Existence of global weak solution for compressible fluid models with a capillary tensor for

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Existence of Homoclinic Solutions

Advances in Dynamical Systems and ApplicationsISSN0973-5321,V olume5,Number1,pp.75–85(2010)/adsaExistence of Homoclinic Solutions for a Class of Second-Order Differential Equations with Multiple LagsChengjun GuoGuangdong University of TechnologySchool of Applied Mathematics,510006,P.R.Chinaguochj817@Donal O’ReganNational University of IrelandDepartment of Mathematics,Galway,Irelanddonal.oregan@nuigalway.ieRavi P.AgarwalFlorida Institute of TechnologyDepartment of Mathematical SciencesMelbourne,Florida32901,U.S.A.agarwal@AbstractThis paper is concerned with the existence of homoclinic orbits for second-order differential equations with multiple lags.By using Mawhin’s continuationtheorem,a nontrivial homoclinic orbit is obtained as a limit of a certain sequenceof periodic solutions of the equation.AMS Subject Classiﬁcations:34K15,34C25.Keywords:Homoclinic orbit,multiple lags,Mawhin’s continuation theorem.1IntroductionIn recent years several authors studied homoclinic orbits for Hamiltonian systems via critical point theory.In particular second-order systems were considered in[1,2,4–6, Received December4,2009;Accepted December11,2009Communicated by Martin Bohner76Chengjun Guo,Donal O’Regan and Ravi P.Agarwal 12–16,19]andﬁrst-order systems in[3,7–9,11,17,18].In this paper we consider the existence of homoclinic orbits for FDE by using Mawhin’s continuation theorem.In particular we discuss the existence of homoclinic orbits for the equationx (t)+a1(t)x (t)−a2(t)x(t)=g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f(t),(1.1) whereτi(i=1,2,···,n)are constants,a1(t)and a2(t)are real continuous functions deﬁned on R with positive period T,f:R→R is a continuous and bounded function, g(t,x1,x2,···,x n)∈C(R×R×R×···×R,R),g(t,0,0,···,0)=0,and is T-periodic in t.A solution x of(1.1)is said to be homoclinic(to0)if x(t)→0as t→±∞.In addition,if x≡0then x is called a nontrivial homoclinic solution.This paper is largely motivated by the work of Rabinowitz[15]in which the exis-tence of nontrivial homoclinic solutions for the second-order Hamiltonian system¨q+V q(t,q)=0was proved.For the sake of completeness,weﬁrst state Mawhin’s continuation theorem [10].Assume X and Y are two Banach spaces,L:Dom L⊂X→Y is a linear mapping and N:X→Y is a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dim Ker L=codim Im L<+∞and Im L is closed in Y.If L is a Fredholm mapping of index zero,then there exist continuous projections P:X→X and Q:Y→Y such that Im P=Ker L and Im L=Ker Q=Im(I−Q). It follows that L|Dom L∩Ker P:(I−P)X→Im L has an inverse which will be denoted by K P.IfΩis an open and bounded subset of X,the mapping N will be called L-compact onΩif QN(Ω)is bounded and K P(I−Q)N(Ω)is compact.Since Im Q is isomorphic to Ker L,there exists an isomorphism J:Im Q→Ker L.Theorem1.1(Mawhin’s continuation theorem[10]).Let L be a Fredholm mapping of index zero,and let N be L-compact onΩ.Suppose(1)for eachλ∈(0,1)and x∈∂Ω,Lx=λNx;(2)for each x∈∂Ω∩Ker(L),QNx=0and deg(QN,Ω∩Ker(L),0)=0.Then the equation Lx=Nx has at least one solution inΩ∩D(L).2Main ResultNow we make the following assumptions on a1(t),a2(t)and f(t):(H1)0≤m1≤|a1(t)|≤M1;(H2)M2=maxt∈[0,T]a2(t)≥a2(t)≥m2=mint∈[0,T]a2(t)>0;Existence of Homoclinic Solutions77(H3)f:R→R is continuous and bounded,f≡0andR |f(t)|2dt12≤η,whereη>0is a positive constant.Our main result is the following theorem. Theorem2.1.Suppose(H1)–(H3)and assume(H4)|g(t,x1,x2,···,x n)|≤rni=1|x i|and m2−M214−2rn>0.Then system(1.1)possesses a nontrivial homoclinic solution x∈C2(R,R)such that x (t)→0as t→±∞.In order to prove the main theorem we need some preliminaries.For each k∈N, setX k:={x|x∈C1(R,R),x(t+2kT)=x(t),∀t∈R}and x(0)(t)=x(t),deﬁne the norm on X k byx =maxmaxt∈[−kT,kT]|x(t)|,maxt∈[−kT,kT]|x (t)|,and setY k:={y|y∈C(R,R),y(t+2kT)=y(t),∀t∈R}.We deﬁne the norm on Y k as y 0=maxt∈[−kT,kT]|y(t)|.Thus both(X k, · )and(Y k, · 0) are Banach spaces.Remark2.2.If x∈X k,then it follows that x(i)(0)=x(i)(2kT)(i=0,1).In the works of Izydorek and Janczewska[12]and Tanaka[19],a homoclinic so-lution of(1.1)is obtained as a limit,as k→±∞,of a certain sequence of functions x k∈X k.So here we will consider a sequence of systems of functional differential equationsx (t)+a1(t)x (t)−a2(t)x(t)=g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t),(2.1) where for each k∈N,f k:R→R is a2kT-periodic extension of the restriction of f to the interval[−kT,kT]and x k is a2kT-periodic solution of(2.1)obtained via Mawhin’s continuation theorem.Deﬁne the operators L k:X k→Y k and N k:X k→Y k byL k x(t)=x (t),t∈R,(2.2) andN k x(t)=−a1(t)x (t)+a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t),t∈R.(2.3)78Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Clearly,Ker L k={x∈X k:x(t)=c∈R}(2.4)andIm L k=y∈Y k:kT−kTy(t)dt=0(2.5)is closed in Y k.Thus L k is a Fredholm mapping of index zero.Let us deﬁne P k:X k→X k and Q k:Y k→Y k/Im(L k)byP k x(t)=12kTkT−kTx(t)dt=x(0),t∈R,(2.6)for x=x(t)∈X andQ k y(t)=12kTkT−kTy(t)dt,t∈R(2.7)for y=y(t)∈Y k.It is easy to see that Im P k=Ker L k and Im L k=Ker Q k=Im(I k−Q k).It follows that L k|Dom Lk∩Ker P k :(I k−P k)X k→Im L k has an inversewhich will be denoted by K Pk.LetΩk be an open and bounded subset of X k.We can easily see that Q k N k(Ωk)isbounded and K Pk (I k−Q k)N k(Ωk)is compact.Thus the mapping N k is L-compact onΩk.That is,we have the following result.Lemma2.3.Let L k,N k,P k and Q k be deﬁned by(2.2),(2.3),(2.6)and(2.7)respec-tively.Then L k is a Fredholm mapping of index zero and N k is L-compact onΩk,where Ωk is any open and bounded subset of X k.In order to prove our main result,we need the following lemma[15].Lemma2.4(See Remark2.2and[15]).There is a positive constant such that for each k∈N and x∈X k the following inequality holds:max t∈[−kT,kT]|x(t)|≤kT−kT(|x(t)|2+|x (t)|2)dt12.Now,we consider the auxiliary equationx (t)+λ[a1(t)x (t)−a2(t)x(t)]=λ[g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)],(2.8) where0<λ<1.Existence of Homoclinic Solutions79 Lemma2.5.Suppose that the conditions of Theorem2.1are satisﬁed.If x k(t)is a2kT-periodic solution of Eq.(2.8),then there are positive constants D i,i=0,1,which are independent ofλ,such thatx(i)k0≤D i,t∈[−kT,kT],i=0,1.(2.9) Proof.Suppose that x k is a2kT-periodic solution of Eq.(2.8).We have from(2.8)thatkT −kT [xk(t)+λa1(t)xk(t)−λa2(t)x k(t)]x k(t)dt=λkT−kT[g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))+f k(t)]x k(t)dt.