1. INTRODUCTION In the present paper, we consider the following compressible Euler equations

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1 ON THE FREE BOUNDARY PROBLEM OF THREE-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS IN PHYSICAL VACUUM CHENGCHUN HAO ABSTRACT. In hs paper, we esabsh a pror esmaes for he hree-dmensona compressbe Euer equaons wh movng physca vacuum boundary, he γ-gas aw equaon of sae for γ = and he genera na densy ρ H 6 (. Because of he degeneracy of he na densy, we nvesgae he esmaes of he horzona spaa and me dervaves and hen oban he esmaes of he norma or fu dervaves hrough he epc-ype esmaes. We derve a mxed space-me nerpoaon nequay whch pay a va roe n our energy esmaes. The resus mprove he known resus (Comm. Mah. Phys. 96 ( on a pror esmaes for he case γ =. CONTENTS 1. Inroducon 1. Premnares 6.1. Noaons and weghed Soboev spaces 6.. Hgher-order Hardy-ype nequay and Hodge-ype epc esmaes 8.3. Properes of J, A and a 1 3. A mxed space-me nerpoaon nequay 1 4. A pror assumpon and he zero-h order energy esmaes The cur esmaes The esmaes for he horzona dervaves The esmaes for he me dervaves The esmaes for he mxed me-horzona dervaves The epc-ype esmaes for he norma dervaves The a pror bound 5 References 5 1. INTRODUCTION In he presen paper, we consder he foowng compressbe Euer equaons ρ dv(ρu = n ( (,T ], (1.1a (ρu dv(ρu u p = n ( (,T ], (1.1b p = on Γ 1 ( (,T ], (1.1c u 3 = on Γ (,T ], (1.1d Γ 1 ( = V (Γ 1 ( = u N n (,T ], (1.1e (ρ,u = (ρ,u n { = }, (1.1f Dae: January 6, 13. Key words and phrases. Compressbe Euer equaons, physca vacuum, free boundary, a pror esmaes. Ths research was paray suppored by he Naona Naura Scence Foundaon of Chna (Gran No and by he Youh Innovaon Promoon Assocaon, Chnese Academy of Scences. 1

2 CHENGCHUN HAO ( =, Γ 1 ( = Γ 1, (1.1g where ρ denoes he densy, he vecor fed u = (u 1,u,u 3 R 3 denoes he Eueran veocy fed, and p denoes he pressure funcon. = ( 1,, 3 and dv are he usua graden operaor and spaa dvergence where = / x. The open, bounded doman ( R 3 denoes he changng voume occuped by he fud, Γ 1 ( := ( denoes he movng vacuum boundary, V (Γ 1 ( denoes he norma veocy of Γ 1 (, N denoes he ouward un norma vecor o he boundary Γ 1 (, and Γ s a fxed boundary. The equaon of he pressure p(ρ s gven by p(x, = C γ ρ γ (x,, (1. where γ s he adabac ndex and we w ony consder he case γ = n hs paper; C γ s he adabac consan whch we se o uny,.e., C γ = 1; and ρ > n ( and ρ = on Γ 1 (. (1.3 Equaon (1.1a s he conservaon of mass, (1.1b s he conservaon of momenum. The boundary condon (1.1c saes ha pressure (and hence densy vanshes aong he vacuum boundary, (1.1d descrbes ha he norma componen of he veocy vanshes on he fxed boundary Γ, (1.1e ndcaes ha he vacuum boundary s movng wh he norma componen of he fud veocy, (1.1f and (1.1g are he na condons for he densy, veocy, doman and boundary. To avod he use of oca coordnae chars necessary for arbrary geomeres, for smpcy, we assume ha he na doman R 3 a me = s gven by = { (x 1,x,x 3 R 3 : (x 1,x T, x 3 (,1 }, where T denoes he -orus, whch can be hough of as he un square wh perodc boundary condons. Ths perms he use of one goba Caresan coordnae sysem. A =, he reference vacuum boundary s he op boundary whe he boom boundary Γ 1 = {(x 1,x,x 3 R 3 : (x 1,x T, x 3 = 1}, Γ = {(x 1,x,x 3 R 3 : (x 1,x T, x 3 = } s fxed wh he boundary condon (1.1d. We se he un norma vecors N = (,,1 on Γ 1 and N = (,, 1 on Γ. We use he sandard bass on R 3 : e 1 = (1,,, e = (,1, and e 3 = (,,1. Smary, he un angen vecors on Γ = Γ Γ 1 are gven by T 1 = (1,, and T = (,1,. Throughou he paper, we w use Ensen s summaon convenon. The k h -para dervave of F w be denoed by F,k = F x k. Then we have (dv(ρu u j =(ρu u j, = dv(ρuu j ρu u j, whch yeds ha (1.1b can be rewren, n vew of (1.1a, as Thus, he sysem (1.1 can be rewren as ρ( u u u p =. ρ dv(ρu = n ( (,T ], (1.4a ρ( u u u p = n ( (,T ], (1.4b p = on Γ 1 ( (,T ], (1.4c u 3 = on Γ (,T ], (1.4d

3 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 3 Γ 1 ( = u N n (,T ], (1.4e (ρ,u = (ρ,u n { = }, (1.4f ( =, Γ 1 ( = Γ 1. (1.4g Wh he sound speed gven by c := p/ ρ and N denong he ouward un norma o Γ 1, he sasfacon of he condon < c N < (1.5 n a sma neghborhood of he boundary defnes a physca vacuum boundary (cf. 1], where c = c = denoes he na sound speed of he gas. In oher words, he pressure acceeraes he boundary n he norma drecon. The physca vacuum condon (1.5 for γ = s equvaen o he requremen ρ N < on Γ 1. (1.6 Snce ρ > n, (1.6 mpes ha for some posve consan C and x near he vacuum boundary Γ 1, ρ (x Cds(x,Γ 1, (1.7 where ds(x,γ 1 denoes he dsance of x away from Γ 1. The movng boundary s characersc because of he evouon aw (1.1e, and he sysem of conservaon aws s degenerae because of he appearance of he densy funcon as a coeffcen n he nonnear wave equaon whch governs he dynamcs of he dvergence of he veocy of he gas. In urn, weghed esmaes show ha hs wave equaon ndeed oses dervaves wh respec o he unformy hyperboc non-degenerae case of a compressbe qud, wheren he densy akes he vaue of a srcy posve consan on he movng boundary ]. The condon (1.7 voaes he unform Kress-Lopansk condon 9] because of resonan wave speeds a he vacuum boundary for he nearzed probem. The mehods deveoped for symmerc hyperboc conservaon aws woud be exremey dffcu o mpemen for hs probem, wheren he degeneracy of he vacuum creaes furher dffcues for he nearzed esmaes. Now, we ransform he sysem (1.4 n erms of Lagrangan varabes. Le η(x, denoe he poson of he gas parce x a me. Thus, η = u η for >, and η(x, = x, (1.8 where denoes he composon,.e., (u η(x, := u(η(x,,. We e v = u η, f = ρ η, A = ( η 1, J = de η, a = JA, where A s he nverse of he deformaon ensor η 1 η = (η,1 η 1, η 1,3, j = η,1 η, η,3, η 3,1 η 3, η 3,3 J s he Jacoban deermnan and a s he cassca adjon of η,.e., he ranspose of he cofacor marx of η, expcy, η, η 3,3 η,3 η 3, η 1,3 η 3, η 1, η 3,3 η 1, η,3 η 1,3 η, a = η,3 η 3,1 η,1 η 3,3 η 1,1 η 3,3 η 1,3 η 3,1 η 1,3 η,1 η 1,1 η,3. (1.9 η,1 η 3, η, η 3,1 η 1, η 3,1 η 1,1 η 3, η 1,1 η, η 1, η,1

4 4 CHENGCHUN HAO Snce on he fxed boundary Γ, η 3 = x 3 = so ha accordng o (1.9, he componens a 3 1 = a3 = on Γ, and v 3 = on Γ due o v (,, 1 = where (,, 1 s he ouward un norma vecor o Γ, hen he Lagrangan verson of (1.4 can be wren n he fxed reference doman as f f A j v, j = n (,T ], (1.1a f v A j f, j = n (,T ], (1.1b f = on Γ 1 (,T ], (1.1c a 3 1 = a3 =, v 3 = on Γ (,T ], (1.1d (ρ,v,η = (ρ,u,e n { = }, (1.1e where e(x = x denoes he deny map on. From he dervave formua of deermnans, we have I foows from (1.1a and (1.11 ha J =JA j v, j = a j v, j. (1.11 f f J 1 J =, or ( f J =,.e., f = ρ J 1, (1.1 hus, he na densy funcon ρ can be vewed as a parameer n compressbe Euer equaons. Usng he deny A j = J 1 a j, we wre (1.1 as ρ v a j ( ρ J =, j n (,T ], (1.13a ρ = on Γ 1, (1.13b a 3 1 = a3 =, v 3 = on Γ (,T ], (1.13c (v,η = (u,e n { = }, (1.13d wh ρ (x Cds(x,Γ 1 for x near Γ 1. To undersand he behavor of vacuum saes s an mporan probem n gas and fud dynamcs. In parcuar, he physca vacuum, n whch he boundary moves wh a nonrva fne norma acceeraon, nauray arses n he sudy of he moon of gaseous sars or shaow waer 8]. Despe s mporance, here are ony few mahemaca resus avaabe near vacuum. The man dffcuy es n he fac ha he physca sysems become degenerae aong he vacuum boundary. The exsence and unqueness for he hree-dmensona compressbe Euer equaons modeng a qud raher han a gas was esabshed n 11] where he densy s posve on he vacuum boundary. Trakhnn provded an aernave proof for he exsence of a compressbe qud, empoyng a souon sraegy based on symmerc hyperboc sysems combned wh he Nash-Moser eraon n 14]. The oca exsence for he physca vacuum snguary can be found n he recen papers by Jang and Masmoud 7, 8] and by Couand and Shkoer 5, 6] for he one-dmensona and hree-dmensona compressbe gases. Couand, Lndbad and Shkoer 3] esabshed a pror esmaes based on me dfferenaed energy esmaes and epc esmaes for norma dervaves for γ = wh ρ H 4 ( where he energy funcon was gven by Ē( := 4 = η( 4 4 = 3 = ρ 4 η( ρ 4 ] v( ρ J ( 4 cur η v( 3 ρ 4 cur η v(.

