Christian Merkle 1 and Christian Rohde 2

Transcription

1 ESAIM: MAN Vol. 41, N o 6, 007, pp DOI: /man: ESAIM: Mathematical Modelling and Numerical Analysis THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE: RIEMANN PROBLEMS AND GHOST FLUID TECHNIQUES Christian Merkle 1 and Christian Rohde Abstract. Systems of mixed hyperbolic-elliptic conservation laws can serve as models for the evolution of a liquid-vapor fluid with possible sharp dynamical phase changes. We focus on the equations of ideal hydrodynamics in the isothermal case and introduce a thermodynamically consistent solution of the Riemann problem in one space dimension. This result is the basis for an algorithm of ghost fluid type to solve the sharp-interface model numerically. In particular the approach allows to resolve phase transitions sharply, i.e., without artificial smearing in the physically irrelevant elliptic region. Numerical experiments demonstrate the reliability of the method. Mathematics Subject Classification. 35M10, 76T10. Received August 1st, 006. Revised April 0, 007. Introduction We consider the flow of a compressible fluid that can appear in two phases, lets say, in a liquid and a vapor phase. Mathematical models for phase transition problems split up into so-called sharp-interface models and phase field models. While in the latter approach phase changes are modeled as steep but continuous solutions of the underlying evolution equations the sharp-interface ansatz leads to discontinuous solutions with exactly localized phase transitions. In this paper we concentrate on a sharp-interface model in one space dimension where phase transitions can be represented as a type of shock wave. More precisely our model consists of the equations of hydrodynamics for an isothermal fluid described by the two conservation laws for density ρ : R [0, 0, 1/b and momentum m := ρv : R [0, R given by ρ t + m x =0 m m t + ρ + pρ =0 x in R 0,. 1 Keywords and phrases. Dynamical phase transitions in compressible media, van-der-waals pressure, kinetic relations, Riemann solver, ghost fluid approach. 1 Abteilung für Angewandte Mathematik, lbert-ludwigs-universität Freiburg, Hermann-Herder Str. 10, Freiburg, Germany. christian@mathematik.uni-freiburg.de Institut für Angewandte Mathematik und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, Germany. c EDP Sciences, SMAI 007 Article published by EDP Sciences and available at or

2 1090 C. MERKLE AND C. ROHDE pρ vapor liquid pτ ρ ellipt ρ ellipt max ρ min 1 ρ ρ τ 8 τ τ τ 7 τ Figure 1. Graph of the van-der-waals pressure function p = pρ and the associated Lagrangian pressure p = pτ with pτ = p1/τ for sufficiently low temperature. We consider the Cauchy problem for 1 and, for initial density ρ 0 : R 0, 1/b andvelocityv 0 : R R, enforce the conditions ρ., 0 = ρ 0, m., 0 = ρ 0 v 0 in R. Here b is a positive number and for the smooth pressure function p :0, 1/b 0, we suppose that there are numbers ρ ellipt min,ρellipt max 0, 1/b withρellipt min <ρ ellipt max such that p is monotone increasing in the admissible range A := 0,ρ ellipt min ρellipt max, 1/b and monotone decreasing in the elliptic region 0, 1/b \A=[ρ ellipt min,ρellipt max ]. We refer to Figure 1 for the typical graph of p and to Section 1 for precise assumptions. On this general level it is only important that the non-monotone shape of p allows to define phases. A state ρ, ρv inthe physical state space A R is called a liquid vapor state if ρ 0,ρ ellipt min ]ρ [ρellipt max, 1/b holds. As a consequence of the non-monotone shape of p the first-order system 1 is hyperbolic for states if and only if the states are liquid or vapor. The system or more exactly its linearized version is elliptic for density values in 0, 1/b \A. Altogether we obtain a mixed-type system. Mixed-type systems are frequently used as models for phase transitions in compressible media cf. [1, 1]. Our ultimate goal is to design a numerical method to solve the multidimensional version of the sharp-interface model 1. In this paper we restrict ourselves to the 1D-case such that curvature effects play no role. Usually in many phase transition and phase separation problems the correct treatment of curvature effects is one of the main challenges. We would like to stress that here due to the interaction of flow and phase transition effects even the 1D-case is by far not trivial. In the rest of the introduction we give an outline of the paper s content and indicate the specific difficulties in this sharp-interface model. The basis of our numerical method is the exact treatment of the Riemann problem for the conservation law 1, that is the special initial data choice { ρ L ρ m L:=ρ Lv L : x<0 x, 0 = ρ m ρ L,ρ R A,v L,v R R. 3 R : x>0 m R:=ρ Rv R System 1 has the form of a conservation law, however the standard solution theory for hyperbolic conservation laws does not apply see e.g. [11,3,36]. This is first due to the fact that the elliptic region separates the state

3 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1091 space into two disjoint sets where 1 is hyperbolic. Secondly, the system is not genuinely nonlinear in the vapor phase anymore which requires to build Riemann patterns with attached-wave structures. Moreover, phase transitions static as well as dynamical ones are naturally modeled as shock waves and thus appear automatically as sharp interfaces. But they are partly not Laxian-type shock waves but of the nonclassical undercompressive type. Let us also stress as a striking property of the system 1 that the Clausius-Duhem inequality is not enough to ensure the unique solvability of the Riemann problem see [3]. To overcome these difficulties we follow the approach developed by LeFloch and co-workers [5, 17, 4] which relies on the concept of the kinetic relation [1, 39, 40]. The main existence result for a thermodynamically admissible Riemann problem solution is then presented at the end of Section 3 Thm The preceding sections serve to introduce the model Sect. 1 and to present a careful analysis for all possible wave types and their thermodynamical admissibility including shock waves, rarefaction waves, attached waves and phase transitions Sect.. In this context we mention the publications [9,10,16,30] on the existence and stability of solutions for the Riemann problem for related systems. A well-developed class of methods to solve the Euler equations for compressible one-phase fluids is provided by upwind finite volume schemes based on approximate Riemann solvers. In Section 4 we propose a similar class of methods for the Cauchy problem 1,. However standard averaging techniques can not be used in our case since this due to the non convex state space might lead to approximate cell average states in the elliptic region. To circumvent this difficulty we suggest a ghost-fluid type algorithm, motivated but different from the algorithm in the seminal paper [13] see [, 4, 14, 0, 6] for other applications. The crucial analytical basis of the ghost fluid algorithm is the Riemann solver which has been developed in the first part of the paper cf. Sects. and 3. Finally, we test the algorithm on problems with known exact solution and report on a number of numerical experiments. We have not been able to prove rigorously the convergence of our algorithm as the mesh parameter tends to zero but we present a convergence result for simple phase transition solutions with Proposition 4.5. In forthcoming papers we extend the approach to multiple space dimensions see in particular [7] and address curvature effects and the non-isothermal situation. Another direct numerical method for the mixed-type Cauchy problem 1, apart from the one presented here is not known to us. However we have to refer to several related publications. A different approach to avoid averaging effects is the DEM-method due to Abgrall and Saurel [3] which has been used in [1] to simulate evaporation fronts. For the numerical solution of the sharp interface model with additional kinetic relations there has been proposed the Glimm-Scheme [8,], front-tracking schemes [41], artificial dissipation methods [5] and a level-set scheme [18,8]. The latter one is different from the one presented here since the authors extend the kinetic relation to the whole computational domain. Recently Chalons has developed a new deterministic method to treat nonclassical shocks with kinetic relations in hyperbolic equations [7]. 1. Basic properties of the mixed-type system In this section we detail all assumptions on the system 1 and introduce fundamental notations. Furthermore we review the spectral properties of the nonlinear flux in 1 and the structure of the associated acteristic fields. Assumption 1.1 Pressure function. i The function p C 0, 1/b, 0, is supposed to satisfy p ρ > 0, p ρ < 0 for ρ 0,ρ ellipt min, p ρ > 0, p ρ > 0 for ρ ρ ellipt max, 1/b, p ρ < 0 for ρ ρ ellipt min,ρellipt max. 4

