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

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 to 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 question is whethe concenttion could fom t the oigin. In this ppe, we develop vnishing viscosity method to constuct ppoximte solutions, develop elted estimte techniques, nd estblish the convegence of the ppoximte solutions to globl finite-enegy entopy solution to the compessible 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 estblish the globl existence of smooth solutions of ppopite initil-boundy vlue poblems fo the Eule equtions with designed viscosity tems nd boundy conditions, nd we then estblish the stong convegence of the vnishing viscosity solutions to finite-enegy entopy solutions of the Eule equtions. 1. Intoduction We e concened with the existence theoy fo spheiclly symmetic globl solutions to the Eule equtions fo multidimensionl compessible isentopic fluids: { t ρ + x ρv =, 1.1 ρv t + ρv v + p =, whee ρ is the density, p the pessue, v R n the velocity, t R, x R n, nd 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,, ρvt, x = mt, x, = x, 1. 1

GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA whee m is scl function. Then the functions ρt, nd mt, 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. 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. The simility solutions of such poblem hve been discussed in lge litetue cf. [9,, 3, 5], 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 to the compessible Eule equtions my blow up ne the oigin t cetin time unde some cicumstnce. A longstnding open, fundmentl question is whethe concenttion could fom t the oigin, tht is, the density becomes delt mss t the oigin, especilly when focusing spheicl shock is moving inwd the oigin cf. [9,, 5]. A globl entopy solution with spheicl symmety with solid bll t the oigin hs been constucted in BV in [19] nd in L in [7] fo the isotheml cse γ = 1. The locl existence of such wek solution fo genel cse 1 < γ 5/3 ws lso discussed in []. A shock cptuing scheme ws intoduced in [6] fo constucting ppoximte solutions to spheiclly symmetic entopy solutions fo genel γ > 1, whee the convegence poof ws limited to be loclly. A fist globl existence of entopy solutions including the oigin ws estblished in [5] fo clss of L Cuchy dt of bitily lge mplitude, which model outgoing blst wves nd lge-time symptotic solutions. Also see LeFloch-Westdickenbeg [16] 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 yield indeed the globl existence of globl finite-enegy entopy solutions to the compessible Eule equtions with spheiclly symmety nd lge initil dt fo 1 < γ??, bsed on ou elie esults in Chen-Peepelits [8]. To estblish the existence of globl entopy solutions to 1.3, we develop the viscosity method to constuct ppoximte solutions, develop elted estimte techniques, nd estblish the convegence of the ppoximte solutions to globl entopy solution. To chieve this, we fist estblish the existence of globl solutions of ppopite initil-boundy vlue poblems fo the Eule equtions with designed viscosity tems nd boundy conditions, nd then estblish the stong convegence of the vnishing viscosity solutions to entopy solutions of the Eule equtions 1.3, which is equivlent to 1.1 vi eltion 1.. The viscosity tems e designed to be dded to the Eule equtions e s follows: { 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. 1.4

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 3 new![ Notice tht in the bove equtions the pessue pρ = κρ γ is ppoximted by p δ ρ = δρ + κρ γ, whee the positive tem δρ is dded to void fomtion of vcuum in the viscous system. The ppmete δ will be chosen to vnish s the viscosity ε. ] We conside 1.4 on the cylinde Q ε = [,, b, with := ε, b := bε R + :=,, nd lim ε =, lim ε nd with the boundy conditions: bε = +, ε ρ, m = =,, ρ, m =b = ρ,, fo t >, 1.5 fo some ρ := ρε >, nd the initil conditions: ρ, mt = = ρ, m fo < < b. 1.6 **WOLG, we my ssume tht lim x vt, x = by the Glilen invince. A pi of mppings η, q : R + := R + R R is clled n entopy-entopy flux pi o entopy pi fo shot of system 1.3 if the pi stisfies the hypebolic system: qu = ηu F U. 1.7 Futhemoe, ηρ, m is clled wek entopy if η =. 1.8 ρ= u=m/ρ fixed An entopy pi is sid convex if the Hessin ηρ, m is nonnegtive in the egion unde considetion. Fo exmple, the mechnicl enegy sum of the kinetic nd intenl enegy nd the mechnicl enegy flux η ρ, m = 1 m ρ + κργ γ 1, q ρ, m = 1 m 3 ρ + κγ γ 1 mργ 1, 1.9 fom specil entopy pi; η ρ, m is convex fo ny γ > 1 in the egion ρ. Any wek entopy-entopy flux 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, 1.1 m ρ + θρθ sψ m ρ + ρθ s[1 s ] λ + ds, 1.11 Assume tht ρ, m L 1 loc R +, with ρ, hs finite totl-enegy: Let γ 1, 3] nd δ = ε?. m + κργ n 1 L 1 R +. ρ γ 1 Theoem 1.1. Thee exist ρ,, b = ρε, ε, bε, with lim ε ρ,, b =,, +, such tht, if ρ ε, mε is sequence of smooth functions with the popeties:

