VANISHING VISCOSITY SOLUTIONS OF THE COMPRESSIBLE EULER EQUATIONS WITH SPHERICAL SYMMETRY AND LARGE INITIAL DATA

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1 Repot no. OxPDE-15/ VANISHING VISCOSITY SOLUTIONS OF THE COMPRESSIBLE EULER EQUATIONS WITH SPHERICAL SYMMETRY AND LARGE INITIAL DATA by Gui-Qing Chen Univesity of Oxfod nd Mikhil Peepelits Univesity of Houston Oxfod Cente fo Nonline PDE Mthemticl Institute, Univesity of Oxfod Andew Wiles Building, Rdcliffe Obsevtoy Qute, Woodstock Rod, Oxfod, UK OX 6GG Febuy 15

2 VANISHING VISCOSITY SOLUTIONS OF THE COMPRESSIBLE EULER EQUATIONS WITH SPHERICAL SYMMETRY AND LARGE INITIAL DATA GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Abstct. We e concened with spheiclly symmetic solutions of the Eule equtions fo multidimensionl compessible fluids, which e motivted by mny impotnt physicl situtions. Vious evidences indicte tht spheiclly symmetic solutions of the compessible Eule equtions my blow up ne the oigin t cetin time unde some cicumstnce. The centl fetue is the stengthening of wves s they move dilly inwd. A longstnding open, fundmentl poblem is whethe concenttion could fom t the oigin. In this ppe, we develop method of vnishing viscosity nd elted estimte techniques fo viscosity ppoximte solutions, nd estblish the convegence of the ppoximte solutions to globl finite-enegy entopy solution of the isentopic Eule equtions with spheicl symmety nd lge initil dt. This indictes tht concenttion does not fom in the vnishing viscosity limit, even though the density my blow up t cetin time. To chieve this, we fist constuct globl smooth solutions of ppopite initil-boundy vlue poblems fo the Eule equtions with designed viscosity tems, ppoximte pessue function, nd boundy conditions, nd then we estblish the stong convegence of the viscosity ppoximte solutions to finite-enegy entopy solutions of the Eule equtions. 1. Intoduction We e concened with the existence theoy fo spheiclly symmetic globl solutions of the Eule equtions fo multidimensionl isentopic compessible fluids: { t ρ + x ρv =, 1.1 ρv t + x ρv v + x p =, whee ρ is the density, p the pessue, v R n the velocity, t R, x R n, nd x is the gdient with espect to x R n. The constitutive pessue-density eltion fo polytopic pefect gses is p = pρ = κρ γ, whee γ > 1 is the dibtic exponent nd, by scling, the constnt κ in the pessuedensity eltion my be chosen s κ = γ 1 /4γ without loss of genelity. Fo the spheiclly symmetic motion, ρt, x = ρt,, vt, x = ut, x, = x Mthemtics Subject Clssifiction. 35Q31; 35L65; 76N1; 35D3; 76M45. Key wods nd phses. Viscosity solutions, globl solutions, spheicl symmety, Eule equtions, compessible fluids, isentopic, multidimensionl, vnishing viscosity, finite enegy, concenttion, stong convegence, ppoximte solutions, compctness fmewok. 1

3 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Then the functions ρ, m = ρ, ρu e govened by the following Eule equtions with geometicl tems: { t ρ + m + n 1 m =, t m + m ρ + pρ + n 1 m ρ =. 1.3 The existence theoy fo spheiclly symmetic solutions ρ, vt, x to 1.1 though fom 1. is equivlent to the existence theoy fo globl solutions ρ, mt, to 1.3. Fo ny poblem with constnt velocity v t infinity, i.e., lim x vt, x = v, we my ssume without loss of genelity tht v =, o equivlently, lim ut, =, by the Glilen invince. The study of spheiclly symmetic solutions cn dte bck 195s, which e motivted by mny impotnt physicl poblems such s flow in jet engine inlet mnifold nd stell dynmics including gseous sts nd supenove fomtion. In pticul, the simility solutions of such poblem hve been discussed in lge litetue cf. [9, 13, 3, 4, 6], which e detemined by singul odiny diffeentil equtions. The centl fetue is the stengthening of wves s they move dilly inwd. Vious evidences indicte tht spheiclly symmetic solutions of the compessible Eule equtions my blow up ne the oigin t cetin time unde some cicumstnce. A longstnding open, fundmentl poblem is whethe concenttion could fom t the oigin, tht is, the density becomes delt mesue t the oigin, especilly when focusing spheicl shock is moving inwd the oigin cf. [9, 3, 6]. Some pogess hs been mde fo solving this poblem in ecent decdes. The locl existence of spheiclly symmetic wek solutions outside solid bll t the oigin ws discussed in Mkino-Tkeno [1] fo the cse 1 < γ 5 3 ; lso see Yng [7]. A shock cptuing scheme ws intoduced in Chen-Glimm [6] fo constucting ppoximte solutions to spheiclly symmetic entopy solutions fo γ > 1, whee the convegence poof ws limited to be loclly in time. A fist globl existence of entopy solutions including the oigin ws estblished in Chen [5] fo clss of L Cuchy dt of bitily lge mplitude, which model outgoing blst wves nd lge-time symptotic solutions. Also see Slemod [4] fo the esolution of the spheicl piston poblem fo isentopic gs dynmics vi self-simil viscous limit, nd LeFloch-Westdickenbeg [17] fo compctness fmewok to ensue the stong compctness of spheiclly symmetic ppoximte solutions with unifom finite-enegy noms fo the cse 1 < γ 5 3. The ppoch nd ides developed in this ppe cn yield indeed the globl existence of finite-enegy entopy solutions of the compessible Eule equtions with spheicl symmety nd lge initil dt fo the genel cse γ > 1, bsed on ou elie esults in [8]. To estblish the existence of globl entopy solutions to 1.3 with initil dt: ρ, m t= = ρ, m, 1.4 we develop method of vnishing viscosity nd elted estimte techniques fo viscosity ppoximte solutions, nd estblish the convegence of the viscosity ppoximte solutions to globl finite-enegy entopy solution. To chieve this, we fist constuct globl smooth solutions of ppopite initil-boundy vlue poblems fo the Eule equtions with designed viscosity tems, ppoximte pessue function, nd boundy conditions, nd then we estblish the stong convegence of the viscosity ppoximte solutions to n entopy solution of the Eule equtions 1.3, which is equivlent to 1.1 vi eltion