4 109 C. MERKLE AND C. ROHDE ii Furthermore, we have lim pρ = lim ρ 0 ρ 0 p ρρ =0, lim pρ =. 5 ρ 1/b The typical shape of such a pressure function is shown in Figure 1 left graph. Due to the fundamental laws of thermodynamics the associated energy density function W C 1 0, 1/b is given up to an unimportant free constant by the relation pρ =ρw ρ W ρ. 6 Furthermore, from 4 it is clear see also Fig. 1 that there are unique densities ρ 1 0,ρ ellipt min andρ ρ ellipt max, 1/b such that we have pρ 1 =pρ ellipt max andpρ =pρ ellipt min. 7 The Maxwell states are defined as the two points ρ M vapor and ρ M liquid such that W ρ M vapor =W ρ M liquid = W ρm liquid W ρm vapor ρ M liquid 8 ρm vapor That means the straight line connecting ρ M vapor,wρm vapor and ρm liquid,wρm liquid has the same slope as W at ρ M vapor and ρm liquid cf. Fig. 1. A short calculation using 6 and 8 leads to the property pρ M vapor = pρm liquid. 9 In our Riemann solution for 1, 3 later on static equilibria occur if and only if the connecting states are the Maxwell states. Remark 1.. For appropriate choices of the constant temperature T>0 and constants a, R > 0 a pressure function satisfying 4 and 5 can be realized as the van-der-waals function pρ = ρrt 1 bρ aρ. 10 Actually all figures and numerical calculations in the following are performed with the constants a =3, b = 1/3, R=8,T=98/300. The eigenvalues and corresponding eigenvectors of the Jacobian of the flux in 1 are λ 1/ ρ, m = m ρ p 1 ρ, r 1/ ρ, m = m ρ p. 11 ρ We observe that the system 1 is hyperbolic for states ρ, m A R, if and only if p ρ > 0 holds, that is, if and only if the state is a liquid or vapor state. We compute for ρ, m A R the acteristic fields of the flux and get the following expression independent of momentum: p ρ λ 1/ ρ, m r 1/ ρ, m = p ρ + p ρ 1 =: Λ ρ ρ

5 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1093 Rewriting the last equations in terms of τ =1/ρ we obtain Λ τ = τ p τ+τ 3 p τ p τ Here we used the Lagrangian pressure p :b, 0, defined by + τ p τ. 1 pτ :=p1/τ τ b,. We refer to Figure 1, right graph, for a visualization of p. From 4, 5 we see that p 1/ρ ellipt min =0and lim τ p τ = 0 hold. Therefore, there existsaninflectionpointτ 1/ρ ellipt min, and another inflection point τ of the function p for which ρ := 1 τ is located in the elliptic region. Define also ρ := 1 τ 0,ρ ellipt min. Note, that p is convex in b, τ τ, and concave in τ,τ. We refer to Figure 1 for the graph of p. Returningto1weseethatΛ =Λ τ vanishes for τ b, if and only if we have 1 τ p τ+τ 3 p τ τ p τ = p τ =0. Thus both acteristic fields of the flux in 1 fail to be genuinely nonlinear for liquid states with ρ = ρ.. Basic elementary waves and thermodynamical admissibility In this section we consider all elementary waves which we need in the construction of the weak solution for the Riemann problem 1, 3 in Section 3 below. We start with shock waves and general phase boundaries. To detect the physically admissible shock waves we analyze the entropy dissipation function and to solve the problem of the failure of genuine nonlinearity we carefully consider the location of associated end states in the phase plane. Finally we discuss rarefaction waves resp. attached waves and last but not least non-laxian shock waves that satisfy a given kinetic relation..1. Rankine-Hugoniot conditions, shock waves, and phase transitions Basic elementary waves for the solution of the Riemann problem for 1 are shock waves. A shock wave 1 with speed s R connecting a state ρ l,m l A R with a state ρ r,m r A R we also write ρ l,m l s ρ r,m r is a discontinuous function of the type ρ = m { ρx, t ρl,m = l T : x<st, mx, t ρ r,m r T : x>st, that fulfills the Rankine-Hugoniot jump conditions m s ρ = m and s m = ρ + pρ The notation is sloppy since usually discontinuous traveling waves satisfying the Rankine-Hugoniot conditions but having zero entropy dissipation are not called shock waves. For the sake of simplicity we include them here. 13

6 1094 C. MERKLE AND C. ROHDE In 14 is and henceforth will be used the notation a := a l a r for some general variable a. The conditions 14 imply in particular that 13 is a weak solution of 1. Now, fixing a state ρ l,m l A R we can determine the set of states ρ r,m r hat can be connected to ρ l,m l A R by a shock wave, the so called Rankine-Hugoniot set for ρ l,m l. For first-order conservation laws the Rankine-Hugoniot set can be acterized easily provided there is an open set containing ρ l,m l where the system is hyperbolic. In our case this is true within each phase and we have the following result see e.g. [15], Thm Theorem.1. Let ρ l,m l A R. There exists a number ε 0 > 0, functionsφ k : ε 0,ε 0 A R, k =1,, and a neighborhood of ρ l,m l such that the set {Φ k ε ε ε 0,ε 0,k=1, } coincides with the Rankine-Hugoniot set for ρ l,m l in this neighborhood. Moreover we have for ε ε 0,ε 0 and k =1, and Φ k ɛ =ρ l,m l T + ɛ r k ρ l,m l +Oɛ 15 s = sρ l,m l, Φ k ɛ = λ k ρ l,m l +Oɛ. 16 According to the numbering of the curves in Theorem.1 we speak of k-shock waves. In the case of a 1-shock wave a specific parameterization as in 15 is given by It follows for ε ε 0,ε 0 ρr ρl =Φ m 1 ɛ = + ɛ r m l 1 m l ρ l p ρ l v r v l p ρ l = + Oɛ. ρ r ρ l ρ r + Oɛ. Therefore we have v r v l 0 0 for ρ r ρ l ρ r ρ l. 17 Returning to the global situation system 14 consists of two equations with three unknowns ρ r, v r and s. These can be reformulated for v r and s depending on ρ r : ρr ρ l v r ρ r = v l pρ r pρ l, 18 ρ r ρ l ρ r pρ r pρ l ρ l pρ r pρ l sρ r = v l = v r 19 ρ l ρ r ρ l ρ r ρ r ρ l We observe that globally the Rankine-Hugoniot set consists of two curves also called k-shock wave curves. Comparing 18 with the local parameterization 17 we conclude: The 1-shock wave curve is given by v r v l = + ρr ρ l ρ l ρ r pρ r pρ l : ρ r ρ l, ρr ρ l ρ l ρ r pρ r pρ l : ρ r ρ l. 0 Equivalently, -shock wave curve is given by v r v l = ρr ρ l ρ l ρ r pρ r pρ l : ρ r ρ l, + ρr ρ l ρ l ρ r pρ r pρ l : ρ r ρ l. 1