4 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA i ρ ε, mε stisfies 1.5 nd m ε, + n 1 m ε nd, t = b, m ε, = ε ρ ε, + n 1 ρ ε m ε,, ρ ε = =, 1.1 + p δ ρ ε = ε m ε, + n 1 m ε ; 1.13 ii ρ ε, mε ρ, m.e. R +, whee we undestnd ρ ε, mε s the zeo extension of ρ ε, mε outside, b; iii m ε ρ ε + ρ ε γ n 1 d m ρ + ρ γ n 1 d s ε, then, fo ech fixed ε, thee exists unique globl clssicl solution ρ ε, m ε t, of 1.4 1.6 with initil dt ρ ε, mε. Futhemoe, thee exists subsequence still lbeled ρ ε, m ε tht conveges.e. t, Ω := [, [, nd in L p loc Ω Lq loc Ω, p [1, γ + 1, q [1, 3γ+1 γ+3, to globl finite-enegy entopy solution ρ, m of the Eule equtions 1.3 in the following sense: i Fo ny ϕ C Ω with ϕ t, =, ρϕt + mϕ n 1 ddt + Ω ρ ϕ, n 1 d = ; ii Fo ll ϕ C Ω, with ϕt, = ϕ t, =, mϕt + m ρ + κργ ϕ n 1 ddt + m ϕ, n 1 d = ; Ω iii Fo.e. t, m ρ + κργ t, n 1 d γ 1 m + κργ n 1 d; 1.14 ρ γ 1 iv Fo ny convex function ψs with subqudtic gowth t infinity nd ny entopyentopy flux pi η ψ, q ψ defined in 1.1 1.11, η ψ n 1 t + q ψ n 1 + n 1 n mη ψ,ρ + m ρ η ψ,m q ψ 1.15 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 in R n, which is of finite enegy 1.14 nd obeys the entopy inequlity 1.15.. Globl Existence of Unique Clssicl Solution of the Eule Equtions with Atificil Viscosity The equtions in 1.4 fom qusiline pbolic system in ρ, m. In this section we show the existence of unique solution ρ, u in clss of smooth functions nd mke some unifom estimtes of the solution independent of the pmete ε >. Fo β, 1, let

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 5 C +β [, b] nd C +β,1+ β Q T be the usul Hölde nd pbolic Hölde spces, whee Q T = [, T ], b cf. [13]. Let γ 1, 3]. Theoem.1. Let ρ, m C +β [, b] with inf b ρ > nd stisfy 1.5, 1.1, nd 1.13. Then thee exists unique globl solution ρ, m of poblem 1.4 1.6 such tht ρ, m C +β,1+ β QT, inf ρ > fo ll T >. Q T The nonline tems in 1.4 hve singulities t ρ =, nd t ρ, m with m =. To estblish Theoem.1, we deive pioi estimtes fo geneic C,1 Q T solution in the nom ρ, 1 ρ, m ρ L Q T, showing by this tht the solution tkes vlues in egion detemined pioi wy fom the singulities, fo fixed ε >. 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 5 nd Theoem 7.1 of [13]. A pioi estimtes e obtined by the following guments: Fist we deive the estimtes bsed on the blnce of totl enegy. Then, in Lemm.1, 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. to close highe enegy estimtes fo ρ, m. With tht, we obtin pioi bound of the L nom of ρ nd, by using gin Lemm.1, pioi bounds of the 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 1.4 1.6 with 1.5, 1.1, nd 1.13..1. Enegy Estimte. As usul, we denote by η = m ρ + eρ, q = m3 ρ + me ρ, ρ p δ s eρ = ρ s ds. Note tht ρ, is the only constnt equilibium stte of the system. Fo the mechnicl enegy-enegy flux pi η, q in 1.9, we denote Eρ, m = η ρ, m η ρ, η ρ ρ, ρ ρ,.1 s the totl enegy eltive to the constnt equilibium stte ρ,. Poposition.1. Let sup t [,T ] E := sup ε> Q T Eρ ε, m ε n 1 d <. Then, fo the viscous solutions ρ ε, m ε = ρ ε, ρ ε u ε detemined by Theoem.1, we hve ρu / + e ρ, ρ d + ε δ + κγρ γ ρ + ρ u + ρu n 1 ddt E,. whee e ρ, ρ = ρeρ ρe ρ ρe ρ + e ρ ρ ρ c 1 ρρ θ ρ θ, θ = γ 1,.3

6 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA fo some constnt c 1 = c 1 ρ, γ >. Futhemoe, fo ny t [, T ], the mesue of the set {ρt, > 3 ρ} is less thn c E fo some c = c ρ, γ >. Poof. We multiply the fist eqution in 1.4 by E ρ n 1, the second in 1.4 by E m n 1, nd then dd them up to obtin nd then E n 1 t + q n 1 η ρ ρ, m n 1 = ε n 1 ρ + n 1 ρ η ρ ηρ ρ, + ε n 1 m + n 1 m η m, E n 1 t + q n 1 η ρ ρ, m n 1 + n 1εmη m n 3 = ερ n 1 η ρ η ρ ρ, + εm n 1 η m..4 Integting both sides of.4 ove Q T nd using the boundy conditions 1.5, we hve sup t [,T ] + ε Q T Eρt,, mt, n 1 d Eρ, mρ, m, ρ, m + m ρ n 1 ddt = E..5 Note tht Eρ, mρ, m, ρ, m is positive definite fom tht domintes ρ γ ρ nd ρ u so tht ε Q T Estimte.5 lso implies δ + κγρ γ ρ + ρ u + ρu n 1 ddt E..6 sup t [,T ] ρu + e ρ, ρ n 1 d E..7 The function e ρ, ρ is positive, qudtic in ρ ρ fo ρ ne ρ, nd gowing s ρ γ fo lge vlues of ρ. In pticul, thee exists c 1 = c 1 ρ, γ > such tht.3 holds. Thus, fo ny t [, T ], the mesue of the set {ρt, > 3 ρ} is less thn c ρe fo some c ρ >. In cse the mesue of the set {ρt, > 3 ρ} be zeo, then we hve the unifom uppe bound 3 ρ fo ρt,. Othewise, fo, b, let Ω be the closest to the point such tht ρt, = 3 ρ. Clely, c ρe. With such choice of, we estimte

