ANALYSIS OF A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED BY OIL WELL CONTROL OPERATIONS

Available online at www.sciencedirect.comMalhemaclaLtientia“ScienceDirect数学物理学报Acta Mathematica Scientia 2012,32B(1):295-314http:// actams.wipm.ac.cnANALYSIS OF A COMPRESSIBLE G AS-LIQUIDMODEL MOTIVATED BY OIL WELLCONTROL OPERATIONS*Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthdaySteinar ErjeFaculty of Science and Technology, University of Stavanger, NO-4068 Stavanger, NorwayE-mail: steinar. evje@uis.noK.H. KarlsenCentre of Mathematics for Applications, University of Oslo,P.O. Box 1053, Blindern, NO -0316 Oslo, NorwayE-mail: kennethk@math. uio.noAbstract We are interested in a viscous two- phase gas- liquid mixture model relevantfor modeling of well control operations within the petroleum industry. We focus on asimplified mixture model and provide an existence result within an appropriate class ofweak solutions. We demonstrate that upper and lower limits can be obtained for the gasand liquid masses which ensure that transition to single phase regions do not occur. This isused together with appropriate a prior estimates to obtain convergence to a weak solutionfor a sequence of approximate solutions corresponding to mollified initial data. Moreover,by imposing an additional regularity condition on the initial masses, a uniqueness resultis obtained. The framework herein seems useful for further investigations of more realisticversions of the gas- liquid model that take into account different flow regimes.Key words gas liquid two phase model; weak solution; existence; uniqueness2000 MR Subject Classification 76T10; 76N10; 65M12; 35L601 IntroductionWe are interested in a one-dimensional two-phase liquid (e) and gas (g) model in the formmt + (vem)x = 0,中国煤化工nt+ (ugn)x= 0,(1.1)MHCNMH G(mve+nug)t+(mv2+mvg)x+p(m,n)x=qF+9gG+M(Vmixx,."Received November 12, 2011. The research of Steinar Evje has been supported by A/S Norske Shell.296ACTA MATHEMATICA SCIENTIAVol.32 Ser.Bwhere μ> 0 andm = AtePe,n=QgPg,.qF= - fUmix, .9G =一9Pmix,Vmix = ceVe + QgUg,Pmix = aePe + QXgPg,where f and g are nonnegative constants. The variables involved are as follows: ae,Qg arevolume fractions, Pe, Pg are fuid densities, and Ve, Vg are fAuid velocities. This model is supple-mented with the following constraints (algebraic relations):p= Cpg,ρ!= constant, γ≥ 1,(1.2)ae +ag=1,(1.3)Vg= K'vmix + S,K, S are constants.(1.4)The model (1.1)-(1.4) is often referred to as the drift-Aux model. Note that by combining (1.2)and (1.3), we get Pg = plo-m. Consequently,p=Cp{(pl-m,n= p(m,n).(1.5)The drift- fAux model is highly relevant in modeling of various well operations [7]. Inparticular, the model has been used for the study of driling operations. Currently, there ismuch focus on development of safe and optimal drilling methods in the context of deepwaterwells. In this setting a typical model problern involves an interesting but complicated interactionof different physical mechanisms like the balance between the pressure gradient induced byfrictional forces qr and the hydrostatic pressure qG, transition from mixture to single-phaseregions, free gas-liquid interface behavior, and various compressible effects like compressionand decompression. One aspect that requires special attention is the possibility of having agas-kick. A gas-kick refers to a situation where gas flows into the well from the surroundingformation. As this gas ascends in the well it will typically experience a lower pressure. Thisleads to decompression of the gas, which can provoke blowout-like scenarios. Clearly, for thestudy of such Aow scenarios we need a two-phase gas-liquid model that takes into accountcompressible effects.The purpose of this paper is to focus on one aspect of the model (1.1) by making severalsimplifying assumptions. More precisely, we first neglect the acceleration terms in the mixturemomentum equation. That is, we consider the simplified momentum balance given by thefollowing static force balancep(m,n)x= qF + qG + μ(Umnix)xx.