(2.10)From(2.10),we havekT −kT {|xk(t)|2+λa2(t)|x k(t)|2}dt=−λkT−kT[g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))+f k(t)−a1(t)xk(t)]x k(t)dt≤λkT−kT |g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))|2dt12×kT−kT |x k(t)|2dt12+λkT−kT|f k(t)|2dt12+M1kT−kT|xk(t)|2dt12×kT−kT |x k(t)|2dt12≤2rnλkT−kT |x k(t)|2dt+λη+M1kT−kT|xk(t)|2dt12 kT−kT|x k(t)|2dt12,soλm2−M214kT−kT|x k(t)|2dt≤λkT−kT|xk(t)|2dt12−M12kT−kT|x k(t)|2dt12 2+λa2(t)−M214kT−kT|x k(t)|2dt+1λ−λkT−kT|xk(t)|2dt ≤2rnλkT−kT|x k(t)|2dt+ηλkT−kT|x k(t)|2dt12,80Chengjun Guo,Donal O’Regan and Ravi P .Agarwalwhich givesm 2−M 214−2rnkT−kT |x k (t )|2dt ≤ηkT−kT|x k (t )|2dt 12.(2.11)From (H 4)and (2.11),there exists a positive constant C 1such thatkT−kT |x k (t )|2dt ≤η2(m 2−M 214−2rn )2=C 1.(2.12)From (2.8),we havekT−kT[x k (t )+λa 1(t )xk (t )−λa 2(t )x k (t )]x k (t )dt=λkT−kT[g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))+f k (t )]x k (t )dt,(2.13)som 1kT−kT|x k (t )|2dt≤kT−kT |a 1(t )||x k (t )|2dt ≤(M 2+2rn )kT−kT |x k (t )|2dt 12+ηkT−kT|x k (t )|2dt12≤[(M 2+2rn )C 1+η]kT−kT|x k (t )|2dt 12,and as a result there exists a positive constant C 2such thatkT−kT|x k (t )|2dt ≤C 2.(2.14)Moreover,for x ∈X k and t,τ∈[−kT,kT ],we have|x (t )|≤x (τ)+ t τx (s )ds .(2.15)Integration of (2.15)over t −12,t +12shows|x (t )|≤ t +12t −12|x (τ)|dτ+t +12t −12t τx (s )dsdτ≤2t +12t −12(|x (τ)|2+|x (t )|2)dτ12.(2.16)Existence of Homoclinic Solutions 81Hence (2.15)and (2.16)implymax t ∈[−kT,kT ]|x (t )|≤kT−kT(|x (t )|2+|x (t )|2)dt 12,x ∈X k ,(2.17)where is given in Lemma 2.4.From Lemma 2.4,(2.12)and (2.14),we havemax t ∈[−kT,kT ]|x k (t )|≤kT−kT(|x k (t )|2+|x k (t ))|2dt 12≤ (C 1+C 2)12=D 0.(2.18)On the other hand,we have from (2.8)thatkT−kT[x k (t )+λa 1(t )x k (t )−λa 2(t )x k (t )]xk (t )dt=λkT−kT[g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))+f k (t )]x k (t )dt,(2.19)so we have kT−kT|x k (t )|2dt≤kT−kT|x k (t )|2dt12M 1 kT−kT|x k (t )|2dt 12+M 2kT−kT|x k (t )|2dt12+ kT−kT |g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))|2dt 12×kT−kT |x k (t )|2dt12+ kT−kT|f k (t )dt12kT−kT|x k (t )|2dt12≤[(2rn +M 2)C 1+M 1 C 2+η]kT−kT|x k (t )|2dt12,and as a result there exists a positive constant C 4such thatkT−kT|x k (t )|2dt ≤[(2rn +M 2) C 1+M 1C 2+η]2=C 4.(2.20)From Lemma 2.4and (2.20),we havemaxt ∈[−kT,kT ]|x k (t )|≤kT−kT(|x k (t )|2+|x k (t )|2)dt 12≤ (C 2+C 4)12=D 1.(2.21)The proof iscomplete.82Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Lemma2.6.Let k∈N.If(H1)–(H4)hold,then the system(2.1)possesses a2kT-periodic solution.Proof.Suppose that x is a2kT-periodic solution of Eq.(2.8).By Lemma2.5,there exist positive constants D i(i=0,1)which are independent ofλsuch that(2.9)is true.Consider any positive constantαk>max0≤i≤1{D i}+ξ,whereξ=maxt∈R|f(t)|.SetΩk:={x∈X k: x <αk}.We know that L k is a Fredholm mapping of index zero and N k is L-compact onΩk (see[2]).RecallKer(L k)={x∈X k:x(t)=c∈R}and the norm on X k isx =maxmaxt∈[−kT,kT]|x(t)|,maxt∈[−kT,kT]|x (t)|.Then we havex=αk or x=−αk for x∈∂Ωk∩Ker(L k).(2.22) From(H4),we have(ifαk is chosen large enough)a2(t)αk+g(t,αk,αk,···,αk)− f k 0>0t∈[−kT,kT](2.23) andx (t)=0,∀x∈∂Ωk∩Ker(L k).(2.24) Finally from(2.3),(2.7)and(2.22)–(2.24),we have(Q k N k x)=12kTkT−kT[−a1(t)x (t)+a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)]dt=12kTkT−kT[a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)]dt=0,∀x∈∂Ωk∩Ker(L k).Then,for any x∈Ker L k∩∂Ωk andη∈[0,1],we havexH(x,η)=−ηx2−x2kT(1−η)kT−kT[−a1(t)x (t)+a2(t)x(t)+f k(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))]dt =0.Existence of Homoclinic Solutions 83Thusdeg {Q k N k ,Ωk ∩Ker(L k ),0}=deg −12kT kT−kT[−a 1(t )x (t )+a 2(t )x (t )+f k (t )+g (t,x (t −τ1),x (t −τ2),···,x (t −τn ))]dt,Ωk ∩Ker(L k ),0}=deg {−x,Ωk ∩Ker(L k ),0}=0.From Lemma 2.5,for any x ∈∂Ωk ∩Dom(L k )and λ∈(0,1)we have L k x =λN k x .By Theorem 1.1,the equation L k x =N k x has at least one solution in Dom(L )∩Ωk .So there exists a 2kT -periodic solution x k of the system (2.1).The proof iscomplete.Lemma 2.7.Let {x k }k ∈N be the sequence given by Lemma 2.6.Then there exists x 0anda subsequence of {x n }n ∈N (again we call it {x n }n ∈N )such that x k →x 0in C 1loc (R ,R )as k →+∞.Proof.By (2.18),(2.21)and the Arzel`a –Ascoli theorem,we obtain that a subsequenceof {x k }k ∈N converges in C 1loc (R ,R )to a solution x 0of (1.1)satisfying∞−∞(|x 0(t )|2+|x 0(t )|2)dt <∞.(2.25)To see this note from (2.1)thatlim k →∞[x k (t )+a 1(t )xk (t )−a 2(t )x k (t )−g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))]=x 0(t )+a 1(t )x0(t )−a 2(t )x 0(t )−g (t,x 0(t −τ1),x 0(t −τ2),···,x 0(t −τn ))=lim k →∞f k (t )=f (t ),so x 0is a solution of (1.1).Also,we have∞−∞[|x 0(t )|2+|x 0(t )|2]dt =limk →∞kT−kT[|x k (t )|2+|x k (t )|2]dt <∞.This shows that (2.25)holds.Lemma 2.8.The function x 0determined by Lemma 2.7is the desired homoclinic solu-tion of (1.1).Proof.The proof will be divided into two steps.Step 1:We prove that x 0(t )→0,as t →±∞.By (2.25),we havelim j →∞|t |≥j[|x 0(t )|2+|x 0(t )|2]dt =0.(2.26)84Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Hence(2.18)and(2.26)shows that our claim holds.Step2:We now show that x 0(t)→0as t→±∞.By(2.16),(2.18)and(2.26),it sufﬁces to prove thatj+1 j |x(t)|2dt→0,as j→+∞.(2.27)On the other hand,we obtain from(1.1)thatj+1 j |x(t)|2dt=j+1j|−a1(t)x(t)+a2(t)x0(t)+f(t)+g(t,x0(t−τ1),x0(t−τ2),···,x0(t−τn))|2dt.Since g(t,0,0,···,0)=0for all t∈R,x0(t)→0as t→±∞,j+1j |x(t)|2dt→0and j+1j|f(t)|2dt→0as j→±∞,so(2.27)follows.Proof of Theorem2.1.The result follows now from Lemma2.8.AcknowledgementSupport by grant10871213from NNSF of China and by grant093051from Guangdong University of Technology of China is acknowledged.References[1]A.Ambrosetti,V.Coti Zelati,Multiple homoclinic orbits for a class of conserva-tive system,Rend.Sem.Mat.Univ.Padova.89(1993),177–194.[2]P.C.Carri˜a o,O.H.Miyagaki,Existence of homoclinic solutions for a class oftime-dependent Hamiltonian systems,J.Math.Anal.Appl.230(1999),157–172.[3]V.Coti Zelati,I.Ekeland,E.S´e r´e,A variational approach to homoclinic orbits inHamiltonian systems,Math.Ann.228(1990),133–160.[4]V.Coti Zelati,P.H.Rabinowitz,Homoclinic orbits for second order Hamiltoniansystems possessing superquadratic potentials,J.Amer.Math.Soc.4(1991),693–727.[5]Y.H.Ding,Existence and multiplicity results for homoclinic solutions to a classof Hamiltonian systems,Nonlinear Anal.25(1995),1095–1113.Existence of Homoclinic Solutions85 [6]Y.H.Ding,M.Girardi,Periodic and homoclinic solutions to a class of Hamilto-nian systems with the potentials changing sign,J.Math.Anal.Appl.189(1995), 585–601.[7]Y.H.Ding,L.Jeanjean,Homoclinic orbits for a nonperiodic Hamiltonian system,J.Diff.Eqns.237(2007),473–490.[8]Y.H.Ding,S.J.Li,Homoclinic orbits forﬁrst order Hamiltonian systems,J.Math.Anal.Appl.189(1995),585–601.[9]Y.H.Ding,M.Willem,Homoclinic orbits of a Hamiltonian system,Z.Angew.Math.Phys.50(1999),759–778.[10]R.E.Gaines,J.L.Mawhin,Coincidence degree and nonlinear differential equa-tion,Lecture Notes in Math,V ol.568,Springer-Verlag,1977.[11]H.Hofer,K.Wysocki,First 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APEC会议中英对照