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 7 Int[α, β] the intevl with endpoints α, β: ρ γ t, ρ γ t, γ ρ γ 1 t, y ρ t, y dy Int[, ] C, ρ ρ γ t, yy n 1 dy Int[, ] 1 1 ρ γ t, y ρ t, y y n 1 dy 1 C, ρ, E ρ γ t, ρ t, n 1 d C <..8 Then estimte.6 yields T ρ γ t, L dt C, ρ, E, T..9.. Mximum Pinciple Estimtes. Futhemoe, we hve Lemm.1. Thee exists C = C, T, E such tht, fo ny t [, T ], whee θ = γ 1. u L C u + ρ θ L,b + u ρ θ L,b + ρ θ L,.1 Poof. Conside the Riemnn invints fo the one-dimensionl Eule equtions. Unde the ssumption on κ = θ /γ 1, they e given by the fomuls: The coesponding chcteistic speeds e: w = m ρ + ρθ, z = m ρ ρθ. λ 1 = m/ρ γρ θ, λ = m/ρ + γρ θ. The poof will be bsed on the qusi-convexity popety of w nd z : w w, w, z z, z,.11 whee nd denote the gdient, the Hessin with espect to ρ, m, nd = m, ρ. Let us multiply the fist eqution in 1.4 by w ρ ρ, m, the second eqution in 1.4 by w m ρ, m, nd dd them to obtin w t + λ 1 w + whee λ 1 is s bove. We obtin n 1θ ρ θ u = ε ρ w ρ + m w m + εw + w t + λ 1 n 1ε w εw n 1ε w 1 mw m, = ε wρ, m, ρ, m n 1θ ρθ u n 1ε u.

8 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA We wite with Then we cn futhe wite whee λ = λ 1 By setting ρ, m = α wρ, m + β wρ, m, α = w t + λw εw w w, β = ρ w m m w ρ w. = εβ w w, w n 1θ ρθ u n 1ε u,.1 n 1ε + εα w w w, w wt, = wt, n 1 t w + εβ w w w θρθ uτ, +, w w. εuτ, L,b dτ, using the qusi-convexity condition.11 nd the clssicl mximum pinciple pplied to the pbolic eqution.1, we obtin: o mx w mx { mx w, mx w},,b [,t] {} {b} mx w mx w + ρ θ L,b + C ρ, Similly, we hve mx z mx z + ρ θ L,b +C ρ, Since ρ, it follows tht By.9, since θ < γ, t t mx u mx w + mx z + ρ θ L,b,b + C T t Then we conclude.1 fom.13. 1 + ρτ, θ L,b uτ, L,b dτ. 1+ ρτ, θ L,b uτ, L,b dτ. 1 + ρτ, θ L,b uτ, L,b dτ..13 ρτ, θ L dτ C, ρ, T, E.

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 9.3. Lowe Bound on ρ. Now we estimte lowe bound on ρ. Lemm.. Thee is C = Cε,, ρ, T, ρ, u L,b, ρ, m H 1,b, which veifies the inequlity sup t [,T ] ρ + m d + T ρ + m ddt C..14 Poof. We multiply the fist eqution in 1.4 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 n 1 + p m ρu m + n 1ε We integte this ove : ρ ρ + n 1ε m m. ρ + m t d + ε ρ + m ddt n 1 = m ρ + mρ ddt + ρu Q + p m ddt t ρu + n 1 Qt m ε ρ m ρ ddt n 1ε m ddt..15 We now estimte the tem Q T ρu + p m ddt fist. Conside ρ γ 1 ρ m ddτ δ m ddτ + C δ ρ γ ρ ddτ t δ m ddτ + C δ ρτ, γ L Ω Using estimte.8, fo t [, T ], we cn wite ρt, γ L C, ρ, E 1 + Using this in the pevious estimte, we obtin ρ γ 1 ρ m ddτ t δ m ddτ + C δ 1 + ρ τ, d ρ γ ρ d dτ. ρ t, d..16 ρ γ ρ τ, d dτ..17

1 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Conside ρu m = u ρ m + ρuu m. We estimte u ρ m ddτ t δ m ddτ + C δ uτ, 4 L ρ τ, ddτ t δ m ddτ + C δ uτ, 4 L ρτ, γ L ρ γ τ, ρ τ, ddτ, whee, in the lst inequlity, we hve used the fct tht γ 1, ]. Using the unifom estimtes.1, we obtin uτ, 4 L sup u 4 C ρ,, u, ρ L 1 + sup ρ γ. Q τ Q τ Inseting this into the bove inequlity, we hve u ρ m ddτ δ m ddτ + C δ Using.16, we obtin u ρ m ddτ δ m ddτ +C δ t t 1 + sup s [,τ] 1 + sup ρs, γ L s [,τ] ρ s, d Futhemoe, we hve ρuu m ddτ δ m ddτ Aguing s in.3, we obtin +C δ t ρu τ, L ρ γ τ, ρ τ, d dτ. ρ γ τ, ρ τ, d dτ..18 ρu τ, L C1 + sup ρs, γ L s [,τ] C 1 + sup s [,τ] ρτ, u τ, d dτ..19 ρ s, d..

