Boundary Layer to a System of Viscous Hyperbolic Conservation Laws

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1 Acta Mathematcae Applcatae Snca, Englsh Seres Vol. 24, No. 3 (28) DOI: 1.17/s Acta Mathema ca Applcatae Snca, Englsh Seres The Edtoral Offce of AMAS & Sprnger-Verlag 28 Boundary Layer to a System of Vscous Hyperbolc Conservaton Laws Xao-hong Qn Department of Appled Mathematcs, Nanjng Unversty of Scence and Technology, Nanjng 2194, Chna (E-mal: xqn@amss.ac.cn) Abstract In ths paper, we nvestgate the large-tme behavor of solutons to the ntal-boundary value problem for n n hyperbolc system of conservaton laws wth artfcal vscosty n the half lne (, ). We frst show that a boundary layer exsts f the correspondng hyperbolc part contans at least one characterstc feld wth negatve propagaton speed. We further show that such boundary layer s nonlnearly stable under small ntal perturbaton. The proofs are gven by an elementary energy method. Keywords Vscous hyperbolc conservaton laws, boundary layer, asymptotc stablty 2 MR Subject Classfcaton 36L65, 36L6 1 Introducton Consder the ntal-boundary value problem (IBVP) for n n hyperbolc system of conservaton laws of the form u t + f(u) x = u xx, x R +, t >, (1.1) wth ntal-boundary data { u (x) :=u(,x) u +, as x, u(t, ) = u. (1.2) Here u =(u 1,u 2,,u n ) t R n, the flux f =(f 1,f 2,,f n ) t : R n R n s assumed to be a smooth functon of u, andu ± are prescrbed constants. Here ( ) t means the transpose of the vector ( ). We assume that f has n real egenvectors whch are λ 1 (u) λ 2 (u) λ n (u) wth correspondng rght egenvectors r 1 (u),r 2 (u),,r n (u). We further assume that the hyperbolc part of the system (1.1) has a strctly entropy/entropy-flux par (E,F),.e. E(u) t + F (u) x =, (1.3) f u s a smooth soluton of and u t + f(u) x =, (1.4) (E j ) m >. (1.5) Here (E j ):= 2 ue means a n n matrx. Manuscrpt receved Aprl 16, 28. Supported by the Natonal Natural Scence Foundaton of Chna (No )

2 524 X.H. Qn In present paper, we are nterested n the large-tme behavor of solutons to the IBVP (1.1) and (1.2). For Cauchy problem of general vscous conservaton laws u t + f(u) x =(B(u)u x ) x, (1.6) t s conjectured by many people that ts soluton s asymptotcally domnated by the Remann soluton of the correspondng nvscd hyperbolc system of conservaton laws, whch conssts of three elementary hyperbolc waves,.e., shock, rarefacton and contact dscontnuty waves. Ths conjecture was verfed n varous cases, for nstance, see [2,5,8,1,11,13,15,16] and the reference theren. However, the large-tme behavor of solutons of the IBVP for (1.6) s much more complcated than those of Cauchy problem due to the presence of boundary. Roughly speakng, a statonary soluton, denoted by boundary layer, s formed due to the nteracton of the boundary and hyperbolc waves, and the soluton s expected to tend to the lnear superposton of boundary layer and the hyperbolc waves. The boundary layer was frstly nvestgated for scalar conservaton laws by Lu, Matsumura and Nshhara n [6] and [7]. Then many authors focused attenton on the compressble Naver-Stokes equaton, see [3,4,9,11, 12], and the references theren. It was showed that the boundary layer soluton s nonlnear stable provded that the perturbaton s small. Recently, we [14] studed the boundary layer for p-system wth unformly artfcal vscosty and proved the boundary layer s stable under arbtrarly large perturbaton. In ths paper, we consder the boundary layer for general system wth artfcal vscosty,.e., (1.1), and hope to establsh a general stablty theorem for boundary layer. We now formulate our man results. Motvated by [6] and [7], we expect that the boundary layer U(x) for the system (1.1) s a statonary soluton satsfyng { f(u)x = U xx, x >, (1.7) U() = u, U(+ ) =u +. Integratng (1.7) on (x, + ) mples { Ux = f(u) f(u + ), x >, U() = u, (1.8) whch s a n n ordnary system. We assume λ (u + ) <, 1 n whch means the matrx f(u + )has negatve egenvalues. From [1], there exsts a -dmensonal stable manfold u + Maround u +,sothatfu M, there exsts a unque soluton U(x) Msolvng (1.7),.e., Theorem 1.1 (Exstence of boundary layer). Assume λ (u + ) <, 1 n, and let u M, then there exsts a unque soluton U(x) Mto (1.7), whch connects u and u +, and x k (U u +) Cδe cx for k =, 1,, (1.9) where δ = u + u. We further have Theorem 1.2 (Stablty of boundary layer). Assume the condtons n Theorem 1.1 hold, and u (x) U(x) H 1 (R + ). (1.1)