第二次世界大战之后，大学的数目不断增长，本科、甚至是博士毕业生也与日俱增，从而催生了大批的技术人员、助手、演讲稿撰写人，徽章生产商，调查人员和政策研究员，效力于各类研究所中。

这些人员主要解决符号和概念问题，与工程师一样高度专业化，所以人们称他们为知识分子。

然而，这些人员与工程师还有所不同——他们仅承担少量教学工作，同时享受政府研究津贴或补贴，但不需从事繁重的日常例行工作。

教授项目预算是其最初的经济基础，使得教授得以在大学里保持独立。

研究生是否存在，大学是否具有资金偿还能力，这些都取决于项目预算。

大学的资金偿还在整个项目预算中占有重要比重。

项目资金不论来自政府或是商界都是基于合同之上的, 不同的项目签订不同的合同, 这一重要特点决定了以下事实：如果调研人员要取得独立，就必须能够取得连续的研究项目。

获得研究项目是知识分子的一种真正才能。

APECThe 9th Informal APEC Leaders’ Meeting was held at Shanghai in China on October 21, 2001. It has taken place at a crucial juncture. The major world economies are experiencing a slowdown more severe than anticipated. Most economies in the Asia-Pacific region have experienced an economic downturn, with some emerging economies particularly affected by unfavorable external market conditions. In addition, the terrorist attacks on the United States have undermined some industries as well as consumer and investor confidence. In the long run, a major challenge for the Asia-Pacific community is to manage the profound changes brought forth by globalization and the New Economy and to benefit from the opportunities that abound. Trends of the world developmentThe new situation, trends and technologies have helped expand productive forces, given full play to creativity, and enhanced exchanges and cooperation among countries and forged closer global ties. All people who love peace and long for development are working hard to create a more progressive world with a bright future. However, mankind is still faced with severe challenges to peace and development. The world is far from being tranquil as regional conflicts have kept cropping up. The objective of democracy in international relations is not yet to be fulfilled. The inequitable and irrational international economic order remains unchanged. The gap between the North and the South is still widening. The establishment of a new world of lasting peace and common prosperity calls for unremitting efforts of people of all countries. Peace and development complement each other. Peace is the prerequisite for development, which is in itself one of the important foundations for peace. Developed countries should earnestly help developing countries with their economic and cultural development so as to lift them out of poverty and backwardness as soon as possible. Developing countries, on the other hand, should follow the trend of the times; seize opportunities, seeking advantages and avoiding disadvantages, so as to set development rolling.East China FairEvery spring, hundreds of thousands of domestic and foreign customers and traders gather in Shanghai to attend the annual East China Fair. There, all kinds of goods made in Shanghai compete with the goods from other parts of China. The advantage of Shanghai lies in its scientific and technological power. Therefore, the export of high-tech products should take the lead in increasing Shanghai’s foreign trade and export. Professor Yang Zhenning, a famous physicist, once said, “The high-tech battle field is the main as well as the last battle ground for China to surpass the developed counties.” Taking the road of promoting high-tech trade is the future need as well as historical necessity.The Third WorldFor a long time, developing countries have suffered greatly from inequitable exchanges due to irrationality in the existing international economic relations. For them, the recent years have witnessed worsening terms of trade, plummeting export earnings, dwindling production investment and heavy debt burdens. There has been even the phenomenon of reverse capital flow from poor to rich countries for many years running. Therefore, more pronounced than a decade ago is the tendency of poor countries becoming poorer and rich countries richer. This situation, if allowed to continue, will bring in its wake of even more serious consequences for the whole world, including both developing and developed countries. The economic development of the Third World has become a pressing issue affecting the global scene. The current international economic relations should not and can not remain completely unchanged in the time to come, and the establishment of a new international economic order is the need of the times.Environmental protectionChina now enjoys sustained economic growth, political stability and national harmony and unity, and the living standards of the people keep improving. We in China pay much heed to environmental protection in the course of economic development, establishing the environmental protection as one of our basic state policies and making unremitting efforts towards this end. In the light of the country’s actual conditions, we have devised a strategy of synchronizing the planning, implementation and progress of economic development, urban and rural construction and environmental benefits. In this connection, we have adopted the three major principles, i.e. to put prevent first, to hold those who cause pollution responsible for cleaning up, and to strengthen environmental control and management. We have improved the legal system in respect of environment protection and set up relevant organs at various levels and an inter-ministerial coordination body at the national level. We have pooled the efforts of various quarters to address the problem of urban environment and to prevent and control industrial pollution. We have conducted extensive education in environmental protection to awaken the whole nation to its importance.Shanghai in the new centuryHuman society is now at the threshold of the 21st century which promises full hopes. Shanghai is not only China’s but the world’s. In the great course of modernization drive of the new century, the people of Shanghai, imbued with more heroic spirit, will unswervingly implement the strategy of building Shanghai into an international economic, financial and trade center as quickly as possible by grabbing new opportunities and meeting new challenges. The Shanghai of tomorrow is bound to create new glory and spring up as another center of international economy.ChinaChina is one of the largest countries in the world, her territory being about the size of the whole Europe. Our sources of food and clothing are from the large areas of fertile land in the vast country. Our extensive forests and rich mineral deposits are found in the mountain ranges across the country’s length and breadth. Many rivers and lakes provide us with water transport and irrigation, and a long coastline facilitates our communication with nations beyond the seas. Our forefather have worked, lived and multiplied on this vast territory since ancient times.The Gap between the South and NorthThe countries of the South are largely bypassed by the benefits of prosperity and progress owing to their existence on the periphery of the developed countries of the North. Most of the people of the South are poor while most countries of the North are opulent. The economies of the South are chiefly weak and defenseless while those of the North are generally strong and resilient. The countries of the South vary greatly in size, in natural resources endowment, in the structure of their economies, and in the level of economic, social and technological development. They also differ in their cultures, in their political systems, and in the ideologies they profess. Their economic and technological diversity has become more marked in recent years, making the South of today even less homogeneous than the South of yesterday.The United NationsOver the previous year the United Nations has been actively engaged in an unprecedentedly wide range of problems. There has never been a more active and more arduous year than this one in terms of hours worked, meetings held or journeys made. The Organization has achieved notable progress this year on some extremely difficult problems thought the purpose of some of its most important activities may be rather to inhibit the worst from happening than to attain a spectacular solution. The pragmatic and realistic approach, which began to emerge a few years ago, has now given rise to practical results which provide an opportunity for the U.N. to show its potential and to develop its capability. This is encouraging in respect of the problems themselves and also for the future of the United Nations.The global issuesWe must join hands in meeting the challenges to human survival and development since we all share one and the same planet. The issues of global proportion affecting the very survival and development of mankind are such problems as ecological degradation, poverty and unemployment, overpopulation, outbreaks of epidemic diseases, rampant drug abuse and transnational crime and lack of protections of the rights and interests of women and children. The developed countries should naturally make greater contributions to environmental protection, for they owe the rest of the world a great deal for having brought about environmental degradation in their industrialization and modernization. It requires not only efforts at the national level but also close coordination and cooperation at the international level to gradually solve these global issues.The future of ShanghaiThe per capita GDP in Shanghai is expected to reach US $ 7500 by 2007. The common people should benefit most directly from the attainment of the goal. The will live more comfortably in bigger space/ more spacious houses. Among the basic necessities of life __ food, clothing, shelter and transportation __ housing is an important element, accounting for a big portion in the money they spend.Interior decoration In 2007, the per capita living space in Shanghai will increase by a big margin/largely/ substantially. In addition, the common people’s spending on such services as education, information and traveling will grow substantially/greatly. To put it in a nutshell, in the future the common people will live a better life, a life of the same standard as enjoyed by those in the middle-ranking developed countries.The development of the western region of ChinaThanks to the rapid economic development in the past 20-plus years, a relatively solid foundation in terms of material wealth and technology was laid in the western region of China, with the society enjoying a rather satisfactory stability, and the market econ omy system being established and improved on a daily basis, providing a great market environment for the western region to maintain the sustained economic growth. Whil e focusing on its policy of expanding the western region, the Chinese government atta ches rather great importance to the restructuring of the economy, the promotion of the scientific and technological development, the deepening of the opening-up as well. W ith the stable advance in the development of the western region, the advantages of the region in resources and economic development will be given full play, thus further ac celerating the economic growth.The Problem of aging population　In the last 50 years，especially since the implementation of the policy of reform and opening up, the life expectancy of the Chinese has increased while the death rate has decreased, as a result of the development of national economy and the improvement of living standards and medical care. The life expectancy of the Chinese was 35 in the early years of the People's Republic, and at present, it is 71. The death rate was 20 per thousand in those days and is 6 per thousand now While we are happy about the increase in our life expectancy, it has also resulted in a rapid increase of the elderly population. Presently, the elderly population in China has exceeded 100 million with an increase of 24 million at an annual growth rate of 3%. By the year 2040, the elderly population will add up to 380 million, or 25.3% of the total, which means that I out of 4 Chinese will be a senior citizen. The aging of population has occurred when China's economy is still underdeveloped, and the growth rate of the elderly population is uneven in different parts of the country.Take a new road to industrialization and implement the strategy of rejuvenating the country through science and education and that of sustainable development. It remains an arduous historical task in the process of our modernization drive to accomplish industrialization. IT application is a logical choice if industrialization and modernization of our country are to be accelerated. It is, therefore, necessary to persist in using IT to propel industrialization, which will, in turn, stimulate IT application, blazing a new trail to industrialization featuring high scientific and technological content, good economic returns, low resources consumption, little environmental pollution and a full display of advantages in human resources.We must press ahead to optimize and upgrade the industrial structure so as to bring about an industrial pattern with high and new technology industries as the leader, basic and manufacturing industries as the kingpin and the service industry developing in all areas. We must give priority to the development of the information industry and apply IT in all areas of economic and social development. We must develop high and new technology industries to provide breakthroughs in stimulating economic growth. It is necessary to transform traditional industries with high and new technology and advanced adaptive technology and invigorate the equipment manufacturing industry. We should continue to strengthen infrastructure. We should accelerate the development of the modern service sector and raise the proportion of the tertiary industry in the national economy. We must correctly handle the relationships of development between the high and new technology industries and traditional industries, between capital-and-technology-intensive industries and labor-intensive industries and between virtual economy and real economy.WTOAfter 15 years of difficult negotiations, we finally came to this historical moment. Hereby, on behalf of the Chinese government, I would like to appreciate the WTOMinisterial Conference for its adoption of the Decision on China’s Accession to the WTO.The WTO accession and full participation in the multilateral trading system are strategic decisions made by the Chinese leaders under accelerated economic globalization. China has made longstanding and unremitting efforts for resuming its GATT（General Agreement on Tariffs and Trade)）Contracting Party status and for acceding to the WTO, which fully demonstrates the resolve and confidence of China to deepen its reform and to open further to the outside world. The WTO accession is not only in the interests of China, but also in the interests of all WTO Members and conducive to the development ofthe multilateral trading system. It will inevitably exert widespread and far-reaching impact on China’s economy and on the world economy in the new century. After its accession, China will, on the basis of the balance between rights and obligations, abide by the WTO rules and honor its commitments while enjoying its rights. China will, as always, value and strengthen equal and mutually beneficial economic andtrade relations with countries and regions in the world, and play an active and constru ctive role in the multilateral trading system and make its positive contribution to the progress of world economy and trade together with other WTO Members.China first participated in the Uruguay Round Negotiations in September 1986 and witnessed the historical transformation of the multilateral trading system from GATTto the WTO. Meanwhile, China’s negotiations for the accession to the multilateral trading system over the past 15 years have been an integral part of the process of its reform and opening up from the beginning to the end. We have drawn the following three basic conclusions from our own experiences:First, only by participating in the multilateral trading system with a positive stance, can various countries in the world better share the benefits of economic globalization;Second, during the course of economic globalization, only by establishing an economic and trade regime in line with international practices and the specific situation of the country, can they avoid their disadvantages and fully play out advantages, and therefore effectively safeguard their respective economic security while opening to the outside world; andThird, only by constantly adapting itself to the development and changes of world economy and by fully reflecting the interests and requirements of all parties including developing countries, can the multilateral trading system maintain its vigor and vitality.World ExpoIn an editorial published on Wednesday, the People's Daily said China won the bid to host the exposition following friendly competition with Russia, Mexico, Polandand South Korea at the assembly held in Monte Carlo, Monaco, on Tuesday.To host the World Expo 2010 is an aspiration of the 1.2 billion Chinese people, the editorial said. The Chinese government, since its announcement in December 1999 to apply for hosting the event, has made solemn commitments to the BIE and has given full support to the bidding process. The Chinese people have enthusiastically rendered support for bidding for World Expo Shanghai 2010.It said that Shanghai was awarded the right to host the event was the result of China's improved international status and increasing influence, the firm support from the Central Committee of the Communist Party of China (CPC), the State Council and people throughout China, and the wide recognition of the host city of Shanghai in the international community.The editorial said the Chinese government is a trustworthy and responsible government, which will honor its commitments and follow rules in the International Convention of the BIE. The Chinese government will make thorough preparations to provide all participants and their exhibits with complete facilities and satisfactory services, and pledges to give financial support to the World Expo Shanghai 2010.It said that China is fully confident in, and capable of, hosting a most successful, splendid and most unforgettable World Expo.China's successful bid is a breakthrough in the history of the event, the editorial said. In a history of over 150 years, the World Expo 2010 will become the first to be held in a developing country, which will expand the BIE's influence in China and in the whole world, and contribute to the promotion of world expositions in China and across the world.The 2010 World Exposition provides China with an important opportunity in the new century, which will greatly promote the building a well-off society in the country, and expedite its socialist modernization drive, said the editorial.Since the adoption of the reform and opening up policy, particularly over the past 13 years after the Fourth Plenary Session of the 13th CPC Central Committee, China has turned into one of the most dynamic countries in the world with its economy rapidly growing, its comprehensive national strength remarkably boosted and people's living standard continuously improved, it said.It said that hosting the exposition is also a great event in Shanghai's efforts to build the city into a world center of economy, finance, trade and transportation, which will tremendously help push forward its modernization and internationalization drive, and the economic development of the Yangtze River triangle region and lower reaches.The editorial said that China is confident that through the 2010 World Exposition, the international community will surely be able to see a prosperous China and a vigorous Shanghai.The World Expo Shanghai 2010 will surely become a landmark in the history of world expositions, it said.In the first 50 years of the 20th century, the Chinese people made unflagging efforts to change the tragic fate that had been theirs ever since the Opium War. After arduous and valiant struggle, they finally ended their semi-colonial and semi-feudal history, won independence for the nation and freedom for the people, and founded the People's Republic of China under the leadership of the Communist Party of China.In the latter half of the 20th century, relying on their own efforts, the Chinese people went all out to make their country strong and embarked on socialist construction.On the poor and deprived foundation of old China, they built a relatively complete industrial structure and national economic framework, raising the levels of their cultural and material lives significantly.China started marching forward steadily toward prosperity.Opening Remarks at China Hi-tech FairWe are delighted to welcome friends, old and new from around the world to the fourth China High-tech Fair on the beautiful coast of South China Sea in the golden fall. Hereby, on behalf of the Chinese government, I would like to extend warm welcome to all the distinguished guests from home and abroad attending this grand event and express heartfelt thanks to all the friends who have been caring for and supporting the Hi-tech Fair.High and new technology is the main embodiment and symbol of cutting-edge productivity and a strong driving force for the progress of human society. Constant innovation of knowledge and rapid progress in science and technology have substantially promoted global economic development, and transformed and enriched people’s life in a more and more profound way.Science and technology development in China is inseparable from that in the rest of the world and the progress of the world science and technology also needs China’s contribution. As a WTO member, China is actively moving with the tide of globalization with a positive stance of all-round opening up. As an important force in the international high-tech field, China will bring into play its rich resources of science, technology, conduct more extensive and profound exchanges and cooperation in the international high-tech area so as to contribute further to the cause of world science and technology development and human progress. China High-tech Fair is an important measure for the opening up of China’s high-tech sector. So far we havesuccessfully organized/hosted three such fairs, showcasing a large number of China’s high-tech achievements and the state-of-the-art products and technology from other world renowned high-tech companies. All these have helped to disseminate the up-to-date concept and perception heralding global scientific and high-tech development, and effectively facilitating the conversion of scientific and technological achievements into tangible productive forces. The high-tech Fair has become a window of opening-up and a stage of interaction/exchange between China and the rest of the world in the field of high and new technology.I’m deeply convinced that this grand event will bring benefit to all the participating friends, and I wish the forth China High-tech Fair score more fruitful results.。

Existence of Infinitely Many Solutions for a Quasilinear Elliptic Problem on Time Scales

Let f : T → R and t ∈ Tk (assume t is not left-scattered if t = sup T). We deﬁne f △(t) to be the number (provided it exists) such that given any ǫ > 0 there is a neighborhood U of t such that
arXiv:0705.3674v1 [math.AP] 24 May 2007
Existence of Inﬁnitely Many Solutions for a Quasilinear Elliptic Problem on Time Scales
Moulay Rchid Sidi Ammi sidiammi@mat.ua.pt
2 Preliminary results on time scales
We begin by recalling some basic concepts of time scales. Then, we prove some preliminary results that will be needed in the sequel.
Delﬁm F. M. Torres delfim@mat.ua.pt
Department of Mathematics
University of Aveiro 3810-193 Aveiro, Portugal
Abstract
We study a boundary-value quasilinear elliptic problem on a generic time scale. Making use of the ﬁxed-point index theory, suﬃcient conditions are given to obtain existence, multiplicity, and inﬁnite solvability of positive solutions.