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 11 Inseting this into.19, we obtin ρuu m ddτ δ m ddτ +C δ t 1 + sup s [,τ] ρ s, d Combining.17,.18, nd.1, we obtin ρu + p m ddτ δ m ddτ t +C δ Φτ 1 + sup s [,τ] whee Φτ = ρτ, u τ, d dτ..1 ρτ, γ 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.5. Conside now ρu Qt t m ddτ δ m ddτ + C δ ρu τ, L ρu τ, d t δ m ddτ + C δ 1 + sup ρ s, d dτ, s [,τ] whee, in the lst inequlity, we hve used.5 nd.. All the othe tems in.15 cn be estimted by simil guments. Thus, we obtin sup ρ τ, s + m τ, s d + ε ρ + m ddτ τ [,t] δ ρ + m ddτ t + C δ 1 + Φτ 1 + sup ρ s, + m s, d dτ. s [,τ] This completes the poof. Lemm.3. Thee exist n pioi bound fo ρ, u L Q T in tems of the pmetes of the poblem, E, nd ρ, u L,b. The estimte fo ρ L Q T follows diectly fom.. Fom this nd Lemm.1, we obtin the estimte on u L Q T. Let ρ, ρ, nd define φρ = { 1 ρ ρ ρ 1 ρ +, ρ ρ < ρ,, ρ > ρ.

1 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Lemm.4. Thee exists C > depending on u L Q T, φρ L 1,b, nd the othe pmetes of the poblem such tht sup t [,T ] φρt, d + T ρ ρ 3 ddt C.. Poof. Indeed, multiplying the fist eqution in 1.4 by φ ρ, we hve φ t + uφ ε φ + n 1ε ρ = ρ 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}. Integting the bove eqution in t, nd using the boundy conditions 1.5, we hve sup t [,T ] T + {ρ< ρ} T T φρ d + ε {ρ< ρ} ρ ρ {ρ< ρ} ρ ddt ρ 3 u ddt + T n 1ε 1 ρ ρ 1 ρ {ρ< ρ} ddt n 1 1 ρu ρ 1 ρ ddt = I 1 + I + I 3..3 Integting by pts, we hve T I 1 ρ u ρ ε 8 {ρ< ρ} T {ρ< ρ} ρ ρ 3 ddt + C ε 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, the lst tem in the bove inequlity is bounded by Thus, we hve I 1 ε 8 T T {ρ< ρ} C 1 + {ρ< ρ} T ρ u ddt ρ ρ ρ 3 ddt + C ε Also, by the simil guments, T n 1 ρu I = ρ u ddt ρ C {ρ< ρ} φρ ddt. T 1 + T 1 + φρ ddt..4 φρ ddt,.5

nd GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 13 I 3 C ε 8 T T {ρ< ρ} {ρ< ρ} ερ ρ ddt ρ ρ 3 ddt + C ε T 1 + φρ ddt..6 Combining the lst thee estimtes in.3 nd using the Gonwll inequlity, we obtin the pioi estimte we need. Then we hve the following estimte: T 1 ρt, dt C 1 + QT ρ L Ω ρ 3 ddt 1 φρ ddt 1/ Q T C 1 + QT ρ ρ 3 ddt 1..7 Lemm.5. Thee exists C depending on φρ L 1 Ω nd the othe pmetes s in Lemm.4 such tht T m t, ρt,, ρ t, ρt,, u t, L,b dt C..8 Poof. Indeed, by the Sobolev embedding nd.7, we hve T m ρ L,b dt T C, ρ, T m L,b ρ 1 L,b dt T which is bounded by.14 nd.. The estimte fo ρ ρ u follows fom u = m ρ uρ ρ m d 1 1 + ρ ρ 3 d 1 dt, is the sme. The estimte fo, the estimte bove, nd Lemm.3. Now we cn obtin unifom estimte fo v = 1 ρ. Notice tht v veifies the inequlity: v t + u ε v εv u + u v. By the mximum pinciple, we hve mx v C mx{ v L Q,b, v}e C T by Lemm.3 nd.8. T u,uτ, L,b dτ C mx{ v L,b, v},.9 The estimtes in Lemm.3 nd.9 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. The constnt M is univesl constnt, unifom in ε >, which my depend on T >, γ, the compct set K, nd othe pmetes independent of ε; we will specify its dependence when needed.

