematik, Universitat Bonn, D{53115 Bonn, Germany
We consider a nonlinear degenerate parabolic or nonlinear hyperbolic system arising from porous media ow modelling. We prove an L1 contraction principle and uniqueness for weak solutions only requiring the gradient of the solution to be in L2 . Keywords: degenerate parabolic system, uniqueness, porous media ow. AMS classi cation code: 35K65, 35M10, 76S05
(1), (2) can be degenerate with various degrees of degeneracy: Firstly (3) is only an ordinary di erential equation, but this alone does not produce any signi cant problem. Typical examples for the nonlinearities are e.g. the Freundlich isotherm (u) = Aup; A > 0; 0 < p < 1
1 Introduction
In this paper we prove an L1 -contraction principle for weak solutions and thus also the uniqueness of weak solutions for a degenerate parabolic system, modelling solute transport through porous media, where the chemical species undergoes adsorption or exchange processes on the surface of the porous skeleton. Such models are of fundamental importance in soil science or subsurface hydrology, e.g. for risk assessment or remediation design for polluted soils. A macroscopic model reads as follows (see 1] for derivation): Assume that the underlying uid ow is described by the water content and the volumetric ux q, due to the conservation of mass and the incompressibity of the uid ful lling the equation

志坤要求翻译文档-整理版1

志坤要求翻译文档-整理版1The U.S. government has endured several painful rounds of scrutiny as it tries to figure out what went wrong on Sept. 11, 2001. The intelligence community faces radical restructuring;the military has made a sharp pivot to face a new enemy; and a vast new federal agency has blossomed to coordinate homeland security. 试图为了找出2001年9月1号那天所犯的错误，美国政府已经经历了好几轮的精密调查。

情报团体面临着激进的重组；军事方面已经将重点转向面对新的敌人；而且一个新的巨大的联邦机构已经成立，以协助配合国内安全。

But did Septem ber 11 signal a failure of theory on par with the failures of intelligence and policy? 但911事件是否标志着一个理论的失败和标志着情报和决策的失败呢？Familiar theories about how the world works still dominate academic debate.关于世界如何运作类似的理论依然在学术争论中处于主导地位。

Instead of radical change, academia has adjusted exist ing theories to meet new realities.学术界已经调整了现存的理论去适应新的现实，而不是进行彻底的改革。

Has this approach succeeded? Does international relations theory still have something to tell policymakers?采取这样的途径能够成功吗？那些国际关系理论还依然能够为那些策略者提供某些提示吗？Six years ago, political scien tist Stephen M. Walt published a much-cited survey of the field in these pages (“On e World, Many Theories,”Spring1998). 六年前，政治科学家Stephen M. Walt发表了关于这个领域的一个调查。

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