一类抛物型方程弱解的正则性

一类抛物型方程弱解的正则性狄华斐;武虔虔;龙群飞【摘要】对一类PDE抛物型方程初边值问题，在一定条件假设下弱解的正则性问题的研究，通过一些技巧和方法。

描述了方程弱解的正则性．这些技巧和方法包括：Galerkin逼近法，解得弱收敛，sobolev不等式，内插不等式等等．%In this paper, we discuss the regularity of the weak solution u to theinitial/boundary value problem for second-order parabolic equations under certain Condition assumption. Via some techniques and methods,we also show the regularity of the weak solution. These techniques and methods include Galinkin-approximation, the weak convergence of the solutions, Sobolev-inequalities, interpolation-inequality, etc.【期刊名称】《贵州师范大学学报（自然科学版）》【年(卷),期】2012(030)002【总页数】5页(P28-32)【关键词】抛物型方程;弱解;弱解的正则性;Galerkin逼近【作者】狄华斐;武虔虔;龙群飞【作者单位】云南民族大学数学与计算机科学学院,云南昆明650500;云南民族大学数学与计算机科学学院,云南昆明650500;云南民族大学数学与计算机科学学院,云南昆明650500【正文语种】中文【中图分类】O29现代抛物型方程问题的可解性的研究总是在合适的泛函空间中考虑其弱解的性质，所以sobolev空间的引进(参看文献［1－2］)为这一研究提供了有效地途径.通过sobolev空间，我们可以在更广泛的函数类中寻求问题的解，这样可解性的问题变得就容易多了，这种解往往就称为“弱解”或“广义解”.本文讨论了抛物型方程:在一定条件假设下弱解的正则性问题.近年来，这类抛物型方程很多学者已经对它进行了广泛的研究，例如文献［2－4］中，作者假设算子L的系数 aij，bi，c是光滑的，并且与时间 t无关.而在本文中我们对L的系数的范围进一步放宽，即aij，bi，c是与时间t有关的，并且只要求c∈L∞ ，要求，则得到与文献［2］中相同的结果.在这里我们假设:U⊂Rn是一个有界开集，∂U光滑;UT=U ×(0，T］(其中T为固定的时间)，在这里f:UT→R，g:U→R为已知函数，u:是未知函数u=u(x，t).偏微分算子L是一致椭圆型算子即存在常数θ＞0，使得:并且L拥有散度形式:并且对方程中的系数aij，bi，c进一步做出额外的假设是Laplace算子－Δ 在空间上的特征函数(光滑)的完备集合(参考文献［2］中的§6.51)，m是一个正整数，在这里我们找函数:其中系数dkm(t)(0≤ t ≤ T ，k=1，…，m)使得:(在这双线性形式表示上的内积).下面我们叙述本文的主要结果:定理1(i)假设算子L的系数，函数同时也假设H－1(U))是方程:的一个弱解，则可得:且其中常数C与U，T及算子L系数有关．(ii)如果假设算子L的系数，函数则可得1)固定m≥1，用乘以等式(4)两边，则可得:在这其中而注记:这里(aij=aji)，所以可得:更进一步得:(这里C为常数，ε＞0且2)综合以上不等式得:我们取且时，由是 L2(U)上的标准正交基，同时是上的标准正交基，所以有:得: 又由得:综合(6)(7)式得:接着用乘以等式(4)两边得:这里因所以存在常数γ≥β≥0 s．t:所以有:(这里 C1，C2为常数)．3)现在记:由 (11)得:η'(t)≤C1η(t)+C2ζ(t)，a．e．0≤t≤T又由Gronwall，s不等式得:所以有:现在对(11)式进行时间t的积分，并利用以上的不等式得:又利用(14)式得:由，所以有:综合以上不等式可得:综合以上不等式可得:取极限m=ml→ ∞ ，可得到u∈ L∞(0，T;H10(U))，u'∈ L2(0，T;L2(U)).4)特别是对任何的时间t，我们有:将上面的不等式重新写为:B[u，v]=(h，v)，其中h=f－ u'，因为h(t)∈L2(U)，a.e.0≤t≤T，所以由椭圆型方程的正则性理论(参考文献[2] §6.32中的定理4)，可得u(t)∈H2(U)，a.e.0≤ t≤ T且有:两边取积分得:综合步骤2)和3)中的不等式，我们即可得到结论(i).5)接下来建立弱解的高阶正则性，假设g∈固定m ≥ 1 ，对方程(u'm，wk)+B[ um，wk;t]=(f，wk)进行时间t求导可得: 注记:这里令有，用，去乘上式的两边，并相加得:其中更进一步得:在式子 (21) 中，有由于因而存在常数γ≥β≥0 s.t:综合(21)(23)(24)式可得:取时，上式可转化为:6)现在记由 (25)式得: η'(t)≤ C1η(t)+C2ζ(t)，a.e.0≤ t≤ T又由Gronwall，s不等式得:所以有对(25)式进行时间t的积分，并利用以上的不等式得:7)我们必须估计(29)式最后一项是Laplace算子－Δ在空间H10(U)上的特征函数(光滑)的完备集合，特别的－Δum=λmum=0 on∂U上，这样:因为且 ( um(0)，wk)=，所以有因此且，所以(29)暗示:8)现在，且－Δwk=－λkwk，用λkdtm(t)乘以上式并将k=1，…，m相加，则可得到:因Δum=0，on∂U，所以B[um，－Δum]=(Lum，－Δum)，接下来引用不等式: 我们从(31)式可得到:综合上式，(16)，(30)和参考文献［2］§5．92 中定理3得:取极限m=ml→∞ ，我们得到对u同样的有界性．9)在这还需证明的是u″∈ L2(0，T;H－1(U))，为了达到目的，取v∈ (U)且，在这里v1∈，因为是的标准正交基所以因为u″m = ~u'm，所以可得 :因为这样，所以u ″m在空间L2(0，T;H－1(U))是有界的．综合以上不等式得:【相关文献】［1］王元明，徐君祥．索伯列夫空间讲义［M］．南京:东南大学出版社，2003:67-83．［2］Lawrence C Evance．Partial Differential Equations［M］．Rhode:Rhode Island，1988:71-73．［3］Lawrence C Evance．A survey of entropy methods for partial differential equations ［J］．Bulletin AMS，2004，41(4):409-438．［4］Zhang Yunzhang，Yang Ganshan．Existence and regularity of weak solutions for the biharmonic equation with complete second order derivative［J］．Dynamics of Continuous，Discrete and Impulsive Systems Series A:Mathematical Analysis，2010(17):215-232．［5］Struwe Michael．On the continuity of bounded weak solutions of quasilinear parabolic systems［J］．Maun math，1981(35):124-145．［6］Boccardo L．Nonlinear elliptic and parabolic equations involving measuredata ［J］．Funct Anal，1989(87):149-169．［7］Lin F H．On the Dirichlet problem forminimal graphs in hyperbolic space ［J］．Inventiones Mathematicae ，1989，96(3):592-612．［8］张德翔．关于热平衡方程的计算和教学［J］．贵州师范大学:自然科学版，1986(1):105-106．［9］F Duzaar，A Gastel，and J F Grotowski．Optimal partial regularity for nonlinear elliptic systems of higher order［J］．Math Sci Tokyo，2001(3):463-499．［10］罗李平，王智慧．具连续分布滞量的偶数阶非线性中立型偏微分方程的振动准则［J］．贵州师范大学学报:自然科学版，2006(02):66-70．［11］J Kinnunen and J L Lewis．Higher integrability for parabolic systems of plaplacian type［J］．Duke Math J，2000(102):253-271．。

Existence of initial data satisfying the constraints for the spherically symmetric Einstein

ing the Vlasov equation and it will be seen that it gives rise to new mathematical
features compared to those cases studied up to now. The second is connected
1 Introduction
The global dynamical behavior of self-gravitating matter is a subject of central importance in general relativity. A form of matter which has particularly nice mathematical properties is collisionless matter, described by the Vlasov equation. It has the advantage that it lacks the tendency observed in certain other models, such as perfect ﬂuids, that solutions of the equations of motion of the matter lose diﬀerentiability after a ﬁnite time. These singularities of the mathematical model form an obstacle to further analysis and prevent the study of the global dynamical properties of the solutions. Collisionless matter is free from these diﬃculties and there is a growing literature on global properties of solutions of the Einstein-Vlasov system [1], [8].

一类拟线性Choquard方程非平凡解的存在性

[3] COLIN M,JEANJEAN L.SolutionsforaQuasilinearSchrödingerEquation:A DualApproach [J].Nonlinear Anal:Theory MethodsAppl,2004,56(2):213-226.
[4] LIUJQ,WANG Y Q,WANGZQ.SolutionsforQuasilinearSchrödingerEquationsviatheNehariMethod[J]. Comm PartialDifferentialEquations,2004,29(5/6):879-901.
则存在常数 C(N,μ,r,t)>0,使得对任意的u∈Lr(ℝN )和v∈Lt(ℝN ),有
∬ℝ2N u(xx)-·yvμ(y)dxdy ≤ C(N,μ,r,t)‖u‖r‖v‖t.
方程 (3)对应的能量泛函为
∫ ∫ J(u)∶=p1 ℝN (1+2p-1 u p) ∇u pdx-21q ℝN (Iμ* u q)u qdx.
Abstract:Weprovedtheexistenceofnontrivial weaksolutionforaclassofquasilinear Choquard equationswithp-Laplacianoperatoras wellastheconvolutiontermsbyusingthe mountainpass lemma. Keywords:Choquardequation;p-Laplacianoperator;mountainpasslemma;nontrivialsolution
2)方程 (1)中的卷积项导致紧性条件不再成立 ,本文利用一些精细的分析技巧解决了该问题 .

N-S方程球对称弱解唯一性

any smooth solution to the Cauchy problem of the full system of compressible Navier–Stokes
equations without heat conduction blows up in a ﬁnite time. The same blowup phenomenon
vacuum initially. However, it still remains open whether there exist global strong solutions which
are spherically symmetric in a ball. The main difﬁculties lie on the lack of estimates of the
P (ρ) = aργ
(1.2)
with the adiabatic constant γ > 1 and a positive constant a. The shear viscosity μ and the bulk one λ are constants satisfying the physical hypothesis
1342
X. Huang / J. Differential Equations 262 (2017) 1341–1358
ρt + div(ρU ) = 0,
(1.1)
(ρU )t + div(ρU ⊗ U ) + ∇P = μ U + (μ + λ)∇(divU ),
where t ≥ 0, x ∈ ⊂ RN (N = 2, 3), ρ = ρ(t, x) and U = U (t, x) are the density and ﬂuid velocity respectively, and P = P (ρ) is the pressure given by a state equation

2019年度广东省科学技术奖公示信息

2019年度广东省科学技术奖公示信息项目名称液体-气体两相流模型的适定性理论主要完成单位单位1：华南理工大学单位2：西北大学单位3：华中师范大学主要完成人（职称、完成单位、工作单位）1.朱长江（职称：教授、工作单位：华南理工大学、完成单位：华南理工大学、主要贡献：提出了完成本项目研究的主要思路，并实施了本项目所有创新点的主要科研工作。

10篇代表作全部都有他的署名并在研究中做出了决定性的贡献。

先后主持完成国家杰出青年科学基金项目和国家自然科学基金重点项目等项目。

）2.姚磊（职称：教授、工作单位：西北大学、完成单位：西北大学、主要贡献：参与本项目主要科研工作。

10篇代表作中有5篇都有他的署名并在研究中做出了重要贡献。

先后主持国家自然科学基金青年项目和面上项目。

基于本项目的主要研究工作，于2012年获全国百篇优秀博士学位论文奖）3.温焕尧（职称：教授、工作单位：华南理工大学、完成单位：华中师范大学、主要贡献：重要科学发现第二项的代表性论文2。

提出了梯度平方分解恒等式。

）项目简介刻画流体之间相互作用的液体-气体两相流模型是石油工业中描述管道和深井中油和气的生产和输运的常见数学模型。

该模型不仅具有深刻的物理意义，而且也具有重要的数学理论价值。

关于其研究是近二十多年来本领域的热点问题之一,有许多关于该模型及其相关模型的数值结果，但对于该模型的适定性理论，即存在性、唯一性和稳定性等结果却很少。

该项目系统地研究了液体-气体两相流模型的自由边界问题、初边值问题和Cauchy问题解的适定性等问题。

挪威应用数学家Steinar Evje 教授在其同一篇论文中提到了我们的其中1篇代表作11次并作为后续研究。

我们在研究爆破机制时提出了速度的梯度平方分解恒等式，从而代替了传统方法中的梯度平方分解不等式。

美国《数学评论》(MR3457694)对我们的一篇代表作进行了评论，认为是文章有趣且非常有技巧。

本项目第一完成人朱长江教授曾获国家杰出青年科学基金资助以及入选了万人计划“国家教学名师”，所领导的两相流模型研究团队被同行专家称为国内外该领域的两个团队之一“Zhu’s group” (具体请见[Lizhi Ruan, Proceedings of the Royal Society of Edinburgh, 144A,351-362, 2014]第352页)，另外一个团队为挪威应用数学家Steinar Evje教授团队。

(完整版)俞敏洪精选100句背7000单词(排版+校对)

1.Typical of the grassland dwellers of the continent is the American antelope, or pronghorn.美洲羚羊，或称叉角羚，是该大陆典型的草原动物。

2.Of the millions who saw Haley's comet in 1986, how many people will live long enough tosee it return in the twenty-first century.1986年看见哈雷慧星的千百万人当中，有多少人能够长寿到足以目睹它在二十一世纪的回归呢？3.Anthropologists have discovered that fear, happiness, sadness, and surprise are universallyreflected in facial expressions.人类学家们已经发现，恐惧，快乐，悲伤和惊奇都会行之于色，这在全人类是共通的。

4.Because of its irritating effect on humans, the use of phenol as a general antiseptic has beenlargely discontinued.由于苯酚对人体带有刺激性作用，它基本上已不再被当作常用的防腐剂了。

5.In group to remain in existence, a profit-making organization must, in the long run,produce something consumers consider useful or desirable.任何盈利组织若要生存，最终都必须生产出消费者可用或需要的产品。

6.The greater the population there is in a locality, the greater the need there is for water,transportation, and disposal of refuse.一个地方的人口越多，其对水，交通和垃圾处理的需求就会越大。

三维Stokes近似系统弱解的全局存在性

第 31 卷第 1 期 2011 年 1 月
高师理科学刊 Journal of Science of Teachers′College and University
Vol. 31 No.1 Jan. 2011
文章编号：1007-9831（2011）01-0022-04
三维 Stokes 近似系统弱解的全局存在性
因为 S 是 Lipschitz 连续，积分方程（9）至少是局部可解的，即在空间 C ([0, T ], X n ) 中，由 Banach 压缩映像原理，很容易得出（9）在 [0, T (n)], 0 < T (n) ≤ T 有唯一局部解 u n ∈ C ([0, T ], X n ) ．反之，若 u n 是（8）的解，定义 ρ n = S (u n ) ，则 ρ n 是方程（3）在区间 [0, T (n)] 上的解．因此初边值问题（3）~（6）在 [0, T (n)] 上存在局部解 ( ρ n , un ) ．
第1期
郭蒙，等：三维 Stokes 近似系统弱解的全局存在性
23
远离真空时解的行为，由于这个模型相对简单，但却能从一定程度揭示流体的相对性质，故引起愈来愈多数学工作者的关注．文献[1]证明了周期边界条件时解的全局存在性．文献[2]给出了在二维柯西问题中，对于较大光滑初始值弱解和强解的全局存在性．文献[3]证明了当初始值是螺旋对称时，三维 Stokes 近似系统柯西问题全局强解的存在唯一性．文献[4]给出了二维 Stokes 近似系统全局强解的一些一致估计和爆破法则．文献[5]证明了三维 Stokes 近似系统初边值问题局部强解的存在唯一性. 本文主要讨论三维有界光滑区域上的 Stokes 近似系统弱解的全局存在性．为了讨论更加简单，取 ρ = 1 ， γ = 1 ， μ = 1 ， a = 1 ， λ + μ = 0 ，则方程组（1）变为