5 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 5 We w no aemp o address exhausve references n hs paper. For more reaed references, we refer he neresed reader o 6, 8] and references heren for a nce hsory of he anayss of compressbe Euer equaons. In he presen paper, we w mprove he resus gven n 3, 6] by usng a smar argumen heren and fuy expong he degeneracy of he na densy ρ and he physca vacuum condon (1.7, especay focusng many on he foowng mprovemen comparng wh 3]. Frsy, he a pror esmaes were gven n 3] based on a proof for a speca case ρ = 1 x 3 of he densy, nsead of genera denses, from whch one can no easy oban he exac Soboev space ha he densy shoud beong o from hs speca densy. Because w appear a erm r ρ v evdeny for genera ρ n he r-h order horzona energy esmaes, we mus assume ρ H r1 ( a eas n vew of he hgher order Hardy nequay gven n Secon., for nsance, ρ H 5 ( f r = 4, see more deas n Sep 4 of he proof of Proposon 6.1. Secondy, we w prove, n deas, a mxed space-me nerpoaon nequay under he framework of Lebesgue spaces whch w pay a va roe n our energy esmaes, raher han he framework of Lebesgue spaces for me bu Soboev spaces for spaa varabes n bounded doman used n 3]. Fnay, s no enough o be conroed for he fourh order energy esmaes, whch can no cose hemseves, for genera ρ a eas, especay n deang wh some ower order erms n he argumen for he me dervaves. Thus, we have o nvesgae he ffh order energy esmaes whch are cosed wh hemseves. We now derve he physca energy of he sysem (1.13. From (1.11, he Poa deny (.16 gven n Secon.3, we ge 1 (ρ v =ρ v v = a j v( ρj (, j = a j ( v ρj J ρj, j = ρ J 1 (a j v ρ J, j. Snce ρ = on he boundary Γ 1 and a 3 v = on Γ, negrang over yeds, wh he hep of Gauss heorem, ha ( 1 E ( := ρ (x v(x, ρ(xj 1 (x, dx = 1 ρ v ρ J 1/ = E (. conserves for a. Ahough he physca energy s a conserved quany, s far oo weak for he purposes of consrucng souons. Insead, we nroduce he foowng ffh order energy funcon E( := 5 = 4 = η( 5 ρ1/ ρ J ( 5. Now, we sae our man resu as foows. 5 v( ρ 5 ] η( Theorem 1.1. Le γ =. Suppose ha (η(,v( s a smooh souon of he sysem (1.8 and (1.13 on a me nerva,t ] sasfyng he na bounds E( < and V := 4 m D 4 m v( L 3 ( <, m= and ha he na densy ρ > n and ρ H 6 ( sasfes he physca vacuum condon (1.7. Then here exss a T > so sma ha he energy funcon E( consruced from he

6 6 CHENGCHUN HAO souon (η(,v( sasfes he a pror esmae sup E( M,,T ] where boh M and T are funcons of E( and V. Remark 1.. The same argumens and he resus hod rue f he boom boundary Γ s aso a movng vacuum boundary,.e., by changng he boundary condon (1.1d no p = on Γ ( (,T ], whch w no cause any addona dffcues excep for he ransformaon of coordnaes. Remark 1.3. For he genera cases γ > 1 wh genera denses, we gve some furher remarks. We hnk ha hey are much more dfferen from he speca case γ =. They need o reform he energy funcon n order o ge a pror esmaes. For he cases γ >, seems o be smar o he case γ = due o γ 1 > γ/ n vew of he exponen of he wegh ρ γ/ and ρ γ 1 and weghed Soboev embeddng reaons gven n Secon.1, bu s no easy o dea wh he wegh ρ γ/ n energy esmaes n vew of he hgher order Hardy nequay. For he cases 1 < γ <, one have o use he wegh ρ γ 1 nsead of ρ γ/ n consrucng he energy funcon accordng o he physca vacuum condon, he hgher order Hardy nequay and weghed Soboev embeddng reaons, especay for he cases 3/ γ <, however one mus dea wh many exra, mporan and dffcu remander negras n he esmaes of every horzona, me or mxed dervaves. For he cases 1 < γ < 3/, mgh be dfferen from and dffcu han he above cases. We w dscuss he above genera cases n forhcomng papers f possbe. The res of hs paper s organzed as foows. We w gve some premnares n Secon. Precsey, we nroduce some noaons and weghed Soboev spaces n Secon.1; we reca he hgher-order Hardy-ype nequay and Hodge s decomposon epc esmaes n Secon.; we gve he properes of he deermnan J, he nverse of he deformaon ensor A and he ranspose of he cofacor marx a n Secon.3. Then, we gve a mxed space-me nerpoaon nequay n Secon 3, and derve he zero-h order energy esmaes n Secon 4 and he cur esmaes n Secon 5. Snce he sandard energy mehod s very probemac due o he degeneracy of ρ, we frs derve he esmaes of he horzona and me dervaves n Secons 6-8 and hen oban he esmaes of norma or fu dervaves hrough he epcype esmaes n Secon 9. Fnay, we w compee he proof of he a pror esmaes n Secon 1.. PRELIMINARIES.1. Noaons and weghed Soboev spaces. Throughou he paper, we w use he foowng noaon: wo-dmensona graden vecor or horzona dervave = ( 1,, he H s ( neror norm s, and H s (Γ boundary norm s. The componen of a marx M a he h row and he j h coumn w be denoed by M j. Somemes, we w use o sand for C wh a generc consan C. We make use of he Lev-Cva permuaon symbo ε jk = 1 ( j( j k(k = 1, even permuaon of {1,,3}, 1, odd permuaon of {1,,3},, oherwse, and he basc deny regardng he h componen of he cur of a vecor fed u: (curu = ε jk u k, j,

7 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 7 where means ha we have aken he sum wh respec o he repeaed scrps j and k. Snce v, j = u / η k η k / x j, foows ha u / η k = x j / η k v, j,.e., u,k = A j k v, j. The chan rue shows ha (curu(η = (cur η v := ε jk v k,s A s j, where he rgh-hand sde defnes he Lagrangan cur operaor cur η. Smary, we have dvu(η = dv η v := v j,s A s j, and he rgh-hand sde defnes he Lagrangan dvergence operaor dv η. We aso use he noaon for any vecor fed F ( η F j = F,s A s j. (.1 For any vecor fed F, we have F 3, j A j F, j A j 3 cur η F =cur(fa = F 1, j A j 3 F3, j A j 1, F, j A j 1 F1, j A j and hen namey, cur η F =(F 3, j A j F, j A j 3 (F 1, j A j 3 F3, j A j 1 (F, j A j 1 F1, j A j =F 3, j A j F3,k A k F, j A j 3 F,k A k 3 F1, j A j 3 F1,k A k 3 F 3, j A j 1 F3,k A k 1 F, j A j 1 F,k A k 1 F1, j A j F1,k A k F 3, j A j F,k A k 3 F1, j A j 3 F3,k A k 1 F, j A j 1 F1,k A k =F, j A j nf,k A k n F, j A j nf n,k A k =F, j A j n(f,k A k n F n,k A k = ηf F, j A j nf n,k A k, η F = cur η F ( η F ( η F, (. where he superscrp denoes he ranspose of he marx. As a generazaon of he sandard nonnear Gronwa nequay, we nroduce a poynomaype nequay. For a consan M, suppose ha f (, f ( s connuous, and f ( M C P( f (, (.3 where P denoes a poynoma funcon, and C s a generc consan. Then for aken suffceny sma, we have he bound (cf. 3, 4] f ( M. For negers k and a smooh, open doman of R 3, we defne he Soboev space H k ( (H k (;R 3 o be he compeon of C ( (C (;R 3 n he norm ( 1/ u k := D α u(x dx, α k for a mu-ndex α Z 3, wh he sandard convenon α = α 1 α α 3. For rea numbers s, he Soboev spaces H s ( and he norms s are defned by nerpoaon. We w wre H s ( nsead of H s (;R 3 for vecor-vaued funcons. In he case ha s 3, he above defnon aso hods for domans of cass H s. Our anayss w ofen make use of he foowng subspace of H 1 (: Ḣ 1 = {u H 1 ( : u = on Γ,(x 1,x u(x 1,x s perodc},