4 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY Fo simplicity of pesenttion, we focus ou nlysis on the physicl egion 1 < γ 3 thoughout the ppe, though the convegence gument lso woks fo ll γ > 1. The viscosity tems nd ppoximte pessue function e designed to ppoximte the Eule equtions e s follows: whee { ρt + m + n 1 m t + m ρ m = ε ρ + n 1 ρ + p δρ + n 1 m ρ ε n 1 n 1 ρ, = ε m + n 1 m ε n 1 n 1 m, 1.5 p δ ρ = κρ γ + δρ, δ = δε >, with ε, 1] nd δε s ε in n ppopite ode. Notice tht the positive tem δρ is dded into p δ ρ to void the possibility of fomtion of cvittion of the solutions to the viscous system 1.5. Fo shllow wte flow, γ =, so tht the tem δρ cn be dopped. We conside 1.5 on cylinde Q ε = R +, b, with R + = [,, := ε, 1, b := bε > 1, nd with the boundy conditions: lim ε =, lim ε bε =, ε ρ, m = =,, ρ, m =b = ρ, fo t > 1.6 fo some ρ := ρε >, nd with ppopite ppoximte initil functions: ρ, m t= = ρ ε, m ε fo < < b, 1.7 stisfying the conditions in Theoem 1.1 below. A pi of mppings η, q : R + R R is clled n entopy-entopy flux pi o entopy pi, fo shot of system 1.3 if the pi stisfies the line hypebolic system: qu = ηu m ρ m + pρ, 1.8 whee = ρ, m is the gdient with espect to U = ρ, m fom now on. Futhemoe, ηρ, m is clled wek entopy if η =. 1.9 ρ= u=m/ρ fixed An entopy pi is sid to be convex if the Hessin ηρ, m is nonnegtive in the egion unde considetion. Fo exmple, the mechnicl enegy η ρ, m sum of the kinetic nd intenl enegy nd the mechnicl enegy flux q ρ, m: η ρ, m = 1 m ρ + κργ γ 1, q ρ, m = 1 m 3 ρ + κγ γ 1 mργ 1, 1.1 fom specil entopy pi of system 1.3, nd η ρ, m is convex in the egion ρ.

5 4 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Any wek entopy pi fo the Eule system 1.3 cn be expessed by with λ = 3 γ γ 1 η ψ ρ, m = ρ q ψ ρ, m = ρ nd the geneting function ψs. ψ m ρ + ρθ s[1 s ] λ + ds, 1.11 m ρ + θρθ sψ m ρ + ρθ s[1 s ] λ + ds, 1.1 Theoem 1.1. Assume tht ρ, m L 1 loc R +, with ρ, is of finite enegy: m + κργ n 1 L 1 R ρ γ 1 Let δ, ρ = δε, ρε, ε, 1 with lim ε δ, ρ =, stisfy ρ γ b n + δ ε bn M, 1.14 fo some M < independent of ε, 1]. If ρ ε, mε is sequence of smooth functions with the following popeties: i ρ ε > ; ii ρ ε, mε stisfies 1.6 nd n 1 m ε = =, 1.15 nd, t = b, m ε, = ε n 1 m n 1 ρ ε ε,, ρ ε + p δ ρ ε = ε n 1 n 1 m ε ; 1.16 iii ρ ε, mε ρ, m.e. R + s ε, whee we undestnd ρ ε, mε s the zeo extension of ρ ε, mε outside, b; iv b m ε + κρε γ γ 1 n 1 d m ρ + κργ γ 1 n 1 d s ε, ρ ε then, fo ech fixed ε >, thee is unique globl clssicl solution ρ ε, m ε t, of with initil dt ρ ε, mε so tht thee exists subsequence still lbeled ρε, m ε tht conveges.e. t, R + := R + R + nd in L p loc R + L q loc R +, p [1, γ + 1, q [1, 3γ+1 γ+3, s ε, to globl finite-enegy entopy solution ρ, m of the Eule equtions 1.3 with initil condition 1.7 in the following sense: i Fo ny φ C R + with φ t, =, ρφt + mφ n 1 ddt + R + ii Fo ll φ C R +, with φt, = φ t, =, mφt + m ρ φ + pρφ + n 1 φ n 1 ddt + R + ρ φ, n 1 d = ; m φ, n 1 d = ;

6 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 5 iii Fo.e. t t 1, η ρ, mt, n 1 d η ρ, mt 1, n 1 d η ρ, m n 1 d; 1.17 iv Fo ny convex function ψs with subqudtic gowth t infinity nd ny entopy pi η ψ, q ψ defined in , η ψ n 1 t + q ψ n 1 + n 1 n mη ψ,ρ + m ρ η ψ,m q ψ 1.18 in the sense of distibutions. Remk 1.1. Theoem 1.1 indictes tht thee is no concenttion fomed in the vnishing viscosity limit of the viscosity ppoximte solutions to the globl entopy solution of the compessible Eule equtions 1.3 with initil condition 1.7, which is of finite-enegy with monotonic decesing in time 1.17 nd obeys the entopy inequlity Remk 1.. To chieve 1.14, it suffices to choose δ = εb k 1 nd ρ = b k fo ny k 1 n nd k n γ.. Globl Existence of Unique Clssicl Solution of the Appoximte Eule Equtions with Atificil Viscosity The equtions in 1.5 fom qusiline pbolic system fo ρ, m. In this section, we show the existence of unique smooth solution ρ, m, equivlently ρ, u with u = m ρ, nd mke some estimtes of the solution whose bounds my depend on the pmete ε, 1] except the enegy bound E below. Fo β, 1, let C +β [, b] nd C +β,1+ β Q T be the usul Hölde nd pbolic Hölde spces, whee Q T = [, T ], b cf. [14]. Fo simplicity, we will dop the ε dependence of the involved functions in this section. Theoem.1. Let ρ, m C +β [, b] with inf b ρ > nd stisfy 1.6 nd Then thee exists unique globl solution ρ, m of poblem fo γ 1, 3] such tht ρ, m C +β,1+ β QT with inf t, Q T ρt, > fo ll T >. The nonline tems in 1.5 hve singulities when ρ = o m =. To estblish Theoem.1, we deive pioi estimtes fo geneic solution in C,1 Q T with ρ, 1 ρ, m ρ L Q T <, showing by this tht the solution tkes vlues in egion detemined pioi wy fom the singulities. With the pioi estimtes, the existence of the solution cn be deived fom the genel theoy of the qusiline pbolic systems, by suitble lineiztion techniques; see Section 5 nd Theoem 7.1 in Ldyzhenskj- Solonnikov-Ultsev [14]. The pioi estimtes e obtined by the following guments: Fist we deive the estimtes bsed on the blnce of totl enegy. Then, in Lemm., we use the mximum pinciple fo the Riemnn invints nd the totl enegy estimtes to show tht the L nom of u = m ρ depends linely on the L nom of ρ γ 1/. This is in tun used in Lemm.3 to close the highe enegy estimtes fo ρ, m. With tht, we obtin the pioi uppe bound ρ in L nd, by using Lemm. gin, the pioi bounds of the