中央财经大学

10
Children born to serve the country (cont.d)
• Maintained the birth rate to maintain military power
Until 30 years old all men lived in one house and were not full-fledged citizens free to live with their wives, to make clandestine marriage instill an early burning love Physical training of bare boys and girls together in a place to draw and allure young men to marry while men who would not marry walked bare outside. Fathers of 3 exempt from military service The state would find another younger mate for a childless wife. • No jealousy • Children were not private to any men but common to the common
中国经济与管理研究院

March 28 & April 6 & 13, 2008 2008年3月28日和4月6和13日
1
Socrates (苏格拉底)
Taught, but not for money like the Sophists. Like the Sophists’, his pupils were from aristocratic party. Taught and sought knowledge by the method of dialectic (question & answer, first practiced by Zeno 芝诺) in dialogues

主要学术贡献

主要学术贡献主要从事非线性偏微分方程的研究，着重探讨流体动力学等领域中的数学理论及其应用。

多次应邀到美国、香港和新加坡等地学术访问或演讲。

主持或参加了多个科研项目，在国内外学术刊物发表论文60余篇。

主讲高等数学，数学物理方程，偏微分方程泛函方法，流体动力学的数学理论，非线性偏微分方程的某些理论。

指导博士研究生11人，其中9人已经毕业并获得博士学位。

指导硕士研究生26人，其中20人已经毕业并获得硕士学位。

合作编写教材《数学物理方程》一部，已由高等教育出版社出版。

获奖励情况(1) 1999年获得教育部科学技术进步一等奖；(2) 2000年获得教育部首届青年教师奖；(3) 2002年获得吉林省第七届青年科技奖；(4) 2006年获得吉林省长春市政府特殊津贴。

科研项目1) “相变和图像处理等领域中的某些非线性扩散方程”，教育部优秀青年教师教学与科研奖励基金项目，项目负责人,2000—2004；2) “图像处理中的非线性扩散模型”，国家自然科学基金项目青年基金项目，项目负责人,2001—2003；3) “流体动力学等领域中的具有退化性或奇异性的某些数学模型”，国家自然科学基金面上项目，项目负责人,2006—2008；4）“数学与其它领域交叉的若干专题”，国家重点基础研究发展计划973计划，参加者，2006—2011；5）“带有奇异性的某些流体动力学模型”,国家自然科学基金面上项目，项目负责人，2010年～2012年。

发表论文目录1.Yuan, Hongjun; Xu, Xiaojing Existence and uniqueness of solutions for a classof non-Newtonian fluids with singularity and vacuum. J. Differential Equations 245 (2008), no. 10, 2871--2916.2.Lian, Songzhe; Yuan, Hongjun; Cao, Chunling; Gao, Wenjie; Xu, Xiaojing On theCauchy problem for the evolution $p$-Laplacian equations with gradient term and source. J. Differential Equations 235 (2007), no. 2, 544--585.3.Lei, Yutian; Wu, Zhuoqun; Yuan, Hongjun Radial minimizers of a Ginzburg-Landaufunctional. Electron. J. Differential Equations1999, No. 30, 21 pp.4.Wu, Zhuoqun; Yuan, Hongjun; Yin, Jingxue Some properties of solutions for asystem of dynamics of biological groups. Comm. Partial Differential Equations22 (1997), no. 9-10, 1389--1403.5.Yuan, Hong Jun, Hölder continuity of interfaces for the porous medium equationwith absorption. Comm. Partial Differential Equations 18 (1993), no. 5-6, 965--976.6.Yuan, Hongjun; Wang, Changjia Unique solvability for a class of fullnon-Newtonian fluids of one dimension with vacuum. Z. Angew. Math. Phys. 60 (2009), no. 5, 868--898. 35Q357.Yin, Li; Xu, Xiaojing; Yuan, Hongjun Global existence and uniqueness ofsolution of the initial boundary value problem for a class of non-Newtonian fluids with vacuum. Z. Angew. Math. Phys. 59 (2008), no. 3, 457--474.8.Xu, Xiaojing; Yuan, Hongjun Existence of the unique strong solution for a classof non-Newtonian fluids with vacuum. Quart. Appl. Math. 66 (2008), no. 2, 249--279.9.Wang, Changjia; Yuan, Hongjun Global strong solutions for a class ofheat-conducting non-Newtonian fluids with vacuum. Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3680–3703,10.Lining, Tong; Hongjun, Yuan Classical solutions to Navier-Stokes equationsfor nonhomogeneous incompressible fluids with non-negative densities. J. Math.Anal. Appl. 362 (2010), no. 2, 476–504.11.Lian, Songzhe; Gao, Wenjie; Cao, Chunling; Yuan, Hongjun Study of thesolutions to a model porous medium equation with variable exponent ofnonlinearity. J. Math. Anal. Appl. 342 (2008), no. 1, 27--38.12.Lian, Songzhe; Yuan, Hongjun; Cao, Chunling; Gao, Wenjie The limiting problemof the drift-diffusion-Poisson model with discontinuous $p$-$n$-junctions.J. Math. Anal. Appl. 347 (2008), no. 1, 157--168.13.Yuan, Hongjun; Chen, Mingtao Positive solutions for a class of $p$-Laplaceproblems involving quasi-linear and semi-linear terms. J. Math. Anal. Appl.330 (2007), no. 2, 1179--1193.14.Xin, Zhouping; Yuan, Hongjun Vacuum state for spherically symmetric solutionsof the compressible Navier-Stokes equations. J. Hyperbolic Differ. Equ. 3 (2006), no. 3, 403--442.15.Yuan, Hongjun; Tong, Lining; Xu, Xiaojing BV solutions for the Cauchy problemof a quasilinear hyperbolic equation with $\sigma$-finite Borel measure and nonlinear source. J. Math. Anal. Appl. 311 (2005), no. 2, 715--735.16.Yuan, Hongjun; Xu, Xiaojing; Gao, Wenjie; Lian, Songzhe; Cao, ChunlingExtinction and positivity for the evolution $p$-Laplacian equation with $L^1$ initial value. J. Math. Anal. Appl. 310 (2005), no. 1, 328--337.17.Hongjun, Yuan; Songzhe, Lian; Wenjie, Gao; Xiaojing, Xu; Chunling, CaoExtinction and positivity for the evolution $p$-Laplacian equation in $R^n$.Nonlinear Anal. 60 (2005), no. 6, 1085--1091.18.Hongjun, Yuan; Xiaoyu, Zheng Existence and uniqueness for a quasilinearhyperbolic equation with $\sigma$-finite Borel measures as initial conditions.J. Math. Anal. Appl. 277 (2003), no. 1, 27--50.19.Yuan, Hongjun The Cauchy problem for a singular conservation law with measuresas initial conditions. J. Math. Anal. Appl. 225 (1998), no. 2, 427--439.20.Hongjun, Yuan Source-type solutions of a singular conservation law withabsorption. Nonlinear Anal. 32 (1998), no. 4, 467--492.21.Yuan, Hong Jun Extinction and positivity for the evolution $p$-Laplacianequation. J. Math. Anal. Appl. 196 (1995), no. 2, 754--763.22.Yuan, Hong Jun The Cauchy problem for a quasilinear degenerate parabolicsystem. Nonlinear Anal. 23 (1994), no. 2, 155--164.23.Yuan, Hong Jun Finite velocity of the propagation of perturbations for generalporous medium equations with strong degeneracy. Nonlinear Anal. 23 (1994), no. 6, 721--729.24.Yuan, Hongjun Continuity of weak solutions for quasilinear parabolicequations with strong degeneracy. Chin. Ann. Math. Ser. B 28 (2007), no. 4, 475--498.25.Yuan, Hong Jun; Lian, Song Zhe; Cao, Chun Ling; Gao, Wen Jie; Xu, Xiao JingExtinction and positivity for a doubly nonlinear degenerate parabolic equation.Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 10, 1751--1756.26.Yuan, Hong Jun; Tong, Li Ning BV solutions for a quasilinear hyperbolicequation with nonlinear source and finite Radon measures as initial conditions.(Chinese) Acta Math. Sci. Ser. A Chin. Ed. 30 (2010), no. 1, 54–70,27.Yuan, Hongjun; Wang, Shu The zero-Mach limit of a class of combustion flow.J. Partial Differ. Equ. 22 (2009), no. 4, 362–375,28.Ren, Chang Yu; Guan, Jin Rui; Yuan, Hong Jun A class of general-form parabolicMonge-Ampère equations. (Chinese) Chinese Ann. Math. Ser. A 30 (2009), no.3, 421--432. 35K9629.Yuan, Hongjun; Yan, Han Existence and uniqueness of BV solutions for a classof degenerate Boltzmann equations with measures as initial conditions. J.Partial Differ. Equ. 22 (2009), no. 2, 127--152. 35F2530.Yuan, Hongjun; Xu, Xiaojing Some entropy inequalities for a quasilineardegenerate hyperbolic equation. J. Partial Differential Equations18 (2005), no. 4, 289--303.31.Yuan, Hong Jun; Wu, Gang Quasilinear degenerate parabolic equation with Diracmeasure. (Chinese) Chinese Ann. Math. Ser. A 26 (2005), no. 4, 515--526;translation in Chinese J. Contemp. Math.26 (2005), no. 3, 291--30232.Yuan, Hongjun; Jin, Yang Existence and uniqueness of BV solutions for theporous medium equation with Dirac measure as sources. J. Partial Differential Equations 18 (2005), no. 1, 35--58.33.Yuan, Hong Jun; Xu, Xiao Jing Existence and uniqueness of BV solutions fora quasilinear degenerate hyperbolic equation with local finite measures asinitial conditions. (Chinese) Chinese Ann. Math. Ser. A 26 (2005), no. 1, 39--48; translation in Chinese J. Contemp. Math.26 (2005), no. 1, 43--5434.Yuan, Hongjun Instantaneous shrinking and localization of functions in$\roman Y_\lambda(m,p,q,N)$ and their applications. Chinese Ann. Math. Ser.B 22 (2001), no. 3, 361--380.35.Yuan, Hongjun Cauchy's problem for degenerate quasilinear hyperbolicequations with measures as initial values. J. Partial Differential Equations12 (1999), no. 2, 149--178.36.Yuan, Hongjun Localization condition for a nonlinear diffusion equation.Chinese J. Contemp. Math. 17 (1996), no. 1, 45--58.37.Yuan, Hongjun Existence and nonexistence of interfaces of weak solutions fornonlinear degenerate parabolic systems. J. Partial Differential Equations 9 (1996), no. 2, 177--185.38.Yuan, Hongjun Extinction and positivity for the non-Newtonian polytropicfiltration equation. J. Partial Differential Equations 9 (1996), no. 2, 169--176.39.Yuan, Hong Jun A localization condition for a class of nonlinear diffusionequations. (Chinese) Chinese Ann. Math. Ser. A 17 (1996), no. 1, 47--58.40.Zhao, Junning; Yuan, Hongjun The Cauchy problem of some nonlinear doublydegenerate parabolic equations. Chinese J. Contemp. Math. 16 (1995), no. 2, 173--192.41. Zhao, Jun Ning; Yuan, Hong Jun The Cauchy problem for a class of nonlinear doubly degenerate parabolic equations. (Chinese) Chinese Ann. Math. Ser. A 16 (1995), no. 2, 181--196.42. Wu, Zhuo Qun; Yuan, Hong Jun Uniqueness of generalized solutions for a quasilinear degenerate parabolic system. J. Partial Differential Equations 8 (1995), no. 1, 89--96.43.Yuan, Hong Jun Regularity of free boundary for certain degenerate parabolic equations. Chinese J. Contemp. Math. 15 (1994), no. 1, 77--86.44.Zhao, Jun Ning; Yuan, Hong Jun Uniqueness of the solutions of $u_t=\Delta u^m$ and $u_t=\Delta u^m-u^p$ with initial datum a measures: the fast diffusion case. J. Partial Differential Equations 7 (1994), no. 2, 143--159.45.Yuan, Hong Jun Regularity of the free boundary for a class of degenerate parabolic equations. (Chinese) Chinese Ann. Math. Ser. A15 (1994), no. 1, 89--97.。