3 Boundary Layer to a System of Vscous Hyperbolc Conservaton Laws 525 Then there exsts postve constants δ and ɛ such that f δ = u + u δ and u U H 1 ɛ,theibvp(1.1) (1.2) has a unque global soluton u(x, t) satsfyng u(x, t) U(x) C([, ); H 1 ), u(x, t) U(x) L 2 (, ; H 1 ). (1.11) Furthermore, sup u(x, t) U(x), as t. (1.12) x R + Notatons. Throughout ths paper, several postve generc constants are denoted by c, C wthout confuson, and C(, ) stand for some generc constants dependng only on the quanttes lsted n the parenthess. For functon space, L p (Ω), 1 p denotes the usual Lebesgue space on Ω R := (, ). W k,p (Ω) denotes the k th -order Sobolev space, and f p =2,we note H k (Ω) := W k,2 (Ω), := L 2 (Ω), and k := Hk (Ω) for smplcty. The doman Ω wll be often abbrevated wthout confuson. 2 Proof of Theorem 1.2 We put the perturbaton φ(x, t) by φ(x, t) =u(x, t) U(x), (2.1) where U(x) s the boundary layer defned n (1.7), then the reformulated problem s φ t +[f(u) f(u)] x = φ xx, x R +,t>, φ (x) :=φ(x, ), as x, φ(,t)=. (2.2) We defne the soluton space X(,T)by X(,T):={φ C([,T]; H 1 ); φ x L 2 (,T; H 1 )}, (2.3) and set N(t) := sup φ 1. τ t Snce the proof for the local exstence of the soluton to (2.2) s standard, the detals are omtted. To prove Theorem 1.2, we only need the followng aprorestmates. Proposton 2.1 (A pror estmates). Let φ X(,T) be a soluton to the IBVP (2.2) for some postve T, and the condtons n Theorem 1.2 hold. Then there exst small postve constants δ < 1 and ɛ < 1 such that f δ = u + u δ,andn(t ) ɛ, φ satsfes φ(t) φ x 2 1 dτ C φ 2 1. (2.4) Proof. Step 1. (1.1) can be wrtten n the followng form where we use the Ensten symbol,.e., a b := n a b. u t + f ju j x = u xx, (2.5) =1

4 526 X.H. Qn Defne { P = E(u) E(U) E (U)(u U ), then And Q = F (u) F (U) E (U) [ f (u) f (U) ], (2.6) P t + Q x = [ E (u) E (U) ] u xx E j (U)U j x[ f (u) f (U) ] =: I 1 + I 2. (2.7) I 1 = [( E (u) E (U) ) u ] x x [ E j (u)u j x E j (U)Ux] j u x = [( E (u) E (U) ) u ] x x [ E j (u)(φ j x + Ux) j E j (U)Ux]( j φ x + Ux ) = [( E (u) E (U) ) u ] x E x j(u)φ x φj x [ E j (u) E j (U) ] U j xφ x E j (u)u xu j x + E j (U)U xu j x = [( E (u) E (U) ) u ] x x E j(u)φ x φj x [ E j (u) E j (U) ] Ux j φ x [( E (u) E (U) ) Ux ] x + [ E (u) E (U) E j (U)φ j] U xx + E j (U)U xxφ j, (2.8) so P t + Q x = [( E (u) E (U) ) φ ] x x E j(u)φ xφ j x + [ E (u) E (U) E j (U)φ j] Uxx [ E j (u) E j (U) ] Uxφ j x + E j (U)Uxxφ j E j (U)Ux j [ f (u) f (U) ] = [( E (u) E (U) ) φ ] x x E j(u)φ x φj x + [ E (u) E (U) E j (U)φ j] Uxx [ E j (u) E j (U) ] U j xφ x E j (U) [ f (u) f (U) f j(u)φ j] U j x. (2.9) Integratng (2.9) over [,t] [, ), we get t Pdx + E j (u)φ xφ j x dxdτ,j { [E = (u) E (U) E j (U)φ j] Uxx [ E j (u) E j (U) ] Uxφ j x E j (U) [ f (u) f (U) fj(u)φ j] } Ux j dxdτ. (2.1) We estmate each term of (2.11) by a Poncaré nequalty x g(x) = g() + g y (y)dy g() + x 1/2 g x. (2.11)

5 Boundary Layer to a System of Vscous Hyperbolc Conservaton Laws 527 We have, C [ E (u) E (U) E j (U)φ j] U xx dxdτ U xx φ 2 dxdτ Cδ φ x 2 dτ, (2.12) whereweusethefactthat u U + φ C and E (u) E (U) E j (U)φ j C φ 2. The other two terms can be treated smlarly, so we have Pdx+ E j (u)φ xφ j x dxdτ C φ 2 + Cδ,j φ x 2 dτ. (2.13) Usng (1.5) and choosng δ sutably small such that Cδ m 2,then Step 2. φ 2 + Multplyng (1.1) by φ xx,thenwehave φ x 2 dτ C φ 2. (2.14) 1( φx 2) 2 t (φt x φ t) x [f(u) f(u)] t x φ xx + φ xx 2 =. (2.15) Integratng t over [,t] R +, and usng Cauchy nequalty and (2.14) gve φ x (t) 2 + φ xx 2 dτ φ x 2 + {[f(u) f(u)] x } 2 dτ φ x 2 + C φ x 2 dτ + C U x φ 2 dxdτ R + C φ 2 1. (2.16) Combnaton of (2.14) and (2.16) mples Proposton 2.1. Proof of Theorem 1.2. Theorem 1.2 s proved by the local exstence and Proposton 2.1. Acknowledgements. The author wshes to thank Prof. F. Huang for hs gudance and stmulatng encouragements. References [1] Chou, S., Hale, J. Method of bfurcaton, Sprnger-Verlag, Berln, 1986 [2] Goodman, J. Nonlnear asymptotc stablty of vscous shock profles for conservaton laws. Arch. Rat. Mech. Anal., 95: (1986) [3] Huang, F., Matsumura, A., Sh, X. Vscous shock wave and boundary layer soluton to an nflow problem for compressble vscous gas. Comm. Math. Phys., 239: (23) [4] Kawashma, S., Nshbata, S., Zhu, P. Asymptotc stablty of the statonary soluton to the compressble Naver-Stokes equatons n the half space. Comm. Math. Phys., 24: (23) [5] Lu, T.P. Nonlnear stablty of shock waves for vscous conservaton laws. Mem. Amer. Math. Soc. 56, no. 328, v+18 (1985)

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