Digital Object Identifier (DOI) 10.1007/s00220-002-0705-4
Mathematical Physics
Weak Solutions of General Systems of Hyperbolic Conservation Laws
Tai-Ping Liu1,2,, Tong Yang3,
1 Department of Mathematics, Stanford University, Stanford, CA 94305, USA 2 Institute of Mathematics, Academia Sinica, Nankang, Taipei, Taiwan, R.O.C.
3 Department of Mathematics, City University of Hong Kong, Hong Kong, P.R. China Received: 16 October 2001 / Accepted: 8 May 2002
Published online: 4 September 2002 – © Springer-Verlag 2002
Abstract: In this paper, we establish the existence theory for general system of hyper- bolic conservation laws and obtain the uniform L1 boundness for the solutions. The existence theory generalizes the classical Glimm theory for systems, for which each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax. We construct the solutions by the Glimm scheme through the wave tracing meth- od. One of the key elements is a new way of measuring the potential interaction of the waves of the same characteristic family involving the angle between waves. A new anal- ysis is introduced to verify the consistency of the wave tracing procedure. The entropy functional is used to study theL1boundedness.
1. Introduction
Consider the Cauchy problem for a general system of hyperbolic conservation laws
ut +f (u)x =0, (1.1)
u(x,0)=u0(x), (1.2)
hereu=u(x, t)=(u1(x, t), . . . , un(x, t))andf (u)aren-vectors.
The system is assumed to be strictly hyperbolic, that is, the eigenvalues of then×n matrixf(u)are real and distinct:
f(u)ri(u)=λi(u)ri(u), li(u)f(u)=λi(u)li(u), li(u)·rj(u)=δij, i, j=1,2, . . . , n,
λ1(u) < λ2(u) <· · ·< λn(u).
(1.3)
The research was supported in part by NSF Grant DMS-9803323.
The research was supported in part by the RGC Competitive Earmarked Research Grant CityU 1032/98P.
By a linear transformation, if necessary, we may assume that theith componentui of the vectoruis strictly increasing in the direction ofri. This can be done at least for a small neighborhood of a given state. In the following we will useui to measure the wave strength of ani-wave.
It is well-known that, because of the dependence of the characteristicsλi(u)on the dependent variablesu, waves may compress and smooth solutions in general do not exist globally in time. One therefore considers the weak solution:
Definition 1.1.A bounded measurable functionu(x, t)is a weak solution of (1.1), (1.2) if and only if
∞
0
∞
−∞[φtu+φxf (u)](x, t)dx dt+ ∞
−∞φ(x,0)u0(x)dx=0 (1.4) for any smooth functionφ(x, t)of compact support in
(x, t)|(x, t)∈R2 .
As a consequence of the weak formulation, a discontinuity (u−, u+)in the weak solution with speedssatisfies the Rankine-Hugoniot (jump) condition
s(u+−u−)=f (u+)−f (u−), (1.5) whereu−andu+are the left and right states of the discontinuity respectively.
This prompts the introduction of the Hugoniot curvesH (u0)passing through a given stateu0as follows:
H(u0)≡ {u:σ(u0−u)=f (u0)−f (u)}, (1.6) for some scalarσ =σ(u0, u).
The Rankine-Hugoniot condition says thatu+∈H (u−)and thats=σ(u−, u+). It follows easily from the strict-hyperbolicity of the system that in a small neighborhood of a given stateu0, the setH(u0)consists ofnsmooth curvesHi(u0),i=1,2, . . . , n, throughu0, such thatσi(u0, u)tends toλi(u0)as u moves alongHi(u0)towardu0. Here we use the notationσi(u0, u)to denote the scalarσ (u0, u)inHi(u0). A discontinuity (u−, u+),u+∈Hi(u−), is called ani-discontinuity.
In general, weak solutions to the initial value problem (1.1) and (1.2) are not unique.
A certain admissibility condition, the entropy condition, needs to be imposed on the weak solution to rule out non-physical discontinuities as follows.
Definition 1.2 (Liu, [20]).A discontinuity(u−, u+)is admissible if
σ (u−, u+)≤σ(u−, u), (1.7) for any stateuon the Hugoniot curveH(u−)betweenu−andu+.