可压磁流体方程组弱解能量的有界性

可压磁流体方程组弱解能量的有界性高真圣【摘要】We consider the global behavior of weak solutions of the equations of compressible magnetohydrodynamic(MHD) flows in time in a bounded three-dimesion domain-arbitrary forces. To achieve our goal for the MHD problem, we also need to develop estimates to deal with the magnetic field and its coupling and interacting with the fluid variables. The nonlinear term (▽×M)×M will be dealt with by the idea arising in compressible Navier-Stokes equations. We use Yong inequality,Holder inequality and Soblev inequality for energy estimate. Under certain restrictions imposed on the adiabatic constant,we proved that total energy of finite energy weak solution is still bounded when the external force is bounded.%研究可压缩磁流体(MHD)方程组的弱解在三维有界区域上关于时间的全局行为.为了解决MHD方程组的这一问题,需要对磁场项、耦合项以及流体项进行估计.对非线性项(▽×M)×M的处理方式是受可压Navier-Stokes方程组的启发.利用Yong不等式、H(o)lder不等式以及Soblev不等式等对弱解进行能量估计.对绝热指数进行适当限制,证明了在有界外力作用下,总能量是有界的.【期刊名称】《厦门大学学报（自然科学版）》【年(卷),期】2012(051)004【总页数】6页(P651-656)【关键词】可压磁流体方程组;弱解;能量估计【作者】高真圣【作者单位】厦门大学数学科学学院,福建厦门 361005【正文语种】中文【中图分类】O175.29考虑下面等熵情形下的黏性可压的磁流体方程组［1－2］：其中t∈I，x∈Ω，并给出如下的Dirichlet边界条件这里，我们假定IR为一个区间且是一个有界的Lipschitz区域，外力函数ρ＝ρ（t，x）≥0表示流体的密度，u＝［u1（t，x），u2（t，x），u3（t，x）］表示流体的速度，M＝［M1（t，x），M2（t，x），M3（t，x）］表示磁场，μ、λ是粘性系数，v是磁场的磁扩散系数，在这里满足a＝eS＞0是由墒S决定的常数，γ＞1是绝热指数.γ的值有着自身的物理意义［3］，例如，单原子气体γ＝ 5/3，双原子气体γ＝7/5，多原子气体γ1.γ也对静态解的存在性、唯一性以及稳定性起着很重要的作用.本文假定首先我们给出在假设（5）～（8）下关于问题（1）～（4）的重整化有限能量弱解的定义.定义1 设q＝ρu，记称（ρ，u，M）为问题（1）～（7）的一个弱解是指：●ρ，u，M满足●方程（1）～（3）在D′（I×Ω）意义下成立；●方程（1）在重整化解意义下成立，即在D′（I×Ω）意义下有其中b满足且满足无穷远处的增长条件●能量不等式在D′（I）意义下成立.注1 由文献［4］的引理7.18和引理7.19知，问题（1）～（7）的任一弱解都满足并且由文献［4］命题7.21及M∈C（I（L2weak（Ω））3）知，总能量ε（t）是下半连续的.文献［5－6］首先给出了系统（1）～（4）的弱解的存在性定理.本文考虑三维可压磁流体方程的弱解关于时间的全局行为，推广了文献［4，7－8］中的关于可压流体N－S方程的相关结论以及文献［9］中N－S－P系统的结论，在他们的文章中，都只针对γ＞5/3进行了证明.最近，当绝热指数γ∈（3/2，5/3］，文献［10］给出了可压流体N－S－P方程组弱解的总能量关于时间有界性的证明，其中包含N－S方程组情况.考虑磁流体（MHD）方程，由于增加了磁场，估计非线性项（× M）×M比较困难，通常的处理可压N－S方程的方法已经失效.另外，我们还需估计耦合项以及混合流体变量项.本文主要得到下面的定理：定理1（能量有界）设γ＞3/2，给定ε0＞0，若存在常数ε∞（与f的范数K：＝｜f｜L∞（I×Ω））、ε0及总质量有关），使得则问题（1）～（7）的任一弱解满足进一步地有，若绝热指数γ∈（5/3，＋∞）则ε∞与ε0无关.首先介绍几个结论：引理1［4］设f满足式（6），ρ、u满足式（10），b（s）满足式（11）～（13）.记及其中k＞0，则在D′（I×Ω）意义下有引理2 设f满足式（6），若ρ，u，M满足式（10）且符合能量不等式（14），那么总质量m是守恒的，即对a.e.t∈R＋，ρ满足式（15）；进一步有，（有必要的话，在一个零测度集上重新定义），ε可表示成任一有界区间I上的非增函数及绝对连续函数之和，且有证明请参考文献［4］，练习6.3，［12］，命题2.1，［9］，引理1.3及［7］，引理2.2.引理3［13］设ΩR3是一个有界区域，则存在一个常数c1使得及1＜p＜∞，有下面的能量估计对定理的证明起着至关重要的作用.命题1 设α，β＞0，在定理1的假设下，若存在一个T∈I，使得则存在常数c2＝c2（K，m），使得下面先为命题的证明作准备工作，命题的证明会在文章的第3部分给出.引理4 设定理1的假设及式（20）成立，则存在一个常数c3＝c3（K，m），使得证明利用式（18）、（5）可得对因此有再利用式（19）、H10（Ω）→L6（Ω）、式（23）、引理3、div M＝0以及Young不等式，可得引理5 设定理1的假设及式（20）成立，则存在一个常数c4＝c4（K，m），使得证明由式（23），可得令t2＝T＋1，式（25）对t1积分可得另一方面最后，式（25）中取t1＝T再结合式（26），可得到利用Hlder不等式和引理4，可得再利用插值不等式可推出之后使用式（9）、引理4、式（27）以及式（28）便可完成证明.类似于文献［7］，进一步的估计需要用到更好的技巧.下面介绍算子Bi［14］：引理6［15］设ΩR3是有界的Lipschitz区域，p、r∈（1，∞）是给定的数，那么存在一个有界的线性算子使得v＝B（f）是问题divv＝f，a.e.在的解；进一步地有，若f∈Lp（Ω）且存在一个满足f＝div g及注2 引理6证明的思路来源于Gladi文献［16－17］，详细的证明过程见文献［15］.引理7 设定理1的假设及式（20）成立，则存在一个常数c6＝c6（K，m），使得这里证明只给出证明的框架，细节可以参考文献［4］，第7.9.5节.使用下面的磨光算子将方程（18）关于变量t进行正则化这里ε＞0，ωε是标准的一维的Friedrichs 磨光核.对任意固定的区间及任意的0＜α＜ε0（I′）（ε0充分小），可得考虑具有如下形式的检验函数这里利用式（30），可得取φi作为式（2）的检验函数，再利用式（32），经过直接计算可得再利用引理6、引理4、Hlder不等式、Sobolev嵌入定理以及引理2，上式右边的各项积分可处理成：这里在式（34）中，令ε→0＋，再利用式（31）得最后式（33）的左边可变形为利用式（31）、（22）、Hlder不等式以及Sobolev嵌入定理可得这里q3＝max｛3γ/（2γ－3），2｝.结合式（35）～（37），式（33）可变成对区间［T，T＋1］上的特征函数我们用一个序列ψm去逼近，式（38）中用ψm代替ψ并令m→∞，再令k→∞，由单调收敛定理得到从而完成结论的证明.利用在空间L1和Lγ＋θ上的插值不等式有再使用Young不等式、式（24）、（29）以及（39）得选择适当c2即可证明命题1.首先利用不等式（21）可知，若1，则存在足够小的正数α＊及足够大的正数β＊使得对任意的α∈（0，α＊）、β∈（β＊，∞）有这里c，α＊和β＊都与m，K及γ有关.选择δ满足和满足以及可推出存在使得事实上若不是此情形，即，对任意的那么对任意的正整数另一方面，利用式（17）及，可推出当T0充分接近inf｛t∈I｝，k充分大时，能量ε［（T0＋k）－］是负数，产生矛盾.因此，结合命题1、式（40）我们得到下一步，我们假定并通过数学归纳法证明对任意整数有再结合命题1、式（40）以及有或者最后，取这里，δ仅与γ有关，β＊与m，K及ε0有关.至此定理1的前半部分已证出.对于后半部分，利用引理3中关于M的估计，参考文献［7］，可推出当γ∈（5/3，∞）时，ε∞与ε0无关，这里我们不再赘述.【相关文献】［1］ Cabannes H.Theoretical magnetofluid－dynamics［M］.New York：Academic Press，1970.［2］ Kulikovskiy A，Lyubimov G.Magnetohydrodynamics［M］.New York：Addison－Wesley，1965.［3］ Chandrasekhar S.An introduction to the study of stellar structures［M］.Chicago：University 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theory of the Navier－Stokes equations［M］.New York：Springer－Verlag，1994.［17］ Lions P.Mathematical topics in fluid mechanics：compressible models［M］.Oxford：Oxford University Press，1998.。

耗散系统弱解整体存在性

耗散系统弱解整体存在性黄金红【摘要】研究带耗散项的双曲守恒律方程组弱解的整体存在性.对于此类问题,Dafermos和肖玲已经有过具体的研究,但要求非齐次项线性部分为严格对角占优矩阵.本文将条件改为只要求非齐次项对应矩阵的特征值大于零,但要求矩阵为可对角化矩阵.%We address the global existence of weak solutions to hyperbolic system of balance laws with dissipation.The issue has been discussed by Dafermos and Hsiao.However,in their study,the main condition to guarantee the existence of the solution is that the matrix of the linear part of the dissipative terms is a diagonally dominant matrix.In this paper,the condition proposed is that the eigen values of the linear part of the dissipative terms（which is an n × n constant matrix） are all positive,but the matrix is a diagonalizable one.【期刊名称】《西安文理学院学报（自然科学版）》【年(卷),期】2012(015)002【总页数】6页(P16-21)【关键词】耗散性;严格双曲方程组;整体解【作者】黄金红【作者单位】南京航空航天大学理学院,江苏南京211106【正文语种】中文【中图分类】O175.28本文考虑平衡律方程组其中，U为n维向量，流函数F(U)及源函数G(U)均为B到Rn的光滑函数，B是Rn中以原点为圆心，ρ为半径的球.当F，G不仅仅依赖于解U，且依赖于变量x，t时，方程组(1)具有如下形式Dafermos和肖玲在文［5］中利用Glimm格式讨论了问题(3)(2)整体解存在性问题，他们引入矩阵其中R(U，x，t)=(R1(U，x，t)，R2(U，x，t)，…，Rn(U，x，t))是一个n×n矩阵，R1(U，x，t)，R2(U，x，t)，…，Rn(U，x，t)为FU(U，x，t)的n个线性无关的右特征向量，GU(U，x，t)为G(U，x，t)关于U的Jacobi矩阵.在原点U=0邻域内，源函数G的耗散性体现在矩阵A的性质上.特别地，称源函数G为强耗散的，如果可以适当地选择矩阵R(U，x，t)使得矩阵A成为严格对角占优矩阵，即当G(U，x，t)具有强耗散性，并且初值满足一定的条件时，Dafermos和肖玲利用随机取点的方法证明了在BV空间中问题(3)(2)解的整体存在性.本文，我们将证明条件(5)可以改变为，矩阵GU(0，x，t)可对角化且特征值实部均大于零.我们只考虑F，G不依赖于变量x，t的情形.记矩阵GU(0)的对角型为B，此时存在可逆矩阵P满足本文主要结果:定理1 假定G(U)=0，方程组(1)为严格双曲的，所有特征要么是真正非线性的，要么是线性退化的.初值U0全变差有界，并将其定义为初值U0取值于Br(0)(Rn中一开球)，是绝对可积的，记其积分值为δ矩阵GU(0)可对角化且特征值的实部均为正的.在以上假定均成立的情况下，存在δ0，ρ0，当δ＜δ0，ρ＜ρ0时，在BV空间中Cauchy问题(1)(2)的弱解U(x，t)整体存在，且对任意的t∈(0，+∞)，U(x，t)绝对可积，且全变差有界.1 近似解的构造利用改进的Glimm格式中随机取点法构造方程组(1)的基本解Uh(t，x).取数列{α0，α1，…，αm，…}⊂(-1，1)为随机数列，固定h＞0为空间步长，选取s为时间步长，s，h满足CFL条件，保证了同时由相距为2h的点发出的两个波不会相交.用t=ms将(x，t)坐标系的上半平面分成一系列带状区域:并将网格点记为(kh，ms).对每个满足m+k=odd的k，选取随机点(xk，m，ms)(xk，m=(k+αm)h)，在t= 0上定义Uh(x，t)=U(x，0).假定在带状区域内，Uh(x，t)已有定义，我们将其定义域扩充到下一带状区域Sm.首先定义t=ms上Riemann问题(1)(2)的近似解为在带状区域Sm内解Riemann问题并记其解为由于非齐次项的影响，将问题(1)(2)在Sm内的近似解定义为可以证明Uh(x，t)满足平衡律方程组.为方便起见引入如下定义2 基本波相互作用估计及Glimm泛函单调性解Riemann问题(11)(12)的过程中，我们得到n个基本波，其中包含激波，疏散波和接触间断.用α=(α1，α2，…，αn)表示方程组的解，α1，α2，…，αn为对应的n个基本波.下文中β=(β1，β2，…，βn)，γ= (γ1，γ2，…，γn)也有同样的意义.由Riemann问题解的惟一性，波的左右状态决定了振幅向量，同样的，振幅向量和左(右)状态决定了右(左)状态.并且，如果我们用U-，U+表示波的左右状态，用ε表示包含有n个基本波的Riemann问题的解，则存在二次连续可微函数Φ(U-，ε)，Ψ(U+，ε)和γ(U-，U+)满足以下关系式:一般来讲我们利用它们的绝对值大小表示波的强度.例如|αi|表示基本波αi的强度，α的强度可定义为事实上，定义波的强度的方式很多.这里我们利用可逆矩阵的性质重新定义一种新的波的强度.我们指出，对任意可逆矩阵Q，Qα同样可以等价地表示波的强度.引入如下记号只需证明与|α|是等价的.即证明存在实数c满足本文选取矩阵Q=P-1，P为(6)给定的矩阵，即表示α的强度.其等价性是显然的.由近似解的构造可知这里联立等式(17)、(18)和(19)，可以得到由隐函数定理知，在(α，β，s)=(0，0，0)的小邻域内，γ是关于α，β和s的二次连续可微函数.记之为为了得到基本波的相互作用估计，我们引入如下引理:引理1 对任意的二次连续可微函数f(x，y)由连续函数性质此引理易证，证明过程从略.引理2 在定理1的条件下此处c为某一常数，证明如前所述，αi，βj称为是相互接近的，如果i＞j或者两个中至少有一个是激波.两次利用引理1可以得到由传统的Glimm格式中波的局部相互作用估计可知其中当α=0，β=0时故由γ(α，0，0)=α，γ(0，β，0)=β可知当β=0时，方程(20)变为方程两边关于s求偏导令α=0，s=0我们得到方程(31)两边关于α求偏导，并且令α=0，s=0可以得到0＜θ＜1是对应中值定理的常数.由基本波的性质及可以将等式(32)化简为当α=0，s=0时可以得到同理可得因为(此时O(1)表示n×n×n的张量)，所以(34)和(35)即为αs，βs的系数由初值条件的充分小，可知GU(θUk，m-1)与GU(0)的可任意接近，当Uk，m-1充分小时，R-1(Uk，m-1)，R(Uk，m-1)可充分接近R-1(0)，R(0)，由Taylor展开式结合以上分析，等式(24)变为等式(36)两边左乘矩阵P-1，利用波的强度的新的定义方法，可知，存在一个正常数τ满足其中Mδ为P-1R-1(0)▽R(Uk，m-1)G(Uk，m-1)P在U=0邻域内的上确界，因为Uk，m充分小，故Mδ是一个小量，因此此处我们利用了下面的不等式c1，c2为正常数.由(40)可证引理结论.定义Glimm泛函为这里K为待定的充分大的正数.引理3 可以选取K充分大，使得L(J)充分小时，对任意的网格曲线J，满足J＞1时，F(J)可由F(I)控制，即证明由归纳法，我们只需考虑当J是I的直接后继时，结论成立.此时J和I之间夹有一个四边形我们定义由(22)，容易证明因此，L(J)充分小时联合以上各式，可得由(46)知可以选取K充分大，可使(44)成立.L(J)的充分小性可以归结为L(0)充分小，L(0)表示夹在t=0和t=s之间的网格曲线，可以进一步转化为对初值U0(x)的要求.引理3即为Glimm泛函的单调性.利用此性质，可以证明方程组(1)整体解的存在性.证明过程和传统的证明过程一样，算法收敛性及弱解定义证明也类似于传统的证明过程，这里从略.［参考文献］［1］ AMADORI D，GUERRA G.Global weak solutions for system of balance laws［J］.Appl.Math.Lett.，1999，12:123-127.［2］ DATERMOS C M.Hyperbolic conservation laws in continuum physics ［M］.2nd edn.，Springer:Heidelberg，2005.［3］ DAFERMOS C M.Hyperbolic systems of balance laws with weak dissipation［J］.Hyper.Differ.Equ.，2006，3:505-527.［4］ DING X，CHANG T，WANG J.et al.A study of the global solutions for a quasilinear hyperbolic system of conservation laws［J］.Scientia Sinica，1973，16:317-335.［5］ DAFERMOS C M，HSIAO L.Hyperbolic system of balance laws with inhomogeneity and dissipation［J］.Indiana U.J.Math.，1982，31:471-491. ［6］ GLIMM J.Solutions in the large for nonlinear hyperbolic system ofequations［J］.Comm.Pure Appl.Math.，1965，18:697 -715.［7］ GLIMM J，LAX P D.Decay of solutions of systems of nonlinear hyperbolic conservation laws［J］.Memoirs of America Mathmatical Society，1970，101:358-371.［8］ JENSSEN H K.Blowup for system of conservation laws，Society for Industrial and Applied Mathmatics［J］.J.Math.A-nal.，2000，31:894-908.。