7 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1095 Remark.. The parameterizations of the Rankine-Hugoniot sets 0 and 1 with v r seen as function of the right-hand state ρ r Asatisfy the monotonicity property To prove this for a 1-shock wave consider dv r dρ r ρ r < 0> 0 for a 1-shock wave a -shock wave. dv r ρ r dρ r = 1 ρr ρ l ρr ρ l pρ r pρ l ± ρ pρ r pρ l + ρ r p ρ r, r ρ l ρ r ρ r ρ l ρ l }{{} >0 where the + sign is valid if ρ r ρ l and the sign if ρ r >ρ l. Hence, dvr dρ r ρ r < 0fora1-shockwave. Fora -shock wave the calculation works the same replacing ± by... Entropy dissipation and acteristic structure Physically meaningful shock waves ρ l,m l s ρ r,m r also have to satisfy the entropy dissipation inequality or Clausius-Duhem condition, that is, s Eρ, v F ρ, v 0. The entropy E :0, 1/b R R and the entropy flux F :0, 1/b R R are given by Eρ, v = 1 1 ρv + W ρ, Fρ, v =v ρv + W ρ+pρ, 3 where W = W ρ is the free energy of the system cf. 6. A shock wave that satisfies is called entropydissipative. Later on in this section we need a reformulation of the entropy dissipation which we derive now. We get from 14,, 3, and 6 1 s Eρ, v F ρ, v = s ρ lvl + W ρ l 1 ρ rvr W ρ r v l pρ l + 1 ρ lvl + W ρ l + v r pρ r + 1 ρ rvr + W ρ r = j = j vl vr s p j W ρ v v l + v r s+ W ρ. Here we used j := ρ l v l s =ρ r v r s. A straightforward computation using 18, 19 leads for ρ l ρ r to s Eρ, m F ρ, m = j ρ 3 ρ l ρ r ρ l + ρ r W ρ ρ 3 ρ lw ρ l +ρ r W ρ r ρ =: j ρ 3 ρ l ρ r ρ l,ρ r. 4

8 1096 C. MERKLE AND C. ROHDE Moreover, defining τ l,τ r := 1 τ l, 1 τ r and using relation 6 yields ρ 3 ρ l ρ r ρ l,ρ r = ρ 3 ρ l ρ r τl,τ r = pτ l+ pτ r = τ r τ l τ r τ l τ r τ l psds pτ l + pτ r pτ l s τ l ps τ r τ l ds. 5 This formulation allows a nice geometrical interpretation to determine the sign and the zero level set of the function. Let τ l,τ r b,. Then we have τ l,τ r = 0 if and only if τ r τ l pτ l + pτ r pτ l s τ l ps τ r τ l ds =0, 6 i.e. if the signed area between the straight line connecting pτ l and pτ r and the graph of p is zero. As we are interested in the zero dissipation level set we define the set Γ := { ρ l,ρ r 0, 1/b ρ l,ρ r =0,ρ l ρ r } ρ,ρ ρ,ρ. 7 Note that the function is only defined for ρ l ρ r but there is a continuous extension since lim ρ ρ ρ, ρ = 1 6 p ρ+p ρ = 1 6 τ 4 p ττ τ. The latter expression vanishes exactly for τ = τ,τ and therefore we added the points ρ,ρ andρ,ρ in definition 7. We observe that Γ is symmetric with respect to the axis {ρ, ρ ρ 0, 1/b} and also deduce from 8 ρ M vapor,ρm liquid, ρm liquid,ρm vapor Γ. 8 Moreover, the set Γ is a closed curve in 0, 1/b, which follows from the geometrical interpretation 6 and Assumption 1.1 on the pressure function p. For the further analysis it is important to classify shock waves according to the following standard system. For k {1, } a k-shock wave is called a Laxian or Lax-compressive shock wave if the condition λ k ρ l,m l s λ k ρ r,m r 9 is satisfied. If one of the relations in 9 holds with equality the corresponding shock wave is called acteristic. Furthermore, in the context of this paper a shock wave is called an undercompressive shock wave if λ ρ l,m l >s>λ 1 ρ r,m r and λ ρ r,m r >s>λ 1 ρ l,m l 30 is satisfied. The definitions are illustrated in Figure. In the framework of the first-order system 1 it is natural to view phase jumps as shock waves: a shock wave ρ l,m l s ρ r,m r is called a phase transition, if the left and the right states are located in different phases see Fig. 1. In this case we also use the notation ρ l,m l pt ρ r,m r. We will see that in our case undercompressive shock waves will only appear as phase

9 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1097 t t s = x t s = x t ρ r,v r ρ r,v r ρ l,v l x ρ l,v l x Figure. Laxian -shock wave left and undercompressive shock wave right. Shock lines for shocks with speed s and some acteristic curves dashed lines in the x, t-half plane. transitions while Laxian waves can be purely hydrodynamical shock waves or phase transitions. Equation 9 is using τ =1/ρ and pτ =p1/τ equivalentto for Laxian 1-shock waves and p τ τ l pτ r pτ l τ r τ l p τ τ r 31 p τ τ l pτ r pτ l τ r τ l p τ τ r 3 for Laxian -shock waves. In the hyperbolic case in each phase in our case, it can be shown that shock waves 13 that satisfy the Lax condition 9 are entropy solutions, i.e. holds. To account for the possible difference between Laxian shocks and entropy dissipative Laxian shocks in the two-phase case we define the following sets: M 1 := {ρ l,ρ r A m l,m r,s R 3 : 14 and 31 hold}, M := {ρ l,ρ r A m l,m r,s R 3 : 14 and 3 hold}. The sets M 1 and M are displayed in Figure 3. Since we are interested in entropy dissipative Laxian shock waves we introduce the subsets M 1 diss := {ρ l,ρ r A m l,m r,s R 3 : 14, 31 and hold}, M diss := {ρ l,ρ r A m l,m r,s R 3 : 14, 3 and hold}. Thus M 1 diss and M diss are bounded by parts of the boundary of A A, the curves Γ,Γ 1,l,Γ1,r,Γ,l,whiegivenfork =1, by Γ,r, Γ k,l := { ρ l,ρ r A,ρ l ρ r m l,m r,s R 3 : 14, and λ k ρ l,m l =s } ρ,ρ, Γ k,r := { ρ l,ρ r A,ρ l ρ r m l,m r,s R 3 : 14, and λ k ρ r,m r =s } ρ,ρ. 33 We note, that the conditions λ k ρ l/r,m l/r =s in 33 are equivalent to pτ τ l = pτ r pτ l τ r τ l or p τ τ r = pτ r pτ l τ r τ l 34

