Axisymmetric solutions of the Chaplygin gas for initial negative radial velocity

J Shanghai Univ(Engl Ed), 2010, 14(5): 380-386Digital Object Identifier( DoI): 10.1007/ s11741-010-0663Axisymmetric solutions of the Chaplygin gas for initialnegative radial velocityGUO Li-hui(郭俐辉)12, SHENG Wan-cheng(盛万成)}1. Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, P. R. China2. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P. R. China@Shanghai University and Springer-Verlag Berlin Heidelberg 2010Abstract In this paper, a three-parameter family of self-similar and weak solutions are constructed rigorously indimensions for all positive time to the euler equations with axisymmetric and radial negative initial velocity for thegas. Under the axisymmetry and self-similar assumptions, the equations are reduced to a system of three ordinary diequations, from which we obtain structures of solutions besides their existence. The solutions exhibit some phenomena, suchas formation and evolution of black hole, expansion and explosive expansion, in the evolution of universe.Keywords Chaplygin gas, axisymmetric, concentration2000 Mathematics Subject Classification 35L65, 35L67, 76L05, 76N10IntroductionBeing different from the Euler equations of polytiOnsh this paper, we consider the axisymmetric solu- tropic gas, the Euler equations of Chaplygin gas haveIs of the Chaplygin gas where initial radial velocitythree eigenvalues and each of them is linear degeneris negative for the 2D isentropic Euler equationsate. For the Chaplygin gas, Brenier studied the onedimensional Riemann problems and obtained the solu-tions with concentration when initial data belong to aPt+(puz+(ou)y=0,(pu)t+(pu+p)r+(puu)v=0,(1) certain domain in phase plane. In 2009, Serrel9)studied(p0)++(pu)x+(p2+p)y=0,irrotational Chaplygin gas. He constructivelythe existence of the transonic solutions for twowith the state equationsaddle and vortex of 2D Riemann problem. Recently,in(10, we solved the lD Riemann problem completely(2) and systematically studied the 2D Riemann problemsfor isentropic Chaplygin gas dynamic system. Then weThe state equation(2)was introduced by Chaplygin!)ric solutions of the ChaplygiTsien(2 and von Karman(3 as a suitable mathematicalgas where the initial radial velocity was positive and obtained detailed structures of the solutions besides theirexistenceof an airplane in aerodynamics. The Chaplygin gas hasIn this paper, we reduce the initial-value problem ofbeen advertised as a possible model for dark energy to the partial differential equations to an infinite boundaryexplain an accelerated expansion of the universe.value problem of the ordinary differential equations byThe study of 2D Riemann problem to the Euler the special three parameter family of data, namely, theystem for polytropic gas was initiated by Zhang andZheng 4. For the polytropic gas, a special Riemann negative. By studying this infinite boundary problemproblem was studied by Zhang and Zheng 5-7. They we get a global solution which consists of three nontriv-obtained the detailed structures of the solutions besides ial connecting orbits chained together at most.their existence. Our main task is to consider the specialWe find that our solutions show some typical properRiemann problem for the Chaplygin gasReceived Nov. 24. 2009: Revised Jan 6. 