FRONT TRACKING FOR A MODEL OF IMMISCIBLE GAS FLOW WITH LARGE DATA

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1 FONT TACKING FO A MODEL OF IMMISCIBLE GAS FLOW WITH LAGE DATA HELGE HOLDEN, NILS HENIK ISEBO, AND HILDE SANDE Abstract. In this aer we stdy front tracking for a model of one dimensional, immiscible flow of several isentroic gases, each governed by a gammalaw. The model consists of the -system with variable gamma reresenting the different gases. The main reslt is the convergence of a front tracking algorithm to a weak soltion, thereby giving eistence as well. This convergence holds for general initial data with a total variation satisfying a secific bond. The reslt is illstrated by nmerical eamles. 1. Introdction We want to describe the one dimensional, immiscible flow for several isentroic gases. The different gases are initially searated, and the ressre is for all gases given by a γ-law, that is, = ρ γ, where ρ is the density and γ is the adiabatic gas constant for each gas. We assme γ(, t) > 1. In Lagrangian coordinates γ only deends on becase the different gases cannot mi. Ths, the flow of these gases is described for and t (0, ) by the system (1.1) v t = 0, t + (v, γ) = 0, γ t = 0, where v = 1/ρ is the secific volme, is the velocity, and (v, γ) = v γ is the ressre fnction. This 3 3 system of hyerbolic conservation laws is strictly hyerbolic for v <. We consider the Cachy roblem for this system, that is, system (1.1) with general initial data (1.) v(, 0) = v 0 (), (, 0) = 0 (), γ(, 0) = γ 0 (),. Glimm [13] roved global eistence of a weak soltion of the Cachy roblem with initial data of small total variation for strictly hyerbolic systems where each family is either geninely nonlinear or linearly degenerate, ths inclding the resent system. This soltion is fond as a limit of the Glimm scheme [13] or of the front tracking method [15, 4]. In [16] we etended the eistence reslt to large initial data for (1.1) by sing the Glimm scheme. In this aer we rove that a front tracking algorithm converges to a weak soltion, thereby giving an alternative eistence argment. System (1.1) is an etension of the system (1.3) v t = 0, t + (v) = 0, which describes the flow of one isentroic gas. The arameter γ is constant, and the ressre, still given by a γ-law, is a fnction of v only. For the -system with γ = 1, Date: November 5, Mathematics Sbject Classification. Primary: 35L65, 76N15; Secondary: 35A05. Key words and hrases. -system, gamma law, mitre of gases. 1

2 HOLDEN, ISEBO, AND SANDE Nishida [0] showed eistence of a global weak soltion for arbitrary bonded initial data. For γ > 1, Nishida and Smoller [1] roved eistence of a weak soltion for initial data where (γ 1) times the total variation of the initial data is sfficiently small. The case with large initial data for systems is also discssed in [5, 9]. The system (1.1) does not have a coordinate system of iemann invariants, only a -iemann coordinate. Therefore we do not have the advantage of changing variables to iemann invariants as for the -system and other systems. Li [17] roved eistence of a soltion for the fll Eler system with large initial data, another 3 3 system withot a coordinate system of iemann invariants. Li s change of variables is insired by the se of iemann invariants, bt a similar aroach does not simlify system (1.1) becase γ is a fnction of. The general reslts by Temle [5] inclde both the reslts of [1] and [17]. In [5] one considers the fl fnction as a smooth one-arameter family of fnctions where one has eistence of a soltion for initial data in B.V. when ɛ = 0. Then the system with 0 ɛ 1 has a niqe soltion if ɛ times the total variation of the initial data is sfficiently small. Letting ɛ = γ 1 for the -system and the Eler eqations, one obtains similar reslts as in [1] and [17]. However, this aroach cannot be sed for system (1.1) since γ is one of the variables. Wissman roved in [9] a large data eistence theorem for the 3 3 system of relativistic Eler eqations in the ltra-relativistic limit. Alying a change of coordinates the shock waves become translation invariant and a Nishida-tye of analysis is sed. For 3 3 systems with a -iemann coordinate, Temle and Yong [6] showed eistence of a soltion for initial data with arbitrary large total variation, rovided that the oscillations are small. This reslt alies to (1.1) as well, bt we want to avoid this restriction on the oscillations. Peng [3, ] also considered certain 3 3 systems (Lagrangian gas dynamics for a erfect gas and a model originating in mltihase flow modeling) with large initial data. All these eistence reslts are roved sing the Glimm scheme. Asakra shows the convergence of front tracking for the -system [3] and for the Eler eqations [] with large initial data. The conditions on the initial data are the same as obtained in [1] and [17]. In [7, 8] front tracking is sed to stdy systems of conservation laws whose fl fnctions deend on a arameter vector, µ, similar to those in [5]. An aroach for establishing L 1 -estimate ointwise in time between entroy soltions for µ 0 and µ = 0 is given. In articlar, letting µ = γ 1, the L 1 -estimate between entroy soltions in the large for the isentroic Eler eqations and the isothermal Eler eqations is established in [7] and between entroy soltions in the large for the the Eler eqations and the isothermal Eler eqations in [8]. Amadori and Corli [1] etend the -system with an etra eqation, λ t = 0, to model mltihase flow, and se front tracking to rove eistence of a weak soltion for large data. As for system (1.1), the ressre fnction in [1] is a fnction of both v and the new variable, λ, making the two systems similar. However, since the adiabatic gas constant, γ, is eqal to one in [1], vacm can never occr for their system as it can for system (1.1). Frthermore, the wave crves in [1] are monotone in λ, reslting in a considerably simler analysis of the wave interactions comared with the analysis necessary for the model considered here. The system treated in [1] is a simlified version of the model discssed by Fan in [1]. Similar models, bt with a rather different ressre law, are also considered in [11] and [19] alying comletely different methods. A model in the contet of the Navier Stokes eqation with finitely many indeendent ressre laws has been stdied in [6].