(1.6)The resulting “vanishing Mach number” model, often with μ = 0 and sometimes called the :no-pressure wave model, has been demonstrated to be highly relevant in the context of wellflow scenarios, see [19] and [18] and references therein.中国煤化工3 the simnplermomentum equation (1.6) is that pressure (acoustic) Wrer, for manyapplications the main interest is the slow transport ofMHCNMH(mass waves),and not a detailed study of the pressure waves. In the present model the pressure waves areapproximated by infinite velocity waves.No.1S. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED297In this paper we want to explore the viscous dominated model where we also neglect thefriction term qF and assume horizontal flow, i.e., qc = 0. In addition, we restrict to the specialAlow case of no slip between the two phases, i.e, Ue = Ug = u. This represents a situation wherethe gas is dispersed in the liquid phase such that the two phase mixture moves with the samevelocity, more or less.Previous work on the model (1.1)Development of good discrete methods for solving the compressible gas-liquid model (1.1)has been a topic for many papers during the last decade [1-3, 6, 7, 11-13, 18- 21]. However, itis only recently that the mathematical properties of this model have been investigated. In [8] asimplified version of (1.1) was studied. More precisely, it was assumed that the gas and liquidvelocities were equal, no external forces were taken into account in the momentum equation,and certain gas terms were neglected in the momentum equation taking advantage of the factthat pr/pg 》1. The existence of global weak solutions was then obtained under suitableassumptions on the initial data. In particular, the result showed that when the initial massesno and mo do not vanish or blow up (n = QgPg and m = a1ρr), then n and m remain bounded:CT'≤n(t, :),m(t,)≤Cr;t∈[0,T],T> 0,for a positive constant Cr. The 1D results [8] are extended to a 2D version of the model in [25].The main assumption in [25] is that the initial energy is small in a certain sense. The providedestimates are also strong enough to give the large time asymptotic behavior of the solution. Wealso refer to [26] for a result on the blow-up behavior of the 2D gas-liquid model in Euleriancoordinates.Studies have also been carried out with the model (1.1) considered in Lagrangian variableswith free boundaries and a viscosity term depending on the masses. A first work in thisdirection can be found in [9, 10]. More recently, these studied have been extended to includethe possibility of different fuid velocities [4], well- reservoir interaction [5], and external forceslike friction and gravity [14].Contributions of the present workThe main purpose of this paper is to explore some other aspects of the two-phase model(1.1) concerning the existence and uniqueness within an appropriate class of weak solutions.The model we investigate becomes different than the one studied in [8, 25, 26] since we considera steady state mixture momentum equation (1.6). A main motivation for this work is toestablish a framework that possibly can allow for inclusion of important physical mechanismsthat currently are neglected in the model studied in the works [8, 25, 26]. Such investigationsare left for future work.An important feature of the model we study is that although we apply a simplified linearEOS for the gas phase (isothermal flow) by choosing γ= 1 in (1.2), the resulting pressure lawp(m, n) for the two-phase mixture (1.5) becomes a nonlinear function. This reflects some of