If a characteristic field of the system (1.1) is genuinely nonlinear, [14], in the sense that
∇λi(u)·ri(u)=0. (g.nl.), (1.8) then the entropy condition is reduced to Lax’s entropy condition
λi(u+) < σi(u−, u+) < λi(u−). (1.9)
If a characteristic field of the system (1.1) is linearly degenerate, i.e.
∇λi(u)·ri(u)≡0. (l.dg.), (1.10) then the entropy condition is reduced to the one for linear waves
λi(u+)=σi(u−, u+)=λi(u−). (1.11) When each characteristic field is either genuinely nonlinear or linearly degenerate, there is the classical existence theory of James Glimm, [12]. An important physical ex- ample of such a system is the Euler equations in gas dynamics. Other physical systems, such as those in elasticity and magneto-hydrodynamics, for instance, are not necessarily genuinely nonlinear or linearly degenerate.
The goal of the present paper is to study, particularly to establish the existence tho- ery, for such a general system. Thus for a given characteristic fieldλi(u), we allow the linearly degenerate manifoldLGi ≡ {u: ∇λi(u)·ri(u)=0}to be neither the empty space, as in the case of genuine nonlinearity, nor the whole space, as in the case of linear degeneracy.
Theorem 1.1.Suppose that system (1.1) is strictly hyperbolic with flux functionf (u)∈ C3, and that for each characteristic fieldλi(u)the linear degeneracy manifoldLDi ei- ther is the whole space or consists of a finite number of smooth manifolds of codimension one, each transversal to the characteristic vectorri(u). Then for the initial data (1.2) with sufficiently small total variationT .V., there exists a global weak admissible solu- tionu(x, t)to the Cauchy problem (1.1) and (1.2) satisfyingtotal variation u(·, t)= O(1)T .V..
Remark 1.1.In this paper, we only prove the existence of the weak solution to (1.1) and (1.2). The admissibility of the weak solution has been established in [19]. It is shown, cf. Theorem 15.1 in [19], that there exist subsets"1and"2of {(x, t) : −∞ < x <
∞, t ≥ 0}with the following properties. "1consists of countable Lipschitz con- tinuous curves and"2consists of countable points. Each curve#in"1 represents a curve of jump discontinuity in the weak solution satisfying the entropy condition (1.7) except for countable points. Each point in"2 represents a point of interaction in the weak solution. And outside"1∪"2, the weak solution is continuous. In fact, for each shock wave in the weak solution, there exists a corresponding approximate shock wave in the approximate solution when the mesh sizes are sufficiently small. Consequently, the admissibility of the shock waves in the weak solution follows from the admissibility of the shock waves in the approximate solutions as the consequence of the design of the scheme.
The Glimm theory for systems with genuinely nonlinear or linearly degenerate fields is based on the study of the interactions of elementary waves in the solutions of the Riemann problems solved by Peter Lax, [14]. The random choice method, the Glimm scheme, is introduced to construct the general solutions using the Riemann solutions as building blocks. A nonlinear functional, the Glimm functionalF[u], is constructed to bound the total variation of the approximate solutions. The functional yields a global measure of the total wave interactions, [13], and allows for the consistency study of the wave tracing method, [19].
For systems, which are not necessarily genuinely nonlinear or linearly degener- ate, there are richer phenomena for nonlinear wave interactions, [19]. We adopt the
Glimm quadratic functional for the interaction of waves in different families. How- ever, for the interaction between waves of the same family, the quadratic functional in general does not exist, and a cubic functional is needed. A cubic functional was introduced in [19], which, however, fails to take into account some aspects of wave interactions. Here we revise the cubic functional in [19] so that it depends global- ly on the wave patterns in the solution. It is defined by the product of the strengths of two waves times the angle between them, when that angle is negative. This is so that such a pair of waves of the same family will interact in general at a later time.
This new cubic functional is an effective measure of the wave interactions in that the functional decreases only due to the interaction of the waves next to each other and that the decrease is exactly of the same order of the waves produced by the interac- tion.
With the present existence theory and the qualitative theory of regularity and large- time behavior of solutions in [19], there is the open problem of theL1stability of the solutions with respect to the initial data. We study the stability problem here, but only for the stability of the constant solutions. For the stability analysis we make use of the classical entropy functional, which is shown to yield the estimate to control the bifur- cation of the Hugoniot curve from the rarefaction wave curve in the general setting. To construct a generalized entropy functional, as for the case when each characteristic field is either genuinely nonlinear or linearly degenerate, to control the estimates of the same cubic order as mentioned above would be the main task to study the stability of the weak solution to this general system.