3 FONT TACKING FO A MIXTUE OF GASES 3 System (1.1) can also be rewritten as a system with discontinos fl. We get v t = 0, t + (v, γ()) = 0, where the adiabatic gas constant of the different gases is given by the discontinos fnction γ(). This rest of this aer is organized as follows: In Section we discss the wave crves of the system. The variable γ is constant along the rarefaction and shock waves of the first and third family, therefore these crves are similar to the wave crves of the -system. However, these crves are not monotone in γ, which considerably comlicates the interactions of waves with different vales of γ. The second family is linearly degenerate and gives rise to a contact discontinity along which and are constant. Ths, by changing variables to, and γ, the iemann roblem is easy to describe. The invariant region for the iemann roblem incldes vacm. This is a roblem since the interaction estimates are not valid when tends to zero, see [18]. Section 3 is the main art of this aer where we first resent the front-tracking algorithm. The soltion of any iemann roblem is made iecewise constant by aroimating rarefaction waves as ste fnctions. In addition, a simlified iemann solver generating non-hysical fronts is introdced in order to ensre that the nmber of fronts remains finite. The simlified solver is only sed for interactions where one or more fronts of the same family collide with a contact discontinity and the sm of the strengths of the incoming fronts times the strength of the contact discontinity is less than some threshold arameter. This solver generates nonhysical fronts, traveling either to the left or the right, with absolte seed larger than any other front. Moreover, when these non-hysical fronts collide with other fronts, they jst ass throgh withot changing strength. In Section 3. we define a Glimm fnctional and by considering all ossible interactions we rove that it is decreasing nder the conditions given in Proosition 3.7. We se this to show that there is a finite nmber of interactions to any given time, hence, and ths the front-tracking algorithm is well-defined. Frthermore, we introdce a generation concet in order to bond the total amont of non-hysical fronts resent at any time. The aroimate soltion fond sing front tracking has bonded total variation and is bonded away from vacm whenever the conditions on the initial data given in Lemma 3.18 and Lemma 3.19 are satisfied. We end Section 3 by roving that the seqence of aroimate soltions converges to a weak soltion of the system. This roves the main theorem: Theorem 3.0. Assme that (s(γ(, 0)) 1)T.V.((, 0), (, 0)) and T.V.(γ(, 0)) are sfficiently small. The the front tracking algorithm is well-defined and gives a seqence which converges to a weak soltion of (1.1). Observe that by redcing the total variation of γ and redcing its sremm, one can allow for arbitrary large total variation of and. De to Wagner [7], this reslt translates into eistence for the system (3.58) in Elerian coordinates. In the last section we stdy some eamles nmerically. In the first eamle we have one gas confined to an interval, srronded by another gas. The two gases have distinct bt constant gammas. The constants that limit the total variation of the initial data are comted, and the initial data are chosen so that they satisfy the conditions in the theorem. The Glimm fnctional is elicitly comted, and we observe decay in accordance with the theorem. In the second eamle the initial data are iecewise constant, while γ is continosly varying in the third eamle.

4 4 HOLDEN, ISEBO, AND SANDE For these two eamles, the total variation of the chosen initial data do not satisfy the theorem, nevertheless we still observe that the Glimm fnctional is decaying.. The system It is well-known that systems of hyerbolic conservation laws sch as (1.1) do not in general have smooth soltions, even for smooth initial data. Ths, by a soltion of (1.1) with the initial data (1.) we mean a weak soltion in the distribtional sense with (v,, γ) L 1 loc ( [0, )) so that (vφ t φ ) ddt + v 0 ()φ(, 0) d = 0, (.1) [0, ) [0, ) (φ t + φ ) ddt + [0, ) γφ t ddt + 0 ()φ(, 0) d = 0, γ 0 ()φ(, 0) d = 0, for all test fnction φ C 0 ( [0, )). If the secific volme, v, becomes infinite, which corresonds to zero density and zero ressre, we have vacm. At vacm, the roerties of the system change and the methods sed here do not aly, therefore we only consider system (1.1) for v(, t) <. Frthermore, we assme γ(, t) > 1. We write U(, t) = (v(, t), (, t), γ(, t)). Often we will work with instead of v, and then also write U(, t) = ((, t), (, t), γ(, t)). For v <, or eqivalently, > 0, system (1.1) is strictly hyerbolic with eigenvales (.) λ 1 = λ, λ = 0, λ 3 = λ, where λ := v = γv γ 1, and corresonding eigenvectors (.3) r 1 = (1, λ, 0), r = ( γ, 0, v ), r 3 = ( 1, λ, 0). Note that the eigenvales and eigenvectors do not deend on. The first and the third family are geninely nonlinear, while the second family is linearly degenerate. Moreover, the system does not ossess a coordinate system of iemann invariants, bt γ is a iemann coordinate for the second family. Before we trn to solving system (1.1) with general initial data, we need to solve the iemann roblem for (1.1), that is, when the initial data consists of two constant states searated by a jm, cf. (.1). The soltion of the iemann roblem consists of to three elementary waves, one from each family, and to two intermediate constant states searating these waves. Ths, we start by looking at the wave crves..1. Wave crves. For the geninely nonlinear families there are two tyes of waves; rarefaction waves which are continos waves of the form U(, t) = w(/t) satisfying (.4) ẇ(/t) = r j (w(/t)), λ j (w(/t)) = /t, j = 1, 3, where λ j is increasing along the wave, and shock waves which are soltions { U l, if < σ j t, (.5) U(, t) = U r, if > σ j t, satisfying the ankine Hgoniot condition (.6) σ j (U r U l ) = f(u r ) f(u l ), j = 1, 3,

5 FONT TACKING FO A MIXTUE OF GASES 5 for a shock velocity σ j. The admissible shock waves are those satisfying the La entroy conditions (.7) λ j 1 (U l ) < σ < λ j (U l ), λ j (U r ) < σ < λ j+1 (U r ), j = 1, 3. For the linearly degenerate family j = there is only one tye of waves called contact discontinities. These waves are soltions of the form (.5) which satisfy the ankine Hgoniot condition (.6) with σ = λ. Fi a left state U l. For each family the wave crve consists of all states U that can be connected to the given left state by a wave of this family. The rarefaction soltion is of the form U l, if < λ j (U l )t, (.8) U(, t) = w(/t), if λ j (U l )t < < λ j (U)t, U, if > λ j (U)t. The rarefaction wave crve is the set of all right states U that can be connected to the left state by a rarefaction wave. For system (1.1) these are ( 1 (v, U l ) := v, l ( ) ) γ l v 1 γ l v 1 γ l l, γ l, v > v l, γ l 1 ( 3 (v, U l ) := v, l + ( ) ) γ l v 1 γ l v 1 γ l l, γ l, v < v l. γ l 1 The shock crves of all right states which can be connected to U l by an admissible shock wave are ( S 1 (v, U l ) : = v, l ( (v l v)(v γ l v γ l l ) ) ) 1/, γl, v < v l, ( S 3 (v, U l ) : = v, l ( (v l v)(v γ l v γ l l ) ) ) 1/, γl,, v > v l, with the shock velocities v γ l l v σ 1 (U l, U) = γ l l (.9) =, v v l 1/γ 1/γ l l l v σ 3 (U l, U) = γ l v γ l l l (.10) =, v l v 1/γ l 1/γ l l resectively. Note that the shock velocities do not deend on. The crve of all right states that can be connected to U l by a contact discontinity is ( ) C (γ, U l ) : =, l, γ, γ > 1, v γ l/γ l with the velocity σ = λ = 0. Note that γ only changes along the contact discontinities. Frthermore, both and = v γ are constant along a contact discontinity, and we therefore choose to work with, and γ. A shock or a rarefaction crve throgh U l lies in the lane γ = γ l and is eqal to the corresonding wave crve for the -system (1.3) with γ = γ l. We roceed by defining the wave crves sing,, and γ, as deicted in Figre 1, { (, l r(, l, γ l ), γ l ), < l, (.11) Φ 1 (, U l ) := (, l s(, l, γ l ), γ l ), > l, (.1) (.13) Φ (γ, U l ) := ( l, l, γ), γ > 1, { (, l + r(, l, γ l ), γ l ), > l, Φ 3 (, U l ) := (, l s(, l, γ l ), γ l ), < l,