14 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA 3.1. A Pioi Estimtes Independent of ε. We will need the following estimte. Lemm 3.1. Let l =,, n 1, nd 1, b. Thee exist M = Mγ, 1, E nd P = P b, ρ such tht, fo ny T >, sup t [,T ] 1 ρt, γ l d M1 + P. 3.1 In this estimte, P = P b, ρ is sum of the poducts of positive powes of b nd ρ. Poof. The poof is bsed on the enegy estimte.5. Let êρ = ρ γ ρ γ γ ρ γ 1 ρ ρ. Since êρ is qudtic in ρ ρ fo ρ ne ρ, it follows tht thee exists Mγ > such tht êρ ρ γ 1 ρ ρ Mγ ρ γ ρ fo ρ, ρ. Moeove, thee e c = c γ nd Mγ such tht, if ρ > c γ ρ, then ρ γ Mγ êρ + ρ γ. Using this constnt M, we wite ρ γ l d 1 = ρ γ l d + ρ γ l d + ρ γ l d 1,b { ρ <ρ<m ρ} 1,b {ρ ρ } 1,b {ρ M ρ} = ρ γ ρ γ γ ρ γ 1 ρ ρ l d + ρ γ l d 1,b { ρ <ρ<m ρ} 1,b { ρ <ρ<m ρ} +γ ρ γ 1 ρ ρ l d + ρ γ l d 1,b {ρ ρ,m ρ} 1,b {ρ ρ } + ρ γ l d. 3. 1,b {ρ M ρ} Using the guments befoe 3. nd the Young inequlity, we hve ρ γ l b d M êρ l d + P b, ρ, 1 1 whee M = Mγ nd P b, ρ s climed. Since 1 êρt, l d 1 l+1 n sup τ [,t] by the fist enegy estimte, then the clim follows. Eρτ,, mτ, n 1 d 1 l E, Lemm 3.. Let K be compct subset of, nd T >. Then thee exists M = MK, T, E, independent of ε >, such tht T ρ γ+1 ddt C. 3.3 K

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 15 Poof. We divide the poof into five steps. 1. Let ω be smooth positive, compctly suppoted function on, b. We multiply the momentum eqution in 1.4 by ω to obtin t ρuω + ρu + pω + n 1 ρu ω ε ωm + n 1 m = ρu + p εm + n 1 m ω. 3.4 Integting 3.4 in ove, b yields t ρuω d + n 1 ρu ω d + εω m + n 1 m = ωρu + p + f 1, 3.5 whee f 1 = ρu + p εm + n 1 m ω d.. Multiplying 3.5 by ρ nd using the continuity eqution 1.4, we hve t ρ ρuω d + ρu + n 1 ρm ερ + n 1 ρ ρuω d nd whee Notice tht +ρ n 1 = ρ u + κρ γ+1 ω + ρf 1, ρ + ε ρu ω d + ερω m + n 1 m ρuω d t + ρu ρ + n 1 ρ ρuω d ρuω d + ρω m + n 1 m = κρ γ+1 + f, 3.6 f = ρf 1 n 1 ρu ρuω d ρ n 1 ρu ωd. ρ + n 1 ρ ρuω d + ρω m + n 1 m n 1 b = ρ ρuω d ρ ρuω ρ ρuω d n 1 ρ uω + n 1 ρ ρuω d + ρ u ω + ρ ρuω + n 1 ρ uω = ρ ρuω d n 1 b ρ uω ρ ρuω d +ρ u ω + n 1 ρ ρuω d.

16 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA It then follows tht whee ρ ρuω d t + ρu n 1 ε ρ ρuω d ρuω d ε ρ + ερ u ω ρuω d ερ uω = κρ γ+1 ω + f 3, 3.7 3. We multiply 3.7 by ω to obtin whee ρω ρuω d t + ρuω f 3 = f n 1 ρ ρuω d ε ωρ ρuω d. ρuω d + ε ρω ρuω d n 1 b ερ uω ε ρω ρuω d + ερ u ω + ερ uωω = κρ γ+1 ω + f 4, 3.8 f 4 = ωf 3 + ρuω ρuω d + ερω We integte 3.8 ove [, T ] [, b] to obtin ρuω d n 1 ρω ρuω d. T ρ γ+1 ω ddt = T + ε + ε ρω T T ερ u ω + ερ uωω ddt ρuω d T T d ρ 3 ω ddt + ε T b ρω ρuω d T T d f 4 ddt ρ u ω + ρ u ω ddt f 3 ddt ρ 3 ω ddt + Msupp ω, T, E. 3.9 The lst inequlity follows esily fom.5,.6, nd the fomul fo f 3. 4. Clim: Let γ > 1. Thee exists C = Csupp ω, T, E such tht ε t ρ 3 ω ddt M + Mε t ρ γ+1 ω ddt. 3.1

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 17 If γ, the clim is tivil. Let γ < α 3. We estimte ε T ρ α ω 3.11 ε sup ρ α γ ω T ddt ρ γ ddt supp ω supp ω εm sup ρ α γ ω supp ω T εm ρ α γ γ γ T ρ ω ddt + εm ρ α γ ω ω ddt T T T εm ρ γ ddt + ρ γ ω ddt + ρ α 3γ ω ddt supp ω T εm 1 + ρ α 3γ ω ddt. 3.1 If α 3γ γ + 1, the estimte of the clim follows. Othewise, since α 3γ < note tht α 3, we cn itete 3.11 with α eplced by α 3γ nd impove 3.11: T T ε ρ α ω ddt εm 1 + ρ 4α 9γ ω ddt. 3.13 If 4α 9γ is still lge thn γ + 1, we itete the estimte gin. In this wy, we obtin ecuence eltion α n = α n 1 3γ, α = α 3, nd the estimte T T ε ρ α ω ddt εmn 1 + ρ αnγ ω ddt. Solving the ecuence eltion, we obtin α n = n α 3γ n 1 1. Fo some n, the expession is less thn γ+1 note tht α 3. Then the expected estimte is obtined. 5. Now etuning to 3.9, we hve fo suitbly smll ε >. T ρ γ+1 ω ddt Msupp ω, T, E The following lemm holds fo wek entopies η; lso cf. [1]. Lemm 3.3. Let η ρ, m be the mechnicl enegy nd η ψ, q ψ be n entopy pi 1.1 1.11 with the geneting function ψs stisfying sup ψs <. s Then, fo ny ρ, m R + nd ny vecto ā = 1,, ηρ, mā, ā M ψ η ρ, mā, ā fo some M ψ >. 3.14