In the next section we sketch the construction of the solution to the Riemann problem and some basic estimates on the Hugoniot curves. These estimates allow us to study the local wave interactions in Sect. 3. In Sect. 4 we study the nonlinear functional and there- by establish the convergence of the approximate solutions. The wave tracing mechanism of [17, 19] is refined here. Previous consistency analysis, [17], requires the boundedness of the quadratic functional. For a non-genuinely nonlinear system, a quadratic functional for interactions of waves of the same characteristic family does not exist in general. Our consistency analysis for the wave tracing method here uses only the estimates resulting from the cubic functionals. The cubic estimates are weaker and there is a new, interesting consistency analysis here.
In the last section of this paper, we study theL1stability of constant state solutions to the system (1.1). There has been much progress on the well-posedness, inL1topology, problem when each characteristic field in the system is either genuinely nonlinear or linearly degenerate. There are two approaches. One starts with [4] on the comparison of infinitesimally close solutions, see [5] and [6]; the other approach [22] is based on the construction of the robust functional, see also [7, 23]. For the more general system (1.1), there is the recent result for the case of one reflection point in [1]. To our knowl- edge, there is no general well-posedness theory without assuming genuine nonlinearity or linear degeneracy on characteristic fields, beyond that of [1].
The purpose of Sect. 6 is to study theL1stability of the constant state solutions of the general systems. We adopt the general approach of [22] and construct a new time- decreasing nonlinear functionalH (t)=H[u(·, t)], which is equivalent to||u||L1 of a weak solutionu(x, t). It also depends explicitly on the wave pattern of this solution. The functionalH[u(·, t)] consists of three parts: the first part is the product of the Glimm’s functional and a linear functionalL(t); the second part is a quadratic functionalQd(t);
and the third part is the convex entropy functional. HereL(t)represents theL1-norm of u(x, t).Qd(t)registers the effect of nonlinear coupling of waves in different families on||u(x, t)||L1(x) by making use of the strict hyperbolicity of the system, andE(t)
captures the nonlinearity of the characteristic fields. The existence of such a functional immediately yields the following theorem.
Theorem 1.2.Suppose that the total variation of the initial data is sufficiently small and is inL1, then theL1norm of the weak admissible solution to the Cauchy problem (1.1) and (1.2) constructed by the Glimm scheme is bounded by a constant times theL1norm of the initial data.
2. Riemann Problem
The solution to the Riemann problem u(x,0)=
ul, x <0,
ur, x >0, (2.1)
for the general system (1.1) was solved in [16, 19]. We enclose the following lemmas on the properties on wave curves in [19] for the self-containedness of the paper.
Thei-rarefaction wave curve from a stateu0, denoted byRi(u0), is the integral curve of the right eigenvectorripassing throughu0,i=1,2, . . . , n. In general the Hugoniot curveHi(u0)and the rarefaction wave curveRi(u0)have second order contact at the initial stateu0, [14]. In general no higher-order contact is expected when the charac- teristic field is genuinely nonlinear. However, as we will see in the following lemmas, the situation is more interesting for non-genuinely nonlinear characteristic fields. The following lemmas are needed for the construction of the wave curveWi(u0)through the stateu0. As mentioned before, the strength of thei-wave is measured by the difference of the parameterui between the right and left states.
Lemma 2.1.For anyu∈Hi(u0)in a small neighborhood ofu0, we have (i)λi(u) > σ (u0, u)(orλi(u) < σ(u0, u)) if and only if
d
duiσ (u0, u) >0, (or d
duiσ(u0, u) <0);
(ii)Hi(u0)is tangent toRi(u)atuonHi(u0)ifσ(u0, u)=λi(u). Proof. Let
u−u0= n j=1
αjrj(u), du
dui =n
j=1
βjrj(u).
Then for weak waves the second order contact betweenHi(u0)andRi(u0)implies αiβi >0 foru=u0,
|αj|
|u−u0|2 is bounded f or j =i. (2.2) By differentiating
σ (u0, u)(u0−u)=f (u0)−f (u),
with respect toui, we have αj d
duiσ(u0, u)=(λj(u)−σ (u0, u))βj, j =1,2, . . . , n. (2.3) Thus (i) follows from (2.2) and(2.3)i. Sinceσ(u0, u)is close toλi, by strict hyperbolicity and(2.3)j, we have (ii).