8 8 CHENGCHUN HAO where, as usua, he vanshng of u on Γ s undersood n he sense of race. For funcons u H k (Γ, k, we se ( 1/ u k := α u(x dx, α k Γ for a mu-ndex α Z. For rea s, he Hber space H s (Γ and he boundary norm s s defned by nerpoaon. The negave-order Soboev spaces H s (Γ are defned va duay: for rea s, H s (Γ := H s (Γ]. The dervave oss nheren o hs degenerae probem s a consequence of he weghed embeddng we now descrbe. Usng d o denoe he dsance funcon o he boundary Γ 1,.e., d(x = ds(x,γ 1, and eng p = 1 or, he weghed Soboev space Hd 1 p (, wh norm gven by d(x p ( F(x F(x dx for any F H 1 d p (, sasfes he foowng embeddng: H 1 d p( H1 p (; ha s, here s a consan C > dependng ony on and p, such ha F 1 p/ C d(x p( F(x F(x dx. (.4 See, for exampe, Secon 8.8 n Kufner 1]. From hs embeddng reaon and (1.7, we oban F C ρ( F(x F(x dx. (.5.. Hgher-order Hardy-ype nequay and Hodge-ype epc esmaes. We w make fundamena use of he foowng generazaon of he we-known Hardy nequay o hgherorder dervaves, see 6, Lemma 3.1]: Lemma.1 (Hgher-order Hardy-ype nequay. Le s 1 be a gven neger, and d(x be defned as above, and suppose ha hen u d Hs 1 ( and From hs emma, we can oban u H s ( Ḣ 1 (, u C u s. (.6 d s 1 Coroary.. Assume ha p, ] and ρ sasfes he condons (1.6 and (1.7, hen g C g 3 3 p 1, g H 3 3 p 1 ( Ḣ 1 (, (.7 ρ L p ( where s he ceng funcon defned by p = mn{n Z : n p}. Proof. By (1.7 and nocng ha d(x = ds(x,γ 1, we see ha 1 ρ (x C d(x for x near Γ 1. (.8

9 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 9 Snce ρ > n and (1.6, for x, we have 3 ρ > ε for some ε >, hen ρ (x = 3 ρ (x θ(1 x 1 x 3 > εd(x for x far away from Γ 1 wh some θ,1]. Thus, combnng wh (.8, we have 1 ρ (x C d(x for x. (.9 I foows, from Soboev s embeddng nequay and Lemma.1, ha for p, ] g C g L p ( d C g C g L p ( d 3(1/ 1/p 1, 3(1/ 1/p ρ f g H 3(1/ 1/p 1 ( Ḣ 1 (. The norma race heorem provdes he exsence of he norma race w N of a veocy fed w L ( wh dvw L ( (see, e.g., 13]. For our purposes, he foowng form s mos usefu (see, e.g., 1]: Lemma.3 (Norma race heorem. Le w be a vecor fed defned on such ha w L ( and dvw L (, and e N denoe he ouward un norma vecor o Γ. Then he norma race w N exss n H 1/ (Γ wh he esmae ] w N 1/ w C L ( dvw L (, (.1 for some consan C ndependen of w. Lemma.4 (Tangena race heorem. Le w L ( so ha curw L (, and e T 1, T denoe he un angen vecors on Γ, so ha any vecor fed u on Γ can be unquey wren as u α T α. Then ] w T α 1/ w C L ( curw L (, α = 1, (.11 for some consan C ndependen of w. Combnng (.1 and (.11, we have ] w 1/ C w L ( dvw L ( curw L ( (.1 for some consan C ndependen of w. The consrucon of our hgher-order energy funcon s based on he foowng Hodge-ype epc esmae (see, e.g., 3]: Lemma.5. For an H r doman, r 3, f F L (;R 3 wh curf H s 1 (, dvf H s 1 (, and F N Γ H s 1/ (Γ for 1 s r, hen here exss a consan C > dependng ony on such ha F s C ( F curf s 1 dvf s 1 F N s 3/, ( F s C F curf s 1 dvf s 1 α=1 F T α s 3/, where N denoes he ouward un norma o Γ and T α are angen vecors for α = 1,.

10 1 CHENGCHUN HAO.3. Properes of J, A and a. From he dervave formua of marces and deermnans, we have A k,s = Ak η, js A j, (.13 A k = A k = A k v, j A j, (.14 J,s =JA j η, js = a j η, js. (.15 I foows from a = JA, (.13 and (.15 ha he coumns of every adjon marx are dvergencefree,.e., he Poa deny, a k,k whch w pay a va roe n our energy esmaes. We aso have =, (.16 a k,s =Jη, js (A k A j A j Ak = J 1 η, js (a k a j a j ak, (.17 a k =Jv, j (A k A j A j Ak = J 1 v, j (a k a j a j ak. ( A MIXED SPACE-TIME INTERPOLATION INEQUALITY In hs secon, we prove a usefu mxed space-me nerpoaon nequay whch w pay a va roe n our energy esmaes. Proposon 3.1 (Mxed nerpoaon nequay. Le F(, x be a scaar or vecor-vaued funcon for,t ], T > and x R 3. Assume ha F (, L 3 (, F L (,T ];L 6 ( and F L (,T ];L (, hen we have F L 3 (,T ] CT /3 F ( L 3 ( sup F( L 6 ( F ( L (,T ] where C s a consan ndependen of T, and F. Proof. Noce ha F ( F F = ( F F = (F F F F F ], (3.1 mpes ha ( F F F F. Then, by he Fubn heorem, negraon by pars wh respec o me, he fundamena heorem of cacuus, he Höder nequay and he Mnkowsk nequay, we have T T F 3 L 3 (,T ] = F 3 dxd = F F F ddx = ( T = T F (T F (T F(T dx ( T C F(T L 6 ( ( F F d F F Fdxd F F d T F(T dx T F(T dx T F ( F ( F(dx ( F F Fddx ( T F ( F ( F d dx ( F F Fdxd F ( ( T F F L 6/5 ( d C F ( L 3 ( C F L 3 (,T ] F L (,T ] F L 6 (,T ] CT /3 F L 3 (,T ] F d dx T F ( L 3 ( sup F L 6 ( F L (,T ] T F L 3 ( d ], F F F dxd

11 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 11 from boh sdes of he n- whch mpes he desred nequay by emnang F L 3 (,T ] equay. 4. A PRIORI ASSUMPTION AND THE ZERO-TH ORDER ENERGY ESTIMATES We assume ha we have smooh souons η on a me nerva,t ], and ha for a such souons, he me T > s aken suffceny sma so ha for,t ], 1 J( 3. (4.1 Once we esabsh he a pror bounds, we can ensure ha our souon verfes he assumpon (4.1 by means of he fundamena heorem of cacuus. Then, by (1.9, Soboev s embeddng H ( L (, we have for,t ], I foows from a = JA and (4.1 ha η( L ( C η(, (4. a( L ( C η( L ( C η( 3. (4.3 A( L ( J 1 (a( L ( C η( 3. (4.4 Now, we prove he foowng zero-h order energy esmaes. Proposon 4.1. I hods for r 4 sup,t ] ρ 1/ v ρ J 1/ ] M CT P(sup E r (.,T ] Proof. Takng he L -nner produc of (1.13a wh v yeds, by negraon by pars, (.16, (1.13b, (1.13c and (1.11, ha 1 d ρ v dx = a j ( ρ d J, j v dx = ρj a j v, j dx a 3 ρj v dx 1 dx Γ = d ρ d J 1 dx. I foows, from he negraon over,t ] and by Höder s nequay, (4.1 and (1.9, ha 1 ρ v dx ρ J 1 dx = 1 ρ u dx ρ dx = 1 ρ1/ u ρ, whch mpes, by Cauchy s nequay, ha 1 4 sup,t ] ρ 1/ Thus, we compee he proof. v sup,t ] ρ J 1/ M CT P(sup E r (.,T ] 5. THE CURL ESTIMATES Takng he Lagrangan cur of (1.13a yeds ha ε j A s j(ρ v a k (ρ J,k,s =,.e., ε j A s jρ,s v ρ ε j A s jv,s ε ja s ja k,s (ρ J,k ε j JA s ja k (ρ J,ks =.

12 1 CHENGCHUN HAO For he as erm, we can nerchange he ndces k and s, and j o fnd ha vanshes due o he fac ε j = ε j. For he frs erm, we can use he equaon (1.13a. Then usng (.17 for he hrd erm, foows ha ε j ρ,s ρ JA s ja k (ρ J,k ρ ε j A s jv,s ε jja s jη a,bs (A b aa k A k aa b (ρ J,k =. Inerchanged he ndces b and s, and j, he erm ε j JA s j ηa,bs A k aa b (ρ J,k vanshes. Hence, from (.15, we have ρ ε j A s jv,s =ε ρ,s j JA s ρ ja k (ρj,k ε j JA s jη a,bs A b aa k (ρj,k =ε j ρ,s ρ A s ja k ρ ρ,k J 1 ε j ρ,s As ja k ρ J 1 A q pη p,qk ε j A s jη a,bs A b aa k ρ ρ,k J 1 ε j A s jη a,bs A b aa k ρ J 1 A q pη p,qk. The frs and as erms vansh by nerchangng he ndces k and s, and j. The second and hrd erms emnae. Therefore, yeds We can oban he foowng proposon. Proposon 5.1. For a (,T ], we have for r 4 r 1 = cur Proof. From (5.1, we ge η( r 1 r Inegrang over (, yeds ε j A s jv,s =, or cur η v =. (5.1 = ρ r cur (cur η v = ε j A s jv,s. cur η v( = curu and compung he r-h order horzona dervaves of hs reaon yeds cur η r v( = r (cur η v( ε j r A s jv r 1,s ε j Cr r k A s j k v,s Nocng ha η = v, we have η( M CT P(sup E r (. (5.,T ] ε j A s j v,s d, (5.3 k=1 = r curu ε j r A s jv r 1,s ε j Cr r k A s j k v,s (cur η r η = cur η r v ε j A s j r η,s = r curu ε j r A s jv r 1,s ε j Cr r k A s j k v,s ε j A s j r η,s Inegrang over,] yeds cur η r η = r curu k=1 k=1 ε j A s r η,m A m j v,s d r 1 ε j A s v,m A m j r η,s d k=1 C r ε j r (A s j v,s d. ε j r (A s j v,s d. ε j r k A s j k v,s d ε j r (A s pv p,q A q j v,s d d, (5.4