10 1098 C. MERKLE AND C. ROHDE Γ 1/,l/r M Γ p ρ Γ M 1 ρ r M Γ,r ρ Γ 1,r M M Γ 1,l Γ,l M M ρ 1 ρ ρ l Γ 1/,l/r M diss Γ p ρ Γ M diss 1 ρ r M Γ,r diss ρ Γ 1,r M 1 diss M Γ 1,l diss Γ,l M diss M diss ρ 1 ρ ρ l Figure 3. Regions of density values which can be connected by Laxian shock waves left figure and dissipative Laxian shock waves right figure are hatched. Note that Laxian waves can only be ruled out due to non-dissipativity if the states are in different phases. We recall that Γ p denotes the density pairs with identical pressure and Γ the pairs with zero entropy dissipation while point on the curves Γ 1/,l/r correspond to acteristic waves cf. 33. For ρ 1 and ρ we refer to 7. They are in particular independent of momentum. Assumption 1.1 ensures that there are exactly two specific densities τ 7,τ 8 with 0 <ρ 8 <ρ ellipt min <ρellipt max <ρ 7 such that the slope of p in both points and the slope of the straight line connecting τ 8, pτ 8 with τ 7, pτ 7 takes the same value see Fig. 1. In view of the fact that intersection points of the curves defined in 33 have to satisfy both relations in 34 we conclude that there is exactly one intersection point in 0,ρ ellipt min ρellipt max, 1/b as well as one in ρ ellipt max, 1/b 0,ρellipt, namely cf. Fig. 4 In the next step we define the set min ρ 8,ρ 7 and ρ 7,ρ Γ p := {ρ l,ρ r A pρ l =pρ r,ρ l ρ r }. We observe that Γ p is symmetric with respect to the axis {ρ, ρ ρ 0, 1/b}. Due to 34 and 7 the union of Γ 1,l, Γ1,r, Γ,l, Γ,r and the curve Γ p has four intersection points ρ ellipt min,ρ, ρ,ρ ellipt min, ρellipt The strict inequalities in 4 show that the points in 36 are the only ones. max,ρ 1, and ρ 1,ρ ellipt max. 36

11 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1099 Recall from 4 that the entropy dissipation vanishes if and only if either ρ l = ρ r, =0orv l = v r = s hold. The latter case implies for states satisfying the Rankine-Hugoniot shock speed formula 19 that p = 0 37 is valid. Note, that due to 9 and 8 the curves Γ and Γ p intersect in the Maxwell points. These are the only intersection points, which can be seen easily from 6. In the vapor phase the genuine nonlinearity breaks down exactly at states with ρ = ρ.asρ,ρ Γ the curves from 33 and Γ intersect at the point ρ,ρ in the vapor phase. This is the only intersection point in the vapor phase, which follows from the shape of p cf. Assumption 1.1 and 6. There are exactly four further intersection points within A of Γ and the curves from 33 see Fig. 4, namely For simplicity we only show that ρ 4,ρ 3 ρ ellipt max of Γ 1,r and Γ,r ρ 3,ρ 4, ρ 4,ρ 3, ρ 5,ρ 6, and ρ 6,ρ 5. 38, 1/b 0,ρellipt min belongs to Γ while it is also an end point see Fig. 4. Due to Assumption 1.1 there is a point ˆτ b, 1/ρellipt max such that pτ τ = pτ pˆτ τ ˆτ holds, i.e. ˆτ,τ satisfies 34 and due to 5 the relation ˆτ,τ < 0. On the other hand, for ρ 7,ρ 8 Γ k,r see 35 it holds 1/ρ 7, 1/ρ 8 > 0. Therefore, there is an intersection point ρ 4,ρ 3 ρ ellipt max, 1/b 0,ρellipt min ofγ and the curves from 33. With help of monotonicity arguments and p ττ > 0inb, 1/ρ ellipt max and p ττ > 0inτ, 1/ρ 8 one can show the uniqueness of this intersection point. Remark.3 Ordering and end points. i For the intersection points identified above we have the ordering relations ρ 6 <ρ 8 <ρ 3 <ρ ellipt min ρ 1 <ρ M vapor <ρ ellipt min ρ 3 <ρ M vapor <ρellipt min <ρellipt max <ρ 5 <ρ 7 <ρ 4, <ρellipt max <ρ M liquid <ρ, <ρellipt max <ρm liquid <ρ 4. The inequalities ρ 6 <ρ 8 <ρ 3 <ρ ellipt min as well as ρ ellipt max <ρ 5 <ρ 7 <ρ 4 can be verified using the definition of the curves Γ 1/,l/r and Γ cf. 5 and Assumption 1.1. ρ 1 ρ M vapor ρ ellipt min ρellipt max ρ M liquid ρ follows directly from the definition of the points see Fig. 1. The strict inequality < then follows from 6. For ρ 4 >ρ 3 it holds pρ 4 >pρ 3 because ρ 4,ρ 3 Γ 1/,r and therefore satisfies the Rankine-Hugoniot jump condition 14 for some v l,v r R. Moreover, it holds ρ 4,ρ 3 =0aswehaveρ 4,ρ 3 Γ cf. 38. Now suppose ρ 3 >ρ M vapor ρ 4 <ρ M liquid analogously. Then pρ 4 >pρ 3 >pρ M vapor =pρm liquid and due to 6 ρ 4,ρ 3 0. Therefore, the statement follows. ii The endpoints of Γ 1/,r lying in A are exactly ρ 4,ρ 3 andρ 6,ρ 5. Endpoints of Γ 1/,l lying in A are exactly ρ 3,ρ 4 andρ 5,ρ 6.

12 1100 C. MERKLE AND C. ROHDE ρ 4 ρ ρ 5 p l=r ρ r ρ 8,ρ 7 λ,l s ρ r λ 1,l s ρ l ρ r p l=r ρ r ρ 1 λ 1,r s ρ l ρ 3 ρ 7,ρ 8 0. ρ 6 0 ρ 3 ρ 1 ρ 6 ρ 5 ρ ρ ρ l Figure 4. Graphs of the functions λ 1,l s,λ1,r s and λ,l s as defined in Lemma.4 and p l=r from Lemma.5. They consist of parts of the acteristic curves and the equal-pressure curve. To prove this note that for all ρ l,ρ r satisfying 34 there exists m l,m r,s R 3 such that 14 holds. Therefore, the only possible endpoints of Γ 1/,r within A are the points where the sign of the entropy dissipation changes, i.e. intersection points of Γ 1/,r and the set of points where the entropy dissipation is zero. Due to 4 this can happen if pρ l =pρ r cf. 37 or if ρ l,ρ r =0holds. Intersection points of Γ p and Γ 1/,r are not in A cf. 36. Therefore, endpoints of Γ 1/,r within A have to be elements of Γ. Due to 38 this is the case for ρ 4,ρ 3, ρ 6,ρ 5. The second part of the remark follows by symmetry. With the above remarks we obtain the configuration as depicted in Figures 3 and 4. The complexity of the two-phase case becomes visible if one recalls that the one-phase case for a perfect fluid leads to the simple situation in the liquid box ρ ellipt max, 1/b. Parts of the sets Γ 1,l, Γ1,r, Γ,l and Γ,r are graphs of functions depending on ρ l or ρ r. We will define those functions by the subsequent lemma cf. also Fig. 4 as we need them in the remainder of the paper. Lemma.4. There exist functions λ 1,r s : 0,ρ 4 0,ρ ellipt min, λ 1,l s : ρ 3,ρ ellipt min ρ ellipt max, 1/b, λ,l s : 0,ρ 4 0,ρ ellipt min such that i Γ 1,r {ρ l,ρ r ρ r 0,ρ ellipt min } = {ρ l,λ 1,r s ρ l ρ l 0,ρ 4 },

