广义Chaplygin气体Aw-Rascle交通流方程组解奇异性的形成.pdf

第40卷第4期2023年7月新疆大学学报（自然科学版）（中英文）Journal of Xinjiang University(Natural Science Edition in Chinese and English)Vol.40,No.4Jul.,2023广义Chaplygin气体Aw-Rascle交通流方程组解奇异性的形成辛晓庆，郭俐辉(新疆大学 数学与系统科学学院，新疆 乌鲁木齐 830017)摘要：研究了广义 Chaplygin 气体 Aw-Rascle 交通流方程组经典解奇异性的形成.利用特征分解方法,当初值满足一定条件时,证明了交通流模型柯西问题经典解的密度本身在有限时间内会发生爆破.此外,通过数值模拟对该物理现象进行了验证.关键词：Aw-Rascle 交通流方程组;广义 Chaplygin 气体;奇异性的形成;特征分解DOI：10.13568/ki.651094.651316.2022.12.09.0002中图分类号：O175.2文献标识码：A文章编号：2096-7675(2023)04-0422-011引文格式：辛晓庆,郭俐辉.广义 Chaplygin 气体 Aw-Rascle 交通流方程组解奇异性的形成J.新疆大学学报(自然科学版)(中英文),2023,40(4):422-432.英文引文格式：XIN Xiaoqing,GUO Lihui.Formation of singularities in solutions to the Aw-Rascle traffic flowsystem for the generalized Chaplygin gasJ.Journal of Xinjiang University(Natural Science Edition in Chinese andEnglish),2023,40(4):422-432.Formation of Singularities in Solutions to the Aw-Rascle TrafficFlow System for the Generalized Chaplygin GasXIN Xiaoqing,GUO Lihui(School of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830017,China)Abstract：The formation of singularities in the classical solutions to the Aw-Rascle traffic flow system withgeneralized Chaplygin gas is studied.By using the method of characteristic decomposition,we prove that thedensity itself of the classical solutions of the Cauchy problem for the traffic flow model will blow up in finite timewhen the initial data meets certain conditions.Additionally,this physical phenomenon is verified by numericalsimulation.Key words：Aw-Rascle traffic flow system;generalized Chaplygin gas;singularity formation;characteristicdecomposition0引 言本文研究Aw-Rascle(AR)交通流方程组t+(u)x=0(u+p()t+(u(u+p()x=0(1)其中:u是速度,是密度,p=p()表示交通压力.收稿日期：2022-12-09基金项目：国家自然科学基金“相对论欧拉方程组音速-超音速解的存在性及相关问题研究”（12161084）；新疆维吾尔自治区自然科学基金“双曲型偏微分方程组初边值问题解的存在性”（2022D01E42）作者简介：辛晓庆（1997-），女，硕士生，从事偏微分方程及其应用的研究，E-mail:通讯作者：郭俐辉（1979-），男，博士，教授，从事偏微分方程及其应用的研究，E-mail:第4期辛晓庆，等：广义Chaplygin气体Aw-Rascle交通流方程组解奇异性的形成423为了弥补Daganzo1提出的二阶交通流模型的不足,Aw和Rascle2建立了AR模型,此模型被广泛用于描述交通阻塞、交通事故的形成及其它交通现象.此外,AR模型还是多车道交通流模型34和混合交通流模型5的基础.在不同的状态方程下,方程组(1)黎曼解的结构有不同的表现形式.当状态方程为多方气体时,方程组(1)是真正非线性且严格双曲的,其中Aw和Rascle2最早研究了其黎曼问题,并获得了包含真空的黎曼解.Sun67分别考虑了当真空存在和真空不存在时一维交通流方程组波的相互作用.Shen和Sun8研究了包含扰动初值的交通流方程组,随着压力的消失,其黎曼解趋于输运方程组的狄拉克激波解和真空解.本文分别考虑广义Chaplygin气体状态方程p=A(2)和p=A,lim0p(,)=0(3)其中:A 0和0 1是常数.当A=1和=1时,该状态方程被称为Chaplygin气体,最早由Chaplygin9,Tsien10和Karman11引入.之后,为了更好地描述宇宙加速膨胀现象和完善Chaplygin气体模型,很多学者把该模型推广到了广义Chaplygin气体模型,它是经典的暗能量模型1213.