theadditional complexity represented by two-phase over s中国煤化Iirly, a potentialdificulty with the model we consider is the singulariin the pressurelaw (1.5). This corresponds to a situation where tralTYH. NM H CGuid Aow ocu.A main observation is that the assumption of no-slip condition implies that the two masses mand n are related as几= s with 8 controlled. As a consequence, we obtain pointwise control298ACTA MATHEMATICA SCIENTIAVol.32 Ser.Bon m and n (lower and upper bounds) which allows us to verify that the pressure p(m, n) iswell- defined.Equipped with pointwise control on m, n we derive several a priori estimates in LP andSobolev spaces of a sequence of approximate solutions {mk, nk, uk} obtained by applying aregularization of initial data mo, no. These estimates yield some basic (weak) convergenceresults. However, strong convergence of mk, nk is required to recover the nonlinear pressure lawp(m,n). This is obtained by studying various renormalizations of the approximate solutionsand corresponding defect measures.An interesting aspect of the model we study is that a (presumably natural) viscous approx-imation of (2.1) is not easy to analyze using weak compactness and renormalization arguments.This appears to be due to the fact that our analysis relies heavily on using the simple equationsatisfied by the quantity s := m, an equation that is not available in the context of uniformlyparabolic problems. A similar dificulty arises when attempting to prove strong convergence ofupwind-type difference schemes, cf. the discussion in Section 6.The remaining part of this paper is organized as follows: In Section 2 we present thecompressible gas-liquid model and state the main results. Section 3 contains the analysisyielding pointwise control on the masses as well as various LP estimates. In Section 4 thecompactness (convergence) of a sequence of approximate solutions is established. Moreover,the limit functions are identified as weak solutions of the two -phase model in question. InSection 5, a uniqueness result is derived under some additional regularity on the initial masses.Finally, we make some concluding remarks in Section 6.2 A Viscous Two .phase ModelWe focus on a two phase model in the following formmt+ (um)x=0，nt + (un)x= 0,(2.1)p(m,n)x = pUxa,μ> 0,where the pressure function is given bynp(m,n)= Cγ≥1.(2.2)\ρI- mIn what follows, wesetC=1,r= 1, and μ= 1. We may restate the model asmt + umx = -mp(m,n),nt + unz = -np(m,n).The main purpose of this work is to establish the global-in- time existence of weak solutionsto the initial-boundary value problemmt + umx = -mp(m,n)，t> 0,中国煤化工nt + unx= -np(m,n)，t> 0, x:fYHCNMHG(2.3)Ux= p(m,n)，p(m,n)=-ρl- m'u(t,x)lx=o=0，m|t=o= mo(x)，n|t=o = no(x),No.1s. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED299where mo,no∈L∞(R+)∩LP(R+) forp> 1 and R+ := (0,∞).We mention that this system (2.3), along with its analysis, shares some resemblence withthe so-called Hunter- Saxton modelUr+Wx=-2v,Ux= U,see for example [22- 24] and references therein.Definition 1 (Weak solution) We call (m(t, x), n(t, x), u(t, x)) a weak solution of (2.3)provided the following conditions hold: .(i) m(t,x),n(t,x) ∈Loe(R+, LP(R+)), u(t,x)∈C(0,∞)xR+);(i) mt+ (um)x=0, nt + (un)x= 0, p(m,n) = Ux in the sense of distributions;(ili) The function u(t,x) is equal to zero at x = 0 as a continuous function. The functionm(t, x) and n(t, x) take on the initial values mo(x) and no(x) in the sense C([0,∞), L'(R+)). .To establish an existence result for weak solutions, the main challenge is to pass to thelimit in a sequence of approximate solutions in the nonlinear pressure function without relyingon BV or Sobolev-type of estimates.Theorem 1 (Existence result)We assume that(i) mo(x), no(x)∈L∞([0,∞))∩LP(0,∞)) for some appropriate P;(i) there