13 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 13 and hen by he fundamena heorem of cacuus I foows ha r curη = r curu cur η r η = cur r η ε j A s v,m A m j r η,s d ε j A s r η,m A m j v,s d r 1 ε j A s j( d r η,s. (5.5 k=1 ε j A s j( d r η,s d Cr k ε j r k A s j k v,s d ε j r (A s pv p,q A q j v,s d d. Noce ha for k = 1, negraon by pars wh respec o me and he fac η( = mpy = ε j r 1 A s j v,s d = ε j A s p r 1 η p,q A q j v,s d = ε j A s p r 1 η p,q A q j η,s ε j r 1 η p,q η,s (A s pa q j d ε j r (A s p η p,q A q j v,s d r 3 Cr = ε j A s p r 1 v p,q A q j η,s d r 3 Cr = r (A s pa q j 1 η p,q v,s d r (A s pa q j 1 η p,q v,s d. Snce oher erms can be esmaed easy, hus, foows ha sup ρ r curη dx M δ sup E r ( CT P(sup E r (.,T ],T ],T ] The weghed esmaes for he cur of m r m η can be obaned smary. From (5.3, we see ha cur η η = curu ε j A s jη,s d and hen by he fundamena heorem of cacuus, curη = curu ε j A s jd η,s I foows ha and ε j A s jη,s d curη M CT P(sup E r (,,T ] ε j A s j v,s d d, r 1 curη M CT P(sup E r (,,T ] ε j A s j v,s d d. where we have used negraon by pars wh respec o me for he negras nvovng r v n order o conro hem by η r and v r 1. Thus, we ge curη r 1 M CT P(sup E r (. (5.6,T ] From (5.3, we ge, by he fundamena heorem of cacuus, curv( = curu ε j A s jd v,s ε j A s jv,s d,

14 14 CHENGCHUN HAO and, by akng he frs order me dervave, curv ( = ε j A s jd v,s. Snce H r ( s a mupcave agebra for r 4, we can drecy esmae he H r (-norm of curv o show ha curv ( r M CT P(sup E r (. (5.7,T ] The esmaes for cur m η( n H r 1 m ( for m r 1 foow from he same argumens. 6. THE ESTIMATES FOR THE HORIZONTAL DERIVATIVES We have he foowng esmaes. Proposon 6.1. Le r {4,5}. For sma δ > and he consan M dependng on 1/δ, hods sup ρ 1/ r v( ρ r η( ρ r ] dvη( η α r 1/,T ] M δ sup E r ( CT P(sup E r (,,T ],T ] where η α = η T α for α = 1,. Proof. Leng r ac on (1.13a, hen we have r = C r r ρ v r = C r r a j ( ρ J, j =, where Cr s he bnoma coeffcen. Takng he L (-nner produc wh r v, we oban 1 d ρ r v 3 6 dx d I m = I m. (6.1 m= m=4 Here, I 1 := r a j ( ρ J, j r ( v dx, I := a j r ρj r v dx,, j I 3 := r 1 I 5 := ( a j ρ r J r v dx,, j Cr =1 I 4 := r a j ( ρ J, j r v dx, r 1 Cr = r ρ v r v dx, r 1 I 6 := Cr =1 ( a j r ρ J r v dx., j Sep 1. Anayss of he negra I 1. We use he deny (.16 o negrae by pars wh respec o x j o fnd ha I 1 = r a j ρ J r v, j dx r a 3 r v ρj dx 1 dx = r a j r v, j ρj dx, Γ due o he boundary condons (1.13b and (1.13c. I foows from (.17 ha I 1 = r 1 ( η,k J 1 (a j ak ak a j r v, j ρj dx = r η,k A k r v, j A j ρ J 1 dx (6.

15 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 15 r η,k A k r v, j A j ρ J 1 dx (6.3 r 1 Cr 1 s s=1 r s η,k s ( J 1 (a j ak ak a j r v, j ρ J dx. (6.4 Snce v = η, we ge (6. = 1 d dv η r η ρ d J 1 dx r η,k A k r η, j A j ρ J 1 dx 1 dv η r η ρ J 1 dx = 1 d dv η r η ρ d J 1 dx 1 r η,k r η, j (A k A j ρ J 1 dx 1 dv η r η ρ J 1 dx. For he negra (6.3, snce v = η, hods r η,k A k A j r v, j = ( r η,k A k r η, j A j r v,k A k r η, j A j r η,k r η, j (A k A j. I foows from (. ha r η,k A k A j r v, j = 1 Thus, we have I foows ha T I 1 d = 1 and I s cear ha (6.3 = 1 d d T T η r η cur η r η ] 1 r η,k r η, j (A k A j η r η cur η r η ] ρj 1 dx η r η cur η r η ] ρj 1 dv η vdx r η,k r η, j (A k A j ρ J 1 dx. η r η(t dv η r η(t cur η r η(t ] ρ J 1 (T dx T. (6.5 η r η dv η r η cur η r η ] ρ J 1 dv η vdxd r η,k r η, j (A k A j A j Ak ρ J 1 dxd T (6.4d. η r η dv η r η cur η r η ] ρj 1 dv η vdxd CT sup ρ r η v 3 η 4 3 CT P(sup E r (,,T ],T ] T r η,k r η, j (A k A j A j Ak ρ J 1 dxd CT sup ρ r η v 3 η 6 3,T ] CT P(sup E r (.,T ]

16 16 CHENGCHUN HAO Now, we anayze he negra T (6.4d. We w use negraon by pars wh respec o me for he cases s = 1 and s =, whe we have o use negraon by pars wh respec o spaa varabes for he case s = r 1. Case 1: s = 1. From negraon by pars wh respec o me, we ge T (r 1 r 1 ( η,k a j Ak Ak a j r v, j ρj dxd (6.6 = (r 1 (r 1 (r 1 (r 1 T T T r 1 η,k r 1 v,k r 1 η,k r 1 η,k ( a j Ak Ak a j ( a j Ak Ak a j ( a j Ak Ak a j ( a j Ak Ak a j r η, j ρj dx (6.7 =T r η, j ρ J dxd (6.8 r η, j ρ J dxd (6.9 r η, j ρ J dxd. (6.1 I s cear ha, by Höder s nequay and he fundamena heorem of cacuus wce, T (6.7 C ρ r 1 ηd ρ r η(t η(t 7 4 ( CT sup ρ r 1 ηd ρ r 1 η( ρ r η(t η(t 7 4,T ] ( and CT sup,t ] ρ r 1 η M CT P(sup E r (,,T ] =1 ρ r 1 η( ρ r η(t η(t 7 4 ( T (6.8 CT sup ρ r 1 ηd ρ r 1 η( ρ r η η 7 4,T ] ( CT sup,t ] ρ r 1 η M CT P(sup E r (,,T ] =1 ρ r 1 η( ρ r η η 7 4 (6.1 CT ρ sup r 1 η P( η 4 v ρ r η M CT P(sup E r (.,T ],T ] We can rewre (6.9 as, for β {1,} T (6.9 =3 r 1 η,β 3 T r 1 η,k ( a 3 A β A β a3 J r η,3 ρdxd (6.11 ( a β Ak Ak a β J r η,β ρdxd. (6.1 Obvousy, we have from he Höder nequay and Soboev s embeddng heorem ha (6.11 CT sup ρ r η 1 P( η 4, v 3, v 1 ρ r η,t ]

17 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 17 CT sup ( ρ 3 η r ρ r η P( η 4, v 3, v 1 ρ r η,t ] M CT P(sup E r (.,T ] By negraon by pars, yeds T (6.1 = 3 r 1 η,kβ I s easy o see ha 3 3 T T ( a β Ak Ak a β r η J ρdxd (6.13 r 1 ( η,k a β Ak Ak a β r 1 η,k ( a β Ak Ak a β r η J ρdxd (6.14,β r η ( J ρ dxd. (6.15,β and (6.13 CT sup ρ r η v 1 P( η 4 ρ r η 1 M CT P(sup E r (,,T ],T ] (6.15 CT sup r 1 η v 1 P( η 4 ρ r η 1 ( ρ L ( ρ L (,T ] M CT P(sup E r (.,T ] In order o esmae (6.14, we frs consder he esmae of he D r 1 v L 3 (,T ] where D r 1 denoes a he dervaves θ for he mu-ndex θ = (θ 1,θ,θ 3 and θ r 1. By Proposon 3.1 wh F = D r 1 η and he Soboev embeddng heorem, we have D r 1 v L 3 (,T ] CT /3 D r 1 v( L 3 ( sup D r 1 v L ( Dr 1 η L 6 (,T ] Thus, we oban M CT /3 sup v r 1 η r,t ] M CT /3 P(sup E r (.,T ] D r 1 v L 3 (,T ] M CT /3 P(sup E r (. (6.16,T ] By he Höder nequay, he Soboev embeddng heorem, he Cauchy nequay and (6.16, we easy ge (6.14 CT /3 ρ v L 3 (,T ] sup 3 η ρ 4 η 1,T ] CT 1/3 v L 3 (,T ] CT ρ sup 3 η ρ 4 η 1 CT (M 1/3 CT /3 P(sup E r (,T ] M CT P(sup E r (.,T ],T ] M CT P(sup E r (,T ] ]