13 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 11 ii Γ 1,l {ρ l,ρ r ρ l 0,ρ ellipt min } = {ρ l,λ 1,l s ρ l ρ l ρ 3,ρ ellipt min }, iii Γ,l {ρ l,ρ r ρ l 0,ρ ellipt min } = {λ,l s ρ r,ρ r ρ r 0,ρ 4 }. Proof. We prove i explicitly and note that ii, iii can be done in the same manner. For ρ l 0,ρ 4 there are two solutions of the equation p τ τ = pτ pτ l τ τ l, τ l = 1 ρ l 39 Define { } 1 λ 1,r s ρ l:=min τ τ solves 39. Then, λ 1,r s ρ l 0,ρ ellipt min, due to the shape of p cf. Assumption 1.1. Moreover we get ρ l,λ 1,r s ρ l Γ 1,r as the condition λ 1ρ, m =s is equivalent to 39 and the corresponding shocks are entropy dissipative. We show the latter statement for a single point ρ l,λ 1,r s ρ l with ρ l 0,ρ ellipt min. Then, the argument can be extended to ρ l 0,ρ 4 as the sign of the entropy dissipation does not change until the intersection point with Γ, i.e. ρ 4,ρ 3 cf. Rem..3ii. Let for simplicity ρ l ρ,ρ ellipt min. Due to 4 we can calculate the sign of the entropy dissipation explicitly. As for ρ l,λ 1,r s ρ l the condition λ 1 ρ l,m l =s holds the corresponding shock wave can due to 9 only be a 1-shock wave. Therefore, j = ρ l v l s > 0 due to 19 with sign for a 1-shock wave. Moreover we have ρ 3 > 0asρ l >λ 1,r s ρ land ρ l,λ 1,r s ρ l > 0 due to 5. Altogether, the entropy dissipation 4 is negative. Accordingly, part of the set Γ p can be seen as the graph of a function depending on ρ r. Lemma.5. There is a function p l=r :ρ 1,ρ ellipt min ρellipt max,ρ ρ 1,ρ ellipt min ρellipt max,ρ such that Γ p = {p l=r ρ r,ρ r ρ r ρ 1,ρ ellipt min ρellipt max,ρ }. Proof. Without loss of generality let ρ ρ 1,ρ ellipt min. Due to Assumption 1.1 there is exactly one ˆρ ρ ellipt max,ρ such that pˆρ =pρ holds. The statement follows with p l=r ρ :=ˆρ. Out of points on the Rankine-Hugoniot curves let us now select the ρ-components of those shock waves that satisfy the Lax entropy condition 9 and are entropy-dissipative. Lemma.6 1-shock waves. Let ρ l,v l and ρ r,v r A R be in one phase, such that 0 holds. Then 31 holds, if and only if we have ρ l,ρ r S 1 := {ρ l,ρ r ρ l 0,ρ,ρ r ρ l,λ 1,r s ρ l]} {ρ l,ρ r ρ l ρ,ρ ellipt min,ρ r [λ 1,r s ρ l,ρ l } { ρ l,ρ r ρ l ρ ellipt max, 1/b,ρ r ρ l, 1/b }. Moreover we have ρ l,ρ r M 1 diss, thus the waves are entropy-dissipative. Proof. We show the statement for ρ l 0,ρ. The other cases are proved similarly. Let ρ l,m l andρ r,m r A Rbein one phase, such that 0 holds. We have to verify 31 for ρ r ρ l,λ 1,r s ρ l. Therefore, first assume ρ l >ρ r, i.e. τ l <τ r.thenasτ l >τ and p ττ τ > 0forτ>τ the following relation holds cf. Fig. 5a: p τ τ l < pτ r pτ l τ r τ l < p τ τ r and therefore, 31 does not hold. Hence, equation 9 does not hold for ρ l >ρ r.

14 110 C. MERKLE AND C. ROHDE pτ pτ 1/ρ min ellipt τ τ l τ r τ 1/ρ min ellipt ˆτ l τ r τ τ l τ a b Figure 5. Auxiliary illustration for the proof of Lemma.6. Now assume ρ r >ρ l, i.e. τ l >τ r,thenasτ l >τ 31 holds, as long as τ r > ˆτ l i.e. ρ r < ˆρ l := 1ˆτ l, where ˆτ l is defined as the unique solution of see Fig. 5b pˆτ l pτ l ˆτ l τ l = p τ ˆτ l. The uniqueness follows from the shape of the function p. Now it remains to prove ˆρ l = λ 1,r s ρ l. This is true due to the definition of the function λ 1,r s. Therefore, 9 is valid for ρ r ρ l,λ 1,r s ρ l. The cases ρ l ρ,ρ ellipt min andρ l ρ ellipt max, 1/b are proved similarly. Finally, we observe from the location of M 1 diss the inclusion S1 M 1 diss which concludes the proof. Equivalently for -shock waves the following lemma is valid: Lemma.7 -shock waves. Let ρ l,v l and ρ r,v r A R be in one phase, such that 1 holds. Then 3 holds, iff ρ l,ρ r S := {ρ l,ρ r ρ r 0,ρ,ρ l ρ r,λ,l s ρ r } {ρ l,ρ r ρ r ρ,ρ ellipt min,ρ l λ,l s ρ r,ρ r } { ρ l,ρ r ρ r ρ ellipt max, 1/b,ρ l ρ r, 1/b }. Moreover we have ρ l,ρ r M diss. The sets S 1 and S are shown in Figures 6 and Rarefactionwavesandattachedwaves Next let us review the structure of rarefaction waves. Let ρ l,m l, ρ r,m r A R with ρ l and ρ r from the same phase be given. For k {1, } a k-rarefaction wave denoted by ρ l,m l rare ρ r,m r is a weak solution ρ, m T : R [0, A R of 1 of the type ρx, t mx, t = ρl m l ρx/t mx/t ρr m r : x ξ k 1 t, : ξ1 k t<x<ξ k t, 40 : x ξ kt.