7 6 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA L noms of m nd u. Finlly, to show the positive lowe bound fo ρ, we obtin n estimte on t u t, L dt. We poceed now with the deivtion of the pioi estimtes. Let ρ, m, with ρ >, be C,1 Q T solution of with Enegy Estimte. As usul, we denote by η δ = m ρ + h δρ, qδ = m3 ρ + mh δ ρ,.1 s the mechnicl enegy pi of system 1.5 with ε =, whee h δ ρ := ρe δ ρ fo the intenl enegy e δ ρ := ρ p δ s ds. s Note tht ρ, is the only constnt equilibium stte of the system. Fo the mechnicl enegy pi ηδ, q δ in.1, we denote η δ ρ, m = η δ ρ, m η δ ρ, η δ ρ ρ, ρ ρ,. s the totl enegy eltive to the constnt equilibium stte ρ,. Poposition.1. Let b E := sup η δ ρε, m ε n 1 d <. ε> Then, fo the viscosity ppoximte solution ρ, m = ρ, ρu detemined by Theoem.1 fo ech fixed ε >, we hve whee sup t [,T ] b 1 ρu + h δ ρ, ρ n 1 d +ε Q T h δ ρ ρ + ρ u + n 1 ρu n 1 ddt E,.3 h δ ρ, ρ = h δ ρ h δ ρ h δ ρρ ρ c 1ρρ θ ρ θ, θ = γ 1,.4 fo some constnt c 1 = c 1 ρ, γ >. Futhemoe, fo ny t [, T ], the mesue of set {ρt, > 3 ρ} is less thn c E fo some c = c ρ, γ >. Poof. We multiply the fist eqution in 1.5 by η δ ρ n 1, the second in 1.5 by η δ m n 1, nd then dd them up to obtin η δ n 1 t + q δ η δ ρ ρ, m n 1 tht is, = ε n 1 ρ + n 1 ρ η δ ρ ηδ ρ ρ, + ε n 1 m + n 1 m η δ m, η δ n 1 t + q δ η δ ρ ρ, m n 1 + n 1εmη δ m n 3 = ερ n 1 η δ ρ η δ ρ ρ, + εm n 1 η δ m..5

8 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 7 Integting both sides of.5 ove fo ny t, T ] nd using the boundy conditions 1.6, we hve b η δ n 1 d + ε ρ, m η δ ρ, m + m ρ n 1 ddt = E. Note tht ρ, m η δ ρ, m is positive qudtic fom tht domintes h δ ρ ρ nd ρ u so tht b η δ n 1 d + ε δ + κγρ γ ρ + ρ u + n 1 ρu n 1 ddt E..6 Q T Estimte.6 lso implies b sup t [,T ] ρu + h δ ρ, ρ n 1 d E. The function h δ ρ, ρ is positive, qudtic in ρ ρ fo ρ ne ρ, nd gows s ρ mx{γ,} fo lge vlues of ρ. In pticul, thee exists c 1 = c 1 ρ, γ > such tht.4 holds. Thus, fo ny t [, T ], the mesue of set {ρt, > 3 ρ} is less thn c E fo some c >. With the bsic enegy estimte.3, we hve Lemm.1. Thee exists C = Cε, T, E > such tht T ρt, mx{,γ} L,b dt C..7 Poof. In the cse tht the mesue of set {ρt, > 3 ρ} is zeo, we hve the unifom uppe bound 3 ρ fo ρt,. Othewise, fo, b, let, b be the closest to point such tht ρt, = 3 ρ. Clely, c ρe. With such choice of, we hve Then estimte.6 yields ρ γ t, ρ γ t,.8 γ ρ γ 1 t, yρ y t, y dy 1 C ρ γ t, yy n 1 b 1 dy ρ γ t, y ρ y t, y y n 1 dy b 1 C ρ γ t, ρ t, n 1 d..9 T ρt, γ L,b dt C,.1 whee C stnds fo geneic function of the pmetes: γ, ε, δ, T, E, nd ρ. Repeting the gument with ρ insted of ρ γ, we conclude.7. Fom now on, the constnt C > is univesl constnt tht my depend on the pmete ε > in..3, while the constnt M > below is nothe univesl constnt independent of the pmete ε s E fom 3, though both of them my lso depend on T >, E, nd othe pmetes; we will lso specify thei dependence wheneve needed.

9 8 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA.. Mximum Pinciple Estimtes. Futhemoe, we hve Lemm.. Thee exists C = C, T, E such tht, fo ny t [, T ], whee u L C u + Rρ L,b + u Rρ L,b + Rρ L,.11 Rρ = ρ p δ s ds..1 s Poof. Conside system 1.5. The chcteistic speeds of system 1.5 without tificil viscosity tems e λ 1 = u p δ ρ, λ = u + p δ ρ, nd the coesponding ight-eigenvectos e [ ] 1 1 =, λ1 = [ 1 λ The Riemnn invints w, z, defined by the conditions w 1 = nd z =, e given by w = m ρ + Rρ, with R defined in.1. They e qusi-convex: ]. z = m ρ Rρ, w w w, z z z,.13 whee is the Hessin with espect to ρ, m nd = m, ρ. Let us multiply the fist eqution in 1.5 by w ρ ρ, m, the second in 1.5 by w m ρ, m, nd dd them to obtin w t + λ w + n 1 u p δ ρ = ε ρ w ρ + m w m + εw + n 1ε w 1 mw m, whee λ is s bove. Then w t + n 1ε λ w εw = ερ, m wρ, m n 1 u p δ ρ n 1ε u. We wite ρ, m = α w + β w, with Then we cn futhe wite α = w w, β = ρ w m m w ρ w. w t + λw εw = εβ w w w n 1 u p δ ρ n 1ε u,.14

10 whee GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 9 λ = λ By setting n 1ε wt, = wt, n 1 + εα w w w w + εβ w w w w. t p δ ρτ, uτ, + εuτ, L dτ,,b nd using the qusi-convexity popety.13 nd the clssicl mximum pinciple pplied to the pbolic eqution.14, we obtin o mx mx w mx w + Rρ L,b +C ρ, Similly, we hve mx z mx z + Rρ L,b +C Since ρ, it follows tht w mx { mx w, mx w},,b [,t] {} {b} t t mx u mx w + mx z + Rρ L,b,b + C t By.7 nd mx{1, γ 1} < 4γ, we hve T Then we conclude.11 fom Lowe Bound on ρ. 1+ ρτ, 1 mx{1,γ 1} L,b uτ, L,b dτ. 1+ ρτ, 1 mx{1,γ 1} L,b uτ, L,b dτ. 1 + ρτ, 1 mx{1,γ 1} L,b ρτ, 1 mx{1,γ 1} L dτ C. uτ, L,b dτ..15 Lemm.3. Thee exists C = C ρ, u L,b, ρ, m H 1,b, γ such tht b ρ + m d + ρ + m ddt C..16 Q T sup t [,T ] Poof. We multiply the fist eqution in 1.5 by ρ nd the second by m to obtin ρ + m t ε ρ + m + ρ t ρ + m t m n 1 = m ρ mρ ρu + p δ m n 1 ρu m + n 1ε ρ ρ + n 1ε m m.

11 1 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA We integte this ove to obtin b ρ + m t d + ε ρ + m ddt Q t = m ρ + n 1 mρ ddt + ρu Q + p δ m ddt t ρu + n 1 Qt m ε ρ m ρ ddt n 1ε m ddt..17 We now estimte the tem Q T ρu + p m ddt fist. Conside p δ ρρ m ddτ m ddτ + C δρ + κγρ γ 1 ρ ddτ t 1 m ddτ + C + ρτ, γ b L ρ d dτ,.18 whee > will be chosen lte. Conside ρu m = u ρ m + ρuu m. We estimte u ρ m ddτ m ddτ + C m ddτ + C t t Using the unifom estimtes.11, we obtin uτ, 4 L b uτ, 4 L b ρ τ, d dτ h δ ρ ρ τ, d dτ. uτ, 4 L,b u 4 L Q τ C ρ,, ρ, u L,b 1 + ρ mx{1,γ 1} L Q τ..19 Inseting this into the bove inequlity, we hve u ρ m ddτ m ddτ + C t 1 + sup ρs, mx{1,γ 1} L s [,τ] b On the othe hnd, using the estimte simil to.8, we cn wite h δ ρ ρ τ, d dτ. ρt, mx{4,γ+} L C 1 + b ρ t, d fo t [, T ]..