are positive constants A1,A2 and B1, B2 such that0 0, there are positive constantsC1 = C1(T),C2 = C2(T) such that0 0. Fori= 1,2, let (mi,ni,u") be a weaksolution on Qr of the system中国煤化工MHCNMHGmi + (u'mi)x=0，ni + (un')x= 0,(2.5)u'(t,x)= | p(m(t, y), n'(t, y))dy, .300ACTA MATHEMATICA SCIENTIAVol.32 Ser.Bwith initial data m'|t=0= mj∈I∞∩BV,n't=0=nj∈L∞c∩BV satisfying0≤m()≤Cmi 0,1 dm1 dnm dt.ndtordlog(m)_ d log(n)In other words, .log(m(t, X()) = lg(n(,X()))+C，i.e, m(t, Xx(x)) = Cn(t, X().In view of the initial data we get the following solution中国煤化工m(t, X:(x)) = n(t, X:(x)(3.2)YHCNMHGwhereA,Br'≤mo()≤A2Br'.(3.3)no(x)No.1S. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED301Along the characteristics the first equation of (2.3) takes the formm(t,X(x))=- mp( m,n| (t, X:(x))=-( nom2(t, X(x))mo) ρl- m(t,X())'since Pl - m(t,Xt(x)) > 0, which implies thatlog(m(t, X())+mLm( mo )t+ C(mo)， C(mo) = log(mo) + PL(3.4)m(t, Xt(x))~ ( mo)moDefine中(m) by中(m) dem log(m) + PimThen we have from (3.4)(m(t,X(x)) =(3.5)nont + C(mo) 'Moreover, we find thatPl-m_(m)=(mog(m)+m)，φ"(m)=-2log(m)pu-mlog(m)+3pl-2m.(m log(m) + pl)3According to (2.4), we have that0< inf, mo(x)≤mo(x)≤supmo(x)< PI.[0,1[0,1]Hence, C(mo) > 0 so we can conclude that the right hand side of (3.5) is always positive anddoes not blow up for any t > 0. Moreover, We can check that φ(m) is strictly increasing in[0,p],φ(0)= 0, φ(ρi)= 0,中"(pr)=-7(Pl log pu+pr)芦< 0 such that m= pl is a maximum point,and中(p1)= log(Po+I. From this we get for0int2+C(mo)= φ(m(t, (x()),which implies thatmo(x) > m(t,Xr:(x)) > m(t2, Xxr2(x)).In other words, m(t, Xt(x)) is decreasing in time (along its characteristics). Since supmo < ρu .[0,i]we conclude that m(t, X(x))∈[0,pi) for allt > 0, and we have the relationpl > supmo > m(t, Xr,(x)) > m(t2,Xt(x))≥0.It is of interest to quantify the rate of the decrease in中国煤化工that purpose,we may argue as follows: Clearly φ- 1 exists on [0, sup[0,i1MYHCNMHGm(t,X:(x)) = φnat + C(mo)302ACTA MATHEMATICA SCIENTIAVoL.32 Ser.BWe may find ψ(x) = Kx for some constant K such thatφ~'(x)≤ψ(x)= Kx,x∈[0,M]，M = φ(supmo) 0 since it is strictly increasing in x; we denote the inverse mapping by Xt +(x). Using thisin (3.2) we see thatn(t, x)m(t, x)= (mo )x-()) =st(t,2),(3.8)Moreover, in view of (3.5),1中(m(t,x)) =(3.9)no(X;2()s(t, x)t + C(mo(X+(x))nolxtn)t + (m(X())Clearly, in view of (2.4) we have the estimate0< BrA21≤s(t,x)≤B2.中国煤化工(3.10)YHCNMHGFor further use we apply (3.8) and definep(m,n) = p(m,ms)= sm_= sg(m) e P(m,s).(3.11)ρ- mNo.1s. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED303As far as the velocity u is concerned, we estimate as follows:u(t,x)=[~P(m,s)dy≤ sup s(t,x)g(m(,))dy≤Br4='[ gg(m(t, y))dy≤CB2A11 | m(t, y)dy≤KCB2A11-dym(x“血)t + C(mo(X; ())mo(X; (u))≤Cu(t).Here we have taken advantage of (3.10) and the fact thatmm(t, x)g(m)= .ρl-m≤ρl - sup(mo(x))≤Cm(t,x),for an appropriate choice of C and by an application of assumption (2.4). We also have used(3.6). Consequently, we can conclude that|lu(t, )L∞(+)≤Cu(), t≥0,(3.12)and hence we have provedLemma 1 For (t,x)∈R+ x R and under the assumnptions (2.4) we have the followingpointwise estimates:0≤m(t,x)≤sup(mo(x)) < PI,0≤n(t,x)≤sup(s)sup(mo(x))≤AI 1 A2B2,|u(t,x)|≤Cu().Note that this is suficient to guarantee that p(n, m) = P(m, s) is well- defined. We maysharpen the above estimates for m and n as follows when we consider a finite timne T> 0.Corollary 1 For any finite time T, there is an ε = e(T) > 0 such that0<ε≤m(t,x),0 0) that this is not the case. This implies thatthere exist points {xk,t} wheret∈[0,T] and a timet∈[0, T] such thatmk = m(thk,xk)→0,tk→t,ask→∞.In light of (3.9), this implies that t =∞, which produces a contradiction. The lower estimate ofn(t, x) follows from the relation n(t,x) = s(t, x)m(t, x) given by (3.8) and its