The following lemma gives an estimate on the interaction of two shock waves in the same direction and shows that the interaction of two admissible shock waves yields an admissible shock plus a cubic order error term.
Lemma 2.2.Suppose that(u0, u1)and(u1, u2)withui2> ui1> ui0are two admissible i-shocks with strengthsα1andα2and speedsσ1andσ2respectively, cf. Definition 1.2.
Letu∗∈Hi(u0)be the state withui2=ui∗, then (i)(u0, u∗)is admissible;
(ii)|u2−u∗| =0(1)α1α2(σ1−σ2);
(iii)σ α=σ1α1+σ2α2+0(1)α1α2(σ1−σ2), whereαandσare the strength and speed of the admissible shock(u0, u∗)respectively. The same estimate holds for the case when ui0> ui1> ui2.
Proof. Set
˜
σα≡σ1α1+σ2α2,
whereα=α1+α2. Then by using the Hugoniot conditions for(u0, u1),(u1, u2)and (u0, u∗), we have
˜
σ (uj2−uj0)−[fj(u2)−fj(u0)]
= ˜σ(uj2−uj0)−[σ2(uj2−uj1)+σ1(uj1−uj0)]
=(σ˜ −σ1)(uj2−uj1)+(˜σ−σ2)(uj1−uj0). (2.4) Chooseu˜0andu˜2on the straight line throughu1with tangentri(u1)such that
˜
ui0=ui0, u˜i2=ui2. Then we have
| ˜u0−u0| =0(1)α12, | ˜u2−u2| =0(1)α22,
(ui2−ui1)(uj1− ˜uj0)=(˜uj2−uj1)(ui1−ui0). (2.5) Combining (2.4) and (2.5) yields
˜
σ (uj2−uj0)−[fj(u2)−fj(u0)]
=(σ˜ −σ2)(α)−1[−(ui2−ui1)(uj1−uj0)+(uj2−uj1)(ui1−ui0)]
=0(1)(σ˜ −σ2)α2(α1+α2)
=0(1)(σ1− ˜σ)α1(α1+α2).
By (2.5) again, we have
˜
σ (u2−u0)−[f (u2)−f (u0)]=0(1)α1α2(σ1−σ2). (2.6)
By comparing the jump condition of theithcomponents for(u0, u2)and(u0, u∗), we have σ− ˜σ =0(1)α1α2(α1+α2)−1(σ1−σ2),
and (iii) follows. From (2.4), (2.6) and the Hugoniot condition for(u0, u∗), we have
˜
σ(u∗−u2)=f (u∗)−f (u2)+0(1)α1α2(σ1−σ2). (2.7) Notice thatσ˜ is close toλi. By consideringu∗−u2in therj direction,j = i, strict hyperbolicity implies (ii).
Finally we prove that the discontinuity(u0, u∗)is admissible. Ifσ1=σ2, then clear- lyu∗ =u2and(u0, u2)is admissible. Ifσ1> σ2, then the admissibility is proved by contradiction as follows. Since
σ2≤σ ≤σ1,
we assume, without loss of generality, thatσ−σ2≥σ1−σ. Under the condition that (u0, u1)is admissible, and assuming that(u0, u∗)is not admissible, then there exists a stateu˜withui1≤ ˜ui < ui∗=ui2, such thatσ (u0,u)˜ =σ. Thusu˜∈Hi(u∗). By (ii), for the stateu¯ ∈Hi(u2)withu˜i = ¯ui, we have
σ(u2,u)¯ −σ(u∗,u)˜ =0(1)α1α2(σ1−σ2). (2.8) Since(u1, u2)is admissible, we have
σ2≥σ(u2,u).¯ (2.9)
Combining (2.8) and (2.9) yields
σ2−σ ≥0(1)α1α2(σ1−σ2). (2.10) Since all the wave strengths are small, (2.10) contradicts the assumption thatσ−σ2≥
1
2(σ1−σ2). Hence(u0, u∗)is admissible and this completes the proof.
For the interaction of a rarefaction wave and a shock of the same family, we have the following lemma. To obtain the precise interaction estimates, we introduce a new infinite step approach by replacing the rarefaction wave by small rarefaction shocks. By doing this, we can apply Lemma 2.2 and show that the limit exists. Without any ambiguity, for any discontinuity waveγ =(u−, u+), we denote its speed byσ (γ )=σ(u−, u+)from now on.