12 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 11 Using. nd γ 1, 3], we obtin u ρ m ddτ m ddτ +C t 1 + sup s [,τ] b ρ s, d b Futhemoe, we hve ρuu m ddτ m ddτ +C t ρu τ, L Aguing s in.19 nd., we obtin ρu τ, L C 1 + sup Inseting this into., we obtin ρuu m ddτ m ddτ +C t 1 + sup s [,τ] b s [,τ] b C 1 + sup s [,τ] h δ ρ ρ τ, d dτ..1 b ρτ, u τ, d dτ.. ρs, mx{,γ} L ρ s, d b Combining.18,.1, nd.4, we obtin ρu + p m ddτ m ddτ t +C Φ 1 τ 1 + sup s [,τ] whee Φ 1 τ = b ρ s, d..3 ρτ, u τ, d dτ..4 b h δ ρ ρ τ, + ρτ, u τ, d ρ s, d dτ, is n L 1, T function with the nom depending on, ε, nd E ; see.3 nd.7.

13 1 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Conside now ρu Qt m ddτ m ddτ + C m ddτ + C t t ρu τ, L 1 + sup s [,τ] b b ρu τ, d dτ ρ s, d dτ, whee, in the lst inequlity, we hve used.3 nd.3. All the othe tems in.17 cn be estimted by simil guments. Thus, we obtin b sup ρ τ, s + m τ, s d + ε ρ + m ddτ τ [,t] ρ + m ddτ + C t 1 + Φτ 1 + sup s [,τ] b ρ s, + m s, d dτ, whee Φτ = Φ 1 τ + ρτ, mx{,γ} L. Choosing smll enough nd using the Gonwll-type gument nd Lemm.1, we complete the poof. As coolly, we cn fist bound ρ L Q T, which follows diectly fom.16 nd., nd then bound u L Q T fom Lemm.. Lemm.4. Thee exists n pioi bound fo ρ, u L Q T in tems of the pmetes T, E, ρ, u L,b, nd ρ, u H 1,b. Define ϕρ = { 1 ρ ρ ρ 1 ρ +, ρ ρ < ρ,, ρ > ρ. Lemm.5. Thee exists C > depending on ϕρ L 1,b nd the othe pmetes of the poblem such tht b sup t [,T ] ϕρt, d + QT ρ ρ 3 ddt C..5 Poof. Indeed, multiplying the fist eqution in 1.5 by ϕ ρ, we hve ϕ t + uϕ ε ϕ + n 1ε ρ = 1 ρ 1 ρ ρ 3 χ {ρ< ρ} u χ {ρ< ρ} + n 1 ρu 1 ρ 1 ρ χ{ρ< ρ} + n 1ε 1 ρ 1 ρ ρ χ {ρ< ρ}.

14 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 13 Integting the bove eqution in t, nd using the boundy conditions 1.6, we hve b ρ sup ϕρ d + εn 1 t [,T ] Q T {ρ< ρ} ρ 3 ddt 1 Q T {ρ< ρ} ρ 1 ρ u ddt + n 1 1 ρu Q T {ρ< ρ} ρ 1 ρ ddt n 1ε 1 + ρ Q T {ρ< ρ} ρ 1 ρ ddt = I 1 + I + I 3..6 Integting by pts, we hve I 1 ρ u ρ ε 8 Q T {ρ< ρ} Q T {ρ< ρ} ρ ρ 3 ddt + C ε Q T {ρ< ρ} u ρ ddt. Since ρ 1 ϕρ fo smll ρ, u is bounded in L, nd {ρt, ρ} is bounded independently of T, then the lst tem in the bove inequlity is bounded by C 1 + ϕρ ddt. Q T Thus, we hve I 1 Q T {ρ< ρ} Q T {ρ< ρ} Also, by the simil guments, n 1 I = nd Q T {ρ< ρ} I 3 C Q T {ρ< ρ} Q T {ρ< ρ} ρ u ddt ρ ρ ρ 3 ρu ρ u ρ ddt + C ερ ρ ddt ρ ρ 3 ddt C ddt + C 1 + ϕρ ddt..7 Q T 1 + ϕρ ddt,.8 Q T 1 + ϕρ ddt..9 Q T Combining the lst thee estimtes in.6, choosing > sufficiently smll, nd using the Gonwll-type inequlity, we obtin the pioi estimte we need. Then we hve the following estimte: T 1 ρt, dt C 1 + ρ L,b Q T ρ 3 ddt 1 ϕρ ddt 1/ Q T C 1 + QT ρ ρ 3 ddt 1..3

15 14 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Lemm.6. Thee exists C > depending on ϕρ L 1,b nd the othe pmetes s in Lemm.4 such tht T m ρ, ρ ρ, u t, dt C,.31 L,b nd C 1 ρt, C..3 Poof. Indeed, by the Sobolev embedding nd.3, we hve T m t, T ρt, dt m t, L L,b,b ρ 1 t, L,b dt T b 1 C m d 1 + b ρ ρ 3 d 1 dt, which is bounded by.16 nd.5. The estimte fo ρ ρ is the sme. The estimte fo u follows fom u = m ρ uρ ρ, the estimtes bove, nd Lemm.4. Now we cn obtin unifom estimte fo v = 1 ρ. Notice tht v veifies the inequlity: v t + εn 1 u v εv n 1u u + v. By the mximum pinciple, we hve mx v C mx { v L Q,b, v } e C T u,uτ, L,b dτ C mx { v L,b, v }, T.33 by Lemm.4 nd.31. The estimtes in Lemm.4 nd.33 e the equied pioi estimtes. The poof of Theoem.1 is completed. 3. Poof of Theoem 1.1 In this section, we povide complete poof of Theoem 1.1. As indicted elie, the constnt M is univesl constnt, independent of ε >, fom now on A Pioi Estimtes Independent of ε. We will need the following estimte. Lemm 3.1. Let l =,, n 1, nd 1, 1]. Thee exists M = Mγ, 1, E such tht, fo ny T >, b sup ρt, γ l d M 1 + ρ γ b n. 3.1 t [,T ] 1 Poof. The poof is bsed on the enegy estimte.3. Let êρ = ρ γ ρ γ γ ρ γ 1 ρ ρ. Using the Young inequlity, we find tht thee exists Mγ > such tht ρ γ Mγ êρ + ρ γ.

