STABILITY OF VISCOUS CONTACT WAVE FOR COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS WITH FREE BOU

Availableonlineatwww.sciencedirect.comMalena cla● ScienceDirectActa Mathematica Scientia. 2010, 30B (6): 1906-1916数学物理学报http://actams.wipm.ac.cnSTABILITY OF VISCOUS CONTACT WAVE FORCOMPRESSIBLE NAVIER-STOKES SYSTEM OFGENERAL GAS WITH FREE BOUNDARYHuang Feimin(黄飞敏) Wang Yong(王勇) f Zhai xiaoyun(翟晓云)Institute of Applied MatheAcademy of Mathematics and SystemsScience Chinese Academy of Sciences, Beijing 100190, ChinaE-mail: huang @amt. accn; yongwang@amss. accn; hairy @amt. accnAbstract In this paper, we study the large time behavior of solutions to the nonisen-tropic Navier-Stokes equations of general gas, where polytropic gas is included as a specialcase, with a free boundary. First we construct a viscous contact wave which approximatesto the contact discontinuity, which is a basic wave pattern of compressible Euler equationin finite time as the heat conductivity tends to zero. Then we prove the viscous contactwave is asymptotic stable if the initial perturbations and the strength of the contact waveare small. This generalizes our previous result 6 which is only for polytropic gasKey words Navier-Stokes equations; contact discontinuity; viscous contact wave2000 MR Subject Classification 35L50; 3560; 35L65: 76R501 IntroductionThe one-dimensional compressible Navier-Stokes equations reads in the Eulerian coordp+(m)正=0,(pu)t +(pu+pa=uuit,(a+"),+(G+2)+0)2=Ab+(n)where i(, t)is the velocity, p(i, t)>0 the density, e(i, t)the absolute temperature, u>0 theviscosity constant and k>0 the coefficient of heat conduction. The pressure p=p(p, 0)andthe internal energy e=e(p, 0) are related by the second law of thermodynamicsider th(t)is a free boundathboundary conditionsdi(t)u((t),t),元(0)=0,(x(t),t)=b->0(1.2)Received June 10, 2010. The research of FMH was supported in part by NSFC(10825102)for distinguishedyouth scholar, NSFC-NSAF (10676037) and 973 project of China(2006CB805902中国煤化工CNMHGNo 6F.M. Huang et al: STABILITY OF VISCOUS CONTACT WAVE1907a(p-1u)|(1.3)which means the gas is attached at the free boundary i=i(t)to the atmosphere with pressurep(see [23)and the initial data(F,0,0)(G,0)=(0,0,60)(),lim(o,o,o)(x)=(p+,0,0+),(1.4)where p+, 0+ are positive constants and 80(0)=0. Here we assumeD-=p+=p(P+,+)Since one dimension Lagrangian coordinate is convenient to study, we transform the Eu-lerian coordinates(i, t)to the Lagrangian coordinates(a, t)byP(y, t)dy, t=t,and our free boundary value problem(1. 1)-(1.5)is changed intox>0,t>ut+p(u, 0)x=pr>0,t>0e(0, 0)(p(v,0))>0,t>0(v,u,6)(x,0)=(v0,u0,60)(x)→(v+,0,0+)asx→+∞whe(x,)=(G,1),0(x,)=0(,1)>0,.+=+andt=0(,1)=m>0 denotesthe specific volume. The pressure p, the internal energy e and the entropy s are the functionsof v and 0. Since there are only two independent thermodynamic variables, p and e are alsoregarded as the functions of v and s. We denoteP=p(v,0)=p(t,s),e=e(v,6)=e(t,s)We assume thatD<0,e0>0,p>0,(1.8)p(u, s) is a convex function of (v, s)(1.9)The study of fuid motion has a very long history and the pioneering work on wave phe-nomena dates back to Riemann in 1860s on gas dynamic. It is well known that for systems ofhyperbolic conservation laws of the formUt+ F(Ux=0,(1.10)there are there basic wave patterns in one-dimensional space. For compressible Euletions, these basic wave pattern are two nonlinear waves, shock wave and rarefaction wave, and中国煤化工CNMHG1908ACTA MATHEMATICA SCIENTIAVol30 Ser. Ba linearly degenerate wave, contact discontinuity. Since(1.10)is an idealization when the dis-sipative effects are neglected, thus it is significant to study the large time asymptotic behaviorof solutions to the corresponding viscous systems in the form ofUt+F(U)x=(B(U)Ur)x(1.11)toward the viscous versions of the three basic wavesSince A Mutsmura and K Nishihara 19 in 1985 proved the stability of viscous shock wavefor isentropic Navier-Stokes system. There have been a lot of works on the asymptotic behaof the solutions for the system(1. 