近年来,关于Chaplygin气体和广义Chaplygin气体的黎曼问题已经有许多研究成果.Brenier14研究了一维Chaplygin气体Euler方程组的黎曼问题,发现当黎曼初值满足一定条件时,黎曼解中会出现狄拉克激波.Cheng和Yang15研究了一维相对论Chaplygin气体Euler方程组的黎曼问题.Wang16构造了广义Chaplygin气体一维交通流方程组的黎曼解,并证明了狄拉克激波解的存在性和唯一性.Chen和Qu17研究了三个常初值Chaplygin气体二维黎曼问题.Serre18研究了二维Chaplygin气体Euler方程组黎曼问题,并得到了两种类型跨音速解的存在性.更多研究成果,请参阅文献19-24.众所周知,对线性双曲方程组来说,如果初值足够光滑,那么其柯西问题在整个被定义的区域上一般存在整体经典解.但是,对非线性双曲方程组来说,不管初值多么光滑多么小,其柯西问题只存在局部的经典解,而在有限时间内,经典解的奇异性一定会发生.Lax25给出了非线性双曲偏微分方程组解奇异性的发展.Pan和Zhu26研究了一维全Euler方程组奇异性的形成.Chen,Pan和Zhu27得到了可压缩Euler方程组奇异性的形成.Athanasiou和Zhu28研究了相对论Euler方程组奇异性的形成.基于以上的研究成果,本文主要考虑方程组(1)带有初值(,u)=(0,u0(x),x(,+)(4)的解奇异性的形成,其中:0是正常数.u0(x)=ua,xb,u(a)=ua,u(b)=ub,u(a)=u(b)=0,当x(a,b)时,u(x)0,使得问题(1),(2),(4)在区域(T):=(x,t)|x(,+),0tk0,则存在一个(x,t)(,+)(0,+),使得柯西问题(1),(2),(4)在区域(t)内有一个经典解,且解满足lim(x,t)(x,t)(x,t)=+(10)证明 令Ca1:x=x1(t;a)和Cb1:x=x1(t;b)分别由dx1(t;a)dt=u(x1(t;a),t),t0 x1(0;a)=a和dx1(t;b)dt=u(x1(t;b),t),t0 x1(0;b)=b(11)来决定.这里Ca1和Cb1是两条分别从点(a,0)和点(b,0)发出的轨迹线,如图1所示.图 1从(a,0)和(b,0)两点发出的轨迹线由质量守恒知ddtZx1(t;b)x1(t;a)(x,t)dx=0(12)426新疆大学学报（自然科学版）（中英文）2023年沿着Ca1对方程组(6)的第一个方程积分得u=ua+kk0uak0(13)因此,由式(11)和(13)知dx1(t;a)dt=u(x1(t;a),t)uak0(14)沿着Ca1对式(14)积分,有x1(t;a)a+(uak0)t,(t0)(15)沿着Cb1对方程组(6)的第一个方程积分得u=ub+kk0ub(16)因此,由式(11)和(16)知dx1(t;b)dt=u(x1(t;b),t)ub(17)沿着Cb1对式(17)积分,有x1(t;b)b+ubt,(t0)(18)用式(18)减去式(15)得x1(t;b)x1(t;a)ba(uaubk0)t,(t0)(19)所以,存在0t(ba)/(uaubk0)使得x2(t,a)x1(t,a)=x1(t,b),且当0tt时,有x2(t,a)x1(t,a)x1(t,b).此外,由式(12)知Zx1(t;b)x1(t;a)(x,t)dx=0(ba),再结合式(19)得maxx1(t;a)x0知,存在0t(ba)/(uaubk0)和x(,+)使得式(10)成立.1.3数值模拟本节将利用Lax-Friedrichs格式的数值模拟来验证定理1的正确性.在数值实验中,A取值为1,取值为1/2.下面给出两个例子.例1给定初值(,u)=(1,u0(x),x(,+),其中u0(x)=8,x32.数值结果如图2所示,存在一个(x,t)(5,6)0.5,使得柯西问题(1),(2),(4)经典解的密度发生爆破.第4期辛晓庆，等：广义Chaplygin气体Aw-Rascle交通流方程组解奇异性的形成427图 2当 t=0.5 时问题(1),(2),(4)中 和 u 的值例2给定初值(,u)=(1,u0(x),x(,+),其中u0(x)=10,x4.数值结果如图3所示,存在一个(x,t)(8,9)0.7,使得柯西问题(1),(2),(4)经典解的密度发生爆破.图 3当 t=0.7 时问题(1),(2),(4)中 和 u 的值由例1和例2可知数值模拟结果与定理1的结论一致.2方程组(1)和(3)奇异性的形成2.1特征方程和特征分解把方程组(1)和(3)化简为t+ux+ux=0ut+(uA)ux=0(20)其特征值为1=u,2=uA,对应的左特征向量为1=(0,1),2=(A1,1).用左特征向量i(i=1,2)左乘方程组(20)得到特征方程1(uA)=02u=0(21)428新疆大学学报（自然科学版）（中英文）2023年其中:1:=t+ux,2:=t+(uA)x.引理3方程组(20)中有以下交换关系1221=(1)11(12)(22)证明 由i(i=1,2)的定义得x=(12)/(A),所以1221=(t+1x)(t+2x)(t+2x)(t+1x)=(1221)x=1(uA)x=(1)11(12).命题2关于变量,我们有特征分解12=n1221=n21(23)其中:n1:=211,n2:=(1)11+(+1)12.证明 由式(22)作用于变量u得12u21u=(1)11(1u2u),又因为2u=0,所以21u=(1)111u,由方程组(21)知1u=A11,因此21=(1)1(1)2+(+1)112(24)由式(22)作用于变量得1221=(1)11(12),把式(24)代入上式得12=2112.2.2奇异性的形成利用特征方法也可得到柯西问题(1),(3),(4)经典解的局部存在性,即存在一个足够小的时间T 0使得问题(1),(3),(4)在区域(T):=(x,t)|x(,+),0tA0,则存在一个(x,t)(,+)(0,+),使得柯西问题(1),(3),(4)在(t)内有一个经典解,且解满足lim(x,t)(x,t)(x,t)=+(25)证明 令Ca1:x=x1(t;a)和Cb1:x=x1(t;b)分别由dx1(t;a)dt=u(x1(t;a),t),t0 x1(0;a)=a和dx1(t;b)dt=u(x1(t;b),t),t0 x1(0;b)=b(26)来决定.