16 Then we hve GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 15 b Since < ε < 1 < bε <, we hve b 1 êρt, l d 1 l+1 n b ρ γ l d M 1 êρ l d + ρ γ b l+1 1. sup τ [,t] b η δ ρτ,, mτ, n 1 d 1 1 n E, by Poposition.1 fo E, independent of ε, which implies tht, fo ll l =,, n 1, b ρ γ l d M 1 n 1 E + ρ γ b n M 1 + ρ γ b n. 1 Lemm 3.. Thee exists M = MT, independent of ε, such tht T b ρ 3 y n 1 dydt M 1 + bn ε Poof. Conside fist the cse γ 1,. We estimte ε T b ρ 3 y n 1 dydt Mε T M + Mε M + Mε = M + Mε M + ε sup ρ 3 γ t, dt,b T b T b T b T b fo ny, b. 3. ρ 3 3γ ρ γ y dydt ρ 6 3γ y n 1 1 dydt ρ 6 3γ y n 1 γ y n 1 γ 3 dydt ρ 3 y n 1 dydt, whee, in the lst inequlity, we hve used the Jensen inequlity. It follows fom the bove computtion tht T b ε ρ 3 y n 1 dydt MT fo ll, b, which ives t 3.. Let now γ [, 3]. Fist, we notice tht since b > 1. sup t [,T ] b b 1 ρ y n 1 dy sup ρ γ n 1 γ b γ 1 d y n 1 γ dy t [,T ] Mb nγ 1 γ Mb n

17 16 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Then we gue s bove: T b ρ 3 y n 1 dydt T sup,b Mb n 1 + = Mb n 1 + ρ t, b T b T b Mb n ε + T Mb n ε ρ y n 1 dy dt ρ ρ dydt ρ γ γ ρ ρy dydt b ρ 4 γ y n 1 1 dydt M bn ε, whee, in the lst inequlity, we hve used the Jensen inequlity with powes nd the enegy estimte.3. γ γ 4 Lemm 3.3. Let K be compct subset of, b. Then, fo T >, thee exists M = MK, T independent of ε such tht T ρ γ+1 + δρ 3 ddt M. 3.3 Poof. We divide the poof into five steps. K 1. Let ω be smooth positive, compctly suppoted function on, b. We multiply the momentum eqution in 1.5 by ω to obtin ρuω t + ρu + p δ ω + n 1 ρu ω ε ωm + n 1 m = ρu + p δ εm + n 1 m ω. 3.4 Integting 3.4 in ove, b yields whee b ρuω dy b + t b f 1 = n 1 y b γ 4 γ nd ρu ω dy + εω m + n 1 m = ωρu + p δ + f 1, 3.5 ρu + p δ εm y + n 1 m ω y dy. y. Multiplying 3.5 by ρ nd using the continuity eqution 1.5, we hve b ρ ρuω dy t + ρu + n 1 ρm ερ + n 1 b ρ ρuω dy +ρ n 1 y = ρ u + ρp δ ω + ρf 1, ρu ω dy + ερω m + n 1 m

18 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 17 nd ρ b ρuω dy t + ρu b ρuω dy + ε ρ + n 1 b ρ ρuω dy + ρω m + n 1 m = ρp δ ω + f, 3.6 whee Notice tht f = ρf 1 n 1 ρm b b ρuω dy ρ n 1 ρu ωdy. y ρ + n 1 b ρ ρuω dy + ρω m + n 1 m b n 1 b = ρ ρuω dy ρuρ ω ρ ρuω dy n 1 ρ uω + n 1 b ρ ρuω dy + ρ u ω + ρuρ ω + n 1 ρ uω = b ρ ρuω dy n 1 b ρ uω ρ ρuω dy +ρ u ω + n 1 b ρ ρuω dy. It then follows tht b ρ ρuω dy t + ρu n 1 ε ρ b b ρuω dy ρuω dy ε ρ + ερ u ω b ρuω dy ερ uω = p δ ρω + f 3, 3.7 whee f 3 = f ε n 1 ρ b ρuω dy. 3. We multiply 3.7 by ω to obtin ρω b b +ε ρω ρuω dy t + ρuω +ερ u ω + ερ uωω b ρuω dy ε ω ρ ρuω dy ερ uω ε b n 1 ρω b ρuω dy ρuω dy = p δ ρω + f 4, 3.8 b n 1 whee f 4 = ωf 3 + ρuω ρuω dy ρω b ρuω dy.

19 18 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA We integte 3.8 ove [, T ] [, b] to obtin δρ 3 + κρ γ+1 ω ddt Q T = ερ u ω + ερ uωω ddt Q T b b T + ρω ρuω dy d f 4 ddt Q T ε ρ 3 ω ddt + εm ρ u ω + ρ u ω ddt Q T Q T b b T + ρω ρuω dy d f 4 ddt Q T ε ρ 3 ω ddt + Msupp ω, T, E. 3.9 Q T The lst inequlity follows esily fom.3.6 nd the fomul fo f Clim: Thee exists M = Msupp ω, T, E such tht ε ρ 3 ω ddt M + Mε ρ γ+1 ω ddt. 3.1 If γ, the clim is tivil. Let γ < β 3. We estimte ε ρ β ω dxdt 3.11 Q T ε sup ρ β γ ω ρ γ ddt supp ω Q T supp ω εm sup ρ β γ ω supp ω εm ρ β γ γ γ ρ ω ddt + εm ρ β γ ω ω ddt Q T Q T εm ρ γ ddt + ρ γ ω ddt + ρ β 3γ ω ddt Q T supp ω Q T Q T M 1 + ε ρ β 3γ ω ddt. 3.1 Q T If β 3γ γ + 1, the estimte of the clim follows. Othewise, since β 3γ < β note tht β 3, we cn itete 3.11 with β eplced by β 3γ nd impove 3.11: ε ρ β ω ddt M 1 + ε ρ 4β 9γ ω ddt Q T Q T If 4β 9γ is still lge thn γ + 1, we itete the estimte gin. In this wy, we obtin ecuence eltion β n = β n 1 3γ, β = β 3, nd the estimte ε ρ β ω ddt Mn 1 + ε ρ βnγ ω ddt. Q T Q T

20 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 19 Solving the ecuence eltion, we obtin β n = n β 3γ n 1 1. Fo some n, the expession is less thn γ+1 note tht β 3. Then the expected estimte is obtined. 5. Now etuning to 3.9, we hve Q T ρ γ+1 + δρ ω ddt Msupp ω, T, E fo ll smll ε >. The following lemm holds fo wek entopies η lso cf. [1]. Lemm 3.4. Let η ρ, m be the mechnicl enegy of system 1.3, nd let η ψ, q ψ be n entopy pi with the geneting function ψs stisfying sup ψ s <. s Then, fo ny ρ, m R + nd ny vecto ā = 1,, ā ηā M ψ ā η ā fo some M ψ > Lemm 3.5. Let K, b be compct. Thee exists M = MK, T independent of ε such tht, fo ny ε >, T Poof. We divide the poof into five steps. K ρ u 3 + ρ γ+θ ddt M 1 + ρ γ b n + δ ε bn. 1. Let ˇη, ˇq be n entopy pi coesponding to ψs = 1 s s. Define ηρ, m = ˇηρ, m ρ,mˇη ρ, ρ ρ, m, qρ, m = ˇqρ, m ρ,mˇη ρ, m, m ρ + p. Note tht the entopy pi ˇη, ˇq is defined fo system 1.3 with pessue p = κρ γ, the thn p δ. Then η, q is still n entopy pi of 1.3. We multiply the continuity eqution in 1.5 by η ρ n 1, the momentum eqution 1.5 by η m n 1, nd then dd them to obtin η n 1 t + q n 1 + n 1 n ˇq + mˇη ρ + m ρ ˇη m + ˇη m ρ, pρ = ε n 1 ρ + n 1 ρ η ρ + m + n 1 m η m δρ η m n