1). We refer to 11, 15, 18, 19, 25 for viscous shock wave, 16-22, 26 for rarefaction wave and [6-10, 27 for viscous contact wave and references theAlthough considerable progress have been obtained, however most of these results are forthe polytropic gas. In particular, all results concerning viscous contact wave are only for thepolytropic gas. Thus a natural question arises: can we show similar stability of viscous contactwave for general gas not only for polytropic gas. In this paper, we give a positive answer tothis question for a free boundary problem, see Theorem 2. 2 below for detailsThe rest of the paper will be arranged as follows. In the next section, we will the viscouscontact wave and state the main result. In Section 3, we proved the main theorem by theelementary energ y estimates2 Viscous contact Wave and main theoremFirst we consider the following Riemann problemat+p(u, 0)x=0p(v,0))0)(x,0)=(v-,0,6-),ifx<0,(v,u,0)(x,0)=(v+,0.,0+),ifm>0where v+ and 0+ are given positive constants. It is known that the Riemann problem(2.1)admits a contact discontinuity(V,U,)(x,t)=(v-,0,6-),x<0(v+,0,O+)provided thatp(U-,6-)=p+=p(+,b+Motivated by [6 and(2. 2)-(2. 3), we begin to construct a viscous contact wave for NavierStokes system. Since in(2. 3) the pressure p is constant, we conjecturp(V, e)=pNote that Pu< 0. If 0 +-0- is small, then the implicit function theorem yields that therexists a C function f(8) such thaf(-),t+=f(0+),中国煤化工CNMHGNo 6F.M. Huang et al: STABILITY OF VISCOUS CONTACT WAVE1909in a small neighborhood N+ of 8+. From(2. 4) and(2.5), we havee)where pe(e)=pe(f(e), 0) and pu(e)=po(f(e), e). It is noted that the third equation of(1.7) can be reducedWe further conjecture that(V, U, e)is governed by the following equationsVt-Ux=0,eo(6+p(=/ne(0,t)=0-,e(+∞,t)=6+,where ee(e)=ee(f(e),e)We compute from(2.6) and(2.8)9()f(ee(0,t)=0-,(+∞0,t)wheeo(e、ep2By the same method as in [1],(2.9)admits a unique self similarity solution e($),5After e is obtained, we definef(e),(210)andf(6)(6-drg()\f(6)9()f(6)f(6)(gObviously, the viscous contact wave we get here satisfies the same properties with those for thepolytropic gas(see [6 for details),i.eL2.1 Let 0+-0=8, the following estimates holddx≤C64(1+t)2dx≤C62(1+t)-+,(212)≤C62(1+t)r(6)dx≤C6(213)It is easy to see(V, U, e) satisfiesU,6-6)川ze(0.+∞)=O(5)(1+t),allp≥1(214)which means the nonlinear wave(V,U,e) approximates the contact discontinuity(v,U, e)inLP norm, p2 1 on any finite time interval as k tends to zero. We call(V, U, e)(a, t) viscous中国煤化工CNMHG1910ACTA MATHEMATICA SCIENTIAVol30 Ser. BTo state our main result, we put the perturbation(, v, S(a, t) b(x,t)=V(x,t)+以(x,t(a, t)=U(, t)+v(a, t)215)(x,t)=(x,t)+(x,t)By the definition of the viscous contact wave, we haveV:-Ux=0+p(v,e)x=u+F,ee(e0+Ope(e)Ux=nwheref()+∞Fg(6)f()a((0)dx)+u(log f(e)(217)G))2=O(1)(62n+6)Substituting(2. 16) into(1.7),(1.7)2 and(2.7) yieldsvt+(p(℃,0)-p(V)Ux+yr Uxealu)U(219)+e(v,6)v)-c(V,.)p(v,6)with the initial data(,v)(x,0)=(c0,vo)(x)∈H1(0,+∞),(x,0)=o(x)∈Hb(0,+∞)The precise statement of our result is as followsTheorem 2.2( Stability theorem) There exist positive constants do and Eo such that forany 0+-0-1=8< 8o, if I(0, 00, So)l1 0∈L2(0,7;2),(0,∈D20.x;H1)}(3.1)for some0