由质量守恒知ddtZx2(t;b)x1(t;a)(x,t)dx=0(27)沿着Ca1对方程组(21)的第一个方程积分得u=ua+AA0uaA0,再结合式(26)有dx1(t;a)dt=u(x1(t;a),t)uaA0(28)沿着Ca1对式(28)积分,有x1(t;a)a+(uaA0)t,(t0)(29)沿着Cb1对方程组(21)的第一个方程积分得u=ub+AA0ub(30)所以,由式(26)和(30)知dx1(t;b)dt=u(x1(t;b),t)ub(31)沿着Cb1对式(31)积分,有x1(t;b)b+ubt,(t0)(32)用式(32)减去式(29)得x1(t;b)x1(t;a)ba(uaubA0)t,(t0)(33)由式(27)知Zx2(t;b)x1(t;a)(x,t)dx=0(ba),再结合式(33)得maxx1(t;a)x0知,存在0t(ba)/(uaubA0)和 x(,+)使得式(25)成立.430新疆大学学报（自然科学版）（中英文）2023年2.3数值模拟本节将利用Lax-Friedrichs格式的数值模拟来验证定理2的正确性.在数值实验中,A取值为1,取值为1/2,取值为0.001.下面给出两个例子.例3给定初值(,u)=(1,u0(x),x(,+),其中u0(x)=8,x.数值结果如图4所示,存在一个(x,t)(7,8)1.0,使得柯西问题(1),(3),(4)经典解的密度发生爆破.图 4当 t=1.0 时问题(1),(3),(4)中 和 u 的值例4给定初值(,u)=(1,u0(x),x(,+),其中u0(x)=7,x3.数值结果如图5所示,存在一个(x,t)(4,5)0.5,使得柯西问题(1),(3),(4)经典解的密度发生爆破.图 5当 t=0.5 时问题(1),(3),(4)中 和 u 的值由例3和例4可知数值模拟结果与定理2的结论一致.参考文献：1DAGANZO C F.Requiem for second-order fluid approximations of traffic flowJ.Transportation Research Part B:Methodological,1995,29(4):277-286.第4期辛晓庆，等：广义Chaplygin气体Aw-Rascle交通流方程组解奇异性的形成4312AW A,RASCLE M.Resurrection of“second order”models of traffic flowJ.SIAM Journal Applied Mathematics,2000,60(3):916-938.3GREENBERG J M.Extensions and amplifications of a traffic model of Aw and RascleJ.SIAM Journal Applied Mathematics,2001,62(3):729-745.4GREENBERG J M,KLAR A,RASCLE M.Congestion on multilane highwaysJ.SIAM Journal Applied Mathematics,2003,63(3):818-833.5MOUTARI S,RASCLE M.A hybrid Lagrangian model based on the Aw-Rascle traffic flow modelJ.SIAM Journal AppliedMathematics,2007,68(2):413-436.6SUN M N.Interactions of elementary waves for the Aw-Rascle modelJ.SIAM Journal Applied Mathematics,2009,69(6):1542-1558.7SUN M N.A note on the interactions of elementary waves for the Aw-Rascle model without vacumJ.Acta Mathematica Scientia,2011,31(4):1503-1512.8SHEN C,SUN M N.Formation of shocks and vacuum states in the vanishing pressure limit of Rirmma solutions to the perturbedAw-Rascle modelJ.Journal of Differential Equations,2010,249:3024-3051.9CHAPLYGIN S.On gas jetsJ.Scientific Memoirs,Moscow University Mathematic Physics,1904,21:1-121.10TSIEN H S.Two dimensional subsonic flow of compressible fluidsJ.Journal of the Aeronautical Science,1939,6:399-407.11KARMAN T.Compressibility effects in aerodynamicsJ.Journal of the Aeronautical Science,1941,8:337-365.12SAHNI V.Dark matter and dark energyJ.Lecture Notes in