21 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA. It cn be checked diectly tht, fo some constnt M = Mγ >, qρ, m 1 M ρ u 3 + ρ γ+θ Mρ + ρ u + ρ γ, 3.16 ˇq + mˇη ρ + uˇη m, 3.17 ˇη m M u + ρ θ, ˇη ρ M u + ρ θ, 3.18 η M ρ + ρ u + ρ γ, ρ η ρ + u η m M ρ + ρ u + ρ γ, 3.19 nd, fo ˇη ρ + uˇη m consideed s function of ρ, u, ˇη ρ + uˇη m ρ M ρ θ 1 u + ρ θ 1, ˇη ρ + uˇη m u M u + ρ θ. 3. Also see [8] fo these inequlities. Moeove, note tht, t = b, q ρ, = ˇq ρ, = c γ ρ γ+θ, ˇη m ρ, = c 1 γ ρ θ, ˇη ρ ρ, =, 3.1 fo some positive c i γ, i =, 1, depending only on γ. 3. We integte eqution 3.15 ove, T, b to find T qτ, n 1 dτ = cθ ρ γ+θ b n 1 T + T +n 1 + n 1 T + + T b b +n 1 b T b b ηt, y η, y y n 1 dy ˇq + mˇη ρ + m ρ ˇη m y n dydτ y n ˇη m ρ, pρ p ρ dydτ εy n 1 ρ yy + n 1 y δρ η m ρ ρ y + η m u u y y n 1 dydτ T b δρ η m y n dydτ ρ y η ρ + m y + n 1 m y η m dydτ y = I I Now we estimte the tems in 3.. Clely, since ρ < 1 nd b > 1 fo smll ε >. I 1 M ρ γ+θ b n 1 M ρ γ b n, Notice tht ηρ, m η ρ, m. It then follows tht I b b By the enegy estimte.6, I t, E. ηρt,, mt, n 1 d η ρt,, mt, n 1 d.

22 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 1 The tem I 3 is nonpositive by 3.17 nd cn be dopped. Using Step, we hve I 4 t, M 1, T 1 + ρ γ b n fo ny t, [, T ] [ 1, b] Conside I 5. We wite n 1 ρ + n 1 ρ η ρ = n 1 ρ η ρ, n 1 m + n 1 m η m = n 1 m η m n 1 n 3 m η m, nd employ integtion by pts note tht η ρ ρ, = η m ρ, = to obtin I 5 = ε +ε t b t ρy η ρ y + m y η m y y n 1 dydτ n 1ε t η τ, n 1 dτ b m η m y n 3 dydτ = J 1 + J + J Using the enegy estimte.3 nd Lemm 3.4, we hve Also, using Step nd 3.1, we hve J 1 t, ME. m η m M ρ u + ρ γ + ρ η m ρ, M η ρ, m + ρ θ ρ. It follows by the enegy estimte.3 tht b J ω d Msupp ω, T 1 + ρ γ b n, fo ny nonnegtive smooth function ω with supp ω, b. We wite η = ρ η ρ + m η m = ρ η ρ + u η m + ρ η m u. Then we conside the integl b J 3 ω d = ε ρ η ρ + u η m ω n 1 ddτ n 1ε ρ η ρ + u η m ω n ddτ Q T Q T ε ρ ρ η ρ + u η m ρ + u η ρ + u η m u η m u ω n 1 ddτ. Q T Noticing tht η ρ + u η m = ˇη ρ + uˇη m + const. nd using Step nd estimtes.3.6 nd 3.1, we obtin b J 3 t, ω d M1, T, ω C ρ u 3 + ρ γ+θ ω n 1 ddτ. Q T To estimte I 6, employing tht η m ρ Mρ θ 1, η m u M, nd the enegy estimte.3, we hve I 6 M δ ε T b ρ 3 n 1 ddτ M δ ε bn M δ ε bn,

23 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA whee we hve used the esult of Lemm 3. nd δ ε < 1 fo smll ε > in the lst inequlity. The lst tem I 7 is estimted in the simil fshion: b I 7 ω d δ Msupp ω ε bn Msupp ω δ ε bn, since δ < 1 fo smll ε >. Finlly, we multiply eqution 3. by the nonnegtive smooth function ω, integte it ove, b, nd use estimte 3.16, togethe with the bove estimtes fo I j, j = 1,, 7, nd n ppopite choice of δ to obtin ρ u 3 + ρ γ+θ ω n 1 ddτ M 1 + ρ γ b n + δ ε bn + 1 ρ u 3 + ρ γ+θ ω n 1 ddτ. This completes the poof. 3.. Wek Entopy Dissiption Estimtes. Let = ε nd b = bε. We choose ρ = ρε nd δ = δε such tht ρ γ b n + δ ε bn M unifomly in ε. 3.5 With this choice ρ, δ, the estimtes on the lemms in 3.1 e unifom in ε. Given sequence of the initil dt functions s in Theoem 1.1, denote ρ ε, m ε by the coesponding solution of the viscosity equtions 1.5 on Q ε = [, [ε, bε] with ρ = ρε s bove. Poposition 3.1. Let η, q be n entopy pi of system 1.3 with fom fo smooth, compctly suppoted function ψs on R. Then the entopy dissiption mesues ηρ ε, m ε t + qρ ε, m ε Poof. We divide the poof into seven steps. e compct in H 1 loc Denote η ε = ηρ ε, m ε, q ε = qρ ε, m ε, nd m ε = ρ ε u ε. We compute ηt ε + q ε = n 1 ρu ε ηρ ε + u ε η ε n 1 m + ε ρ ε ηρ ε + 1 mε ηε m. We notice tht ε ρ ε η ε ρ + m ε η ε m + εη ε δρ η ε m = I ε I ε I ε 1t, Mρ ε u ε 1 + ρ ε θ M ρ ε u ε + ρ ε + ρ ε γ, 3.8 bounded in L 1, T ; L 1 loc,, independent of ε ll of the functions e extended by outside, b. 3. Next, I ε = ε n 1 η ε m ε ηm ε n 1 + ε η ε =: Iε + Ib ε. 3.9

24 then Since GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 3 η ε m ε η ε m M ρ ε + ρ ε u ε, I ε in L 1 loc R + s ε. 3.3 On the othe hnd, if ω is smooth nd compctly suppoted on R +, then ε ωt, ddt = ε n 1 η ε ω ddt Q ε I ε b εmsupp ω ρ ε L γ+1 supp ω ω H 1 R +. Since ρ ε L γ+1 supp ω is bounded, independent of ε see 3.3, the bove estimte shows tht Ib ε in H 1 loc R + s ε Fo I3 ε, we use Lemm.1 to obtin I ε 3 = ε ηρ ε, m ε ρ ε, m ε, ρ ε, m ε M ψ ε η ρ ε, m ε ρ ε, m ε, ρ ε, m ε. Combining 3.3 with Poposition.1 nd Lemm 3.4, we conclude tht I3 ε is unifomly bounded in L 1, T ; L 1 loc, To show tht I4 ε in H 1 loc s ε, we need the following clim, dopting the guments fom [18]. Clim: Let K, be compct subset. Then, fo ny < < 1 nd ε >, T ε 3 ρ ε ddt M ε γ + + ε In pticul, nd K T K ε 3 ρ ε ddt, εη ε in L p, T ; L p loc, fo p := 1,. γ + 1 Now we pove the clim. Fo the simplicity of nottion, we suppess supescipt ε in ll of the functions. Define { ρ ϕρ =, ρ <, + ρ, ρ, so tht ϕ ρ = χ {ρ< } ρ, ρϕ ρ ϕρ = ρ fo ρ <, ρϕ ρ ϕρ = fo ρ, whee χ A ρ is the indicto function tht is 1 when ρ A nd othewise.