15 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE S R A R S 1 ρ 0 1 S λ 1,r s ρ l A 1 11 R ρ r ρ Figure 6. Classical 1-waves: density components of states ρ l,v l andρ r,v r within one phase can be connected by a classical 1-wave. S 1 corresponds to an 1-shock wave cf. Lem..6, R 1 to a 1-rarefaction wave cf. Lem..9 and A 1 to an attached 1-wave cf. Lem..10. Here ρ, m T C 1 ξ1 i,ξi, A R has to be chosen such that ρ, mt is a classical solution of 1 for ξ1 kt< x<ξ k t with the property ρ l ρξ k 1 =ρ l, ρξ k =ρ r, mξ k 1 = ρ vξ k 1 =m l, mξ k = ρ vξ k =m r, 41 wherewehave ξ1 k = λ kρ l,m l,ξ k = λ kρ r,m r. For the hydrodynamical equations 1 the functions ρ and m = ρ v have to satisfy 41 and for ξ ξ1 k,ξk have to be real solutions of the system of ordinary differential equations ρ 1/ ξ = ρξ p ρξ ρξp ρξ + p ρξ, ρ v 1/ ξ = ρ 1/ vξ ξ p ρξ. 4 Eliminating ξ in the above equations we can express v in terms of ρ: ρ p r v ρ = v l dr. 43 ρ l r With 43 we have got a parameterization in the ρ, m-plane of the rarefaction curves, i.e. the set of states ρ r,m r A R that can be connected by a k-rarefaction wave to ρ l,m l.

16 1104 C. MERKLE AND C. ROHDE λ,l s ρ r R S 1 ρ r S A R R A S ρ ρ Figure 7. Classical -waves: density components of states ρ l,v l andρ r,v r within one phase can be connected by a classical -wave. S corresponds to a -shock wave cf. Lem..7, R to a -rarefaction wave cf. Lem..9 and A to an attached wave cf. Lem..10. ρ l Remark.8. The parameterizations of the rarefaction curves given in 4 with v r as a function of ρ r have the monotonicity property dv r dρ r ρ r < 0> 0 for a 1-rarefaction wave -rarefaction wave. This can be seen from dv r p ρ r ρ r =, dρ r ρ r with for a 1-rarefaction wave and + for a -rarefaction wave. It is well-known see [11] that the system of ordinary differential equations 4 has a unique solution if the condition cf. 1 Λ = λ k r i > 0, k {1, } 44

17 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1105 is satisfied along the wave. This leads to: Lemma.9 1/-rarefaction waves. Let ρ l,m l and ρ r,m r A R be within one phase such that 43 holds. Then 44 holds, if and only if ρ l,ρ r R 1 := {ρ l,ρ r ρ l 0,ρ,ρ r 0,ρ l } {ρ l,ρ r ρ l ρ,ρ ellipt min,ρ r ρ l,ρ ellipt min } {ρ l,ρ r ρ l ρ ellipt max, 1/b,ρ r ρ ellipt max,ρ l }, for a 1-rarefaction wave and ρ l,ρ r R := {ρ l,ρ r ρ r 0,ρ,ρ l 0,ρ r } {ρ l,ρ r ρ r ρ,ρ ellipt min,ρ l ρ r,ρ ellipt min } {ρ l,ρ r ρ r ρ ellipt max, 1/b,ρ l ρ ellipt max,ρ r}, for a -rarefaction wave. Proof. Note that for a 1-rarefaction wave cf. 1 Λ τ > 0 p τ < 0 ρ ρ,ρ holds. Now let ρ l,v l andρ r,v r be two states on the rarefaction wave 43 with negative sign for a 1- rarefaction wave. If ρ l ρ,ρ ellipt min ρ,ρ holds then we have λ 1 ρ r,m r >λ 1 ρ l,m l forρ r ρ l,ρ ellipt min since λ 1 r 1 > 0 on the wave. If ρ l 0,ρ ρ, 1/b holds then we have λ 1 ρ r,m r >λ 1 ρ l,m l forρ r <ρ l. The statement for -rarefaction waves is proved analogously. The sets R 1 and R are shown in Figures 6 and 7. Note that we have no difficulties to satisfy the Clausius- Duhem inequality for rarefaction waves as continuous weak solutions. We observe from Figure 6 that we cannot connect all states in a single phase by only Laxian shock waves and rarefaction waves. Furthermore, we need attached waves. Let ρ l,m l, ρ r,m r A R. If there exists a state ˆρ, ˆm such that an 1-shock wave ρ l,m l s ˆρ, ˆm with speed s = λ 1 ˆρ, ˆm and an 1-rarefaction wave ˆρ, ˆm rare ρ r,m r exist we say that ρ l,m l andρ r,m r can be connected by an 1-attached wave with notation ρ l,m l att ρ r,m r. Obviously, the function ρx,t mx,t = ρl m l ρx/t mx/t ρr m r : x<λ 1 ˆρ, ˆmt, : λ 1 ˆρ, ˆmt x<λ 1 ρ r,m r t, : x λ 1 ρ r,m r t is then a weak solution of 1, where ρ, m are defined analogously as in 40 but with ˆρ, ˆm as lefthand state. Similarly, if there exists a state ˆρ, ˆm such that a -rarefaction wave ρ l,m l rare ˆρ, ˆm and a -shock wave ˆρ, ˆm s ρ r,m r with speed s = λ ˆρ, ˆm exist we say that ρ l,m l andρ r,m r can be connected by an -attached wave. Then, the function ρx,t mx,t = ρl m l ρx/t mx/t ρr m r : x<λ ρ l,m l t, : λ ρ l,m l t x<λ ˆρ, ˆmt, : x λ ˆρ, ˆmt is a weak solution of 1, where ρ, m are defined analogously as in 40 but with ˆρ, ˆm as righthand state. The attached waves within one phase are described in the following lemma