18 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Lemm 3.4. Let K, be compct. Thee exists M = MK, T, E independent of ε such tht, fo ny ε >, T whee P is descibed in Lemm 3.1. Poof. We divide the poof into five steps. K ρ u 3 + ρ γ+θ ddt M 1 + P b, ρ, 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 p ρ. Then η, q is still n entopy-entopy flux pi. We multiply the continuity eqution in 1.4 by η ρ n 1, the momentum eqution 1.4 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 It cn be checked diectly tht, fo some constnt C >, ρ η ρ + m + n 1 m η m. 3.15 qρ, m 1 M ρ u 3 + ρ γ+θ Mρ + ρ u + ρ γ, 3.16 ˇq + mˇη ρ + uˇη m. 3.17. Clim: The following inequlities hold pointwisely fo ρ, m R + with suitble choice of M = Mγ nd u = m ρ : ˇη m M u + ρ θ, ˇη ρ M u + ρ θ, 3.18 η Mρ + ρ u + ρ γ, ρ η ρ + u η m Mρ + ρ u + ρ γ. 3.19 Fo ˇη ρ + uˇη m consideed s function of ρ, u, it holds: ˇη ρ + uˇη m The poof is diect nd is omitted. 3. Note tht, t = b, ρ Mρθ 1 u + ρ θ 1, ˇη ρ + uˇη m u M u + ρθ. 3. 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.

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 19 We integte eqution 3.15 ove, T, b, 1, b, b, to find T qτ, n 1 dτ = cθ ρ γ+θ b n 1 T + T +n 1 + n 1 T ηt, η, n 1 d ˇq + mˇη ρ + m ρ ˇη m n ddτ n ˇη m ρ, pρ p ρ ddτ T + ε n 1 ρ + n 1 ρ η ρ + m + n 1 m η m ddτ = I 1 + + I 5. 3. 4. Now we estimte the tems in 3.. By P b, ρ we denote geneic function with the popeties defined in Lemm 3.1. Clely, I 1 MT P b, ρ. Notice tht ηρ, m Eρ, m. It then follows tht I n 1 sup τ [,T ] n 1 sup τ [,T ] ηρτ,, mτ, n 1 d Eρτ,, mτ, n 1 d. By the enegy estimte.6, I t, ME. The tem I 3 is nonpositive by 3.17 nd cn be dopped. Using Step, we hve I 4 t, M 1, T 1 + P b, ρ fo ny t, [, T ] [ 1, b]. 3.3 5. It emins to estimte 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 use the integtion by pts note tht η ρ ρ, = η m ρ, = to obtin I 5 = ε +ε t t ρ η ρ, + m η m, n 1 ddτ n 1ε η τ, n 1 dτ t m η m n 3 ddτ = J 1 + J + J 3. 3.4 Using the fist enegy estimte.6 nd Lemm 3.3, we hve Also, using Step nd 3.1, we hve J 1 t, MγE. m η m M ρ u + ρ γ + ρ η m ρ, M η ρ, m + ρ θ ρ.

GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA It follows by the enegy estimte.5 tht We wite J t, M 1, T, E 1 + P b, ρ. η = ρ η ρ + m η m = ρ η ρ + u η m + ρ η m u. Then, fo positive smooth nd compctly suppoted function ω on 1, b, we conside the integl J 3 ω d = ε t +ε t ρ η ρ + u η m ω n 1 ddτ n 1ε t ρ η ρ + u η m ω n ddτ ρρ η ρ + u η m ρ ρu η ρ + u η m u + ρ η m u ω n 1 ddτ. Noticing tht η ρ + u η m = ˇη ρ + uˇη m + const., using Step, estimtes.5,.6, nd 3.1, we obtin fom the bove eqution tht J 3 t, ω d M 1, T, E, ω C 1, δ + δ t ρ u 3 + ρ γ+θ ω n 1 ddτ. Finlly, we multiply eqution 3. by ω, integte it ove, b, nd use estimte 3.16, togethe with the bove estimtes fo I i s nd J i s, nd n ppopite choice of δ to obtin t ρ u 3 + ρ γ+θ ω n 1 ddτ M 1 + P b, ρ + 1 This completes the poof. t ρ u 3 + ρ γ+θ ω n 1 ddτ. 3.. Wek Entopy Dissiption Estimtes. Let = ε, b = bε, nd ρ = ρε s ε such tht P bε, ρε s ε, whee P b, ρ is function ppeing in Lemms 3.1 nd 3.4. 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 1.4 on Q ε = [, [ε, bε] with ρ = ρε s bove. Poposition 3.1. Let η, q be n entopy pi fom 1.1 nd 1.11 with smooth, compctly suppoted function ψs on R. Then the entopy dissiption mesues ηρ ε, m ε t + qρ ε, m ε x Poof. We divide the poof into five steps. is compct in H 1 loc. 3.5