6 6 HOLDEN, ISEBO, AND SANDE where (.14) (.15) r(, l, γ l ) := γ l γ l 1 s(, l, γ l ) := ( γ l 1 γ l γ l 1 γ l l ), (( 1 γ l l 1 γ l ) ( l )) 1/. ecall that if = 0, we have vacm, therefore, the wave crves are only well- 3 γ Figre 1. The wave crves throgh two left states with different γ. defined for > 0 and l > 0. All reslts are for waves contained in (.16) D = {(,, γ) [ min, ma ], <, γ (1, γ]}, where min > 0, ma < and γ (1, ) are constants. For initial data given by (1.) we will later establish the er and lower bond on and show that (.17) γ := s(γ 0 ()), for all waves. We moreover have an er bond on the wave seed for all waves (or fronts) contained in D, and we define (.18) λ ma = ma U D {λ i, σ i } = ma U D {λ i}, where the last eqality is de to the La entroy condition (.7). Before we discss some imortant roerties of the wave crves, we mention the backward wave crves. These are the crves of all left states U that can be connected to a given right state U r by a wave of the given family. We denote these wave crves by Φ i. The backward 3-wave crve will be sed several times and this is given by { (, r r( r,, γ r ), γ r ), < r, (.19) Φ 3 (, U r ) := (, r + s( r,, γ r ), γ r ), > r, where r and s are given by (.14) and (.15). We now trn to the roerties of the wave crves. Lemma.1. The wave crves in D have the following roerties: (i) The fnction Φ 1 is strictly decreasing and the fnction Φ 3 is strictly increasing when considered as fnctions of. (ii) Given two wave crves, Φ j (, U 1 ) and Φ j (, U ) where j {1, 3}, so that U 1 is not on Φ j (, U ) and U is not on Φ j (, U 1 ). Then the two wave crves never intersect.

7 FONT TACKING FO A MIXTUE OF GASES 7 (iii) Consider the rojections onto the (, )-lane of the wave crves throgh U 1 = ( l, l, γ 1 ) and U 1 = ( l, l, γ ) where γ 1 γ. If r( l, l, γ 1 ) < r( l, l, γ ), then the rojected wave crves going to the right (with resect to ) will never intersect, while the rojected wave crves going to the left will intersect as decreases. If r( l, l, γ 1 ) > r( l, l, γ ), then the rojected wave crves going to the right will intersect, while the rojected wave crves going to the left will not. If r( l, l, γ 1 ) = r( l, l, γ ), then none of the rojected wave crves will intersect. (iv) The sloe of a rarefaction wave in the lane γ = γ l, r/, only deends on and γ l, not on l. Frthermore, there eist two constants r min and r ma only deending on min, ma and γ so that r min r(, l, γ l ) r ma. (v) The sloe of a shock wave in the lane γ = γ l, s/, deends on, γ l and l. Frthermore, there eist two constants s min and s ma only deending on min, ma and γ so that s min s(, l, γ l ) s ma. (vi) The wave crves have a continos derivative at U l, lim l s(, l, γ l ) = r( l, l, γ l ). Frthermore, s(, l, γ l ) r(, l, γ l ), for all l. Hence, a shock wave is always steeer than a rarefaction wave at a given l rovided both waves lie in the lane γ = γ l. (vii) arefaction waves are additive; if a rarefaction wave connects U 1 to U and another rarefaction wave of the same family connects U to U 3, then the rarefaction wave connecting U 1 to U 3 eqals the concatenation of the other two rarefaction waves. (viii) Given two 1-shock waves starting at ( 1,, γ) and (,, γ), resectively, and assme 1 <. Then the shock wave starting at 1 is steeer than the shock wave starting at at any given oint, that is, s(,, γ) < s(, 1, γ), for all > 1. (i) Given two 3-shock waves starting at ( 1,, γ) and (,, γ), resectively, and assme 1 <. Then the shock wave starting at is steeer than the shock wave starting at 1 at any given oint, that is, s(, 1, γ) < s(,, γ), for all 1 <.

8 8 HOLDEN, ISEBO, AND SANDE Proof. All the roerties follows from differentiating the wave crves. γ 1 γ γ 1 γ ( l, l) ( l, l) (a) Becase r( l, l, γ l ) > r( l, l, γ ), the rojected wave crves going to the right intersect. (b) Becase r( l, l, γ l ) < r( l, l, γ ), the rojected wave crves going to the left intersect. Figre. The wave crves throgh U 1 = ( l, l, γ 1 ) (dotted line) and U = ( l, l, γ ), where γ 1 < γ, rojected onto the (, )- lane. The rojection onto the (, )-lane of two wave crves with different γ s are shown in Figre. Note that the rojected wave crves intersect, cf. roerty (iii), becase the sloes of the rojected wave crves deend on γ. The net lemma gives an estimate on how different two waves with different γ s are. Lemma.. Let ɛ 1 and ɛ be 1-waves of the same tye sch that ɛ 1 connects ( 0, 0, γ 1 ) to (, 1, γ 1 ) and ɛ connects ( 0, 0, γ ) to (,, γ ), or let η 1 and η be 3-waves of the same tye sch that η 1 connects (, 1, γ 1 ) to ( 0, 0, γ 1 ) and η connects (,, γ ) to ( 0, 0, γ ). Assme that all waves are contained in D and frthermore that 1 <. Then (.0) 1 c 0 γ γ 1, where c only deends on min, ma and γ. Note that for 1-waves we comare two waves where the rojected waves start at the same oint in the (, )-lane, while we for 3-waves comare two waves where the rojected waves end at the same oint. The roof of this lemma is given in [16] and is based on the techniqes sed in [8]... The iemann Problem. We have the following fndamental definition. Definition.3. The iemann roblem for (1.1) is the Cachy roblem with initial data { (.1) U(, 0) = U l, if < 0, U r, if > 0, where U = (v,, γ) and U l, U r are constants. Lemma.4. The iemann roblem for (1.1) where U l and U r are contained in D, cf. (.16), has a niqe soltion withot vacm if (.) r l < r( r, 0, γ r ) r(0, l, γ l ).