18 18 CHENGCHUN HAO Hence, we oban (6.6 M CT P(sup E r (.,T ] Case : s =. By negraon by pars wh respec o me, yeds T Cr 1 r η,k (A j ak ak A j r v, j ρj dxd (6.17 = Cr 1 r η,k (A j ak ak A j r η, j ρj dx (6.18 =T T Cr 1 r v,k (A j ak ak A j r η, j ρj dxd (6.19 T Cr 1 r η,k (A j ak ak A j r η, j ρj dxd (6. T Cr 1 r η,k (A j ak ak A j r η, j ρ J dxd. (6.1 Appyng he fundamena heorem of cacuus yeds for a sma δ > (6.18 C ρ r η(t 1 P( η(t 4 η(t 1 ρ r η(t ( T C ρ P( η(t 4, η(t r vd η( 1 ρ r η(t 1 M δ sup E r ( CT P(sup E r (.,T ],T ] Smar o (6.14, we can ge (6.19 (6. M δ sup E r ( CT P(sup E r (.,T ],T ] By an L 6 -L 6 -L 6 -L Höder nequay and he Soboev embeddng heorem, we ge Therefore, we have obaned (6.1 M CT P(sup E r (.,T ] (6.17 M δ sup E r ( CT P(sup E r (.,T ],T ] Case 3: s = r 1. We wre he space-me negra as, for β {1, } T η,k r 1 (a j Ak Ak a j r v, j ρj dxd (6. T = η,k r 1 (a β Ak Ak a β r v,β ρj dxd (6.3 T η,β r 1 (a 3 A β A β a3 r v,3 ρ J dxd. (6.4 By negraon by pars wh respec o x β, we have T (6.3 = η,kβ r 1 (a β Ak Ak a β r v ρj dxd (6.5 T T η,k r 1 (a β,β Ak Ak a β,β r v ρ J dxd (6.6 η,k r 1 (a β Ak,β Ak,β aβ r v ρ J dxd (6.7

19 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 19 T T η,k r 1 (a β Ak Ak a β r v (ρ,β J dxd (6.8 η,k r 1 (a β Ak Ak a β r v ρ J,β dxd. (6.9 From (.17, foows ha T (6.5 = η,kβ r η p,q JA β (Aq pa k Ak pa q ] r v ρj dxd (6.3 We can wre T (6.3 = = T T T T η,kβ r ] η p,q JA β (Ak pa q Aq pa k r v ρj dxd (6.31 η,kβ r η p,q JA β p(a k A q Aq Ak ] r v ρ J dxd. (6.3 η,kβ r η p,α JA β (Aα p A k Ak pa α ] r v ρ J dxd η,αβ r η p,3 JA β (A3 pa α A α p A 3 ] r v ρ J dxd η,kβ r 1 η p,α A β (Aα p A k Ak pa α r v ρ J 1 dxd (6.33 r 3 T Cr m m= T I s easy o see ha r 3 T Cr m m= η,kβ m1 η p,α r m JA β (Aα p A k Ak pa α ] r v ρ J dxd (6.34 η,αβ r 1 η p,3 A β (A3 pa α A α p A 3 r v ρ J 1 dxd (6.35 η,αβ m1 η p,3 r m JA β (A3 pa α A α p A 3 ] r v ρ J dxd. (6.33 CT ρ 1/ sup η 1 ρ r η 1 P( η 3 ρ 1/ r v M CT P(sup E r (,,T ] and by an L 6 -L 6 -L 6 -L Höder nequay and he Soboev embeddng heorem, (6.34 (6.36 CT ρ 3/ sup P( η 4, η r ρ 1/ r v M CT P(sup E r (.,T ],T ],T ] (6.36 By usng negraon by pars wh respec o me, we have (6.35 = η,αβ r 1 η p,3 A β (A3 pa α A α p A 3 r η ρj 1 dx (6.37 =T T v,αβ r 1 η p,3 A β (A3 pa α A α p A 3 r η ρj 1 dxd (6.38 T T η,αβ r 1 v p,3 A β (A3 pa α A α p A 3 r η ρ J 1 dxd (6.39 η,αβ r 1 η p,3 A β (A3 pa α A α p A 3 J 1] r η ρ dxd. (6.4

20 CHENGCHUN HAO Obvousy, yeds by usng he fundamena heorem of cacuus wce ( T (6.37 C ρ 3 η 3 ρ 3 ηd P( η(t 3, η(t r ρ r η(t 1 C ρ 3 η 3 ( CT ρ 3 η( ρ M CT P(sup E r (.,T ] T 3 ηd P( η(t 3, η(t r ρ r η(t 1 By he Höder nequay, he Soboev embeddng heorem and he fundamena heorem of cacuus, we ge (6.38 CT sup ρ 3 η 1 r 1 η ρ r η 1 P( η 3,T ] ( CT sup ρ 3 v 3 ρ 3 η( ρ,t ] M CT P(sup E r (.,T ] Smary, we have T 3 η ρ r η 1 P( η 3, η r and (6.39 CT sup 3 η 1 ρ r 1 η ρ r η 1 P( η 3 M CT P(sup E r (,,T ],T ] (6.4 CT ρ sup 3 η 1 r 1 η v 1 ρ r η 1 P( η 3 M CT P(sup E r (.,T ],T ] We can dea wh (6.31 and (6.3 as he same argumens as for (6.3. Thus, we oban We wre (6.6 = T T T (6.5 M CT P(sup E r (. (6.41,T ] η,k r 1 η p,qβ A q p(a β Ak Ak A β r v ρ J 1 dxd (6.4 η,k r 1 η p,αβ A β p(a k A α A α A k r v ρ J 1 dxd (6.43 η,α r 1 η p,3β A β p(a α A 3 A3 A α r v ρ J 1 dxd (6.44 r T Cr 1 m m= r T Cr 1 m m= r T Cr 1 m m= η,k m η p,qβ r 1 m JA q p(a β Ak Ak A β ] r v ρ J dxd (6.45 η,k m η p,αβ r 1 m JA β p(a k A α A α A k ] r v ρ J dxd η,α m η p,3β r 1 m JA β p(a α A 3 A3 A α ] r v ρ J dxd. (6.46 (6.47

21 I s easy o see ha FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 1 (6.4 (6.43 (6.44 CT ρ 1/ sup,t ] M CT P(sup E r (.,T ] η 4 ρ r η P( η 3 ρ 1/ r v By he Höder nequay and he Soboev embeddng heorem, we have Hence, (6.45 (6.46 (6.47 CT ρ 3/ sup,t ] M CT P(sup E r (.,T ] ρ 1/ r v P( η 4, η r (6.6 M CT P(sup E r (. (6.48,T ] By (.13, we have T (6.7 = η,k r 1 ( η p,qβ JA k p(a q Aβ A β Aq r v ρj dxd = T η,k r 1 η p,qβ A k p(a q Aβ A β Aq r v ρ J 1 dxd (6.49 r T Cr 1 m m= η,k m η p,qβ r 1 m ( JA k p(a q Aβ A β Aq r v ρ J dxd. By he Höder nequay and he Soboev embeddng heorem, we have (6.49 CT sup η 4 ρ r η P( η 3 ρ 1/ r v ρ 1/ M CT P(sup,T ] and smar o (6.45 Thus, For (6.8 and (6.9, s easy o have Hence, (6.5 M CT P(sup E r (.,T ],T ] E r (, (6.5 (6.7 M CT P(sup E r (. (6.51,T ] (6.8 (6.9 CT ( ρ 3/ 3 ρ 3/ M CT P(sup E r (.,T ] sup,t ] ρ 1/ r v P( η 4, η r (6.3 M CT P(sup E r (. (6.5,T ] By negraon by pars wh respec o me, hods (6.4 = η,β r 1 (a 3 A β A β a3 r η,3 ρj dx (6.53 =T T v,β r 1 (a 3 A β A β a3 r η,3 ρ J dxd (6.54