Physics,2005,653:141-179.13ZHANG X,WU F Q,ZHANG J.New generalized Chaplygin gas as a scheme for unification of dark energy and dark matterJ.Journal of Cosmology and Astroparticle Physics,2005,2006(1):731-750.14BRENIER Y.Solutions with concentration to the Riemann problem for one-dimensitional Chaplygin gas equationsJ.Journal ofMathematical Fluid Mechanics,2005,7:326-331.15CHENG H J,YANG H C.Riemann problem for the relativistic Chaplygin Euler equationsJ.Journal of Mathematical Analysisand Applications,2011,381(1):17-26.16WANG G D.The Riemann problem for one dimensional generalized Chaplygin gas dynamicsJ.Journal of Mathematical Analysisand Applications,2013,403(2):434-450.17CHEN S X,QU A F.Two dimensional Riemann problem for Chaplygin gasJ.SIAM Journal on Mathematical Analysis,2012,44:2146-2178.18SERRE D.Multi-dimensional shock interaction for a Chaplygin gasJ.Archive for Rational Mechanics and Analysis,2008,191:539-577.19尹淦.AW-Rascle模型的Riemann解的压力消失极限J.新疆大学学报(自然科学版),2013,30(3):289-296.20GUO L H,LI T,PAN X L.The Riemann problem with delta initial date for the one-dimensional Chaplygin gas equations with asource termJ.Nonlinear Analysis:Real World Applications,2018,41:588-606.21LAI G,SHENG W C.Elementary wave interactions to the compressible Euler equations for Chaplygin gas in two dimensionsJ.SIAM Journal on Applied Mathematics,2016,76(6):2218-2242.22SHEN C.The Riemann problem for the Chaplygin gas equations with a source termJ.ZAMM-Journal of Applied Mathematicsand Mechani,2016,96(6):681-695.23宋赟,郭俐辉.带有源项的Chaplygin气体非对称Keyfitz-Kranzer方程组含狄拉克初值的广义黎曼问题J.新疆大学学报(自然科学版),2019,36(3):292-302.24SUN M N.The exact Riemann solutions to the generalized Chaplygin gas equations with frictionJ.Communications in NonlinearScience and Numerical Simulation,2016,36:342-353.25LAX P.Development of singularities of solutions on nonlinear hyperbolic partial differential equationsJ.Journal of MathematicalPhysics,1964,5:611-613.26PAN R H,ZHU Y.Singularity formation for one dimensional full Euler equationsJ.Journal of Differential Equations,2016,261(12):7132-7144.27CHEN G,PAN R H,ZHU S G.Singularity formation for the compressible Euler equationsJ.SIAM Journal on MathematicalAnalysis,2017,49(4):2591-2614.28ATHANASIOU N,ZHU S G.Formation of singularities for the relativistic Euler equationsJ.Journal of Differential Equations,432新疆大学学报（自然科学版）（中英文）2023年2021,284:284-317.29LAI G,ZHU M.Formation of singularities of solutions to the compressible Euler equations for a Chaplygin