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 1 1. 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 + n 1 m ε ηε m. We hve ερ ε η ε ρ εm ε η ε m + εη ε = I ε 1 + + I ε 4. 3.6 I ε 1t, Mρ ε u ε 1 + ρ ε θ M ρ ε u ε + ρ ε + ρ ε γ, 3.7 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. 3.8 Since η ε m ε ηm ε C ψ ρ ε + ρ ε u ε, then I ε is bounded in L 1, T ; L 1 loc,, independently of ε. 3.9 On the othe hnd, if ω is smooth, compctly suppoted on R +, then ε Ib ε ωt, ddt = ε n 1 η ε ω ddt 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 +. 3.3 3. 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 Combining 3.31 with Poposition 3.3, we hve 4. To show tht I3 ε fom [17]. M ψ ε Eρ ε, m ε ρ ε, m ε, ρ ε, m ε. I3 ε is unifomly bounded in L 1, T ; L 1 loc,. 3.31 in H 1 loc, we need the following clim, dopting the guments Clim 3.1. Let K is compct subset of,. Then, fo ny < δ < 1, ε >, T ε 3 ρ ε ddt C εδ γ + δ + ε. 3.3 In pticul, K T K ε 3 ρ ε ddt,

GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA nd εη ε in L p, T ; L p loc, fo p := 1,. γ + 1 Now we pove Clim 3.1. Fo the simplicity of nottion, we suppess supescipt ε in ll of the functions. Define { ρ φρ = ρ < δ, + δρ δ, ρ δ, so tht δ φ ρ = χ {ρ<δ} ρ, ρφ ρ φρ = ρ fo ρ < δ, ρφ ρ φρ = δ fo ρ δ, whee χ A ρ is the indicto function tht is 1 when ρ A nd othewise. Let ω be nonnegtive smooth, compctly suppoted function on,. We compute fom the continuity eqution, the fist eqution in 1.4: φω t + φuω φuω 1 ρ χ {ρ<δ} + δ χ {ρ>δ} ωu + n 1 ρu min{ρ, δ} = εφ ωρ ε min{ρ, δ}ω n 1ε ρ + ω min{ρ, δ}ρ εω ρ χ {ρ<δ}. 3.33 Integting 3.33 ove, T,, we obtin T εω ρ χ {ρ<δ} ddt = φω T T d + φuω ddt + 1 T ρ n 1 χ {ρ<δ} + δχ {ρ>δ} ωu ddt ρu min{ρ, δ} ddt T T n 1ε ε min{ρ, δ}ω ρ ddt + ω min{ρ, δ}ρ ddt = J 1 + + J 6. 3.34 We estimte the integls on the ight: T J 1 Msupp ω δ + δ ρ ddt Msupp ω, T, E, M δ; 3.35 supp ω T J δ ρu χρ<δ + δ + δρ u χ ρ>δ ddt supp ω T δ ρ + ρ u dtdt supp ω MT, supp ω, E, M δ; 3.36

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 3 J 3 δ 3 ε T supp ω ρ + ερ u ddt MT, supp ω, E, M δ ε ; 3.37 T J 4 Msupp ωδ ρ + ρ u ddt Msupp ω, T, E, M δ. 3.38 supp ω J 5 εδ γ ε 4 ε 4 T T T γ T ερ ρ ddt + ε ρ ρ χ {ρ<δ} ω ddt supp ω supp ω T ρ γ ρ ω ddt + ε ρ ω supp ω ω ddt + εδ γ MT, supp ω, E, M ρ ω ddt + εmt, supp ω, E, M + εδ γ MT, supp ω, E, M. 3.39 Moeove, J 6 is estimted in the sme wy s J 5. Thus, estimte 3.3 is poved. Notice tht η M ρ η ρ + uη m + ρ u M ρ 1 + ρ θ + ρ u. 3.4 Let p ]1, [ to be chosen lte on. Compute T T ε p η p ddt C ε p ρ p ddt + K K T δ + C δ + C T K T δ + C δ + ε p 1 M K T ε p ρ ddt K ε p ρ ρ θ + ρ u p ddt ε p ρ p γ ρ ρ p + ρ 1 u p ddt K T T δ + M δ K ε 3 ρ ddt K ερ γ ρ + ρ u + ερ p p ddt ε 3 ρ ddt + ε p 1 MT, K, E, 3.41 povided tht p = γ + 1 p = γ + 1. Combining this with estimte 3.3, we ive t the conclusion of the clim. 5. Combining Steps 1-4, we conclude ηρ ε, m ε t + qρ ε, m ε = f ε + g ε, 3.4

4 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA whee f ε is bounded in L 1, T ; L 1 loc, nd g ε in W 1,p loc R + fo some p 1,. This implies tht ηρ ε, m ε t + qρ ε, m ε x e confined in compct subset of W 1,q 1 loc, 3.43 fo 1 < q 1 <. On the othe hnd, using the fomuls 1.1 1.11 nd the estimtes in Poposition.1 nd Lemm 3.4, 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 x e unifomly bounded in W 1,q loc. 3.44 The intepoltion compctness theoem cf. [3, 11] indictes tht, fo q 1 > 1, q q 1, ], nd p [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,p loc R +, which is geneliztion of Mut s lemm in [1, 4]. Combining this intepoltion compctness theoem fo 1 < q 1 <, q >, nd p = with the fcts in 3.43 3.44, we conclude the esult. 3.3. Stong Convegence nd the Entopy Inequlity. The pioi estimtes nd compctness popeties we hve obtined in Sections 3.1 nd 3. 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 [16] 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 3.3 nd 3.4, 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 enegies: m ε ρ ε m ρ, ρε γ ρ γ in L 1 loc R +. Since the enegy η ρ, m is convex function, by pssing to the limit in.4, we obtin t t 1 m ρ + ργ t, n 1 ddt t 1 t m + ρ γ t, n 1 ddt, ρ fom which 1.14 follows. This implies tht thee is no concenttion fomed in the density ρ t the oigin =. Finlly, the enegy estimtes.5.6 nd the estimtes in Lemms 3.3 3.4 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,

