Available online at www.sciencedirect.comMalhemadlFgintiaScienceDirect數(shù)學(xué)物理學(xué)報Acta Mathematica Scientia 2007,27B(2):347-360http://actams.wipm.ac.cnASYMPTOTIC STABILITY OF RAREFACTIONWAVE FOR HYPERBOLIC-ELLIPTIC COUPLEDSYSTEM IN RADIATING GAS*Ruuan Lizhi (阮立志)Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University,Wuhan 430079, ChinaDepartment of Applied Mathematics, South-Central University for Nationalities, Wuhan 430074, ChinaE-mail: ruandmagon1976@169.comZhang Jing(張晶)Laboratory of Nonlinear Analysis, Departrnent of Mathematics, Central China Normal University,Abstract In this article, authors study the Cauch problem for a model of hyperbolicelliptic coupled system derived from the one dimensional system of the radiating gas.By considering the initial data as a 8mall disturbances of rarefaction wave of inviscidBurgers equation, the global existence of the solution to the corresponding Cauchy problemand asymnptotic stability of rarefaction wave is proved. The analysis is based on a prioriestimates and L2.energy method.Key words Hyperbolic eliptie coupled Bystemn, rarefaction wave, asymptotic stability,L2-energy method I2000 MR Subject Classification 35M10, 35Q351 IntroductionIn this article, we will consider the Cauchy problem for a byperbolic -lliptic coupled systemin radiation hydrodynamicsut+ f(u)x+qx= 0,(1.1)l-qxx+q+4x=0,with initial datau(x,0)= u(x)→u比，as $→士∞, .(1.2)where f(u)∈C3(R) satisfies f"(u)> 0 foru∈R.中國煤化工*Received December 5, 2004. The reeearch was supportedMYHCNMHG,ProjetoftheNatural Science Foundation of China (10431060), the Key Project of Chinese Ministry of Education (104128),and the South-Central University For Nationalities Natural Science Foundation of China (YZY05008)348ACTA MATHEMATICA SCIENTIAVol.27 Ser.BThe first equation is the hyperbolic conservation law and the second is an eliptic equation.Such a hyperbolicelliptic coupled system typically appears in radiation hydrodynamics, cf. [15].The system (1.1) is derived as the third-order approximation of the full system describing themotion of radiating gas in thermo-nonequilibrium, while the 8econd-order approximation givesthe viscous Burgers equation ut + f(u)x = Uxz, and the first-order approximation gives theinviscid Burgers equation ut + f(u)z = 0 (see [5]). Hamer [1] studied these equations in thephysical respect, especially for the steady progressive shock wave solutions. It is also studiedfrom the mathematical point of view for the case f(u) =氣u2 and u_ > u+, cf. [5, 6, 7].Precisely, Kawashima, and Nishibata proved the stability of the traveling wave of the Cauchyproblem (1.1)-(1.2). Recently, Kawashima and Tanaka show the stability of the rarefactionwave and the asymptotic rate. In their proof, the second- order approximation is based on theviscous Burgers equation, cf. [8].The purpose of this article is to study the case u_ < u+ for general f(u).As in {3, 13], according to the idea of asymptotic analysis, we conjecture that the term-9xx of (1.1)2 decays to zero as t→∞, faster than other terms. Therefore, it is natural toexpect that the solution of (1.1) time -asymptotically behaves as that of the following system) + f(@)x +亞=0,(1.3)[q+x=0,or equivalentlyi + f()x= urx,(1.4)( g=-lx.It is well-known that the asymptotic behavior of the solution to (1.4)1 with initial data (1.2)ast→∞is closely related to that of the Riemann problem for the corresponding hyperbolicconservation law, cf. [2, 9]:rt+f(r)x=0,(1.5)with initial datau_，for x<0,r(x,0) =唱(x) =u+，for x>0.