gasJ.AppliedMathematics Letters,2022,129:107978.30DAI Z H,ZHANG T.Existence of a global smooth solution for a degenerate goursat problem of gas dynamicsJ.Archive forRational Mechanics and Analysis,2000,155(4):277-298.31LI J Q,ZHANG T,ZHENG Y X.Simple waves and a characteristic decomposition of the two dimensional compressible EulerequationsJ.Communications in Mathematical Physics,2006,267:1-12.32COURANT R,FRIEDRICHS K O.Supersonic flow and shock wavesM.New York:Interscience,1948.33LI T T,YU W C.Boundary value problem for quasilinear hyperbolic systemsM.North Carolina:Duke University,1985.责任编辑：赵新科(上接第421页)21ZHANG X Y,LYU X X,LI X D.Sampled-data-based lag synchronization of chaotic delayed neural networks with impulsivecontrolJ.Nonlinear Dynamics,2017,90(3):2199-2207.22ABDURAHMAN A,ABUDUSAIMAITI M,JIANG H J.Fixed/predefined-time lag synchronization of complex-valued BAMneural networks with stochastic perturbationsJ.Applied Mathematics and Computation,2023,444:127811.23SADER M,ABDURAHMAN A,JIANG H J.General decay lag synchronization for competitive neural networks with constantdelaysJ.Neural Processing Letters,2019,50:445-457.24LI H L,CHEN S L,YANG J K.Fixed-time synchronization of delayed quaternion-valued neural networks with fractional-orderunder quantized controlJ.Journal of Xinjiang University(Natural Science Edition in Chinese and English),2023,40(1):1-9.25ZHENG C D,ZHANG L L.On synchronization of competitive memristor-based neural networks by nonlinear controlJ.Neuro-computing,2020,410:151-160.26PANG L,HU C,YU J,et al.Fixed/preassigned-time synchronization for impulsive complex networks with mismatched parame-tersJ.Neurocomputing,2022,511:462-476.27冯靓,胡成,于娟.耦合忆阻复值神经网络的固定时间同步J.新疆大学学报(自然科学版)(中英文),2021,38(2):129-143.28XU Z L,PENG D X,LI X D.Synchronization of chaotic neural networks with time delay via distributed delayed impulsivecontrolJ.Neural Networks,2019,118:332-337.29HAN S Y,HU C,YU J,et al.Stabilization of inertial Cohen-Grossberg neural networks with generalized delays:a direct analysisapproachJ.Chaos,Solitons and Fractals,2021,142:110432.30CUI N,JIANG H J,HU C.Global asymptotic and robust stability of inertial neural networks with proportional delaysJ.Neurocomputing,2018,272:326-333.31ZHANG Y K,ZHOU L Q.Stabilization and lag synchronization of proportional delayed impulsive complex-valued inertial neuralnetworksJ.Neurocomputing,2022,507:428-440.32POPOV V M,GEORGESCU R.Hyperstability of control systemsM.Berlin,Heidelberg:Springer-Verlag,1973.责任编辑：赵新科