lower and upperbounds (3.10).DRemark 2 An interesting and relevant aspect o中国煤化工plifed variantsof this) is to obtain a clearer understanding when vacuuase fow) mightappear. The above calculation shows that for the modeMHCNMH(uaranteed thatvacuum will not occur in finite time. However, it's not clear to us to what extent this will bethe case for more general slip relations and/or pressure laws.304ACTA MATHEMATICA SCIENTIAVol.32 Ser.BRemark 3 If transition to single-phase liquid flow occurred in an appropriately modifiedtwo-phase model of the form (2.1) with pressure law (2.2), then this would correspond to asituation where m becomes equal to ρl, i.e, p(m,n) = Ux blows up, say at time t*. In this casewe would have to describe precisely how the solution should be extended beyond the blow-uptime t*. Note that a similar phenomenon is observed for the Hunter- Saxton equation. Thisambiguity gives rise to different notions of weak solution, see [22- -24] and references therein.We now consider the case with general initial data mo, no with compact support. Letjn(x), k= 1,2, . be a standard Friedrich's mollifier satisfyingjk(x)= kj(kx)，j(x)∈C°(R)， j(x)≥0， . j(x)dx=1.We assume 1≤mo() < ρl, 0≤no(), and mo,no∈LP(R). Without loss of generality, weassume that supp(mo), sup(no)C [0,1). Setmb(x) = jk * mo(x),n哈(x) = jk *no(x).For each fixed k, by the preceding calculation we find that (2.3) has a global smooth solutionme(t,x), ne(t,x) with initial data m&(x) and n哈(x), respectively. Clearly, from the thirdequation of (2.3)Uk =p(mk, nk)dy.We also note that in view of (3.12) we havel|ue(t, )∞(a+)≤Cu,supp(me(t, )) C [0, Km()),where Km(t) is determined from the first equation of (3.1) which implies that X(x)≤Cut sothat Km(t)= Cu(t)t + 1 will do the job.The plan now is to prove that the approximations {me(t, x), nn(t,x)}k21 are compact inL9([0,T]x R+) for anyT>0,q> 1.According to (3.8) we have sn(t,x) = 65i), and from (3.11) it followsme(t,a)p(mk,nx) =ρl- mn= 8k9(mx)= P(mk, sk).We have the following estimates:Lemma 2 The approximate solutions {mx(t, x)} satisfy the following estimates:|mx(t, )|L>(R+)≤|mollLp(R+),p≥1,(3.13)ImkP(mk, s)|L2(Qx)S 1mlO0Q+),p> 1,(3.14)|P(me(t, ), 8e(t, ))L(R+)≤|IP(mo, 8o)lex(k+),(3.15)Is≈' P(mk, 8)}IL2(Qw≤|IP(mo, so)lu(R+),(3.16)|IP(me(t, ), 8e(t, )IL(&+)≤|IP(m,中国煤化工(3.17)|P(mk, 8)+1z2(Qx)≤:IP(mo,sMCHCNMHG(3.18)|I8ε' P(mk, 8)+21+Q.w≤IIP(mo, 8o)9ILl(a+)，q≥1.(3.19)No.1S. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED305Proof Multiplying the first equation of (2.3) by f'(mx) we getf(mk)t + uef(mx)x = -f"(mk)mxP(mk,sk).This can also be written in the formf(mx)t + (uxf(mk)x = [f(m) - f(m)m)]P(me,8k).First, by using the choice f(m)= mP forp≥1 we get(m%)t + (ukm%)x = -(p - 1)mP(mk;,sk).That is, by using that the support of ma(t,x) is contained in [0, K(T)] fort∈[0,T] we get。° /。" [mp)+()- ImrP(m ndarat=o，p21. .From this, we get。(m呢)dx+(p - 1)mR P(mk, sk)drdt =(m)dx.In particular, since both terms on the left hand side are non- negative,pK(T)Jm(,)dxs .JR+mo(x)"dx = ImllLola+)and(p-1)mR P(mk, 8k)dxcdt≤|mlLr(R+)，Thus, by letting T→∞, the estimates (3.13) and (3.14) follow. Next, we are interested inderiving an equation for the choicemkf(mz;,sk) = P(mk;,8k) = skg(mx) = 8k-ρ- mkNoting thatmkg'(mk) = g(mk) + g(mx)",(3.20)and from the two mass equations of (2.3) it follows thatne(t,x)(8k)t + ue(8k)x=0,se(t,x)=(3.21)me(t,x)Now we can calculate as follows:P(mk, 8k)t = (sk(m>))t= sk9' (mx)(mn)t + g(mk)(sk)t= -skg'(mk)(mwuk)x - g(mx)u&(Sk)x= -(sk)2g'(mr)mrg(mx) - u中国煤化工:= -(8k)*g'(mk )mkg(mk) - u:MYHCNMHG= -(sk)^[g(mk) + g(mx)]g(mx) - urP(mk, 8k)x= - P(me, 8k)2 - (<&k)9g(mx)3 - upP(mk, 8k)x,(3.22)306ACTA MATHEMATICA SCIENTIAVol.32 Ser.Band(Uk P(mk, 8)x = (uk)xP(mk, 8k) + upP(mk,8k)x = P(mz,sk)2 + urP(mk,8k)x. (3.23)Here we have repeatedly used the third equation of (2.3) together with (3.20) and (3.21).Combining (3.22) and (3.23) we arrive at the following equationP(m;,8k)t + (ukP(mk, 8k)x = -8F'P(mk, 8k)3.