9 FONT TACKING FO A MIXTUE OF GASES 9 Proof. Note that if γ l = γ r, then the iemann roblem for (1.1) redces to the iemann roblem for the -system (1.3). The soltion of this roblem is described in detail in [4, Ch. 17, A], and it is niqe if (.) is satisfied with γ l = γ r. A -wave takes s from one lane, γ = γ 1, to another lane, γ = γ, while and remain constant. Therefore, the iemann roblem has a niqe soltion if the rojections onto the (, )-lane of the 1-wave crve, Φ 1 (, U l ), and the backward 3-wave crve, Φ3 (, U r ), have a niqe intersection oint. From roerty (i) of Lemma.1 we have that the rojection of Φ 1 is strictly decreasing in and it follows that the rojection of Φ 3 is strictly increasing in. Hence, the rojected crves intersect at most once. The only case where the two crves do not intersect is if the rojection of the backward 3-rarefaction wave from U r always lies above the rojection of the 1-rarefaction wave from U l. Ths, if r r( r, 0, γ r ) < l r(0, l, γ l ), then the rojections of Φ 3 (, U r ) and Φ 1 (, U l ) onto the (, )-lane have a niqe intersection oint, and the iemann roblem has a niqe soltion. The soltion of the iemann roblem (U l, U r ) is constrcted as follows: Let (, ũ) be the niqe intersection between the rojections of Φ 1 (, U l ) and Φ 3 (, U r ) onto the (, )-lane. We connect U l = ( l, l, γ l ) to Ũ1 = (, ũ, γ l ) by a 1-crve, then we go from Ũ1 to Ũ = (, ũ, γ r ) along a contact discontinity, and finally connect Ũ to U r = ( r, r, γ r ) by a 3-wave..3. Invariant region and vacm. A region Ω is invariant for the iemann roblem if for any iemann roblem with initial data in Ω, its soltion is also in Ω. For the -system we know from [14, E. 3.5] that the conve region in the (v, )-lane between the integral crves of the eigenvectors is invariant. This region bonds v from below, bt not from above, ths vacm is inclded in the invariant region. In the (, )-lane this corresonds to the region bonded by = 0 and the two integral crves. Since γ cannot take any other vales than those of the initial data, we find the invariant region for the -system for each γ and take the nion of these. This gives s an invariant region for (1.1). Moreover, this gives s the er bond on, ma, which we need, bt is still not bonded away from vacm. 3. The Cachy roblem We now trn to the Cachy roblem and se front tracking to obtain a seqence of aroimate soltions. The goal of this section is to show that a sbseqence converges to a weak soltion of (1.1). In order to do this, we find a sitable Glimm fnctional and show that it decreases in time. This reqires detailed analysis of all ossible interactions and most of this section is devoted to this. First of all we need some notation. We let ɛ define a 1-wave, α a 1-shock wave, µ a 1-rarefaction wave, η a 3-wave, β a 3-shock wave, ν a 3-rarefaction wave, ζ a -wave, θ a 1- or 3-wave, Frthermore, we define the strength of a 1-wave or a 3-wave as the jm in across the wave and the strength of a -wave as the jm in γ across the wave. The strength of a wave or a front is denoted by θ. We are now ready to discss front tracking and to define fronts. Note that we will se the above notation for fronts as well as waves. In addition, we will define non-hysical fronts which will be denoted by θ n and the strength of a non-hysical front will be defined as its jm in.

10 10 HOLDEN, ISEBO, AND SANDE 3.1. Front tracking. The first ste of front tracking is to aroimate the initial data (1.) by a iecewise constant fnction U δinit 0 so that δinit lim U0 U 0 L 1 = 0, δ init 0 where U 0 = ( 0, 0, γ 0 ) and δ init is the distance between the discontinities. Frthermore, the aroimation has to satisfy (.) at every discontinity so that all initial iemann roblems have a niqe soltion. Ths, no vacm forms at t = 0+. We then solve the iemann roblem defined by the discontinities in U δinit 0. All soltions of iemann roblems in front tracking have to be iecewise constant. Since shock waves and contact discontinities are already iecewise constants, we se an aroimate iemann solver where the continos rarefaction waves are aroimated. We relace the rarefaction wave from the left state, U l, to the right state, U r, by a ste fnction. Let k := r l /δ. Then we divide the rarefaction wave into k jms, each with strength ˆδ = θ /k δ. The discontinities move with the seed of their left state. Note that the jms in the aroimated rarefaction wave do not satisfy the ankine Hgoniot condition. It is obvios that this aroimate soltion of the iemann roblem converges to the eact soltion a.e. when δ tends to zero. Solving all iemann roblems resent initially by the aroimate solver, generates an aroimate soltion of the Cachy roblem for small t > 0. The soltion is iecewise constant and a front is one discontinity in the soltion. Hence, a shock wave or a contact discontinity is one front, while an aroimated rarefaction wave consists of k fronts where each front has strength less than or eqal to δ. Note that the two arameters δ init and δ are chosen so that δ init = O(δ). We denote the aroimate soltion U δ. We track all fronts in U δ ntil two or more fronts interact, that is, collide at a collision oint (, τ). The colliding fronts are called incoming fronts. Then we solve the iemann roblem defined by the states immediately to the left and right of the incoming fronts, and the fronts in this aroimate soltion are called otgoing fronts and are sally identifiable by a rime. We kee tracking all fronts and solving iemann roblems each time fronts collide. In order to ensre that front tracking is well-defined for all times, we follow the aroach of Bressan [4] and introdce non-hysical fronts. Ths, an interaction is either solved by the standard aroimate solver as described above, or by a simlified iemann solver. Let ρ > 0 be a fied threshold arameter. Interactions of the form ζ+ i ɛ i, or the symmetric form i η i+ζ, are solved sing the simlified iemann solver if (3.1) ζ ɛ i ρ, or ζ η i ρ, i i resectively, otherwise the aroimate iemann solver is sed. All other interactions are always solved sing the aroimate solver. The simlified iemann solver introdces non-hysical fronts which we denote θ n. By constrction, both and γ are constant across a non-hysical front and its strength eqals the jm in. In order to reserve the symmetry roerty of system (1.1), we introdce non-hysical fronts traveling both to the left and to the right. In either case they travel with the absolte seed λ n > λ ma, hence the name. Note that the ankine Hgoniot condition (.6) is not satisfied for a non-hysical front. Let s first detail the soltion of the interaction between one front and a contact discontinity sing the simlified solver. The soltion consists of two hysical fronts and a non-hysical front: ζ + ɛ ɛ + ζ + θ n.