Lemma 2.3.Suppose that(ul, u1)is ani-rarefaction wave,(u1, ur)is an admissible i-discontinuity, anduil < ui1< uir. Then there existsu∗ ∈ Ri(ul)withuil ≤ ui∗ ≤ui1, andu˜∗∈Hi(u∗)withu˜i∗=uir such that
(i)| ˜u∗−ur| =0(1)α2
ui1
ui∗(λi(u)−σ(u1, ur))dui, where the integral is along theRi(u∗) curve, andα2=uir−ui1.
(ii)σ (u∗,u˜∗)β = ˆλα1+σ(α2)α2+0(1)α2
ui1
ui∗(λi(u)−σ (u1, ur))dui, whereα1 = ui1−ui∗,β = α1+α2, andλˆ is the average speed of the centered rarefaction wave (u∗, u1):
λˆ ≡ ˆλ(u∗, u1)≡ 1 ui1−ui∗
ui 1
ui∗
λi(u)dui. (iii)(u∗,u˜∗)is admissible.
Proof. Ifσ (u1, ur)=λi(u1), then the lemma holds trivially because the linear super- position of the two Riemann solutions yields the solution to the Riemann data(ul, ur). Whenσ (u1, ur) < λi(u1), for any stateu∈Ri(ul)betweenul andu1, letu˜ ∈ Hi(u) withu˜i =uir. Setθ(u)≡λi(u)−σ (u,u). Then we have˜ θ(u1) >0.
Suppose that(u,u)˜ is admissible andθ(u) >0. We claim that forw∈Ri(ul)with ui−wipositive and sufficiently small, thenθ(w) < θ(u)and(w,w)˜ is also admissible.
In fact, by Lemmas 2.1 and 2.2, we know that when1=ui−wi is sufficiently small, (α+1)σ (w,w)˜ =ασ (u,u)˜ +1λi(u)+0(1)1αθ(u)+0(1)12, (2.11) whereα≡ ˜ui−ui >0. By usingλi(w) < λi(u)and the entropy conditionλi(u) >
σ(u,u)˜ , (2.11) impliesθ(w) < θ(u).
That(w,w)˜ is admissible can be proved by contradiction. Suppose that(w,w)˜ is not admissible, then there existswˆ ∈ Hi(w)˜ such thatσ (w,ˆ w)˜ = σ(w,w). If˜ wˆi < ui, then we letw ≡ ˆwand1 ≡ui − ˆwi. Otherwise, we chooseuˆ ∈Hi(˜u)withwˆi = ˆui. When1is sufficiently small, we have
σ (u,ˆ u)˜ =σ(w,ˆ w)˜ +0(1)|ui − ˜ui|θ(u)1. (2.12) Since(u,u)˜ is admissible, the entropy condition yields
σ(u,˜ u)ˆ ≤σ(u,u).˜ (2.13)
Combining (2.11), (2.12) and (2.13) yields
0(1)θ(u)1α≥1[θ(u)+1],
which is a contradiction to the assumption thatθ(u) > 0, given that1and the wave strength are weak.
Now we are ready to prove (i) and (ii). We first divide the rarefaction waveα1into small rarefaction waves with each strength less than1,1a given small positive number.
And then we replace each small rarefaction wave by a small rarefaction shock. Denote all these rarefaction shocks from left to right byα1,k≡u1,k−1−u1,k,k=1,2, . . . , m, with speed
σ (α1,k)= 1
2(λi(u1,k−1)+λi(u1,k)), k=1,2, . . . , m.
Obviously,u1,k∈Ri(u1),u1,0=u1andu1,m=u∗. Now we consider the sequence of interactions betweenβk≡(u1,k−1,u˜1,k−1)andα1,k. By using the fact that the Hugoniot curve and the rarefaction curve have second order contact, an application of Lemma 2.2 yields the following estimate for the interaction ofβkandα1,k:
σ (u1,k,u˜1,k)βk+1 =σ(u1,k−1,u˜1,k−1)βk+α1,kσ (α1,k)
+0(1)βkα1,kθ(u1,k−1)+0(1)12, (2.14)
| ˜u1,k− ˜u1,k−1| =0(1)βkα1,kθ(u1,k−1)+0(1)12. By summing up (2.14) with respect tokfromk=1 tom, we have
σβ=σ2α2+ m k=1
α1,kσ (α1,k)+0(1) m k=1
βkα1,kθ(u1,k−1)+0(1)α11,
| ˜u∗− ˜u| =0(1)m
k=1
βkα1,kθ(u1,k−1)+0(1)α11, (2.15) whereβ =βk+1=(u∗,u˜∗).