(3.24)This implies the identityfrom which we getP(me,8k)dx+sπ' P(mk, sh)*dxdt =P(mo, s)dx.From this equality the estimates (3.15) and (3.16) follow. Finally, consider the choice f(mk,Sk) =P(mk,8k)9 forq≥1. Then we calculate as follows: .(P(mk,8k)9)t = qP(mk, 8k)9-1 P(mk, 8k)t= qP(mk;,sh)9-1| -8π1 P(mk, 8kh)3 - (ukP(mk, 8)|= qP(mk, 8k)9-1[-s7' P(mk, 8k)3 - P(mk, 8k)2 - upP(mk, 8k)z], (3.25)(upP(mk, 8k)x = (uk)xP(mk, 8)9 + ukqP(ms, 8)9-1 P(mk, 8k)x= P(mk, 8k)9+1 + urqP(me, 8k)9-1 P(mk,8k)x.(3.26)Adding (3.25) and (3.26) gives(P(mk, 8k)9)e + (urP(mk,sk)9)z = -(q - 1)P(mk,8k)9+1 - q8z21 P(m,8k)9+2.Hence, forq≥1,|drdt=0,and the estimates (3.17), (3.18), and (3.19) follow.DBy making use of the relation n = sm and the estimates (3.10). we ohtain the followingk-uniform estimates for ne(t, x):中国煤化工Corollary 2 The following estimates hold for theMYHCN MH Ge(t,x):Ine(t, )Ln/Qa+)≤B2Ai lmll+).,No.1s. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED3074 CompactnessWe begin by establishing strong compactness of the velocity {urk(,x)}. By Morrey's in-equality,Wlt2(Q∞)→C1oe*(Qo),for 2 2. We haveue(t,x)=p(mk, nk)(y, t)dy =P(me, se)(y,tdy.In view of the pressure equation (3.24) we getOruk =a,P(me, 8k)(y,t)dy = -(uRP(mk, s)dy) --f。"sr'P(m, se)*dy.Consequently, for (t,x) ∈[0,T]x R+ (using that ue(t,0) = 0)pKm(T)|Brua|≤max(ux)P(m;,8k)+sg'P(me, 8k)3dy ./o≤CuP(mk, sk) + max(s )IP(mo, so)"|L(&+),where we have applied (3.17) with q = 3. This implies thatlaux|"≤c(P(mo,sk)9 + IP(mo, o)":(a+),q> 2.That is, by (3.18) withq- 1 as the exponent we get (note that q> 2)|8ruk|"dxdt≤c( 1。" fh. Poms).drdt + K(TIP(m, o)0%*)≤c(g-2IP(mo, 8)1-1-+ Km[(T)IPomo0)lxa+).Since ux = P(m, s), then (3.18) also implies that|8uxux{9dxrdt =P(m, 8k)9dxdt≤一IP(mo, 80)9 IL心(R+), q>2.Hence, we can conclude (by Ascoli-Arzela and Banach Sakes theorems) that there is some :u(t,x)∈Wioe(Q∞) forq> 2 and a subsequence of {ue(t, x)} such that u<(t, x) converges tou(t, x) uniformly on any compact subset of Q∞. Furthermore, axue(t,x) = P(me(t, x), se(t,x))converges weakly to a limit functiono(t,x) = P(mt(t,x),(,x)) in,中国煤化工YHCNMHGFurthermore, it is clear that1≤δX{(x)≤C，|8rX[(x)|≤C.308ACTA MATHEMATICA SCIENTIAVol.32 Ser.BThe first estimate follows from (3.7) and the upper bound on P(mk, sk) (cf. Lemma 1). Simi-larly, the second estimate follows from the first equation of (3.1) and the pointwise upper boundon Uk, also guaranteed by Lemma 1. Consequently, we have uniform Holder continuity in spaceand time for {Xf(x)}, i.e, the sequence converges to a limit function Xt(x) uniformly on comn-pact sets in Q∞. Clearly, the sarne properties hold for the inverse Y{*()= (Xt)-'(:), such thatY()→Y() uniformly on compact sets in Q∞where for each t, and Y()= x;().Since se(t,x)= "s((Xt)-1(x)), see (3.8), we conclude that se(t, x) converges a.e. to thelimit function s(t.x)= (Xt (x)).To sum up, we have the following lemma.Lemma 3 (Compactness) Regarding the initial data mo and no, we assume0<ε≤mo(x)< PI,0<ε≤no(x)<∞,mo∈L2(R+),P(mo,8o) = p(mo,no)∈L2(R+),for some ε > 0. We have the following basic convergence result towards limit functions(m,n, u,v,w) ask→∞:Uk→1 uniformly in [0,R]X [0,T] for each R> 0 and pointwise in Qr(4.1)and the limit function u belongs to W1,9(Qr)- +Cioc(Qr) forq> 2; .8k→8a.e. in Q∞and .s(t,x)= m(xr(x)), i.e.,dd。s(t, X(x))=0， that is,st + usx= 0;(4.2)momk一m in L"(Qr),p≥1;(4.3)nk一n in LP(Qr),Vk:= P(mk,8k)= sk9(mn) = Axuk(4.4)→xu=U= sg(m) in L"(Qr)，q≥ 1;(Uk)*=(P(mk, 8)3) = (8)*g(mx)8一s32w in L'(Qr);(4.5)Uemk一um in LP(Qr),(4.6)uknk→un in LP(Qr),p≥1.Finally, the limit functions m, u, 8, w are related by the inequalitiesP(m,s)= 8g(m)≤U,v3≤s'w,(4.7)or equivalently,g(m)≤g(m),((m中国煤化工CNMHGProof The limit operations (4.1) and (4.2) follsion, whereas(4.3), (4.4), (4.5), (4.6) follow from the estimates of Lemma 2 and Corollary 2. The relations(4.7) rely on the convexity and continuity properties of g(:) and ()3.CNo.1s. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED309We are now in a position to prove strong convergence of mk, nk by analyzing a particularrenormalization (in the sense of Diperna-Lions) of the approximate solutions and their limits.Strong convergence ensures that the weak limit functions m and n solve the original equations.Lemma 4 (Limit equations) The limit functions (m, n, 8, u, v, w) from Lemma 3 satisfymt + (um)x= 0,nt + (un)x= 0,st+ USx = 0,Ux =U,(4.8)in the sense of distributions on QT, andm∈C([0,T];L(R+))，lim |lm(,t) - mollLo(R+)=0,n∈C(0,T];L'(R+))，lim |n(,t) - nollL>(R+) = 0,for any p≥1. Moreover,Ur+ (uo)x=-s- '[s3w](4.9)in the sense of distributions on Qr andu」。(,) - P(m((),())dx=0.Proof The approximate solutions (mk, nk, un) satisfy the system8xmk + 8x(ukmn)= 0,8xUk = p(mk,2n)= P(mk,sk),ne(t,x) = me(t,x)no(Xk;(x))= me(t,x)se(t,x). .mo(Xx;(x))In view of Lemma 3, it follows that (4.8) holds. Similarly, (4.9) follows from the pressureequation (3.24).Lemma 5 (Identification) Suppose that(i) u(t, x) is bounded and continuous in Qr with u(0,t) = 0fort∈[0,T], m∈L°((0,T);LP(R+)), .and m≥0a.e. in Qr;(i) v∈L°((0,T);L"(R+)) and P(m,s)≤va.e. in Qr;(ii) w∈L∞((0,T);L'(R+), andu3≤s3w a.e. in Qr;iv) Ast→0,f~(0(t,x) - P(m(t, x),s(t, x))dx→0;(4.10)(v) The limit functions u, m, n, s, U, w satisfy the systemmt + (um)x=0，nt + (un)x=0，st+ U8x=0，Ux=0,(4.11)v+ (ww)x=-s-1[s3u],(4.12)中国煤化工in the sense of distributions on Qr.MYHCNMHG.Then P(m,s) = sg(m)=va.e. in Qr.Proof The proof follows along standard lines in the theory of renormalized solutions.Wesetme=m*w*,ne=n*w*,se=8*ws,vE=v*w*,andwe=w*wewhereweisa310ACTA MATHEMATICA SCIENTIAVol.32 Ser.Bstandard molifer acting on the spatial variable. In view of (4.11) together with an applicationof the Diperna- Lions lemma we getmi + um;=-v°me + R,(4.13)where Re = u(m)x- (umx)*we +v°me- (rm)*w* and R°→0in LP(Qr) for anyp≥1.Having this regularized version of the first equation in (4.11), the plan is now to derivefrom this an equation that contains information about P(m, s). First, we multiply (4.13) withg'(me) and rewrite (using ur = ) such that we getg(m°)t + ug(m°)x= -v9m*°g'(m')+ R*g'(me).Then we multiply by sE and get[s9g(m)]t + u[s'g(m')]x - 9(m')[s; + us;]= -v*m's9g'(m9) + R*g'(m)s",orP(m'e,s<)e + (uP(m',8)x - 9(m^)\sf + us] = vP(m",s9)- v*ms9g(m)+ Rfg'(m*)s*.Sendingε→0 we getP(m,8)t + (uP(m, 8)x = vP(m, s) - omsg'(m),in the sense of distributions. Using (3.20) and P(m, s) = 8g(m), we getP(m,s)t + (uP(m, 8) = vP(m,8) - vs!g(m) + g(m)2]= -vsg(m)2 = -s-'vP(m,s)2.(4.14)Taking the diference between (4.12) and (4.14) we get8t[v - P(m, s)] + 8.(u[o - P(m,s)) = s-+(vs2g(m)2 - 832w)= 87 '(vP(m, s)2 - s32w)≤s→2(v3_s3w)≤0,(4.15)using the relations P(m,s) = sg(m)≤0 and v3≤s3w, see (4.7). Recalling that (4.15) holds inthe sense of distributions we can choose a test function 4(t,x) = w1 (t)w2(x) and then let wr(t)be a smooth approximation to X[t,t2] fort1 < t2 whereas w2(x) = 1. Then (4.15) simplifies to-f。" [° (0r.,)- (m(.M,(rldrs0,that is,-f。* (0t.)- P(,(.(0.)+。(0(2,2) - P(r(e:+(),(,)))x≤o.Lettingt1→0 and t2→t and comparing with (4.10), we getv(t,x) - P(m(t,x),8(t,x))≤0中国煤化工which implies that v= P(m,s) a.e. in Qr, in view of (HCNMHGCProof of Theorem 1 The existence result of Theorem 1 now follows as a result ofLemmas 3, 4, and 5.No.1s. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED3115 A Uniqueness Result in the L∞∩BV ClassIt was proved in Section 4 that the smooth solutions (mk, nk, uk) converge to a weaksolution (m, n, u) of (2.3). Now we want to prove that this weak solution, which satisfies0≤m(,)≤Cm<ρ，0≤n(,)≤Cn, |u(，,川≤ Cu,(5.1)possesses spatial BV regularity provided the initial data do so; more precisely,mo,no∈BV(R)=→m,n∈L°(0,T;BV(R))，T>0.(5.2)To this end, it is suficient to establish an estimate of the formJR|8xm(t,x)|+ |8rne(t,x)\dx≤Cr，t∈ (0,T),for some constant CT that is independent of k.Set q呢= 8xmk,吹= Aank. Then8rqh +u8qn" + p(mk,nk)g呢= -q" P(mk, nk) - mk (0m(mk, nk)qm + Pn(mk,nk)q%).Multiplying by sgn(q") yields|q"'t + Uk |q"| + p(mk, nx) |q"'|= - |qT"|p(mk,nk) - mkPm(mk, nk) |q7"| - mePn(mk, n)q"sgn(g#")≤Iml|L lPn(m,n)x |q"I≤C|q"I,where we have used thatnρl-m’Pm =(ρ1- m)2’ρI- mare all (nonnegative) bounded quantities. In divergence form, this equation inequality reads|q%| + (u |q%"|)z ≤Cm |g%|, .where the constant Cm is independent of k.Similarly, it follows that|q}lt + (4k |g於|)。≤Cn |g%^|，for some constant Cn that does not depend on k.Adding the inequalties for |q%^| and |q| yields(|q}|+ |#|)t + (uk (|q#"1+ |))≤C(\g#|+|qg"|)，C:= Cm +Cn,and thus by Gronwall's inequality,中国煤化工MTHCNMHGJ,(Iil8xme(,)I + I8xe(t,x)I )dx≤eCt|0xm;(U,x) + |0xnk(U,x)| )dx,and the claims follows.312ACTA MATHEMATICA SCIENTIAVol.32 Ser.BLet us now turn to the uniqueness of weak solutions in the L∞∩BV class, i.e., the proofof Theorem 2. In view of the assumptions, the solutions (mi ,n, u") satisfy (5.1) and (5.2). Bythe DiPerna-Lions regularization lermma, one can prove that the weak solutions (m;, n;, u) of(2.5) are entropy solutions, i.e., for any convexη:R→R,n(m*)+ (uin(m)。+ p(m',n") [m'n(m')- n(m*)]≤0,n(n')+ (uin(n)。+ p(m',n") [n'n(n")- n(n')]≤0,in the sense of distributions.Now the uniqueness of weak solutions is an immediate consequence of a result proved in[17] regarding continuous dependence of entropy solutions with respect to the fux function. Wemay apply Theorem 1.3 in [17] to conclude that there exists a constant C such that|m'(t,)- m2(,)z≤|m - mlL +," 11(.) - (?()drds[' l(,(2(,)) Im(es,lavds≤|m-mlL+|p(m' ,n1) - p(m2,n2)| dxds≤|lm - mllL2 +Cm[" (mt -m21+|2 -n1)ard. (5.3)Similarly,In'(,)- n2(,)心≤|贴- nL +Cn.。J(Im2- m21+Ih2 -1)adse.(5.4)By adding the two inequalities (5.3), (5.4) and, fllowing this, applying the Gronwallinequality, we arrive at|m'(t,) - m2(t, )，+ In'(t,) - n2(,)心≤eCt (Im] - m1La |In -一n12)，C:=Cm +Cn.This concludes the proof of Theorem 2.6 Concluding RemarksIn this work we have investigated a simplifed no pressure gas liquid model which is com-posed of two continuity equations for the two phases and a steady state momentum equationwhich represents the balance between the pressure gradiel中国煤化工derive point-wise upper and lower bounds on the masses which guararYCN M H Chase mistureremains a two-phase mixture, i.e., no transition to single: ute time. Ex-istence of weak solutions is shown under minimal regularity on the initial masses. Moreover, auniqueness result is derived by requiring that the the initial masses are BV bounded.No.1s. Evje & K.H. Karlsen: A COMPRESSIBLE GAS-LIQUID MODEL MOTIVATED313Interesting extensions of the model studied in this work would be to take into accountthat the two phases can move with different fuid velocities, consider inclusion of more generalpressure laws, as well as take into account terms representing external forces like gravity andfriction.It is dificult to find solutions of the system (2.3) without resorting to numerical methods.Fortunately, it is possible to devise very simple finite difference schemes for computing approx-imate solutions of (2.3). To this end, introduce the spatial grid cells Ij = [xj-1/2, xj+1/2),where xj+1/2 = xj士△x/2,j∈No:= {0,1,}. The forward/backward difference operatorsare denoted by D+/D-, respectively. Let {m} iEN,; {n&}eNo be discrete initial data satisying0≤m°(x)