18 1106 C. MERKLE AND C. ROHDE Lemma.10 1/-attached waves. Let ρ l,v l, ρ r,v r A R with densities in the vapor phase. They can be connected by an 1-attached wave ρ l,m l att ρ r,m r, if and only if we have cf. Fig. 6 ρ l,ρ r A 1 := {ρ l,ρ r ρ l 0,ρ,ρ r λ 1,r s ρ l,ρ ellipt min } {ρ l,ρ r ρ l ρ,ρ ellipt min,ρ r 0,λ 1,r s ρ l}, and by a -attached wave ρ l,m l att ρ r,m r, if and only if we have cf. Fig. 7 ρ l,ρ r A := {ρ l,ρ r ρ r 0,ρ,ρ l λ,l s ρ r,ρ ellipt min } {ρ l,ρ r ρ r ρ,ρ ellipt min,ρ l 0,λ,l s ρ r}. In both cases the Clausius-Duhem inequality is satisfied. Proof. The proof is done for the case of a 1-attached wave and for simplicity we only consider the case of ρ l 0,ρ. Values of ρ r 0,λ 1,r s ρ l can only be reached by a single 1-rarefaction or one-shock wave. Bigger values of ρ r cannot be connected to ρ l by a single shock or rarefaction wave Lems..6 and.9. The only way to connect ρ l,m l toρ r,m r is therefore by a acteristic 1-shock wave ρ l,m l s ˆρ := λ 1,r s ρ l, ˆm with speed s followed by an one-rarefaction wave ˆρ, ˆm rare ρ r,m r. For this construction s = λ 1 ˆρ, ˆm holds which follows from the definition of λ 1,r s. This defines also uniquely the intermediate momentum ˆm by the Rankine-Hugoniot conditions. The location of the set M 1 diss shows that the acteristic shock wave is entropy dissipative and thus is satisfied for the whole attached curve. The corresponding sets A 1 and A are shown in Figures 6 and 7. Remark.11. With the consecutive use of 0, 1 and 43 there is a parameterization of the k-attached waves, k {1, }, in theρ, v-plane and due to the monotonicity properties of the parameterizations of the k-shock and k-rarefaction curves see Rems.. and.8 it holds that v r = v r ρ r is monotone decreasing for a 1-attached wave and monotone increasing for a -attached wave..4. Kinetic relation and subsonic phase transitions Up to now we have clarified the possible wave structure in one phase. To take into account phase transition let us note that the solution of the Riemann problem containing undercompressive shock waves which turns out to be necessary is in contrast to the classical hyperbolic case not unique any more [3]. One remedy of this problem is to admit only those undercompressive shock waves that satisfy an additional single algebraic constraint, the so-called kinetic relation. Note that undercompressive shock waves are automatically phase transitions in our case. The kinetic relation was first proposed by [1] see also [39,40] and motivated by experiments in the case of nonlinear elasticity. For the system 1 mathematically a similar situation occurs [6, 1] so that it suggests itself to carry over this concept. In the following we work with a specific kinetic relation. Definition.1 Kinetic relation. Let ρ kr ρ M vapor,ρ 4. The function Ψ : λ 1,r s ρ kr,ρ ellipt min ρellipt max,λ 1,l s ρ ρ,ρ ellipt min ρellipt max,ρ kr givenby Ψρ := ρ M liquid + ρ + ρ kr ρ M liquid λ 1,r s ρ kr ρ M vapor ρm vapor ρ ρ M liquid λ1,l s ρ ρ ρ M vapor : λ 1,r s ρ kr ρ<ρ M vapor, p l=r ρ : ρ M vapor ρ<ρ ellipt min, p l=r ρ : ρ ellipt max <ρ ρ M liquid, ρ λ 1,l s ρ : ρ M liquid <ρ λ1,l s ρ is called kinetic relation. A phase transition ρ l,m l pt ρ r,m r is called Ψ-admissible if and only if ρ l =Ψρ r holds. 47

19 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1107 As can be checked easily using ρ 4 >ρ M liquid and ρ 3 <ρ M vapor see Rem..3 the kinetic relation from Definition.1 satisfies Ψρ r < 0forρ r λ 1,r s ρ r ρ kr,ρ M vapor ρ M liquid,λ 1,l s ρ. 48 In fact 48 is the crucial property to construct solutions for the Riemann problem see Lem. 3.1 and we can formulate similar existence theorems for more general kinetic relations satisfying 48. Remark.13. i Up to our knowledge explicit formulae for kinetic relations in the case of liquid vapor have not been suggested. In this sense Definition.1 appears to be quite arbitrary. However, note that the Maxwell states ρ M vapor and ρm liquid are connected by the kinetic relation so that the equilibrium configuration connecting the Maxwell states is Ψ-admissible. Then we use Laxian shocks whenever possible which implies that the kinetic relation is chosen to end in the acteristic curves and is not extended further. As the most simple choice we then take a linear function. ii Mathematically one has to study whether the solution of the Riemann problem depends continuously on the choice of the kinetic function. Our conjecture is that the qualitative structure of the solution of the Riemann problem does not depend sensitively on the kinetic function see [7], Sect for Riemann solutions to different kinetic relations. A general theory on continuous dependence with respect to the kinetic relation can be found in [9]. It is out of the scope of this work to apply the theory in [9] to our case but it is likely that continuous dependence can be proven rigorously. iii A Ψ-admissible phase transition ρ l,m l pt ρ r,m r is entropy-dissipative: it is clear that it satisfies the inequality if one of the density states is given by the function p l=r since these jumps have zero entropy dissipation. In the other cases it follows from the location of the curve Γ with respect to the connecting states see Fig Generalized wave curves and solution of the Riemann problem 3.1. Generalized 1-wave curve Before we can state the main theorem the construction of a generalized 1-wave curve we need to define an additional function g which we will use in its proof. With the help of this function we make sure that the speeds of the elementary waves within the generalized 1-wave curve are such that they do not interact. Lemma 3.1. There exists a function g :0,ρ ellipt min ρm liquid,λ1,l s ρ, such that for ρ l 0,ρ ellipt min p g ρ l 1 p Ψg ρ l 1 gρ l 1 Ψgρ l 1 = p ρ 1 l p Ψg ρ l 1 ρ 1 l Ψgρ l 1 Proof. Let us first define the functions h 1,h,l :[1/λ 1,l s ρ, 1/ρ M liquid ] R as h 1 τ := pτ p Ψτ τ Ψτ and h,l τ := pτ l p Ψτ τ l Ψτ, where Ψτ :=1/Ψ1/τ. Then, it holds h 1 1/λ 1,l s ρ = p 1/λ 1,l s ρ p 1/Ψ λ1,l s ρ = p 1/λ 1,l s ρ p 1/ρ 1/λ 1,l s ρ 1/Ψ λ 1,l s ρ 1/λ 1,l = p τ τ, s ρ 1/ρ

20 1108 C. MERKLE AND C. ROHDE λ 1,l s ρ λ 1,l s ρ l 1. 1 λ,l s ρ r kinetic relation ρ r λ 1,r s ρ kr ρ λ 1,r s ρ l ρ kr ρ l Figure 8. The graph of the kinetic relation bold solid line is displayed as given in Definition.1 is displayed. Furthermore the entropy dissipation curve Γ dashed line, and the acteristic curves solid lines are shown which impose some restrictions on the choice of the kinetic relation as discussed in Remark.13. due to the definition of λ 1,l s and p1/ρ M h 1 1/ρ M liquid = liquid 1/Ψρ p M liquid 1/ρ M liquid 1/ΨρM liquid = p1/ρm liquid p1/ρm vapor 1/ρ M liquid =0. 1/ρM vapor A short calculation using 4 cf. Assumption 1.1 and the monotonicity of Ψ 48 shows On the other hand for all ρ l 0,ρ ellipt min wehave dh 1 τ > 0. dτ h,l 1/λ 1,l s ρ = p1/ρ l p 1/Ψλ 1,l s ρ = p1/ρ l p1/ρ 1/ρ l 1/Ψ λ 1,l s ρ 1/ρ l 1/ρ > p τ τ, p1/ρ l p 1/Ψρ M h,l 1/ρ M liquid liquid = 1/ρ l 1/Ψρ M liquid = p1/ρ l p1/ρ M vapor 1/ρ l 1/ρ M < 0, vapor and with the same arguments as above dh,l τ < 0. dτ