Now we estimate the termm
k=1βkα1,kθ(u1,k−1).
We denote
Ekl =βlα1,kθ(u1,l−1), 1≤l≤k, Ek=Ekk =βkα1,kθ(u1,k−1).
ForEkl, noticing that each|α1,k| ≤1, we have the following estimate:
Ekl =Ekl−1+α1,k[βl−1(θ(u1,l−1)−θ(u1,l−2))+(βl−βl−1)θ(u1,l−1)]
=Ekl−1+α1,kβl−1(λi(u1,l−1)−λi(u1,l−2))
−α1,k[βl−1(σ (βl)−σ(βl−1))+α1,l−1(σ(βl)−λi(u1,l−1))]
≤Ekl−1+0(1)α1,kEl+0(1)13. (2.16)
Hence we have
El ≤El0+0(1)α1,k
l i=1
Ei+0(1)12. Using the fact thatm
k=1α1,k≤α1is small, we have m
l=1
El ≤0(1)m
l=1
El0+0(1)1.
Therefore, by letting1tending to zero, (2.15) and (2.17) imply (i) and (ii).
We next construct thei-wave curve from a stateul,i=1,2, . . . , n, with the property that any stateu∈Wi(ul)can be connected toulon the left byi-waves. That is, we will construct a curveWi(ul)throughulsuch that it passes through a single stateuon each hyperplane with fixedui in a small neighborhood oful. For definiteness, we consider the caseuil < ui. The case whenuil > ui can be discussed similarly. First we find a unique stateu1with the following properties:
(i)uil ≤ui1≤ui;
(ii)(ul, u1)is an admissible discontinuity such thatui1−uil is maximum.
Ifui1=ui, then we are done withu=u1. If not, by Lemma 2.2, there is no admissible discontinuity with left stateu1and theui component of the right state lies in(ui1, ui].
Therefore, according to Lemma 2.1, we have∇λi·ri(u1)≥0, and∇λi·ri(u) >0 for statesu ∈ Ri(u1)nearu1with theithcomponent larger thanui1. Thus, there exists a unique stateu2∈Ri(u1)with the following properties:
(i)u1andu2are connected byi-rarefaction wave andui1< ui2≤ui.
(ii)ui2is the maximum in the sense that there is no stateu∗∈Ri(u1)with the property that there exists admissible discontinuity(u∗, u∗∗)withui1< ui∗< ui2andui∗< ui∗∗≤ui.
Ifui2=ui, thenu=u2and we are done. If not, the above procedure can be continued until we finally reach the stateuon the curveWi(ul)with the givenui. Thus(ul, u)
forms an elementaryi-wave described above whenu ∈ Wi(ul). The wave curves are Lipschtz continuous, but have the following basic stability property:
Lemma 2.4.Wave curvesWi(¯u0)andWi(u˜0)through different initial states have the followingC2-like property: Given a stateu¯onWi(¯u0), there exists a stateu˜onWi(˜u0) such that
¯
u− ˜u= ¯u0− ˜u0+O(1)| ¯u0− ˜u0|| ¯u− ¯u0|.
Proof. We first remark that for a genuinely nonlinear field, the wave curveWi consists of Hugoniot and rarefaction curves. For linearly degenerate field the wave curve is the rarefaction curve, which is the same as the Hugoniot curve, [12]. In either case, the depen- dence of a wave curve on its initial state isC2, [12], and the lemma follows immediately by mean value theorem. However, this may not be the case when theithcharacterisitic field in not genuinely nonlinear or linearly degenerate, as in the case we are interested in. For the general case, ani-wave in the Riemann solution may contain both shock and rarefaction waves of the sameithcharacteristic family, called a compositei-wave.