(1.6)It is well-known that the entropy solution r(x, t) of Riemann problem (1.5) and (1.6) can begiven explcitly byu_，r< f'(u_),r(x,t)={ (')-()，f"(u_)t≤τ≤ j"(u4)t,(1.7)x> f'(u+)t.Let u(x, t) be the smooth rarefaction wave of (1.5) and (1.6) defined bv Lemma21 and (x,t) =-ux(x,t). Then, when initial data (1.2) are a small中國煤化工th rarefactionwave of (1.5) and (1.6), we will prove that the solutiorYHC N M H Guchy problem(1.1) and (1.2) exists and tends to (&(x, t), q(x, t)) time asymptotically by applying L2. energymethod.No.2Ruan & Zhang: ASYMPTOTIC STABILITY OF RAREFACTION WAVE349Notations Hereafter, we denote several positive constants by C; (i = 0,1,2,..) or onlyC without confusion. For function spaces, L2(R) and L°(R) be usual Lebesgue spaces on R =(-∞,∞) with norms lfIr2(RB) = (r |f(x)|dx)* and |f|L∞(R) = sup|f(x), respectively, wealso write |HI|L2(R)= l1I|I and !IILx∞(n)=川. HIL∞. H*(R) denotes the usual h. th orderSobolev space with norm |I|k =(12川)For simplicity, |If(, t)|z2 and lf(,t)h aredenoted by |f()|L2 and I()|k, respectively.2 Smooth A pproximation and PreliminariesAs in [10, 11, 16, 20], we first consider the following Cauchy problem, which will be usedto construct the smooth approxinate solution of the Riemann problem (1.5) and (1.6)元+辯x=0,(,)=io(x)=g(r+ +r-)+號(r+-r-)Kody_(2.1)1+y’wherer+= f'(u+) and K is a constant such that K seo科= 1.Let (x,t)= (f)-(x,)). Then }(x,t)|≤C|(x, t)| and (x,t) satisfesin+ f(網(wǎng))x= 0,(2.2)試(x,0) = uo(x) =(f)-(0(x))→u士，a8 x→t∞.For the Cauchy problem (2.2), we have the following fundamental properties (see [10, 11, 16,20]):Lemma 2.1 Letu_ < u+ and f(w) ∈C2(R) satisfy f"(u)> 0 for u∈r. Then theCauchy problem (2.2) admits a unique global smooth solution i(x, t) satisfyingiz> 0，|II∞ ≤C|iIlL∞, .(2.3)and( {i(t)呢?！蹸h(u+-u_ )(1+t)-P+1， 1≤p<∞,|&awx()I|2≤Ch(u+-u-)(1+t)-2-*，k= 0,1,2,3,|150xxx()|2≤Ch(u+-u_ -)(1+t)-4-k， k=0,1,“些≤Ch(u+-u-)(1 +t)一量,(2.4)J-∞Uxdx≤Ch(u+-u >)(1 +t)-3,元[* da≤Ch(4+-u_-)(1 +)-號,where h(u+ -u_) is a function of u+一u_ and satis中國煤化工Let:YHCNMHGI w(x,t) = u(x,t) - u(x,t),(2.5)\z(x,t) = q(x,t) - (x,t),350ACTA MATHEMATICA SCIENTIAVol.27 Ser.Bwhere i(x, t) is the solution of (2.2) obtained by Lemma 2.1 and (x,t)= -ix(x,t).Then the system (1.1) can be rewritten aswt+ (f(w+2)- f(2)); +z+和=0,(2.6)l -zx+z+ Wx = xx.The corresponding initial data arew(x,0) = wo(x) = uo(x) - io(x).(2.7)We seek the solutions of (2.6) and (2.7) in the set of functions X (0, T) defined byX(0,T)= {(w, x)|w∈L°(0,T;H7),wx∈I2(0,T;H),z∈L∞(0,T;H*)∩L2(0,T;H)}.Now we state our main results as follows:Theorem 2.2 Let wo∈H2(R) and Eo = |l and 8 = |u+ - u_| are sficientlysmall. Then the Cauchy problem (1.1)-(1.2) (or equivalently (2.6)- (2.7)) admits a uniqueglobal solution (w(x, t), z(x,t))∈X(0,T) satisfyinglu()1 + I()歸+ f (0u=<()1 + l<()1 + |+(.)dr≤C(1) +()，(2.8)whereHw()2,i2>= | i(w2 +w + w2)(r)dx. .Moreoverim sup (w(x, t)| + |wx(x,t)| + |z(x, t)| + |z<(x,t)| + |zx(,) = 0.(2.9)t- +∞0 x∈R3 Proof of Theorem 2.2First, we give the proof of the estimate (2.8). For this, we consider the estimate of the8olution (w(x, t), z(x, t)) of (2.6)-(2.7) under the a priori assumption|u()1≤q，0≤t<∞,(3.1)where0 0, independent of t, such thatu2(x,t)dx≤C，廣u{(x,t)dx ≤C,(3.50)u2(x,t)dx≤C，吲(r,r)dxdr≤c,thensup |u(x,t)|→0，as t→∞.(3.51)中國煤化工To prove (2.9), the following Lemma is satisfied by Le(2.8):Lemma 3.10 Under the assumptions of Theo1MYHCNM Hit,z(,t) of|we(t)怪+ He()1≤C (1陘+ h()).(3.52)No.2Ruan & Zhang; ASYMPTOTIC STABILITY oF RAREFACTION WAVE359Proof By the first equation of (2.6) and the first equation of (3.12), we havewt=-(f(w+詞)一f().-zx-亞,andwxt=-(f(w+2)- f())xx-Zxx- qIt follows( 崛≤C(w2 +u峪+屆+臨)，(3.53)嵫≤C(w2 +u2 +w2 +碰+站).Integrating (3.53) with respect to x over R, we have from (2.8) and Lemma 2.1|v()≤C (011 + h()).(3.54)Differentiating the second equation of (2.6) with respect to t, we get-xxt+z= xxt - Wxt.(3.55)Multiplying (3.55) by zt and integrating it with respect to x over R, we have(益+z?)dx={_ixxzrdx+ Iwrzxtdx.(3.56)By Cauchy- Schwartz inequality, we have廠~ (強(qiáng)+引)a≤)J u+o/ (:+4)az+號/[。w?dr,2 J-∞which implies from (3.54) and Lemma 2.1(Z+z4)d s≤C(o貶 +h()).(3.57)Similarly, differentiating the second equation of (3.12) with respect to t, we get- xt +zxt = xxxt一Wxxt.(3.58)Multiplying (3.58) by 2zxt, then integrating it with respect to x over R and using Cauchy-Schwartz inequality, we have(3.59)j~ (:+4)d≤j_~ .+嗎，J-xwhich implies by (3.54) and Lemma 2.1f (綠+ e)d ((1 +h().This proves Lemma 3.10.中國煤化工The proof of Theorem 2.2 is completed.MHCNMHGAcknowledgment The authors wish to thank Professor Changjiang Zhu for his encour-agement and guidance.360ACTA MATHEMATICA SCIENTIAVol.27 Ser.BReferences1 Hamer K. 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