22 CHENGCHUN HAO T T η,β r 1 (a 3 A β A β a3 r η,3 ρ J dxd (6.55 η,β r 1 (a 3 A β A β a3 r η,3 ρ J dxd. (6.56 I s easy o see ha by appyng he fundamena heorem of cacuus hree mes (6.53 C η(t 4 ρ r η(t P( η 3 ( T ρ r 1 ηd ρ r T η 1 vd ρ r 3 T η 1 vd L 3 ( (1 sgn(5 r ρ 3 T η 1 r 4 vd 1 M CT P(sup E r (.,T ] By (.17 and (.13, we have T (6.54 = v,β r ( η p,α JA β (A3 A α p A α a 3 p r η,3 ρj dxd (6.57 T T v,β r ( η p,α JA β p(a 3 Aα A α a3 r η,3 ρj dxd (6.58 v,β r ( η p,α JA β (A3 pa α A α p a 3 r η,3 ρ J dxd. (6.59 We sp (6.57 no wo negras,.e., T (6.57 = v,β r 1 η p,α JA β (A3 A α p A α a 3 p r η,3 ρj 1 dxd (6.6 r 3 m= C m r Obvousy, we see ha and T v,β m1 η p,α r m ( JA β (A3 A α p A α a 3 p r η,3 ρj dxd. (6.6 CT sup v 1 ρ r η 1 P( η 3 ρ r η M CT P(sup E r (,,T ],T ] (6.61 CT ρ sup v 1 P( η r ρ r η M CT P(sup E r (.,T ],T ] Boh (6.58 and (6.59 can be dea wh as he same argumen as for (6.57. Thus, we oban (6.54 M CT P(sup E r (.,T ] 1 (6.61 By (.18 and (.14, we have T (6.55 = η,β r 1 ] v p,α JA β (A3 A α p A α a 3 p r η,3 ρj dxd (6.6 T η,β r 1 ] v p,α JA β p(a 3 Aα A α a3 r η,3 ρj dxd (6.63

23 We wre (6.6 = T FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 3 T r T Cr 1 m m= η,β r 1 v p,α JA β (A3 pa α A α p a 3 ] r η,3 ρ J dxd. (6.64 η,β r 1 v p,α A β (A3 A α p A α a 3 p r η,3 ρ J 1 dxd (6.65 η,β m v p,α r 1 m ] JA β (A3 A α p A α a 3 p r η,3 ρj dxd. (6.66 Then, by he same argumens as for (6.44 and (6.47, we can esmae (6.65 and (6.66, and hen (6.55 M CT P(sup E r (.,T ] I s easy o see ha (6.56 has he same bounds. Thus, we oban he esmaes of (6.4 and hen of (6.,.e., (6. M CT P(sup E r (.,T ] Case 4: s = r and r = 5. Inegraon by pars wh respec o me gves T η,k 3 ( J 1 (a j ak ak a j 5 v, j ρj dxd = η,k 3 ( J 1 (a j ak ak a j 5 η, j ρj dx T T v,k 3 ( J 1 (a j ak ak a j 5 η, j ρj dxd T η,k 3 ( J 1 (a j ak ak a j 5 η, j ρj dxd T η,k 3 ( J 1 (a j ak ak a j 5 η, j ρ J dxd, whch can be conroed by M CT P(sup,T ] E 5 ( from he Höder nequay and he fundamena heorem of cacuus. Therefore, we oban T (6.4d M δ sup,t ] E r ( CT P(sup E r (.,T ] Sep. Anayss of he negra I. Smar o hose of I 1, by (1.13b, (1.13c and η = v, we have I = a j r ρj r v, j dx a 3 r ρj r v dx 1 dx = r ρa j J 1 r η, j dx. Γ Inegraon by pars shows ha T I (d = r ρ(a j J 1 r η, j dxd T = r ρ(a j J 1 r η, j dxd T r ρa j J 1 r η, j dx ( r T ρ δ j =T (A j J 1 d r η, j (T dx,

24 4 CHENGCHUN HAO whch yeds, by Höder s nequay and Soboev s embeddng heorem, ha T I (d C r ( ρ T sup ρ r η v 3 η 4 3 ρ r dvη ρ,t ] M δ sup E r ( CT P(sup E r (,,T ],T ] where we requre ρ H max(4,r ( because, for r 5, r ρ r ρ Cr m m ρ r m ρ (r 1/ m= ρ Cr m m ρ r m ρ m= ρ C r ρ ρ C r 1 ρ ρ ρ C r ρ ρ C ρ r C ρ 4 ρ r 1 C ρ 3 ρ r C ρ max(4,r, (6.67 by he hgher order Hardy nequay. Sep 3. Anayss of he negra I 3. Smar o hose of I 1, by (1.13b, (1.13c, (.15 and η = v, we have I 3 = = = Due o v = η, we ge (6.68 = ρ r J a j r v, j dx ρ r J a 3 r v dx 1 dx Γ ρ r 1 (J 3 Ja j r v, j dx r JA j r v, j ρ J dx (6.68 r Cr 1 s s= = d d r Cr 1 s s= r 1 s J 3 s1 Ja j r v, j ρ dx. (6.69 r η k, A k r v, j A j ρ J 1 dx (6.7 r 1 s a k s1 η k, r v, j A j ρ J dx (6.71 r η k, A k r η, j A j ρ J 1 dx (6.71 r η k, r η, j A k A j dv η vρ J 1 dx (6.7 r η k, r η, j (A k A j ρ J 1 dx. (6.73 Nocng ha r dvη( =, negrang over,t ] yeds T (6.68d = dv η r T η(t ρj 1 (T dx (6.7 (6.73 (6.71]d. By Höder s nequay and Soboev s embeddng heorem, we oban T (6.7 (6.73d CT sup ρ r η v 3 η 6 4 CT P(sup E r (.,T ],T ]

25 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 5 By negraon by pars, we have T r (6.71 = Cr 1 s s= r = T T Cr 1 s s= T Cr 1 s s= T Cr 1 s s= r r r 1 s a k s1 η k, r v, j ρ J A j dxd r s a k s1 η k, r 1 v, j ρj A j dxd (6.74 r 1 s a k s η k, r 1 v, j ρj A j dxd (6.75 r 1 s a k s1 η k, r 1 v, j (ρj A j dxd. (6.76 We frs consder (6.74 and sp no four cases. Case 1: s =. By an L -L -L Höder nequay, he Soboev embeddng heorem and he fundamena heorem of cacuus for he norm ρ r 1 v, we can easy ge T r a k ηk, r 1 v, j ρj A j dxd M CT P(sup E r (. Case : s = 1. Inegraon by pars yeds T Cr 1 1 r 1 a k η k, r 1 v, j ρj A j dxd (6.77 T =Cr 1 1 r a k η k, r v, j ρj A j dxd (6.78 T Cr 1 1 r 1 a k 3 η k, r v, j ρj A j dxd (6.79 T Cr 1 1 r 1 a k η k, r v, j (ρj A j dxd. (6.8 By an L -L 6 -L 3 Höder nequay, (6.16 and he Soboev embeddng heorem, we ge (6.78 (6.8 M CT P(sup E r (.,T ] By usng (.17, hods T (6.79 = Cr 1 1 r ] η p,q J(A k Aq p A q k A p 3 η k, r v, j ρj A j dxd =C 1 r 1 T C 1 r 1 C 1 r 1 C 1 r 1 T r 1 η p,β (A k Aβ p A β k A p 3 η k, r v, j ρ J 1 A j r 3 T Cr m m=,t ] dxd (6.81 r 1 η p,3 (A β k A3 p A 3 k Aβ p 3 η k,β r v, j ρj 1 A j dxd (6.8 r 3 T Cr m m= m1 η p,β r m ] J(A k Aβ p A β k A p 3 η k, r v, j ρj A j dxd (6.83 m1 η p,3 r m ] J(A β k A3 p A 3 k Aβ p 3 η k,β r v, j ρj A j dxd. (6.84

26 6 CHENGCHUN HAO By usng he L 6 -L -L 3 Höder nequay for (6.81 and L -L 6 -L 3 Höder nequay for (6.8 on hgher order erms, ogeher wh (6.16 and he Soboev embeddng heorem, we ge (6.81 (6.8 M CT P(sup E r (.,T ] From he L -L 6 -L -L 3 Höder nequay for he case m = and L 6 -L -L -L 3 Höder nequay for he case 1 m r 3 on hgher order erms, ogeher wh (6.16 and he Soboev embeddng heorem, we can ge he same bounds for (6.83 and (6.84. Then, we oban (6.77 M CT P(sup E r (.,T ] Case 3: s =. By negraon by pars, yeds T Cr 1 r a k 3 η k, r 1 v, j ρj A j dxd (6.85 T =Cr 1 r 1 a k 3 η k, r v, j ρj A j dxd (6.86 T Cr 1 r a k 4 η k, r v, j ρj A j dxd (6.87 T Cr 1 r a k 3 η k, r v, j (ρj A j dxd. (6.88 I s cear ha C 1 r 1 (6.86 = C r 1 (6.79 whe he aer has been jus esmaed. By an L6 -L -L 3 Höder nequay for hgher order erms, (6.16 and he Soboev embeddng heorem, we ge (6.87 (6.88 M CT P(sup E r (.,T ] Tha s, (6.85 has he same bound M CT P(sup,T ] E r (. Case 4: s = r wh r = 5. I s easy o see ha T a k 4 η k, 4 v, j ρj A j dxd (6.89 can be bounded by he desred bound n vew of Höder s nequay and he fundamena heorem of cacuus. Nex, we consder (6.75. Snce for he case s = and for he case s = 1 T C 1 r 1 T r 1 a k η k, r 1 v, j ρ J A j dxd = 1 r 1 (6.77, r a k 3 η k, r 1 v, j ρj A j dxd = C1 r 1 (6.85, we have he desred bounds. For he case s =, we have, by Höder s nequay and he Soboev embeddng heorem and he fundamena heorem of cacuus, ha T C r 1 r 3 a k 4 η k, r 1 v, j ρj A j dxd ( CT sup η r P( η 3 ρ 4 T η ρ r 1 ηd ρ r 1 v(,t ] M CT P(sup E r (.,T ] C r 1