25 4 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Let ω be nonnegtive smooth, compctly suppoted function on,. We compute fom the continuity eqution, the fist eqution, in 1.5: ϕω t + ϕuω ϕuω 1 ρ χ {ρ< } + δ χ {ρ> } ωu + n 1 ρu min{ρ, } = εϕ ωρ ε min{ρ, }ω n 1ε ρ + ω min{ρ, }ρ εω ρ χ {ρ< } Integting 3.34 ove, T,, we obtin T εω ρ χ {ρ< } ddt = ϕω T T d + ϕuω ddt + 1 T ρ T n 1 χ {ρ< } + δχ {ρ> } ωu ddt ρu min{ρ, } ddt T T n 1ε ε min{ρ, }ω ρ ddt + ω min{ρ, }ρ ddt = J J We estimte the integls on the ight: T J 1 Msupp ω + ρ ddt Msupp ω, T ; 3.36 supp ω T J ρu χ{ρ< } + + ρ u χ {ρ> } ddt supp ω T ρ + ρ u dtdt supp ω Msupp ω, T ; 3.37 J 3 3 ε T supp ω T J 4 Msupp ω J 5 ε γ ε 4 ε 4 T T T supp ω ρ + ερ u ddt Msupp ω, T ε ; 3.38 supp ω ρ γ ρ ω ddt + ε ρ + ρ u ddt Msupp ω, T ; 3.39 ερ γ ρ ddt + ε T supp ω ρ ω ddt + εmsupp ω, T T ρ ρ χ {ρ< } ω ddt supp ω ρ ω ω ddt + ε γ Msupp ω, T + ε γ Msupp ω, T. 3.4

26 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 5 Moeove, J 6 is estimted in the sme wy s J 5. Thus, estimte 3.33 is poved. Now we pove the second pt of the clim. Notice tht η M ρ η ρ + uη m + ρ u M ρ 1 + ρ θ + ρ u Let q 1, to be chosen lte on. Compute T T ε q η q ddt M ε q ρ q ddt + K + M + M + M T + ε q 1 M + M K T K T K K T T T ε q ρ ddt K ε q ρ ρ θ + ρ u q ddt ε p ρ q γ ρ ρ q + ρ 1 u q ddt K ε 3 ρ ddt K ερ γ ρ + ρ u + ερ q q ddt ε 3 ρ ddt + ε q 1 CT, K, 3.4 povided tht q = γ + 1, which holds if nd only if q = γ+1. Combining this with estimte 3.33, we ive t the conclusion of the clim. 6. Conside the lst tem I5 ε. This tem is bounded in L1, T : L 1 loc,. Indeed, fo compct set K,, using the enegy estimtes.3 nd Lemm 3., we obtin T T I 5 ddt M ψ δρ ρ ddt K K Mψ, K 1 + δ ε Mψ, K 1 + δ T ε + δ ε K T ρ 4 γ ddt K ρ 3 ddt Mψ, K 1 + δ ε + δ ε bn. Fom the choice of δ, the tem on the ight is unifomly bounded in ε. 7. Combining Steps 1 6, we conclude ηρ ε, m ε t + qρ ε, m ε = f ε + g ε, 3.43 whee f ε is bounded in L 1, T ; L 1 loc, nd g ε in W 1,q loc R + fo some q 1,. This implies tht, fo 1 < q 1 <, ηρ ε, m ε t + qρ ε, m ε e confined in compct subset of W 1,q 1 loc. 3.44

27 6 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA On the othe hnd, using fomuls nd the estimtes in Poposition.1 nd Lemm 3.5, we obtin tht, fo ny smooth, compctly suppoted function ψs on R, ηρ ε, m ε, qρ ε, m ε e unifomly bounded in L q loc R +, fo q = γ + 1 > when γ > 1. This implies tht, fo some q >, ηρ ε, m ε t + qρ ε, m ε e unifomly bounded in W 1,q loc The intepoltion compctness theoem cf. [3, 11] indictes tht, fo q 1 > 1, q q 1, ], nd q [q 1, q, compct set of W 1,q 1 loc R + bounded set of W 1,q loc R + compct set of W 1,q loc R +, which is geneliztion of Mut s lemm in [, 5]. Combining this intepoltion compctness theoem fo 1 < q 1 <, q >, nd q = with the fcts in , we conclude the esult Stong Convegence nd the Entopy Inequlity. The pioi estimtes nd compctness popeties we hve obtined in imply tht the viscous solutions stisfy the compensted compctness fmewok in Chen-Peepelits [8]. Then the compctness theoem estblished in [8] fo the cse γ > 1 lso see LeFloch-Westdickenbeg [17] yields tht ρ ε, m ε ρ, m.e. t, R + in L p loc R + L q loc R + fo p [1, γ + 1 nd q [1, 3γ+1 γ+3. This equies the unifom bounds nd the estimte: m q = ρ q 3 u q ρ q 3 ρ u 3 + ρ γ+1 fo q = 3γ+1 γ+3. Fom the sme estimtes, we lso obtin the convegence of the enegy s ε : η ρ ε, m ε η ρ, m in L 1 loc R +. Since the enegy η ρ, m is convex function, by pssing to the limit in.6, we obtin t t 1 which implies tht, fo.e. t, η ρ, mt, n 1 ddt t 1 t R + η ρ, mt, n 1 d η ρ, m n 1 d, η ρ, m n 1 d This implies tht thee is no concenttion fomed in the density ρ t the oigin =. Futhemoe, we multiply both sides of.5 by smooth function φt C 1R + with φ =, integte it ove R +, nd pss to the limit ε to obtin η ρ, mφ t ddt, R + which, togethe with 3.46, concludes 1.17.

28 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 7 Finlly, the enegy estimtes.3.6 nd the estimtes in Lemms imply the equi-integbility of sequence of η ε ψ, qε ψ, mε ρ η ε ψ, m ε ρ ε m η ε ψ, qε ψ, fo ny ψs tht is convex with subqudtic gowth t infinity: lim s ψs s =. Pssing to the limit in 3.7 multiplied by n nd integted ginst smooth compctly function suppoted on,,, we obtin Limit in the Equtions. Let φt, be smooth, compctly suppoted function on [, [, bε, with φ t, = fo ll close to. Assume tht the viscosity solutions ρ ε, m ε e extended by outside of [ε, bε]. Multiplying the fist eqution in 1.5 by n 1 φ nd then integting it ove R +, we hve R + ρ ε φ t + m ε φ + ερ ε φ + n 1 + ρ ε φ, n 1 d =. R + φ n 1 ddt Note tht, by the enegy inequlity, 1 ρε γ n 1 d is bounded, independent of ε, which implies tht thee is no concenttion of mss t =. Pssing to the limit in the bove eqution, we deduce ρφt + mφ n 1 ddt + ρ φ, n 1 d =, R + R + which cn be extended to hold fo ny smooth, compctly suppoted function φt, on [, [,, with φ t, =. Conside now the momentum eqution in 1.3. Let φt, be smooth, compctly suppoted function on [, ε, bε. Multiplying the fist eqution in 1.5 nd then integting it ove R +, we obtin m ε φ t + mε R ρ ε φ + p δ ρ ε φ + n 1 φ + εm ε φ n 1 ddt + + m ε φ, n 1 d =. R + Pssing to the limit, we find mφ t + m ρ φ + pρ φ + n 1 φ n 1 ddt + m φ, n 1 d =. R + R + Note tht the tem contining δρ conveges to zeo by Lemm 3. since δ = δε s ε. This eqution cn be extended fo ny smooth compctly suppoted function φt, on [, [, with φt, = φ t, =, since m ρ + ργ t, n 1 L 1 loc [, [,.