21 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1109 Therefore, there is exactly one ˆτ l 1/λ 1,l s ρ, 1/ρ M liquid such that h 1ˆτ l =h,l ˆτ l. Now the statement follows from gρ l :=ˆτ 1 l ρ M liquid,λ1,l s ρ. Now we proceed to the main theorem recalling that we denote by a classical k-wave a k-shock/rarefaction/attached wave. Theorem 3. Generalized 1-wave curve. Let v l R and ρ l A. Case 1. ρ l 0,ρ ]: For ρ r there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by a 0,ρ ellipt min classical 1-wave. For ρ r ρ ellipt max,gρ l ] : If Ψρ r λ 1,r s ρ l,ρ ellipt min there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by an 1-attached wave ρ l,v l att Ψ ρ r,v for some v R followed by a phase transition Ψ ρ r,v pt ρ r,v r. If Ψρ r ρ,λ 1,r s ρ l : there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r s by a Laxian 1-shock wave ρ l,v l Ψ ρ r,v for some v R followed by a phase transition Ψ ρ r,v pt ρ r,v r. For ρ r g ρ l, 1/b there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by a Laxian 1-shock wave. Case. ρ l ρ,ρ ellipt min : For ρ r 0,ρ ellipt min there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by a classical 1-wave. For ρ r ρ ellipt max,gρ l ] : If Ψρ r ρ l,ρ ellipt min there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by an 1-rarefaction wave ρ l,v l rare Ψ ρ r,v for some v R followed by a phase transition Ψ ρ r,v pt ρ r,v r. If Ψρ r ρ,ρ l there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r s by a Laxian 1-shock wave ρ l,v l Ψ ρ r,v for some v R followed by a phase transition Ψ ρ r,v pt ρ r,v r. For ρ r g ρ l, 1/b there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by a Laxian 1-shock wave. Case 3. ρ l ρ ellipt max, 1/b : For ρ r 0,λ 1,r s ρ kr : If ρ l ρ kr, 1/b there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by an 1-rarefaction wave ρ l,v l rare ρ kr,v kr for some v kr R followed by a phase transition ρ kr,v kr pt λ 1,r s ρ kr,v for some v R followed by an attached 1-rarefaction wave λ 1,r s ρ kr,v rare ρ r,v r. If ρ l ρ ellipt max,ρ kr there exists a unique vr R, such that ρ l,v l can be connected to ρ r,v r by a Laxian 1-shock wave ρ l,v l s ρ kr,v kr followed by a phase transition ρ kr,v kr pt λ 1,r s ρ kr,v for some v R followed by an attached 1-rarefaction wave λ 1,r s ρ kr,v rare ρ r,v r.

22 1110 C. MERKLE AND C. ROHDE Laxian one-shock g ρ l one-wave + phase transition classical one-wave kinetic relation 0.6 one-wave + phase transition one-wave + classical phase transition+ one-wave one-rarefaction ρ l ρ r Figure 9. Structure of the generalized 1-wave curve cf. Def. 3.. Combinations of states ρ l and ρ r can be connected by a combination of rarefaction-waves, Laxian shock-waves and undercompressive shock-waves satisfying the kinetic relation. For ρ r λ 1,r s ρ kr,ρ ellipt min : If ρ l Ψ ρ r, 1/b there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by an 1-rarefaction wave ρ l,v l rare Ψ ρ r,v for some v R followed by a phase transition Ψ ρ r,v pt ρ r,v r. If ρ l ρ ellipt max, Ψρ r there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r s by a Laxian 1-shock wave ρ l,v l Ψ ρ r,v for some v R followed by a phase transition Ψ ρ r,v pt ρ r,v r. If ρ r ρ ellipt max, 1/b there exists a unique v r R, such that ρ l,v l can be connected to ρ r,v r by a classical 1-wave. All shock waves are entropy-dissipative. Proof. The existence and uniqueness and thermodynamical admissibility of the constructions above follows from Lemmas.6,.9,.10, the definition of the kinetic relation Def..1 and Remark.13. For the well-posedness of the constructions above we have to check that in each case the elementary waves do not interact: Case 1. ρ l 0,ρ ]: For ρ r 0,ρ ellipt min the solution consists of a single classical elementary wave cf. Lems..6,.9 and.10.

23 THE SHARP-INTERFACE APPROACH FOR FLUIDS WITH PHASE CHANGE 1111 For ρ r ρ ellipt max,gρ l and Ψρ r λ 1,r s ρ l,ρ ellipt min : The rightmost speed of the 1-attached wave ρ l,v l att Ψρ r,vforsomev R cf. Lem..10 is smaller than the speed of the succeeding phase transition Ψρ r,v pt ρ r,v r because of 30. So this construction is always Ψ-admissible. If ρ r ρ ellipt max,gρ l and Ψρ r ρ,λ 1,r s ρ l : The speed s of the preceding 1-shock wave ρ l,v l Ψρ r,vforsomev R has to be smaller than the speed s ph of the following phase transition Ψρ r,v pt ρ r,v r : s ρl v 1 p Ψ ρ r p ρ l } Ψρ r Ψρ r ρ l {{ s } ρr < v 1 p ρ r p Ψ ρ r } Ψρ r ρ r Ψρ r {{ s ph }. Using the Lagrangian pressure p this is equivalent to pτ l pˆτ τ l ˆτ > pτ r pˆτ, τ r ˆτ where ˆτ =1/Ψρ r. Because of Lemma 3.1 this is true for ρ r <gρ l. For ρ r = gρ l : s = s ph and ρ r,v r s ρ l,v l can be connected by a single Laxian shock with speed s. Finally, for all ρ r >gρ l the left and right hand state are connected by a single Laxian shock wave ρ r,v r s ρ l,v l. Case is checked similarly. Case 3. ρ l ρ ellipt max, 1/b : For simplicity suppose ρ r 0,λ 1,r s ρ kr. The left-hand state ρ l,v l is connected to ρ kr,v kr forsomev kr by a classical 1-wave either a shock or a rarefaction wave. Then, ρ kr,v kr pt λ 1,r s ρ kr,v are joined by a Ψ-admissible phase transition. Finally, λ 1,r s ρ kr,v rare ρ r,v r is connected to each other by a rarefaction wave. Again we have to check the speeds of the waves: Suppose the leftmost wave is a rarefaction wave then the fastest speed of propagation is λ 1 ρ kr,v kr which is less than the speed of the undercompressive shock wave due to 30. If the leftmost wave is a classical shock wave then this property can be calculated with the help of 19. The rightmost wave always is a rarefaction wave. The smallest speed of propagation is λ 1 λ 1,r s ρ kr,v which equals the speed of the preceding undercompressive wave due to the definition of λ 1,r s. With Theorem 3. there is defined the generalized 1-wave curve, i.e., the set consisting of all the right handstates ρ r,v r A R that can be connected to a left hand state ρ l,v l A R by a combination of 1-rarefaction waves, Laxian 1-shock waves, 1-attached waves, and undercompressive shock waves satisfying the kinetic relation 47 cf. Fig. 9. In Figure 10a a the generalized 1-wave curve is depicted for a specific choice of ρ l,v l. From the construction of the generalized 1-wave curve in Theorem 3. it is clear that the righthand velocity state can be computed uniquely and continuously from the given state ρ l,v l and the righthand density state ρ r 0,ρ ellipt min ρellipt max, 1/b. Let us therefore define the mapping { W ρ l,v l 0,ρ ellipt 1 : min ρ ellipt max, 1/b R ρ v r ρ, 49 where v r = v r ρ is this righthand velocity state.