27 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 7 For he case s = r and r = 5, T a k r η k, r 1 v, j ρj A j dxd Cr r 1 ( CT sup η 4 P( η 3 ρ r T η ρ r 1 ηd,t ] M CT P(sup E r (.,T ] ρ r 1 v( For (6.76, by he Höder nequay, he Soboev embeddng heorem and he fundamena heorem of cacuus, we can easy oban he desred bounds. Thus, T (6.71 M CT P(sup E r (. Now, we urn o he esmaes of T (6.69d. Case 1: s =. By (.15 and negraon by pars, we see ha T r 1 J 3 Ja j r v, j ρdxd (6.9 T = 6 r (J 3 η,k A k η p,q A q pa j r v, j ρj dxd T = 6 r 1 η,k η p,q A k Aq pa j r v, j ρj 1 dxd =6 6 T r 3 m= T T C m r T,T ] m1 η,k r m (J 3 A k η p,q A q pa j r v, j ρ J dxd r η,k η p,q r 1 v, j ρ J 1 A k Aq pa j dxd (6.91 r 1 η,k η p,q r 1 v, j ρj 1 A k Aq pa j dxd (6.9 ( r 1 η,k η p,q r 1 v, j 1 T C m m= r 3 T Cr m m= r 3 T Cr m m= r 3 T Cr m m= ρ J 1 A k Aq pa j dxd (6.93 m η,k r m (J 3 A k η p,q r 1 v, j ρj A q pa j dxd (6.94 m1 η,k r 1 m (J 3 A k η p,q r 1 v, j ρj A q pa j dxd (6.95 m1 η,k r m (J 3 A k η p,q r 1 v, j ρj A q pa j dxd (6.96 m1 η,k r m (J 3 A k η p,q r 1 ( v, j ρ J A q pa j dxd. (6.97 By he Höder nequay, he Soboev embeddng heorem and he fundamena heorem of cacuus, we can easy oban he desred bound for (6.91 and (6.93-(6.97. M δ sup E r ( CT P(sup E r (,,T ],T ]

28 8 CHENGCHUN HAO For (6.9, we use negraon by pars wh respec o me o ge (6.9 =6 r 1 η,k η p,q r 1 η, j ρj 1 A k Aq pa j dx =T (6.98 T 6 r 1 v,k η p,q r 1 η, j ρj 1 A k Aq pa j dxd (6.99 T 6 r 1 η,k v p,q r 1 η, j ρj 1 A k Aq pa j dxd (6.1 T 6 r 1 η,k η p,q r 1 ( η, j ρj 1 A k Aq pa j dxd. (6.11 I s cear ha (6.99 = (6.9. Thus, (6.9 = 1 ((6.98 (6.1 (6.11. (6.1 Moreover, negraon by pars yeds T (6.1 =6 r η,k v p,q r η, j ρj 1 A k Aq pa j dxd (6.13 T 6 r 1 η,k 3 v p,q r η, j ρj 1 A k Aq pa j dxd (6.14 T 6 r 1 η,k v p,q r ( η, j ρj 1 A k Aq pa j dxd. (6.15 Snce (6.14 = (6.9, we have from (6.1 (6.9 = (6.98 (6.11 (6.13 (6.15. (6.16 By negraon by pars, we ge for = T (6.98 = 1 r η,k η p,q r 1 η, j ρj 1 A k Aq pa j dx 6 r 1 η,k η p,q r 1 ( η, j ρj 1 A k Aq pa j dx, whch yeds he desred bound for (6.98 by usng he Höder nequay, he Soboev embeddng heorem and he fundamena heorem of cacuus. Inegraon by pars mpes T (6.11 =6 r η,k 3 η p,q r 1 ( η, j ρj 1 A k Aq pa j dxd ( T T Due o (6.17 = (6.11, foows ha r η,k η p,q r ( η, j ρj 1 A k Aq pa j r η,k η p,q r 1 ( η, j ρ J 1 A k Aq pa j dxd (6.18 dxd. (6.19 (6.11 = 1 ((6.18 (6.19, whch mpes he desred bound for (6.11 by usng he Höder nequay and he Soboev embeddng heorem. I s cear ha boh (6.13 and (6.11 have he same bound n vew of he Höder nequay and he Soboev embeddng heorem. Thus, we have obaned (6.9 M δ sup E r ( CT P(sup E r (.,T ],T ]

29 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 9 Case : s = 1. From (.15, foows ha T Cr 1 1 r J 3 Ja j r v, j ρdxd (6.11 T = 6Cr 1 1 r 3 ( ( J 3 A k η,k JA q p η p,q a j r v, j ρdxd = 6C 1 r 1 6C 1 r 1 r 3 m= T Cr 3 m r 3 T Cr 3 m m= r 3 m ( J 3 A k m1 η,k ( JA q p η p,q r v, j ρja j dxd (6.111 r 3 m ( J 3 A k m1 η,k η p,q r v, j ρ J A j Aq pdxd. (6.11 By usng negraon by pars wh respec o for (6.111 and he cases m =,r 3 n (6.11, and negraon by pars wh respec o me for he case m = 1 n (6.11, ogeher wh he Höder nequay, he Soboev embeddng heorem, (6.16 and he fundamena heorem of cacuus, we oban (6.11 M δ sup E r ( CT P(sup E r (.,T ],T ] Case 3: s =. By (.15 and negraon by pars, yeds T Cr 1 r 3 J 3 3 J r v, j ρja j dxd (6.113 = 6C r 1 6C r 1 r 4 T Cr 4 n n= r 4 C n 1 T r 4 C m n= m= =6Cr 1 r 4 T Cr 4 n n= 6Cr 1 r 4 T Cr 4 n n= 6Cr 1 r 4 T Cr 4 n n= 6Cr 1 r 4 C n 1 T r 4 C m n= m= 6C r 1 6C r 1 r 4 C n 1 T r 4 C m n= m= r 4 C n 1 T r 4 C m n= m= n1 η,k r 4 n J 3 3 η p,q r v, j A q pa j Ak ρ J dxd n1 η,k r 4 n J 3 m1 η p,q m ( JA q p r v, j A j Ak ρ Jdxd n η,k r 4 n J 3 3 η p,q r 1 v, j A q pa j Ak ρ J dxd (6.114 n1 η,k r 4 n J 3 4 η p,q r 1 v, j A q pa j Ak ρ J dxd (6.115 n1 η,k 3 η p,q r 1 v, j ( r 4 n J 3 A q pa j Ak ρ J dxd (6.116 n η,k r 4 n J 3 m1 η p,q m ( JA q p r 1 v, j A j Ak ρ Jdxd (6.117 n1 η,k r 4 n J 3 m η p,q m ( JA q p r 1 v, j A j Ak ρ Jdxd (6.118 n1 η,k r 4 n J 3 m1 η p,q 3 m ( JA q p r 1 v, j A j Ak ρ Jdxd (6.119

30 3 CHENGCHUN HAO 6C r 1 r 4 C n 1 T r 4 C m n= m= n1 η,k m1 η p,q m ( JA q p r 1 v, j ( r 4 n J 3 A j Ak ρ Jdxd. (6.1 Due o he case n = n (6.114 s equa o Cr 1 (6.9, we have obaned he desred bound for hs case of ( For (6.115-(6.1 and he case n = r 4 wh r = 5 of (6.114, we can easy use he Höder nequay, he Soboev embeddng heorem and he fundamena heorem of cacuus o ge he desred bounds. Then, so does ( Case 4: s = r wh r = 5. Inegraon by pars wh respec o me yeds T J 3 4 Ja j 5 v, j ρdxd = J 3 4 Ja j 5 η, j ρdx T T ( J 3 a j 4 J 5 η, j ρdxd T J 3 4 Ja j 5 η, j ρ dxd T J 3 4 Ja j 5 η, j ρ dxd, whch can be easy conroed by he desred bound n vew of he Höder nequay and he fundamena heorem of cacuus. Therefore, combnng wh four cases, we oban T (6.69d M δ sup,t ] E r ( CT P(sup E r (. (6.11,T ] Sep 4. Anayss of he remander I 4. For =, we have from he Höder nequay, he Soboev embeddng heorem and he Cauchy nequay ha T r ρ v r v dxd CT ρ 1/ L ( r ρ ρ sup,t ] ρ 1/ r v v ρ ρ 4 r1 δ sup ρ 1/ r v CT 4 v 4,T ] M δ sup E r ( CT P(sup E r (,,T ],T ] where we need he condon ρ H r1 ( n vew of hgher-order Hardy s nequay. For = 1, we have, a a smar way, ha T r 1 ρ v r v dxd CT ρ 1/ r 1 ρ L ( sup ρ 1/ ρ r v v 3,T ] ρ ρ 4 r δ sup ρ 1/ r v CT 4 v 4 3,T ] M δ sup E r ( CT P(sup E r (.,T ],T ] For =, we ge, by he Höder nequay and he Soboev embeddng heorem, T r ρ v r v dxd CT ρ 1/ r ρ L ( sup ρ 1/ ρ r v v 1 1,T ] ρ ρ 4 r δ sup ρ 1/ r v CT 4 v 4 3,T ]