低碳生活英语作文带翻译

低碳生活英语作文带翻译低碳生活英语作文带翻译（通用16篇）在生活、工作和学习中，大家都跟作文打过交道吧，作文是人们把记忆中所存储的有关知识、经验和思想用书面形式表达出来的记叙方式。

还是对作文一筹莫展吗？以下是店铺收集整理的低碳生活英语作文带翻译，仅供参考，大家一起来看看吧。

低碳生活英语作文带翻译篇1Now we are facing two problems of global warming and a shortage of natural resources, both of which are getting more and more serious day by day. Thus, it is important for us to promote a low-carbon lifestyle.如今我们面临着两大环境问题：全球变暖和自然资源匮乏，且情况日益严重。

由此可见，提倡低碳生活的重要性。

Low-carbon lifestyle covers many aspect of our life like transportation, traveling, home-heating, and so on. In order to promote a low-carbon life, some plans must be carried out as follows. Firstly, don’t waste any electricity and water. Remember to turn off the lights and water taps unless we use them. Secondly, we had better take the subways or bus to work, so as to reduce the emissions of carbon dioxide. Thirdly, we should use cloth bags for shopping instead of plastic bags, and use recyclable bowls and chopsticks instead of disposable ones, for it can not only save resources, but also reduce pollution. What’s more, it is of great significance to plant more trees, since trees can balance the carbon dioxide emissions.低碳生活方式包括了我们生活的很多方面，比如交通、旅行、室内取暖等。

具变指数黏弹性波动方程能量解的爆破

具变指数黏弹性波动方程能量解的爆破高云柱;孟秋;郭微【摘要】考虑一类具变指数黏弹性波动方程能量解的爆破性,通过构造能量函数研究能量函数的性质,并利用所得结果和Cauchy不等式、积分估计等,得到具变指数非线性波动方程能量解在有限时刻爆破的性质.%We considered the properties of blow-up of solutions of energy for a class of viscoelastic wave equations with variable-exponents . By constructing an energy function , we studied the properties of the energy function , and used the obtained results , Cauchy inequality and integral estimates to get the properties of blow-up of solutions of energy for a nonlinear wave equations with variable-exponents in finite time .【期刊名称】《吉林大学学报（理学版）》【年(卷),期】2018(056)003【总页数】5页(P503-507)【关键词】变指数;黏弹性波动方程;能量解;爆破性【作者】高云柱;孟秋;郭微【作者单位】北华大学数学与统计学院 ,吉林吉林132013;北华大学数学与统计学院 ,吉林吉林132013;北华大学数学与统计学院 ,吉林吉林132013【正文语种】中文【中图分类】O175.260 引言考虑下列具变指数非线性波动方程的初边值问题:(1)其中: Ω是N(N≥1)上的有界区域, 具有光滑的边界; α为非负常数; 指数函数p(x)和函数g(t)分别满足如下条件:(H1) p(x)是定义在上的可测函数, 使得∀(H2) g: +→+为C1函数, η为正常数, 满足当p为常数时, 关于问题(1)解的存在性和爆破性研究已有许多结果[1-5]. 近年来, 关于电磁流变学方面数学模型的研究受到广泛关注, 特别在变指数研究方面取得了许多结果[6-9]. 此外, 各种物理现象, 如一些波动模型、服从非线性Boltzmann模型的纵向运动控制系统出现的问题等模型, 也取得了一些研究结果[10-13].1 预备知识设p(x)满足条件(H1), 则变指数Legesgue空间Lp(·)(Ω)是指所有可测函数, 使得令则空间Lp(·)(Ω)赋予Luxemburg范数其为可分自反的Banach空间. Lp(x)(Ω)的对偶空间为Lp′(x)(Ω), 其中变指数Legesgue空间是Orlicz-Musielak空间[7]的特殊情形.对任何正整数k, 取Wk,p(x)(Ω)={u∈Lp(x)(Ω): Dαu∈Lp(x)(Ω), |α|≤k},Wk,p(x)(Ω)的范数定义为易知Wk,p(x)(Ω)也是一个Banach空间, 称其为特殊的广义Orlicz-Sobolev空间. 引理1[9] 设Φ∈C2([0,T))满足条件(2)Φ(t)≥0, Φ(0)>0,并且则(3)其中:且Φ(t)满足类似文献[11], 易得如下问题(1)能量解的存在性定理.定理1 设指数p(x)满足条件(H1), 则问题(1)至少存在一个弱解u: Ω×(0,∞)→, 使得2 主要结果首先, 定义解的能量函数如下:◇◇u+‖其中(g ◇u)(t)=g(t-τ)‖u(t)-记下面给出本文的主要结果, 即能量解的爆破性定理. 定理2 设若(H1),(H2)成立, 初始能量E(0)>0, 且满足则有式(3), 其中且Φ(t)满足证明: 对Φt(t)关于t求导得utt将方程(1)第一个式子两边同乘以u, 并在Ω上积分得即u(τ)(4)将方程(1)第一个式子两边同乘ut, 并在Ω上积分有即注意到对式(5)两边在(0,t)上积分得整理得其中◇结合式(4),(6)并注意到及Ψ(τ)dτ≥0, 经计算易得又因为所以从而得(7)比较式(2)和式(7), 可知于是由引理1知, 存在使得式(3)成立, 且满足参考文献【相关文献】[1] Cavalcanti M M, Domingos Cavalcanti V N, Soriano J A. Exponential Decay for the Solution of Semilinear Viscoelastic Wave Equations with Localized Damping [J]. Electron J Diff Equ, 2002, 2002(44): 227-262.[2] Cavalcanti M M, Oquendo H P. Frictional versus Viscoelastic Damping in a Semilinear Wave Equation [J]. SIAM J Control Optim, 2003, 42(4): 1310-1324.[3] 高云柱, 高文杰. 具强阻尼黏弹性波动方程组解的指数衰退 [J]. 吉林大学学报(理学版), 2010,48(3): 347-352. (GAO Yunzhu, GAO Wenjie. Exponential Decay of Solutions of a Viscoelastic Wave Equation with Strong Damping [J]. Journal of Jilin University (Science Edition), 2010, 48(3): 347-352.)[4] Messaoudi S A. Blow-Up of Positive-Intial-Energy Solutions of a Nonlinear Viscoelastic Hyperbolic Equation [J]. J Math Anal Appl, 2006, 320(2): 902-915.[5] Messaoudi S A. General Decay of the Solution Energy in a Viscoelastic Equation with a Nonlinear Source [J]. Nonlinear Anal, 2008, 69(8): 2589-2598.[6] Antontsev S, Zhikov V. Higher Integrability for Parabolic Equations of p(x,t)-Laplacian Type [J]. Adv Diff Equ, 2005, 10(9): 1053-1080.[7] GAO Yunzhu, GUO Bin, GAO Wenjie. Weak Solutions for a High-Order Pseudo-parabolic Equation with Variable Exponents [J]. Appl Anal, 2014, 93(2): 322-338.[8] CHEN Yunmei, Levine S, Rao M. Variable Exponent, Linear Growth Functionals in Image Restoration [J]. SIAM J Appl Math, 2006, 66(4): 1383-1406.[9] Korpusov M O. Non-existence of Global Solutions to Generalized Dissipative Klein-Gordon Equations with Positive Energy [J]. Electron J Diff Equ, 2012, 2012(119): 1-10. [10] GUO Bin, GAO Wenjie. Blow-Up of Solutions to Quasilinear Hyperbolic Equations with -Laplacian and Positive Initial Energy [J]. Compte s Rendus: Mécanique, 2014, 342(9): 513-519.[11] GAO Yunzhu, GAO Wenjie. Existence of Weak Solutions for Viscoelastic Hyperbolic Equations with Variable Exponents [J]. Bound Value Prob, 2013, 2013(1): 1-8.[12] Messaoudi S A, Talahmeh Ala A. A Blow-Up Result for a Nonlinear Wave Equation with Variable Exponent Nonlinearities [J]. Appl Anal, 2017, 96(9): 1509-1515.[13] GUO Bin. An Inverse Hölder Inequality and Its Application in Lower Bound Estimates for Blow-Up Time [J]. Comptes Rendus Mécanique, 2017, 345(6): 370-377.。