2010中国煤化工Project supported by the Natural Science Foundation of China(GraCNMHGai Leading AcademicDiscipline Project(Grant No. J50101)CorrespondingauthorSHENGWan-cheng,PhD,Prof,Email:mathwcsheng@shu.edu.cnhanghai Univ(Engl Ed), 2010, 14(5)ties: There appear discontinuities in the solutionsSince both(4)and initial data(7)remain invariant unsatisfy the Rankine-Hugoniot relations, and theder self-similar transformations, we look for self-similarof back discontinuities is less than that of frontsolutions(p, u, v)which only depend on 5=F. Thus,nuities. This portion of solutions exhibits some phenwe can get the following infinite boundary-value probena in the evolution of universe, such as explosive ex- lem for a system of ordinary differential equationspansion of universe. In addition the solutions with con- dp pe du 12 dvcentration exhibit the formation and evolution of black df54' ds 54' deon The paper is organized as follows: In Section 1, some (00 p, ,0)=(po, o, vo),luminaries are delivered including axisymmetry in-duction, far-field solutions, intermediate equations, and wherethe rankine-Hugoniot relations; The solutions without4:=a2-(u-5)2,6:=v2-u(5-u)swirls while uo <0 are shown in Section 2; In Sections3 and 4. the solutions in the intermediate field and the∑:=(-u)e-u△=2(6-u)-uc2,(10)inner-field solutions for uo <0 are shown respectively.c:=VP(p)=-1 Prreliminariesis the speed of soundWe impose axisymmetry to the system iin the presentIn this paper, the global solutions or their existenceaper, which means that solutions (u, v, p) satisfyare constructed or established for problems(8),(9 )withthe initial data satisfying po>0, u0 <0, Do E R. Sincep(t,r,)=p(t,r,0),the case vo <0 can be transformed by v--v to thecase vo>0, we can assume vo>0u(t, r, 0) cos 8 -sin/u(t, r,0)(3) 1.1 Far-field solutions and intermediatev(t,r,6)e cose/u(t, r, Ofield equationse introduce the variable s =4, the infinitefor allt>0, 8ER, and r>0, where(r, e)are the polar boundary-value problems(8), (9)can be written ascoordinates. Then, for the smooth solutions, system(1)can be written todpp(u(1-u)-82)ds92c2-(1-8u)2Pt+(pu)+du 8uc2-v2(1Pr Uds 92c2(11)dv0,(P, u, v)ls=0=(p0, uo, vo),where p =p(t,r,0),u=u(t, r, 0)and v =v(t, r, 0) which is well-posed and has a unique local solution forrepresent the radial velocities and the pure rotational any initial datum with po>0. Thus the problems(8)velocities in the flow, respectivelyWe consider Riemann problem(4)with the initial(9)has a local solution near E= +oo for any datumdata(Po, uo, vo)(p(0,r,6),u(0,r,6),v(0,r,6)5)I=8u,J=8,K(13)(p0(6),uo(6),to(6)using the state equation(2), system(11)can be rewrit-We impose the axisymmetry condition(3)into(5)ten asand find that the initial data satisfydI2K2-(1-D(2+f(1-m)p(0,r,6)Sds 1-I,u(U,TdK K(K+J+IH中国煤化工where po>0, uo and vo are arbitrary constants. Hence correour data fortionsCNMHG2), we look for solu-PionT0)=p(0,r,0)T(7)(,,K)~(a0,,c),(15)382J Shanghai Univ(Engl Ed), 2010, 14 (5 ) 380-3861.2 Rankine-Hugoniot relation2 Intermediate field solutions for uo <0In self-similar plane(,n)=(, 2), system(1)canwithout swirlsreduced toWe deal with the case uo =0, i.e., the distribution of-EPE +(pu)E-nPn+(pu)n=0,the integral curves on the invariant surface J=0. Weshall assume uo <0 in this section. Hence we have a5(ou)E+(ou+pE-n (pu)n+(puv)n=0, (16) subsystem for(I, K)E(pu)e+(puu)e-n(ou)n+(pu+p)n=0dI,2K2-(1-D)2dsLet U=u-S,V=v-n. The system(16)can bedKwritten as=K2-(1-(PU)E+(pV)n=-2pIntroducing a new parameter T, we can rewrite(23)asPU2+p)+(pUV)n=-3pU,(17)(pUV)E+(pV+p)n=-3pV.dr=1(1-D2-2K2)dKnuous solutions, the Rankine-Hugoniot conId be satisfiedd7=8(1-D2-K(pl)o=pVIf uo also vanishes, then we have a trivial solutionp= po,U=UDenote a: =1-I-K2, b: =(1-1)2-21(pUNo=pv+p,d:=(1-1)2-K. It can be verified that our far-fieldsolutions starting at s=0+ enter the regionlg)=g1-g2, q1, 92 are states on the two sides. When n1:=((I,J,K)I <0, K>0, d>0)axisymmetry is assumed, the discontinity curve has an The stationary points