31 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 31 M δ sup E r ( CT P(sup E r (.,T ],T ] For = 3, we oban, by he Höder nequay and he Soboev embeddng heorem, ha T r 3 ρ 3 v r v dxd CT ρ 1/ r 3 ρ L ( sup ρ 1/ ρ r v 3 v,t ] ρ ρ 4 r δ sup ρ 1/ r v CT 4 v 4 3,T ] M δ sup E r ( CT P(sup E r (.,T ],T ] For = 4 wh r = 5, we have, by he Höder nequay and he Soboev embeddng heorem, ha T r 4 ρ 4 v r v dxd CT ρ 1/ r 4 ρ L ( sup ρ 1/ ρ r v 4 v,t ] ρ ρ 4 r 1 δ sup ρ 1/ r v CT 4 v 4 4,T ] M δ sup E r ( CT P(sup E r (.,T ],T ] Sep 5. Anayss of he remanders I 5 and I 6. By negraon by pars, (.16, (1.13b and (1.13c for = 1,,r 1, we have T r 1 T I 5 d = Cr r a j (ρj r v, j dxd. =1 whch can be wren as, by negraon by pars wh respec o me, r 1 Cr r a j (ρj r η, j dx (6.1 =T =1 r 1 =1 r 1 =1 T Cr T Cr r a j (ρ J r η, j dxd (6.13 r a j (ρ J r η, j dxd. (6.14 Case 1: = 1. By usng he fundamena heorem of cacuus wce and (.13, we ge for (6.1 r 1 a j (T (ρ J (T r η, j (T dx = r 1 a j (T ρ J (T r η, j (T dx r 1 a j = T (T ρ J (T Aq(T p η q,p (T r η, j (T dx r 1 a j dj (T ρ r T η, j (T dx (1 ρ r 1 v ρ r η CT ρ 3 P(sup E r (,T ] M δ sup E r ( CT P(sup E r (.,T ],T ] r 1 a j dj A p q η q,p ρ r η, j (T dx

32 3 CHENGCHUN HAO Smary, we have for (6.13 T r 1 a j (ρ J r η, j dxd M δ sup,t ] E r ( CT P(sup E r (.,T ] For (6.14, s harder o be conroed han (6.1 and (6.13. We wre as T r 1 a j (ρ J r η, j dxd T = r 1 a j ρ J Aqv p q,p r η, j dxd (6.15 T r 1 a j ρ (J Aqv p q,p r η, j dxd (6.16 T r 1 a β ρ J Aq v p q,p r η,β dxd (6.17 T I s easy o see ha (6.15 and (6.16 are bounded by r 1 a 3 ρ J A p q v q,p r η,3 dxd. (6.18 M δ sup E r ( CT P(sup E r (.,T ],T ] By negraon by pars, Höder s nequay, (6.16 and Soboev s embeddng heorem, hods T (6.17 = r 1 a β,β Ap q v q,p r η ρj dxd T T T T r 1 a β Ap q v q,p r η ρ J,β dxd r 1 a β Ap q,β v q,p r η ρ J dxd r 1 a β Ap q v q,pβ r η ρ J dxd r 1 a β Ap q v q,p r η (ρ,β J dxd T ρ r a η v 1 ρ r η 1 d T ρ r 1 a η 4 3 η r v 1 ρ r η 1 d T ρ r 1 a η 3 v 1 ρ r η 1 d T ρ r 1 a η 3 v L 3 ( ρ r η 1 d T ρ 3 r 1 a A v 1 ρ r η 1 d M δ sup E r ( CT P(sup E r (.,T ],T ] For (6.18, we can easy ge he desred bound by an L 6 -L 3 -L Höder s nequay and he Soboev embeddng heorem because each componen of a 3 s quadrac n η due o (1.9.

33 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 33 Case : =. By he fundamena heorem of cacuus, Höder s nequay and he Soboev embeddng heorem, we can see ha r a j (ρj r η, j dx M δ sup E r ( CT P(sup E r (. =T,T ],T ] From (.18, he Höder nequay, he Soboev embeddng heorem and (6.16, yeds T r a j (ρj r η, j dxd T = r ( Jv q,p (AqA j p Ap qa j (ρj 4 η, j dxd T = J r v q,p (AqA j p Ap qa j ρ J A k η,k 4 η, j dxd remanders T C ρ r v L 3 ( η 7 r ρ r η d remanders M δ sup E r ( CT P(sup E r (,,T ],T ] snce he remanders can be easy conroed by he desred bound. Smary, we can ge he bound for he as negra T r a j (ρ J r η, j dxd M δ sup E r ( CT P(sup E r (.,T ],T ] Case 3: = 3. By usng he Höder nequay, he Soboev embeddng heorem and he fundamena heorem of cacuus, he spaa negra (6.1 can be bounded by M δ sup,t ] E r ( CT P(sup,T ] E r (. Smary, we can ge he same bound for he frs space-me negra (6.13. Snce he norm ρ J 3 s conaned n he energy funcon E r (, he as spaceme negra (6.14 have he same bound by he Höder nequay, he Soboev embeddng heorem and he fundamena heorem of cacuus. Case 4: = r 1 and r = 5. They can be easy conroed by he desred bound, especay wh he hep of he fundamena heorem of cacuus for (6.14. We can dea wh he negras n I 6 by usng a smar argumen and we om he deas for smpcy. Sep 6. Summng denes. Inegrang (6.1 over, T ], by Höder s nequay and Cauchy s nequay, we have, for suffceny sma T such ha 1 ρ r v dx 1 = 1 ρ r u dx 1 T η r η dv η r η cur η r η ] ρ J 1 dx T 1 r ρdv η r η(t J 1 dx M δ sup,t ] r η,k r η, j (A k A j A j E r ( CT P(sup,T ] η r η dv η r η cur η r η ] ρ J 1 dv η vdxd T E r (. Ak ρ J 1 dxd T r ρ(a j J 1 r η, j dxd (6.4 (6.73 (6.71 (6.69]d By he fundamena heorem of cacuus, we have η r η ρj 1 dx = ( η r η j ( η r η j ρ J 1 dx = T I 4 I 5 I 6 ]d r η j,s A s r η j,p A p ρ J 1 dx

34 34 CHENGCHUN HAO ( ( = ρ (A s J 1 ( d A s ( r η j,s = ρ r ( ( η dx ρ (A s J 1 ( d r η j,s ( whch yeds ρ ( r η j A p ( d r η j,p dx sup ρ η r η ρ r η,t ] A p ( d A p ( r η j,p dx ρ ( r η j A p ( d r η j,p dx ( (A s J 1 ( d r η j,s dx, CT sup ρ r η v r 1 η 8 r CT sup ρ r η v r 1 η 4 r.,t ],T ] Thus, akng T so sma ha CT v r 1 η 4 r 1/6, we oban 1 sup ρ r η sup ρ η r η 3,T ],T ] sup ρ r η. (6.19,T ] and Smary, we have = = = = dv η r η ρj 1 dx ( r dvη r η k, ρ ρ ρ r dvη dx (J 1 A k d ( r dvη r η, j ( ρ r dvη r η, j A j dx d ( ρ r dvη r η k, (J 1 A k d dx ( ρj 1 r ( η k, Ak d r η, j cur η r η ρj 1 dx ( r curη ε j r η j,s ρ r curη dx ε j ε k ε j ε k A j d dx A j d dx, (6.13 (A s J 1 d ( r curη ε k r η,p ( ρ r curη r η j,s (A s J 1 d dx ( A p k d dx ρ r curη r η,p A p k dx d ( ρ r ( η j,s (A s J 1 d r η,p A p k dx, d (6.131 whch yeds for a suffceny sma T ha 1 sup ρ r dvη sup ρ dv η r η 3,T ],T ] sup ρ r dvη, (6.13,T ] 1 sup,t ] ρ r curη sup,t ] ρ cur η r η 3 sup ρ r curη. (6.133,T ] Thus, we oban he desred nner esmaes wh he hep of cur esmaes.

35 FREE-BOUNDARY 3D COMPRESSIBLE EULER EQUATIONS IN VACUUM 35 Sep 7: Boundary esmaes. By Lemma.4, (.5, he fundamena heorem of cacuus and Höder s nequay, we oban r η α 1/ r η cur r 1 η ρ r η ρ r η cur r 1 η ρ r vd ρ r η cur r 1 η ρ T 1/ sup ρ r v ρ r η cur r 1 η ],,T ] whch mpes he desred esmaes from cur esmaes. We have he foowng esmaes. sup,t ] 7. THE ESTIMATES FOR THE TIME DERIVATIVES Proposon 7.1. Le r {4,5}. Then, for a sma δ > and he consan M depend on 1/δ, we have ] ρ 1/ v ρ r η ρ r dvη r Proof. Acng r where I 1 := M δ sup,t ] E 5 ( CT P(sup E 5 (.,T ] on (1.13a, and akng he L (-nner produc wh r v, we oban 1 d ρ r v dx I 1 I = I 3, d I 3 := r r 1 Cr =1 a j ( ρ J, j r v dx, I := r ( a j ρ a j J r v dx., j ( ρ r J, j r v dx, Sep 1. Anayss of he negra I 1. Inegraon by pars gves, by nocng ha r a 3 r v = on Γ, ha I 1 = r a j r v, j ρ J dx r a 3 r v ρj dx 1 dx Γ = r 1 (v,k J 1 (a j ak ak a j r v, j ρ J dx = r 1 v,k r v, j Ak A j ρ J 1 dx (7.1 r 1 v,k r v, j Ak A j ρ J 1 dx (7. Then, we have (7.1 = 1 d d 1 r 1 Cr 1 s s=1 r 1 s v,k s (J 1 (a j dv η r 1 v ρj 1 dx 1 dv η r 1 v ρ J 1 dx. ak ak a j r 1 r v, j ρ J dx. (7.3 v,k r 1 v, j (A k A j ρ J 1 dx