11 FONT TACKING FO A MIXTUE OF GASES 11 The otgoing front ɛ has the same strength and tye as ɛ, and connects U l = ( l, l, γ l ) to Ũ1 = ( r, ũ, γ l ), as deicted in Figre 3(a). The contact discontinity is, as always, nchanged, connecting Ũ1 to Ũ = ( r, ũ, γ r ). The non-hysical front then connects Ũ to U r = ( r, r, γ r ). Moreover, the non-hysical front has ositive seed traveling to the right. For the symmetric case, η + ζ θ n + ζ + η, the non-hysical front has negative seed. Ũ 1 Ũ Ũ 1 Ũ Ũ 3 ǫ ζ θ n ǫ ζ η θ n U l ζ ǫ U r U l ζ U r i ǫ i U 1 (a) Simlified solver for ζ + ɛ. U 1 (b) Simlified solver for ζ + P i ɛ i. Figre 3. The simlified iemann solver with non-hysical fronts (dashed lines). The soltion we get sing the simlified solver when two or more fronts of the same family interact with a contact discontinity, consists of one hysical front of each family, in addition to a non-hysical front; ζ + i ɛ i ɛ + ζ + η + θ n, see Figre 3(b). In order to determine the otgoing fronts, we introdce two ailiary fronts, ɛ and η. These fronts are the soltion of the iemann roblem (U 1, U r ), ths, ɛ connects U 1 to the intermediate state U = (,, γ r ), and η connects U to U r. Let ɛ be the front that has the same strength and tye as ɛ, bt with γ = γ l, that is, connecting U l to Ũ1 = (, ũ, γ l ). The contact discontinity is nchanged, connecting Ũ1 to Ũ = (, ũ, γ r ). Let η be η shifted in the -direction so that η connects Ũ to Ũ3 = ( r, r +ũ, γ r ). Finally, the non-hysical front connects Ũ3 to U r. The non-hysical front has ositive seed and changes only the vale of, as it is sosed to. This constrction of the soltion sing the simlified iemann solver is insired by the formal tool of slitting an interaction into stes that we will introdce in the net section. More details on the rocess of finding the otgoing fronts sing the simlified solver are inclded in the roof of Lemma 3.11 where we obtain estimates for these interactions. Note that ɛ + ζ + η is the soltion of the iemann roblem (U l, Ũ3), ths, the ankine Hgoniot condition (.6) is satisfied for any shock or contact discontinity. However, it is not satisfied for the nonhysical front or any aroimated rarefaction wave. We resolve the symmetric interaction in a similar manner, and get a non-hysical front with negative seed; η i + ζ θ n + ɛ + ζ + η. i

12 1 HOLDEN, ISEBO, AND SANDE U l α ζ β θ n θ n α β β U l U r θ n β ζ α α θ n U r (a) in the (, t)-lane. (b) Projected onto the (, )-lane. Figre 4. The interaction θ n + β + ζ + α α + ζ + β + θ n. Whenever we have an interaction with an incoming non-hysical front, as in Figre 4, we first let the non-hysical front ass throgh with its strength nchanged. Then we solve the remaining interaction, which is slightly shifted along the -direction, sing the aroimate or simlified solver according to condition (3.1). Note that all wave crves are invariant in the (, ) lane nder a translation in. Before we trn to the discssion of all ossible interactions, we look at the error introdced sing the simlified solver instead of the aroimate solver. The lemma is given for the interactions involving 1-fronts, bt we have the same reslts for the symmetric interactions involving 3-fronts. Lemma 3.1. Consider the interaction ζ + n i ɛ i for n 1. Let n ζ + ɛ i ˆɛ + ζ + ˆη, i be the soltion, with intermediate states Ûi, i = 1,, obtained sing the aroimate solver, and let { n ɛ + ζ + θ n, if n = 1, ζ + ɛ i ɛ + ζ + η + θ n, if n > 1, i be the soltion obtained sing the simlified solver, with intermediate states Ũi, i = 1, and i = 1,, 3, resectively. Then, σˆα σ α = O(1) θ n, if ɛ = α, λˆµ λ µ = 0, if ɛ = µ, and, if ˆη is of the same tye as η for n > 1, σ ˆβ σ β = O(1) θ n, if η = β, λˆν λ ν = O(1) θ n, if η = ν. Moreover, Ûi Ũi = O(1) θ n, i = 1,, and U r Ũj = O(1) θ n where j = if n = 1 and j = 3 if n > 1. Proof. First note that and are eqal for Ũ1 and Ũ, and for Û1 and Û, and we therefore omit the indices. Figre 5 shows the soltions of ζ + ɛ for both solvers, and Figre 6 shows the soltions and the ailiary fronts for an interaction of the tye ζ + i ɛ i. The rarefaction fronts µ and ˆµ have the same left state, and they therefore have the same seed. Likewise, the left state is the same for α

13 FONT TACKING FO A MIXTUE OF GASES 13 θ n U r Ũ μ Û ˆβ U l ˆμ μ U l α α θ n Ũ U r ˆα Û ˆβ (a) ζ + µ. (b) ζ + α. Figre 5. The interaction (dashed lines) solved by the aroimate solver (dash-dotted lines) and by the simlified solver (solid lines). U l U l α 1 α α 3 α 1 α α 4 α 5 ˆα α Û α 6 α 7 ν U ˆν U r θ n α Ũ 3 Ũ ν α α 4 α 5 α ν U (a) When η and ˆη are of the same tye (b) When η and ˆη are not of the same tye. α 3 α 6 Ũ ˆα α 7 Ũ 3 U r θ n Û ˆβ Figre 6. The interaction ζ + 7 i=1 α i (dashed lines), with the ailiary crves (dotted lines), solved by the aroimate solver (dash-dotted lines) and by the simlified solver (solid lines). and ˆα. However, the seed of a shock-front deends on the vale of at the right state as well, where = for α and = ˆ for ˆα. Since this difference in is less than a constant times the jm in across the non-hysical front, that is, ˆ = O(1) θ n, we get σˆα σ α = σ 1 ( l, ˆ) σ 1 ( l, ) σ 1( l, ) ˆ = O(1) θ n, where σ 1 is the derivative with resect to the second argment and ˆ. If n > 1 and η is of the same tye as ˆη, as for the interaction deicted in Figre 6(a), then = r at the right state for both fronts. However, at the left state we have = for η and = ˆ for ˆη. This is the same difference in as