29 8 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Acknowledgements: The esech of Gui-Qing G. Chen ws suppoted in pt by the UK EPSRC Science nd Innovtion Awd to the Oxfod Cente fo Nonline PDE EP/E357/1, the UK EPSRC Awd to the EPSRC Cente fo Doctol Tining in PDEs EP/L15811/1, the NSFC unde joint poject Gnt 17811, nd the Royl Society Wolfson Resech Meit Awd UK. The esech of Mikhil Peepelits ws suppoted in pt by the NSF Gnt DMS The uthos would like to thnk the Isc Newton Institute fo Mthemticl Sciences, Cmbidge, fo suppot nd hospitlity duing the 14 Pogmme on Fee Boundy Poblems nd Relted Topics whee wok on this ppe ws undetken. Refeences [1] Binchini, S. nd Bessn, A. Vnishing viscosity solutions of nonline hypebolic systems, Ann. of Mth., 161 5, [] Chen, G.-Q. Convegence of the Lx-Fiedichs scheme fo isentopic gs dynmics III, Act Mth. Sci. 6B 1986, 75 1 in English; 8A 1988, in Chinese. [3] Chen, G.-Q. The compensted compctness method nd the system of isentopic gs dynmics, Lectue Notes, Pepint MSRI-57-91, Bekeley, Octobe 199. [4] Chen, G.-Q. Remks on R. J. DiPen s ppe: Convegence of the viscosity method fo isentopic gs dynmics [Comm. Mth. Phys , 1 3], Poc. Ame. Mth. Soc , [5] Chen, G.-Q. Remks on spheiclly symmetic solutions of the compessible Eule equtions, Poc. Roy. Soc. Edinbugh, 17A 1997, [6] Chen, G.-Q. nd Glimm, J. Globl solutions to the compessible Eule equtions with geometicl stuctue, Commun. Mth. Phys , [7] Chen, G.-Q. nd Li, T.-H. Globl entopy solutions in L to the Eule equtions nd Eule-Poisson equtions fo isotheml fluids with spheicl symmety, Methods Appl. Anl. 1 3, [8] Chen, G.-Q. nd Peepelits, M. Vnishing viscosity limit of the Nvie-Stokes equtions to the Eule equtions fo compessible fluid flow, Comm. Pue Appl. Mth. 63 1, [9] Count, R. nd Fiedichs, K. O. Supesonic Flow nd Shock Wves, Spinge-Velg: New Yok, [1] Dfemos, C. M. Hypebolic Consevtion Lws in Continuum Physics, Spinge-Velg: Belin, 1. [11] Ding, X., Chen, G.-Q., nd Luo, P. Convegence of the Lx-Fiedichs scheme fo the isentopic gs dynmics I II, Act Mth. Sci. 5B 1985, 483-5, in English; 7A 1987, ; 8A 1989, in Chinese; Convegence of the fctionl step Lx-Fiedichs scheme nd Godunov scheme fo the isentopic system of gs dynmics, Comm. Mth. Phys , [1] DiPen, R. Convegence of the viscosity method fo isentopic gs dynmics, Commun. Mth. Phys , 1 3. [13] Gudeley, G. Stke kugelige und zylindische Vedichtungsstosse inde Nhe des Kugelmittelpunktes bzw. de Zylindechse, Luftfhtfoschung , no. 9, [14] Ldyzhenskj, O. A., Solonnikov, V. A., nd Ultsev, N. N. Line nd Qusi-line Equtions of Pbolic Type, LOMI AMS, [15] Lx, P. D. Shock wve nd entopy, In: Contibutions to Functionl Anlysis, ed. E.A. Zntonello, , Acdemic Pess: New Yok, [16] Liu, T.-P. Qusiline hypebolic system, Commun. Mth. Phys , [17] LeFloch, Ph.G. nd Westdickenbeg, M. Finite enegy solutions to the isentopic Eule equtions with geometic effects, J. Mth. Pues Appl. 88 7, [18] Lions, P.-L., Pethme, B., nd Sougnidis, P. E. Existence nd stbility of entopy solutions fo the hypebolic systems of isentopic gs dynmics in Eulein nd Lgngin coodintes, Comm. Pue Appl. Mth ,

30 GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 9 [19] Lions, P.-L., Pethme, B., nd Tdmo, E. Kinetic fomultion of the isentopic gs dynmics nd p-systems, Comm. Mth. Phys , [] Mkino, T., Mizoht, K., nd Uki, S. Globl wek solutions of the compessible Eule equtions with spheicl symmety I, II, Jpn J. Industil Appl. Mth , [1] Mkino, T. nd Tkeno, S. Initil boundy vlue poblem fo the spheiclly symmetic motion of isentopic gs, Jpn J. Industil Appl. Mth , [] Mut, F. Compcité p compenstion, Ann. Scuol Nom. Sup. Pis Sci. Fis. Mt , [3] Rosselnd, S. The Pulstion Theoy of Vible Sts, Dove Publictions, Inc.: New Yok, [4] Slemod, M. Resolution of the spheicl piston poblem fo compessible isentopic gs dynmics vi self-simil viscous limit, Poc. Roy. Soc. Edinbugh, 16 A 1996, [5] Tt, L. Compensted compctness nd pplictions to ptil diffeentil equtions, Resech Notes in Mthemtics, Nonline Anlysis nd Mechnics, Heiot-Wtt Symposium, Vol. 4, Knops R.J. ed., Pitmn Pess, [6] Whithm, G. B. Line nd Nonline Wves, John Wiley & Sons: New Yok, [7] Yng, T. A functionl integl ppoch to shock wve solutions of Eule equtions with spheicl symmety, I. Commun. Mth. Phys , ; II. J. Diff. Eqs , Gui-Qing G. Chen, Mthemticl Institute, Univesity of Oxfod, Andew Wiles Building, Rdcliffe Obsevtoy Qute, Woodstock Rod, Oxfod OX 6GG, UK; Acdemy of Mthemtics nd Systems Science, Chinese Acdemy of Sciences, Beijing 119, Chin. E-mil ddess: chengq@mths.ox.c.uk Mikhil Peepelits, Deptment of Mthemtics, Univesity of Houston, 651 PGH Building, Houston, Texs , USA E-mil ddess: mish@mth.uh.edu