以色列外长佩雷斯1994年在接受诺贝尔和平奖时的演讲

佩雷斯被认为是中东和平进程的设计师。

1994年，佩雷斯与已故总理拉宾和巴勒斯坦领导⼈阿拉法特⼀起，因推动以巴双⽅达成了具有⾥程碑意义的《关于临时⾃治安排的原则声明》，被授予当年的诺贝尔和平奖。

佩雷斯于1993年还被联合国教科⽂组织授予“博⽡尼和平奖”，1996年与约旦国王侯赛因⼀起被授予美国费城⾃由勋章。

Your Majesties， The Chairman and Members of the Nobel Committee， Prime Minister Brundtland， Prime Minister Yitzhak Rabin， Chairman Arafat， Members of the Norwegian Government， Distinguished Guests， I thank the Nobel Prize Committee for its decision to name me among the laureates of the Peace Prize this year. I am pleased to be receiving this prize together with Yitzhak Rabin， with whom I have labored for long years for the defence of our country and with whom I now labor together in the cause of peace in our region. This is a salute to his daring leadership. I believe it is fitting that the prize has been awarded to Yasser Arafat. His quitting the path of confrontation in favour of the path of dialogue， has opened the way to peace between ourselves and the Palestinian people， to whom we wish all the best in the future. We are leaving behind us the era of belligerency and are striding together toward peace. It all began here in Oslo under the wise auspices and good will of the Norwegian people. It is a privilege for me to say thank you to the Norwegian people for its great auspices. From my earliest youth， I have known that while obliged to plan with care the stages of our journey， we are entitled to dream， and keep dreaming， of its destination. A man may feel as old as his years， yet as young as his dreams. The laws of biology do not apply to sanguine aspiration. I was born in a small Jewish town in White Russia. Nothing Jewish remains of it. From my youngest childhood， I related to my place of birth as a mere way station. My family's dream， and my own， was to live in Israel， and our voyage to the port of Jaffa was a dream that came true. Had it not been for this dream and this voyage， I would probably have perished in the flames， as did so many of my people， among them most of my own family. I went to school at an agricultural youth village in the heart of Israel. The village and its fields were enclosed by barbed wire which separated their greenness from the bleakness of the enmity all around. In the morning， we would go out to the fields with scythes on our backs to harvest the crop. In the evening， we went out with rifles on our shoulders to defend our lives. On Shabbat we would go out to visit our Arab neighbours. On Shabbat， we would talk with them of peace， though the rest of the week we traded rifle fire across the darkness. From the Ben Shemen Youth village， my comrades and I went to Kibbutz Alumot in the Lower Galilee. We had no houses， no electricity， no running water. But we had a magnificent view and a lofty dream： to build a new， egalitarian society that would ennoble each of its members. Not all of it came true， but not all of it went to waste. The part that came true created a new landscape. The part that did not come true resides in our hearts to this very day. For two decades， in the Ministry of Defence， I was privileged to work closely with a man who was and remains， to my mind， the greatest Jew of our time. From him I learned that the vision of the future should shape the agenda for the present； that you can overcome obstacles by dint of faith； that you may feel disappointed —— but never despair. And above all， I learned that the wisest consideration is the moral one. David Ben-Gurion has passed away， yet his vision continues to flourish： to be a singular people， to live at peace with our neighbours. The wars we fought were forced upon us. Thanks to the Israel Defence Forces， we won them all， but we did not win the greatest victory that we aspired to： release from the need to win victories. We proved that aggressors do not necessarily emerge as the victors， but we learned that victors do not necessarily win peace. It is no wonder that war， as a method of conducting human affairs， is in its death throes， and that the time has cometo bury it. The sword， as the Bible teaches us， consumes flesh， but it cannot provide sustenance. It is not rifles but people who triumph， and the conclusion from all the wars is that we need better people， not better rifles —— to avoid wars， to win peace. There was a time when war was fought for lack of choice. Today peace is the “no-choice” option for all of us. The reasons for this are profound and incontrovertible. The sources of material wealth and political power have changed. No longer are they determined by the size of territory won by war. Today they are a consequence of intellectual potential，obtained principally by education. Israel， essentially a desert country， has achieved remarkable agricultural yields by applying science to its fields，without expanding its territory or its water resources. Science must be learned； it cannot be conquered. An army that can occupy knowledge has yet to be built. And that is why armies of occupation are passe. Indeed， even for the defense of the country you cannot rely on the army alone. Territorial frontiers are no obstacle to ballistic missiles， and no weapon can shield a nation from a nuclear device. Today，the battle for survival must be based on political wisdom and moral vision no less than on military might. Science， technology， information are —— for better or for worse —— universal， not national. They are universally available. Their availability is not contingent on color of skin or place of birth. Past distinctions between West and East，North and South， have lost their importance in the face of a new distinction： between those who move ahead in pace with new opportunities， and those who lag behind. Countries used to divide the world into their friends and foes. No longer. The foes now are universal —— poverty，famine， religious radicalization， desertification， drugs， proliferation of nuclear weapons， ecological devastation. They threaten all nations， just as science and information are the potential friends of all nations. Classical diplomacy and strategy were aimed at identifying enemies and confronting them. Now they have to identify dangers， global and local， to tackle them before they become disasters. As we leave a world of enemies， as we enter a world of dangers， the future wars which may break out will not be，probably， the wars of the strong against the weak for conquest， but the wars of the weak against the strong for protest. The Middle East must never lose pride in having been the cradle of civilization. But though living in the cradle， we cannot remain infants forever. Today as in my youth， I carry dreams. I would mention two： the future of the Jewish people and the future of the Middle East. In history， Judaism has been far more successful than the Jews themselves. The Jewish people remained small， but the spirit of Jerusalem —— the capital of Jewish life， the city holy and open to all religions —— went from strength to strength. The Bible is to be found in hundreds of millions of homes. The moral majesty of the Book of Books has been undefeated by the ups and downs of history. Moreover， time and again， history has succumbed to the Bible's immortal ideas. The message that the one，invisible God created man in His image， and hence there are no higher and lower orders of man， has fused with the realization that morality is the highest form of wisdom and， perhaps， of beauty and courage， too. Slings， arrows， gas chambers can annihilate man， but they cannot destroy human values， the dignity and freedom of the human being. Jewish history presents an encouraging lesson for mankind. For nearly four thousand years， a small nation carried a great message. Initially， the nation dwelt in its own land； later， it wandered in exile. This small nation swam against the tide and was repeatedly persecuted， banished， downtrodden. There is no other example in all history —— neither among the great empires nor among their colonies and dependencies —— of a nation， after so long a saga of tragedy and misfortune， rising up again， shaking itself free， gathering together its dispersed remnants， and setting out anew on its national adventure. Defeating doubters within and enemies without. Reviving its land and its language. Rebuilding its identity， and reaching toward new heights of distinction and excellence. The message of the Jewish people to mankind is that faith and moral vision can triumph over all adversity. The conflicts shaping up as our century nears its close will be over the content of civilization， not over territory. Jewish culture has lived over many centuries； now it has taken root again in its own soil. For the first time in our history， some five million people speak Hebrew as their native language. That is both a lot and a little： a lot， because there have never been so many Hebrew-speaking people； but a little， because a culture based on five million people can hardly withstand thepervasive， corrosive effect of the global television culture. In the five decades of Israel's existence， our efforts have focused on re-establishing our territorial center. In the future，we shall have to devote our main effort to strengthen our spiritual center. Judaism —— or Jewishness —— is a fusion of belief， history， land and language. Being Jewish means to belong to a people that is both unique and universal. My greatest hope is that our children， like our forefathers， will not make do with the transient and the sham， but will continue to plow the historic Jewish furrow in the fields of human spirit， that Israel will become the center of our heritage， not merely a homeland for our people； that the Jewish people will be inspired by others， but at the same be to them a source of inspiration. The second dream is about the Middle East. In the Middle East most people are impoverished and wretched. A new scale of priorities is needed， with weapons on the bottom and regional market economy at the top. Most inhabitants of the region —— more than sixty percent —— are under the age of eighteen. The Middle East is a huge kindergarten， a huge school. A new future can be and should be offered to them. Israel has computerized its education and has achieved excellent results. Education can be computerized throughout the Middle East， allowing young people， Arabs and others， to progress not just from grade to grade but from generation to generation. Israel's role in the Middle East should be to contribute to a great， sustained regional revival： * A Middle East without wars， without enemies， without ballistic missiles， without nuclear warheads.p >。

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RN
This choice of capillarity term allows to get solution with jumps, i.e with sharp interfaces. Before tackling the global stability theory for the system (N SK ), let us derive formally the uniform bounds available on (ρ, u). 2
t 1 ( ρ|u|2 + Π(ρ) + Eglobal [ρ(., t)])(x)dx(t) + (µD(u) : D (u) RN 2 0 RN |m0 |2 + Π(ρ0 ) + Eglobal [ρ0 ] dx, + (λ + µ)|divu|2 )dx ≤ 2ρ RN where we have: κ Eglobal [ρ(., t)](x) = φ(x − y )(ρ(y, t) − ρ(x, t))2 dy. 4 RN
1
with:
interfaces, whereas in the sharp interface models, the interfaces represent zone of discontinuity for the density. Coquel, Rohde and their collaborators present an alternative model with a capillarity term which avoids spatial derivatives. The model reads: ∂t ρ + div(ρu) = 0 (N SK ) ∂t (ρ u) + div(ρu ⊗ u) − µ∆u − (λ + µ)∇divu + ∇(P (ρ)) = κρ∇D [ρ] (ρt=0 , ut=0 ) = (ρ0 , u0 ) µ > 0 and λ + 2µ > 0
where ρ denotes the density of the ﬂuid and u ∈ RN the velocity, µ and λ represent the viscositent of capillarity, P is a general pressure function. We are particulary interested by Van der Waals type pressure: P : (0, b) → (0, +∞) RT∗ ρ − aρ2 P (ρ) = b−ρ where a, b, R, T∗ are positive constants, R being the speciﬁc gas constant. For ﬁxed values a, b we choose the constant reference temperature T∗ so small as P to be monotone decreasing in some non-empty interval. Further we impose the conditions: u(t, x) → 0, ρ(t, x) → 0 as |x| → +∞, (1.1)
In last section, we consider also more general situations: monotone pressure, other boundary conditions than boundary condition 1.1, at inﬁnity instead of 1.1: u(t, x) → u∞ , ρ(t, x) → ρ∞ as |x| → +∞, (1.2)
∗ Universit´ e Paris XII - Val de Marne 61, avenue du G´ en´ eral de Gaulle 94 010 CRETEIL Cedex T´ el´ ephone : (33-1) 45 17 16 51 T´ el´ ecopie : (33-1) 45 17 16 49 e-mail : haspot@univ-paris12.fr
where ρ∞ is a given nonnegative constant. The term κρ∇D [ρ] corresponds to the capillarity which is supposed to model capillarity eﬀects close to phase transitions in [13]. The classical Korteweg’s capillarity term is D [ρ] = ∆ρ. Based on Korteweg’s original ideas Coquel, Rohde and their collaborators in [4] and Rohde in [17] choose a nonlocal capillarity term D which penalizes rapid variations in the density ﬁeld close from the interfaces. They introduce the following capillarity term: D [ρ] = φ ∗ ρ − ρ where φ is chosen so that: φ ∈ L∞ (RN ) ∩ C 1 (RN ) ∩ W 1, 1 (RN ), φ(x)dx = 1, φ even, and φ ≥ 0.
Study of compactness for compressible ﬂuid models with a speciﬁc Korteweg tensor
Boris Haspot
∗
arXiv:0803.1930v1 [math.AP] 13 Mar 2008
Abstract This work is devoted to the global stability theory of solutions for a general isothermal model of capillary ﬂuids derived by C. Rohde in [17], which can be used as a phase transition model. This chapter is structured in the following way: ﬁrst of all inspired by the result by P.-L. Lions in [15] on the Navier-Stokes compressible system we will show the global stability of weak solution for our system with isentropic pressure and next with general pressure. Next we will consider perturbations close to a stable equilibrium as in the case of strong solution.
1
1.1
Introduction
Presentation of the model
The correct mathematical description of liquid-vapor phase interfaces and their dynamical behavior in compressible ﬂuid ﬂow has a long history. We are concerned with compressible ﬂuids endowed with internal capillarity. One of the ﬁrst model which takes into consideration the variation of density on the interface between two phases, originates from the XIXth century work by Van der Waals and Korteweg [13]. It was actually derived in his modern form in the 1980s using the second gradient theory, see for instance [12, 18]. Korteweg suggests a modiﬁcation of the Navier-Stokes system to account additionally for phase transition phenomena in introducing a term of capillarity. He assumed that the thickness of the interfaces was not null as in the sharp interface approach. This is called the diﬀuse interface approach. Korteweg-type models are based on an extended version of nonequilibrium thermodynamics, which assumes that the energy of the ﬂuid not only depends on standard variables but on the gradient of the density. In terms of the free energy, this principle takes the form of a generalized Gibbs relation, see [18]. In the present chapter, we follow a new approach introduced by Coquel, Rohde and their collaborators in [4]. They remark that the local diﬀuse interface approach requires more regular solution than in the original sharp interface approach. Indeed the interfaces are assumed of non zero thickness, so that the density vary continuously between the two