14 14 HOLDEN, ISEBO, AND SANDE above, ths, σ ˆβ σ β = σ 3 (ˆ, r ) σ 3 (, r ) σ 3(, r ) ˆ = O(1) θ n, λˆν λ ν = λ(ˆ) λ( ) λ 3( ) ˆ = O(1) θ n, where σ 3 is the derivative with resect to the first argment, λ 3 the derivative with resect to, and ˆ. Moreover, γ is eqal for the two soltions and ũ û θ n, ths, Ûi Ũi = O(1) θ n, i = 1,. Finally, let j = for n = 1 and j = 3 for n > 1. Then, j = r and ũ j r = θ n, hence, U r Ũj = O(1) θ n. In front tracking an interaction is a collision of arbitrarily many fronts at one oint in sace-time. However, in order to collide at the same oint, their seeds mst decrease from left to right. This observation has the immediate conseqence. Lemma 3.. All interactions between hysical fronts in front tracking for system (1.1) is of the general form m n (3.) η i + ζ + ɛ j, i=1 where η i is a 3-front, ζ is a contact discontinity, ɛ j is a 1-front, and two adjacent fronts cannot both be rarefaction-fronts. All interactions with incoming non-hysical fronts are of the same general form with a non-hysical front as the leftmost and/or the rightmost incoming front. Frthermore, all wave families do not need to be resent in an interaction. This is a major difference between front tracking and the Glimm scheme where at most for waves can interact. Frthermore, only the case with two interacting fronts or waves is the same in front tracking and in the Glimm scheme. Still, the following, simle symmetry roerty for system (1.1) roved in [16], is sefl also for the interactions in front tracking. Lemma 3.3. [16, Lemma 3.1] Under the transformation, a 1-wave connecting U l to U r becomes a 3-wave connecting U r to U l, and vice versa. A -wave is nchanged nder this transformation, and a non-hysical front becomes a nonhysical front traveling in the oosite direction. Frthermore, the leftmost wave with resect to will become the rightmost wave with resect to, and so on. One of or main goals is to show that the aroimate soltion can be constrcted at any time in a finite nmber of stes. Therefore we look at which interactions increase the nmber of fronts resent. Firstly, recall that the soltion of a iemann roblem consists of to three waves, one from each family. Hence, the soltion fond by the aroimate iemann solver has for or more fronts if, and only if, a rarefaction wave slits into several fronts. For an interaction between three or more fronts solved by the aroimate solver, the nmber of fronts can therefore only increase de to slitting of rarefaction waves. Frthermore, an otgoing contact discontinity is only resent if there is an incoming contact discontinity. Ths, the nmber of fronts for an interaction between two fronts, none of which are contact discontinities, can only increase de to slitting of rarefaction waves. Whenever the simlified solver is sed for an interaction between two incoming fronts, we get two otgoing hysical waves and one otgoing non-hysical front. If there are three or more incoming fronts, the simlified solver gives three otgoing hysical waves and one non-hysical front. Hence, for an interaction solved by the j=1

15 FONT TACKING FO A MIXTUE OF GASES 15 simlified solver, the nmber of hysical fronts can increase only de to slitting of rarefaction waves. Ecet for slit rarefaction waves, the nmber of fronts increases only for the interaction between a contact discontinity and one other front solved by the aroimate solver. These interactions have at least three otgoing fronts, and we refer to them as γ-collisions. Definition 3.4. A γ-collision is the interaction between a contact discontinity and a 1- or 3-front. The for different γ-collisions, where symmetry redces it to two distinct cases, are discssed discssed in the roof of Lemma 3.8 in Section 3.. If the strength of an otgoing rarefaction wave is larger than δ, it slits into several fronts. The interactions where this might haen are either a new rarefactioncollision or an increasing rarefaction-collision as defined below. Definition 3.5. A new rarefaction-collision is an interaction where there is an otgoing rarefaction wave of a family in which there are no incoming rarefactionfronts. Definition 3.6. An increasing rarefaction-collision is an interaction where the strength of an otgoing rarefaction wave is greater than the sm of the strengths of the incoming rarefaction-fronts of the same family. Note that a γ-collision can also be a new rarefaction-collision, an increasing rarefaction-collision, or even both. Smming the front tracking constrction, we have defined a iecewise constant fnction U δ, so that for all fied t, U δ (, t) is a iecewise constant fnction. Frthermore the constrction gives a seqence of collision times τ 1 < τ <..., and U δ (, t) is defined for all t lim n τ n. We shall show that either {τ n } is a finite seqence or lim n τ n =, i.e., that U δ (, t) can be constrcted for any t > The decreasing Glimm fnctional. Set t n = (τ n +τ n+1 )/, where we have defined τ 0 = 0, and define the fnctional (3.3) G(t n ) := F (t n ) + 3C 1 (γ 1)Q 1 (t n ) + 3C Q (t n ), where C 1 is the constant aearing in the estimates given by (3.16) for the interaction of Tye Bbii, cf. the roof of Lemma 3.8, c (3.4) C := min{r min, s min } = kc, where c is the constant from Lemma. and (3.5) 1 k := min{r min, s min }. Note that both C 1 and C are constants only deending on min, ma and γ. This is the same fnctional as the Glimm fnctional defined in [16], and the two first terms are similar to the Glimm fnctional sed in [1]. The linear fnctional F and the two qadratic fnctionals Q 1 and Q are defined by (3.6) F (t n ) := { θ all shock-fronts θ at t = t n }, (3.7) Q 1 (t n ) := { α β all aroaching 1- and 3-shock-fronts at t = t n }, (3.8) Q (t n ) := { ζ θ all aroaching airs of ζ and θ at t = t n }, where two fronts of different families are aroaching if the front of the lowest family is to the right of the other. Note that F and Q 1 only sm over shock-fronts,