of (24)in n1 are the pointsinfinite slope at the 5-axis. Thus we set o = oo in (18) (I, K)=(0,0),(1,0),(0, 1), (-1, 2)to obtainFrom the equations(24)we know that I=0,K=0and 1-I-K=0 are invariant surfaces, sincelpU2+P9)4-1-K2=(1-r-K0(K2-k+P-1LUV=0.Thus we conclude that all solutions starting in the clo-sure do not leave the closure 3, as T increase. TheThe first solution to(19) is the slip lineaxis K=0 and the axis I=0 in0< K< 1 are trivialsolutionsp1=P2,5=ul=u2Lemma 2.1 All integral curves of (24)from thenegative radial velocity doThe other solutions to(19solutionsorigin(I, K)=(0, 0)withnot always under the curve a=0Proof We only need to prove the integral curve is=u1±c1,ot always under the curve a=0inI∈(-∞,-1).Ifv2=t1( 21) I E(00, 1), there have one integral curve always underthe curve a=0, then we have 1-I>K and(-u2)p2=(6-u1)p1drBy using I 9u, J= sU, K= sc, the Rankine-Hugoniotr(1-)2-2K2)<-(1+1)relation can be rewritten asHence we getl1±K1=1d(1+D1+n2J1=J2中国煤化工(22) and11-I2CNMHG1+lm1+l0J Shanghai Univ(Engl Ed), 2010, 14(5): 380-386383Thus there exist Tm, s.t., To-Tm =1r, which means for some constants 0>0 and To. Thus s approaches a-H<0, it contradicts I E(-00, -1)finite number as T-o0 since(1-1)2-K approachesTheorem 2ll integral curves of(24)from the zero exponentiallyorigin(I, k)=(0, 0)with initial negative radial velocityFor the solutions ending at(0, 1), we extent the solu-go(L, K)=(1,v2)or(0, 1), except the trivial solution tions by the constant states(0,0, p"),where p""is thevalue of p at the ending point and satisfy the relationProof On the curve a=0.i.e,K=1-I, we findnat gk=0 and d=-1(1-I)(1+I), then we havep=8,ie,c=*,p"=sdr <0 if I E(-00,-1)and a>0 if I E(1, 0). On Whent=1, E is the radius of the circle of the constantthe curve b=0.i.e k2= d-d. Hence we find that state in the physical plane. In all, we have constructeddr=0 and dk=K(1-n, and we have k <0 if global solutions to the reduced system(24)in the caser∈(-∞,-1)and$>0ifI∈(-1,0From Lemma 2.1, we know that some integral curves 3 General solutions in the intermediateacross the curve a=0 and b=0 only on I E(1,0)then go to the stationary point (1, 0). Thus all integralcurves go to the stationary(1,0 )or(-1, v 2)except theConsider the case vo>0 and uo <0 for system(14).trivial solution K =0 (see Fig. 1).We find that it is convenient to introduce a new variableFrom equation(25), we know that Ins approaches T to write the system(14)into the forminfinity as the solutions approach the point (I, K)(1, v2), since(1-1)2-K2#0 at the point.Thus8→+∞,ie,5→0+, and the solutions have thedn=(1-D(4-D(2+m1-)-2K)asymptoticd=1-2D(1-n2-x2,u()=-5,p(p)()=22,ass→0+dK=K(1-D(1-I-2-K2)Next, we show that the parameter s approaches finitenumber when solutions of(24)approach to the point s=8(1-D)((1-1)2-K2)0, 1). We can linear the equations(24)at (L,K)(0, 1)to findThis is an autonomous system for(I, J, K, 9), and thefirst three equations(28)form an autonomous subsys-dfor (L,J, K). Fred(K-1)(26)1=1,J=0,K=0,andD=1-I-K=0are=-I-2(K-1)invariant surface. SinceD(1-1)(-k2-2K+P2-1-J2)A:=1-I-J2-KB;=(1-D(J2+I(1-I)-2IK2,a:={(I,K)J>0,K>0,D>0}It can be verified that all far-field solutions withuo<0, vo >0 and po >0 enter the region S2 in g>0.Fig1 Phase portrait of solutions without swirlsFrom the equations(28)we know that I=0,K=0and D=1-I-K=0 are invariant surfaces. Thus weThe eigenvalues of(26)are A1=-1, A2 =-2, corre- conclude that all solutions starting in the closer 2 dosponding to the eigenvectors are(0, 1)and(-1, 1). Then not leave the closure 52 as increasesolutions of (24)near(0, 1)approach(0, 1)exponentiallyasT→+∞. From the equation(25), we find中国煤n2 are J2=I(1-DCN MHGes of (28)from