From the above description, thei-wave curve consists of a finite number of Hugoniot, rarefaction, and a new type of “mixed” curves. A mixed curveMi(u0)is a collection of states u∗, which is related to a fixed rarefaction curveRi(u0)with the following properties: (i)u∗ ∈ Hi(u)for a stateuonRi(u0); (ii)σ (u, u∗)= λi(u); (iii)(u, u∗) satisfies the entropy condition (E); (iv) at the initial stateu0where the mixed curve and the rarefaction wave meet the characteristic is linearly degenerate∇λi ·ri(u0) = 0;
and (v) the wave curve containsRi(u0)andMi(u0), which meet atu0. These properties are used to construct a wave pattern which contains the rarefaction wave followed by a one-sided contact discontinuity(u, u∗). As the one-sided contact discontinuity(u, u∗) grows in strength, the rarefaction waves weaken as its end stateumoves away fromu0
alongRi(u0).
We first show that the aforementioned two curvesMi(u0)andRi(u0)are of second order tangency atu0. Differentiate the jump conditionσ (u∗−u)=f (u∗)−f (u)with respect to the arc length ofRi(u0):
σ(u∗−u)=(f(u∗)−σ)u∗−(f(u)−σ )u.
Note thatu = r(u). Evaluating the above atu = u∗ =u0, using (iv) above, which implies thatσ=0, and thatσ =λi there, we have
(f−λi)u∗=(f−λi)ri =0.
Thusu∗is parallel tori(u)atu∗=u=u0. Setu∗=cri(u). Next, we differentiate the jump condition twice to yield
σ(u∗−u)+2σ(u∗−u)=f(u∗)u∗u∗−f(u)uu
+(f(u∗)−σ )u∗−(f(u)−σ )u. And we evaluate this, again atu=u∗=u0,
(c2−1)friri+(f−λi)u∗=(f−λi)ri.
We now differentiatefri =λirialongri(u)atu=u0and use (iv) above,λi=0, friri =(λi−f)ri.
Sinceri=u, the last two identities yield
(f−λi)u∗=c2(f−λi)u
atu=u∗=u0. Recall thatu∗=criandc=1 in general. Thus we need to renormalize the differentiation alongMi(u0)to be with respect to the arc length as follows:
˙ u∗≡ 1
cu∗. We have
¨
u∗=c2u∗+1 c
1 c
u∗, and so from the previous identity we have, atu=u∗=u0,
(f−λi)¨u∗=(f−λi)u.
Thusu¨∗ =uexcept for a multiple ofri. On the other hand, bothu˙∗anduare unit vectors and sou¨∗anduare perpendicular tou˙∗=u=ri. We therefore conclude that
¨ u∗=u
atu=u∗=u0. ThusMi(u0)andRi(u0)are of second order contact atu0.
On the other hand, a wave curve is in general only Lipschitz when two mixed curves meet. This corresponds to the vanishing of the rarefaction wave between two discon- tinuities to form a single discontinuity. We now concentrate on proving our lemma for this key case. Thus we assume that two wave curves are very close and one of them is a single Hugoniot curveHi(u0)corresponding to an admissible shockβ=(u0, u1)with one-sided contact discontinuityσ(β)=λ(u1), λ≡λi. The other nearby wave curve correspond to a shockα=(u2, u3)followed by a rarefaction waveδ=(u3, u4). Letu5
andu6be states onHi(u3)andHi(u2), respectively, and denote byβ¯=(u2, u6), see the picture below. We assume that the statesu1, u4, u5andu6are on the same hyperplane transversal to theithcurves, as do the statesu0andu2. Thus we have
u1∈H(u0); u3∈H(u2); u6∈H (u2);
u5∈H(u3); u4∈R(u3),
σ (β)=λ(u1), σ (α)=λ(u3).
We want to show that
13≤11+O(1)11(α+δ), where13≡ |u1−u4|, 11≡ |u1−u2|. We have
13≤ |u1−u6| + |u6−u5| + |u5−u4|.
u0
u1
u3
u4
u5
u6
u2
β
β¯ 11
12
13
δ α
For simplicity in notation we denote byf(u)≡ ∇λ·r(u)the change ofλ≡λialong the characteristic directionr≡ri. This measures the degree of genuine nonlinearity at the stateu. The Hugoniot and rarefaction curves are close to each other iffis small:
|u5−u4| =0(1)|f(u3)|δ3.
From the second-order contact of Hugoniot and rarefaction curves and our analysis of Hugoniot curves before,
|u1−u6| =11+0(1)11(α+δ),
|u6−u5| =0(1)δα|σ(α)−σ(δ)| =0(1)δα|λ(u3)−σ (δ)| =0(1)|f(u3)|δ2α.