16 16 HOLDEN, ISEBO, AND SANDE while Q also sms over rarefaction-fronts. Frthermore, none of the terms involve the strength of non-hysical fronts. We call the lines t = t n time lines. The only difference between the fnctionals above and the fnctionals sed for the Glimm scheme in [16] is that the above ones are defined on time lines, while the fnctionals in [16] are defined on mesh crves. We need two more fnctionals, one smming over all shock- and rarefactionfronts at t = t n and one smming over the contact discontinities at t = t n. Note that the sm of all contact discontinities is constant for all time lines. We define (3.9) (3.10) L(t n ) := { θ all θ at t = t n }, F γ := { ζ all ζ}. We will show that G is a decreasing fnctional in time. Let (3.11) C = min{ C, 1}, where the minimm is taken over all the constants C aearing in the estimates for interactions of Tye Ba discssed in the roof of Lemma 3.8. Note that 0 < C 1 deends only on min, ma and γ. The rest of this sbsection will be devoted to roving the following reslt: Proosition 3.7. If (3.1) 3C 1 (γ 1)L(t 0 ) C 3 and 3C F γ C 3. then G defined by (3.3) is decreasing and F (t n ) 5 3 L(t 0). In articlar, G decreases by at least 3q across an increasing rarefaction-collision where the strength of the rarefaction wave increases by q > 0, by at least 3 θ across a new rarefactioncollision where θ denotes the new otgoing rarefaction wave, and by at least 3k θ n for an interaction where a non-hysical front is generated. We rove this roosition throgh a series of lemmas where we start by considering interactions between two fronts, then gradally bild to interactions of the general form given by (3.), inclding incoming non-hysical fronts. For all ossible interactions in front tracking we show that G is decreasing and, in articlar, we identify all new or increasing rarefaction-collisions and all interactions generating a non-hysical front. Before we state and rove the different lemmas, we resent the general idea based on indction on sccessive time lines: First we show that G(t 1 ) G(t 0 ) 0. Then we assme G(t n ) G(t n 1 ) G(t 0 ). The indction ste is to show that := G(t n+1 ) G(t n ) 0. Note that if G is decreasing to t = t n, then we have F (t n ) G(t n ) G(t 0 ) = F (t 0 ) + 3C 1 (γ 1)Q 1 (t 0 ) + 3C Q (t 0 ) (3.13) F (t 0 ) + 3C 1 (γ 1)(F (t 0 )) + 3C L(t 0 )F γ (1 + 3C 1 (γ 1)F (t 0 ) + 3C F γ )L(t 0 ) (1 + 3C 1 (γ 1)L(t 0 ) + 3C F γ )L(t 0 ) ( 1 + C 3 + C ) L(t 0 ) L(t 0). We only give the estimates for here. Estimating G(t 1 ) G(t 0 ) is very similar, giving terms involving F (t 0 ) where the estimate for has terms involving F (t n ). For the more involved interactions we se a comtational trick where we divide the interaction into stes where only a art of the fronts interact at each ste. It is imortant to note that in the front tracking algorithm all fronts in an interaction meet at the same oint and that no seeds are altered. It is jst in the estimation

17 FONT TACKING FO A MIXTUE OF GASES 17 of we do this ste rocedre as a formal trick to go from the incoming fronts to a set of fronts which are comarable to the otgoing fronts. Note also that the otgoing fronts are not altered in this rocess. This method corresonds to the se of inner diamonds for the Glimm scheme in [16]. DiPerna [10] constrcts the otgoing soltion by resolving the interaction into a comosition of binary interactions. This method of decomosition is similar to or formal method of dividing an interaction into stes. Ths, we divide the interaction into l stes where only some of the fronts interact at each ste, the rest is left nchanged. As long as the interaction at one ste is an interaction already analyzed, we know that G decreases across that ste. We contine this ntil we at some oint directly can show that G is decreasing across the last ste, where the last ste is going from some collection of fronts to the otgoing fronts. Formally, the stes are obtained by shifting the seeds of the incoming fronts slightly, so that only the intended fronts meet at a shifted collision oint. This is done for each ste and we introdce intermediate time lines, t = t i, so that the interaction at the ith ste lies between t i 1 and t i where t 0 = t n and t l = t n+1. As long as we have i := G(t i ) G(t i 1 ) 0 for i = 1,..., l, it follows that 0. Note again that this ste rocedre is only a comtational trick, and that the front-tracking algorithm as sch involves no shifting of seeds. t = t n+1 t = t n+1 t = t t = t 1 (a) The original interaction. t = t n t = t n (b) The interaction divided into stes. Figre 7. A tyical interaction of the form i η i + j ɛ j. Figre 7 shows how a tyical interaction of the tye i η i + j ɛ j is divided into two additional stes. First we let all 3-fronts interact at one collision oint whereas all 1-fronts interact at a different oint. Both interactions reslt in a 1-wave and a 3-wave. At the second ste we let the aroaching 3- and 1-wave interact. Ths, at t = t we have a collection of for waves and we comare these to the otgoing fronts. Note that we have not shifted or altered the otgoing fronts at any oint in this ste rocedre. For some cases we se an additional trick to avoid getting too many stes. Instead of letting some fronts interact at a shifted collision oint, we relace the fronts with new fronts connecting the same left and right state. Since this is not a valid interaction, we need to show that i 0 for this ste, and we do that by comaring the new fronts with the relaced fronts. Still this is jst a formal trick and the otgoing fronts are not altered. In Lemma 3.8 throgh Lemma 3.14 we cover all ossible interactions, and we start by the cases with two interacting fronts. ecall that these are the same interactions as for the Glimm scheme, cf. [16], and they are labeled in the same manner as in [16].

18 18 HOLDEN, ISEBO, AND SANDE Lemma 3.8. For all interactions between two fronts we have 0. In articlar, 3q for all increasing rarefaction-collisions where the strength of the rarefaction wave has increased by q > 0 and 3 θ for new rarefactioncollisions where θ denotes the new rarefaction wave. Moreover, 1 9 C ζ θ for all γ-collisions where θ is the incoming front, and 3k θ n for interactions generating a non-hysical front. Proof. The ossible interactions between two fronts are the same as the interactions of Tye B considered when sing the Glimm scheme, cf. [16]. Therefore, we here give the estimates withot roofs. All the estimates for interactions withot a contact discontinity are obtained from the estimates by Nishida and Smoller in [1], while the estimates for interactions with a contact discontinity are fond sing Lemma.. The estimates for the interactions between a contact discontinity and another front solved by the simlified solver are also obtained sing Lemma.. Tye Ba: Two waves of the same family. (i) α 1 + α α + ν, symmetric to β 1 + β µ + β : This is a new rarefactioncollision and we have α α 1 α = ν 3 ν. U l α 1 α α U r ν Figre 8. The interaction α 1 + α α + ν. (ii) α + µ, symmetric to ν + β. There are two ossible otcomes: α + µ µ + β : For this case we have µ µ, β α C β 0. α + µ α + β : We have α + β α C β 0. (iii) µ + α, symmetric to β + ν: There are two ossible otcomes: µ + α µ + β : For this case µ µ, β α C β 0. µ + α α + β. In this case, the interaction is relaced by a new one, (3.14) µ + α 1 β + α α + β, for which we have the estimate α + β α C β 1 0.

19 FONT TACKING FO A MIXTUE OF GASES 19 β U r μ U l U l α β μ α U r α μ (a) α + µ µ + β. (b) α + µ α + β. Figre 9. The interaction α + µ. α μ α μ Ul β β U r μ U l α α Ur β (a) µ + α µ + β. (b) µ + α α + β. Figre 10. The interaction µ + α. Frthermore, we have 0 by estimate (3.16) for β + α below, cf. Tye Bbii. Hence, 0. Tye Bb: Different families, no contact discontinity. (i) ν + µ ν + µ. None of the rarefaction-fronts increase, and we have (3.15) µ µ, ν ν 1 0. (ii) β + α α + β. We have (3.16) α α (γ 1)C 1 α β, β β (γ 1)C 1 α β, ths 1 9 (γ 1)C 1 α β. (iii) ν + α α + ν, symmetric to β + µ µ + β. This is an increasing rarefaction-collision where we for q > 0 have α α = q, ν ν = q 3 q.