GLOBAL SOLUTIONS TO THE EULER EQUATIONS WITH SPHERICAL SYMMETRY 5 fo ny ψs tht is convex with subqudtic gowth t infinity, tht is, ψs s α, α [,, fo s 1. Pssing to the limit in 3.6 multiplied by n nd integted ginst smooth compctly function suppoted on,,, we obtin 1.15. 3.4. Limit in the Equtions. Let ωt, be smooth, compctly suppoted function on [, [, bε, with ω t, =, fo ll close to. Assume tht the viscosity solutions ρ ε, m ε e extended by outside of [ε, bε]. By multiplying the fist eqution in 1.4 by n 1 ω, we obtin R + ρ ε ω t + m ε ω + ερ ε ω + n 1 ω n 1 ddt + ρ ε ω, n 1 d =.3.45 R 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 ρωt + mω n 1 ddt + R + R ρ ω, n 1 d =, which cn be extended to hold fo ll smooth, compctly suppoted function ωt, on [, [,, with ω t, =. Conside now the momentum eqution in 1.3. Let ωt, be smooth, compctly suppoted function on [, ε, bε. By multiplying the fist eqution in 1.4, we obtin R + + R + R m ε ω t + mε ρ ε ω + pρ ε ω + n 1 m ε ω, n 1 d =. ω + εm ε ω n 1 ddt Pssing to the limit, we find mω t + m ρ ω + pρω + n 1 ω n 1 ddt + m ω, n 1 d =. R This eqution cn be extended fo ll smooth compctly suppoted function ωt, on [, [, with ωt, = ω t, =, since m ρ +ργ t, n 1 L 1 loc [, [,. Acknowledgements: The esech of Gui-Qing 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-11848.

6 GUI-QIANG G. CHEN AND MIKHAIL PEREPELITSA Refeences [1] Binchini, S.; Bessn, A. Vnishing viscosity solutions of nonline hypebolic systems, Ann. of Mth., 161 5, 3 34. [] 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, 43 76 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. 91 1983, 1 3], Poc. Ame. Mth. Soc. 15 1997, 981 986. [5] Chen, G.-Q. Remks on spheiclly symmetic solutions of the compessible Eule equtions. Poc. Roy. Soc. Edinbugh, 17A 1997, 43 59. [6] Chen, G.-Q. nd Glimm, J. Globl solutions to the compessible Eule equtions with geometicl stuctue, Commun. Mth. Phys. 18 1996, 153 193. [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, 15-C43. [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, 1469 154. [9] Count, R. nd Fiedichs, K. O. Supesonic Flow nd Shock Wves, Spinge-Velg: New Yok, 1948. [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, 51 54 in English; 7A 1987, 467-48; 8A 1989, 61 94 in Chinese; Convegence of the fctionl step Lx-Fiedichs scheme nd Godunov scheme fo the isentopic system of gs dynmics, Comm. Mth. Phys. 11 1989, 63 84. [1] DiPen, R. Convegence of the viscosity method fo isentopic gs dynmics, Commun. Mth. Phys. 91 1983, 1 3. [13] Ldyzhenskj, O. A., Solonnikov, V. A., nd Ul cev, N. N. Line nd Qusi-line Equtions of Pbolic Type, LOMI AMS, 1968. [14] Lx, P. D. Shock wve nd entopy, Contibutions to Functionl Anlysis, ed. E.A. Zntonello, 63 634, Acdemic Pess, New Yok, 1971. [15] Liu, T.-P. Qusiline hypebolic system, Commun. Mth. Phys. 68 1979, 141 57. [16] LeFloch, Ph.G. nd Westdickenbeg, M. Finite enegy solutions to the isentopic Eule equtions with geometic effects, J. Mth. Pues Appl. 88 7, 386 49. [17] 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. 49 1996, 599 638. [18] Lions, P.-L., Pethme, B., nd Tdmo, E. Kinetic fomultion of the isentopic gs dynmics nd p-systems, Comm. Mth. Phys. 163 1994, 415 431. [19] 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. 9 199, 431 449. [] Mkino, T. nd Tkeno, S. Initil boundy vlue poblem fo the spheiclly symmetic motion of isentopic gs, Jpn J. Industil Appl. Mth. 11 1994, 171 183. [1] Mut, F. Compcité p compenstion, Ann. Scuol Nom. Sup. Pis Sci. Fis. Mt. 5 1978, 489 57. [] Rosselnd, S. The Pulstion Theoy of Vible Sts, Dove Publictions, Inc.: New Yok, 1964. [3] 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, 139-C134. [4] 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, 1979. [5] Whithm, G. B., Line nd Nonline Wves, John Wiley & Sons: New Yok, 1974.

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