Thus we have
13≤11+0(1)11(α+δ)+0(1)|f(u3)|δ2(α+δ). (2.17) With this clearly our estimate13 ≤ 11+0(1)11(α+δ)follows if we can show that Claim.
|f(u3)|δ=0(1)(11+13).
To prove the Claim we have from above that
|f(u3)|δ ≡0(1)|λ(u4)−λ(u3)|
≤0(1)(|λ(u4)−λ(u1)| + |λ(u1)−λ(u3)|)
=0(1)13+0(1)|σ(β)−σ (α)|
≤0(1)(13+ |σ (β)−σ(β)| + |σ (¯ β)¯ −σ(α)|)
=0(1)[(13+11)+ |σ(β)¯ −σ (α)|].
To finish the proof of the Claim we note from simple scalar consideration that
|σ(β)¯ −σ (α)| =0(1) δ2
α+δ|f(u3)| |f(u3)|δ.
This completes the proof of the lemma.
Theorem 2.1 (Liu [16]).Under the same hypotheses as in Theorem 1.1, the Riemann problem (1.1) and (2.1) has a unique solution in the class of elementary waves satis- fying the entropy condition, cf. Definition 1.2, provided that the states are in a small neighborhood of a given state.
Proof. The i-waves, i = 1,2, . . . , n,are the building blocks for the solution of the Riemann problem. Thei-waves take values along the wave curvesWi. Since the wave curvesWi have tangentri at the initial state, it follows from the independency of the vectorsri, i=1,2, . . . , n,and the inverse function theorem that the Riemann problem can be solved uniquely in the class of elementary waves.
3. Wave Interaction
In this section the relation of the waves before the interaction and the scattering data for the completed interaction is studied for the interaction of two sets of solutions of the Riemann problem.
For ani-waveαto the left of ani-waveβ, we define6(α, β)to represent the effective angle between them:
6(α, β)≡θα++θβ−+
θγ. (3.1)
Hereθα+ represents the value ofλi at the right state of αminus its wave speed. It is negative ifαis a shock and is set zero if it is ai-rarefaction wave. Similarly the termθβ− denotes the difference between the speed ofβand the value ofλiat its left end state.θγis the value ofλiat the right state of the waveγminus that at the left state. It is positive ifγ is a rarefaction wave and is negative if it is a shock. The sum
θγis over thei-wavesγ betweenαandβ. Subject to wave interactions of distinct families,−6(α, β)represents the angle betweenαandβ when waves of other characteristic families between them propagate away. When6(α, β)is positive, the two waves will not be likely to meet and should not be included in the potential wave interaction functional. When6(α, β)is negative, the two waves may eventually meet and interact. In this case|α||β||6(α, β)|
reflects accurately the potential interactions of waves of the same characteristic family.
To obtain the estimate for the interaction of two Riemann solutions, we need the following lemmas from [19]. Let(ul, ur)be ani-discontinuity, set
Di(ul, ur)≡ {u:(u−ul)σ(ul, ur)−(f (u)−f (ul))
=c(u)ri(u) f or some scalar c(u)}.
The following lemma is similar to Lemma 2.2, its proof is therefore omitted.
Lemma 3.1.Forul andur close,Di(ul, ur)is a smooth curve throughulandur in a small neighborhood ofur. Moreover, if a stateu˜satisfies
(˜u−ul)σ−(f (u)˜ −f (ul))= ˜cri(˜u)+K,
for some scalarc˜and some vectorK, then there exists a vectoruonDi(ul, ur)such that
|u− ˜u| =0(1)|K|.
To express the stability of a wave pattern in the next lemma, we need the following definition on partition of waves.
Definition 3.1.Letur ∈Wi(ul)so thatulis related tourbyi-discontinuities(uj−1, uj), andi-rarefaction waves(uj, uj+1),j odd,1 ≤j ≤ m−1,u0=ul andum =ur. A set of vectors{v0, v1, . . . , vp}is a partition of(ul, ur)if
(i)v0=ul,vp =ur,vk−i 1≤vki,k=1,2, . . . , p, (ii){u0, u1, . . . , um} ⊂ {v0, v1, . . . , vp},
(iii)vk ∈Ri(uj),j odd, ifuij < vik< uij+1, (iv)vk ∈Di(uj−1, uj),j odd, ifuij−1< vki < uij.
We set