20 0 HOLDEN, ISEBO, AND SANDE Ur Ul ν μ β α ν α Ur Ul μ β ν α α ν Ul (a) ν + µ µ + ν. Ur (b) β + α α + β. (c) ν + α α + ν. Figre 11. The interactions of Tye Bb Tye Bc: With a contact discontinity: These are the for ossible γ-collisions. (i) ζ +µ, symmetric to ν +ζ: There are two ossible otcomes for this γ-collision, and, in addition, we have the case where the simlified iemann solver is sed, introdcing a non-hysical front. ν U r μ μ β U r μ μ U l U l (a) ζ + µ µ + ζ + ν. (b) ζ + µ µ + ζ + β. Figre 1. The interaction ζ + µ. ζ + µ µ + ζ + ν : This interaction is a new rarefaction-collision and an increasing rarefaction-collision with q = ν. We have from which we find µ µ = ν C µ ζ, 8 3 C µ ζ 3 ν 3 q. ζ + µ µ + ζ + β : The rarefaction-front does not increase and µ µ 0, β C µ ζ 10 9 C µ ζ. ζ + µ µ + ζ + θ n : By constrction, µ = µ. Using Lemma., we find (3.17) θ n c µ ζ, from which we get 3C µ ζ 3k θ n,

21 FONT TACKING FO A MIXTUE OF GASES 1 where k given by (3.5) deends only on min, ma and γ. θ n Ũ U l U r θ n Ũ U r μ μ μ μ U l U l U l α α α α U r θ n Ũ Ũ θ n U r (a) ζ + µ µ + ζ + θ n. (b) ζ + α α + ζ + θ n. Figre 13. The interaction ζ + ɛ solved sing the simlified solver. (ii) ζ + α, symmetric to β + ζ. This γ-collision has two ossible otcomes, in addition to the case with a non-hysical front. U l U l α α α α (a) ζ + α α + ζ + ν. U r ν (b) ζ + α α + ζ + β. U r β Figre 14. The interaction ζ + α. ζ + α α + ζ + ν : For this new rarefaction-collision we have α α 0, ν C α ζ, ths, 3 C α ζ 3 ν. ζ + α α + ζ + β : For this case we have α α = β C α ζ 1 9 C ζ β. ζ + α α + ζ + θ n : By constrction, α = α and by Lemma. (3.18) θ n c µ ζ.

22 HOLDEN, ISEBO, AND SANDE Ths, 3C α ζ 3k θ n, where k only deends on min, ma and γ. With the basic interactions between two fronts covered, we are able to consider more involved interactions. First interactions between arbitrary many fronts of the same family are stdied. Two interactions of this kind are given in Figre 15, see also Eamle 3.10 below. Note that no interaction of this form can be an increasing rarefaction-collision. U l α 1 α μ 1 α 3 α 4 α α 5 α 6 α 7 U r α 4 U r β μ U l μ 3 α ν (a) P 7 i=1 α i α + ν. (b) µ 1 + α + µ 3 + α 4 µ + β. Figre 15. Some interactions of the form (3.19). Lemma 3.9. For all interactions between arbitrary many fronts of the same family where two adjacent fronts cannot both be rarefaction-fronts, we have 0, and in articlar, 3 θ for new rarefaction-collisions where θ denotes the new rarefaction wave. Frthermore, there are three ossible otcomes for these interactions; n µ + β, n α + ν, (3.19) ɛ i α i=1 + ν, symmetric to η i µ α + β, i=1 + β, α + β. Proof. We rove the lemma for interactions between three or more 1-fronts, the interactions with n = are already covered by Lemma 3.8. None of the interactions can have two rarefaction waves as otgoing waves de to roerty (i) of Lemma.1. ecall also that increases along a 1-shock wave and decreases along a 1-rarefaction wave. Consider first the case α +ν. Then the interaction is a new rarefaction-collision where U r is to the right of U l and above the 1-shock wave starting at U l. Since only the α i -fronts among the incoming fronts bring s to the right, we have n α α i ν 3 ν. i=1 For the case µ + β, U r is to the left of U l. The only incoming fronts bringing s to the left are the µ i -fronts, ths n µ µ i. i=1

23 FONT TACKING FO A MIXTUE OF GASES 3 Hence, no interaction between fronts of the same family is an increasing rarefactioncollision. Therefore we consider the last two cases together, that is, ɛ + β where ɛ is either a shock or a rarefaction wave. We divide the interaction into several stes where two fronts interact at each ste, hence, j 0 by Lemma 3.8. ecall that two adjacent fronts in the interaction cannot both be rarefaction-fronts. The strategy is as follows: Start with the rightmost front and search for the first lace where two adjacent fronts are of different tyes, i.e., α i + µ i+1 or µ i + α i+1. Let these fronts interact with otcome ɛ k + β k. Whenever there is a 1-shock to the right of β k, we roceed by letting them interact; β k + α α k+1 + β k+1, and we reeat this as long as there is a 1-shock to the right of the 3-shock. Ths, we end with a collection of β-waves as the rightmost waves. Frthermore, whenever this rocess reslts in two adjacent rarefaction waves, we recall from roerty (vii) of Lemma.1 that rarefaction waves (and fronts) are additive and we add them to a new rarefaction wave. We contine this rocess ntil all 1-fronts of different tyes have interacted, and we are left with either µ + βk or α k + βk. For the first case we have (3.0) n i=1 ɛ 1 i µ + k β k µ + β, where we already know that 1 0. By roerty (i) of Lemma.1 it follows that k β k > β, ths, there is a q > 0 so that µ µ = q, β β k = q 0. k For the latter case we have n ɛ 1 i α k + i=1 k k β k α + β, where we already know that 1 0. Frthermore, it follows from the roerties (viii) and (i) of Lemma.1 that α α k 0, β β k 0 0. k k This roves the lemma for the interaction i ɛ i, and the reslts for i η i follows by symmetry. However, we inclde another estimate for the last case discssed above, which will rove sefl later. The nmber of α k -fronts is less than or eqal to the nmber of incoming α i -fronts. Going careflly throgh each stes, we find that each α k has a corresonding incoming α i so that (3.1) α k j i(1 + C 1 (γ 1) ɛ j ) α i 4 3 α i, becase (1 + C 1 (γ 1) ɛ j ) C 1 (γ 1) ɛ j j i j C 1(γ 1)F (t n 1 ) C 1(γ 1) 5 3 L(t 0) 4 3.