theonlIn==/((1-n2-K2)dr(27)[U, U, U) cau uuv always be under thesurface A= 0J Shanghai Univ(Engl Ed), 2010, 14(5 380-386Proof We only need to prove that the integral(22)(see Fig. 2), and then end on E-. This portion ofcurve is not always under the curve A =0 in I E solutions exhibit some phenomena, such as explosive ex-(00,-3). If I E(00, 3), the integral curve always pansion of universe, in the evolution of universe. Fromunder the curve A= 0, then we have J4< 1-L, is the second equation of (14), we obtainK2<1-I. Thus we can obtainds 91-Id=(1-DB<-+m2d J1-2IHence s approaches finite values.Then we getSince the solutions go to the stationary point F, wecontinue the solutions with concentration. This portiond(1+n)(1+1)2of solutions exhibits some phenomena, such as the for-mation and evolution of black hole in the evolution ofwhere exist Tm, s.t. 2mwhich means -n<0. It contE:={(1,JK川=1(1-D),K=1-I}F={(I,K)r=1,K=0}E={44=1(-D.K=1-4<1s4Fig 2 Solutions on D=0E-:={(1,JK)J=(1-D,K=1-,01(1-1)in00 and uo<0 go to the stationary points E and F, and J and K eventually go to infinity. Then the scaledexcept the trivial solution K=0.variables I, J and K are not suitable for this portion ofLinearizing the system(28)at the point e, we find the solutions. We introduce new variablesset of directionsRcb=5,v≈t73;=(-(1-D(4I+1),0,(1-D(I+1)Introducing a new parameter T and using the statewhere0System(33)can be written asE*:={U,v,R")R=U*+1,V"=U*,dX0<2U0.dr=2B2(1-x)(1+V-R2(1-X)=M2Thus we easily know that e>0, cu>0 and B1>0.We first find all stationary points of 33 )in the 523. Theyare given inBy linearizing at the point(X,V2,R2)=(0,0,0),weobtainG1:={(U,v,B)R=U,V=0}dX=-2X+V2G2:={(U,vR)U=0,00 and enter into 3. Note that theJ. SIAM Journal on Mathematical Analysis, 1990,unstable manifold, i. e, the R2 axis, ends on one end of21(3):593630(35). We conclude that nearby integral curves in the [5] ZHANG T, ZHENG Y X. Axisymmetric solutions of thecenter unstable manifold end on (35)(see Fig 3)Euler equations for polytropic gases J]. Archive for Ra-tional Mechanics and Analysis, 1998, 142 (3): 253-279two dimensional compressible Euler equations [J].Dis-crete and Continuous Dynamical System, 1997, 3(1):117-133.[7 ZHENG Y X. Systems of conservation laws,two-dimensional Riemann problems M. BostonBirkhauser. 2001: 119-193[8]BRENIER Y. Solutions with concentration to the Rimann problem for the one-dimensional Chaplygin gasequations J]. Journal of Mathematical Fluid Mechan-ics,2005,7(3):S326S331Fig 3 Unstable manifold9 SERRE D. Multidimensional shock interaction forNow we have solved the problems(33)-(35). By us-Chaplygin gas J. Archive for Rational Mechanics andng the third equation in( 32), we conclude that r is anAnalysis,2009,191(3):539577increasing function of T∈R,→0 as t goes to-∞, O GUo L H, SHENG W C,and→ a as t goes to+ oo for some a∈(0,+∞),dimensional Riemann problemropic ChaplyginFrom equation (34), we know that c is decreasinggas dynamic system J. Communications on Pure andwhen T→∞, since R-U>0ande>0.Thus,byApplied Analysis, 2010, 9(2): 431-458.when f+0+. In addition we can conclude that B>c [11] KELLEY A. The stable, center stable, center, center un-which exhibits the phenomena of the expansion in thetial Equations, 1967, 3: 546-570.evolution of the universeIn all, we have constructed the axisymmetric solu- [12] CARR J. Applications oftions of the Chaplygin gas with initial negative radialplied mathematical sciencevelocity completelySpringer-Verlag, 1981: 119-193References[13 HENRY D. Geometric theory of semilinear parabolicEquations [M]. Berlin, Heidelberg: Springer-Verlag,[1] CHAPLYGIN S A. 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