Geng LAI Sisi XIE

Abstract In order to construct global solutions to two-dimensional (2D for short) Riemann problems for nonlinear hyperbolic systems of conservation laws,it is important to study various types of wave interactions.This paper deals with two types of wave interactions for a 2D nonlinear wave system with a nonconvex equation of state:Rarefaction wave interaction and shock-rarefaction composite wave interaction.In order to construct solutions to these wave interactions,the authors consider two types of Goursat problems,including standard Goursat problem and discontinuous Goursat problem,for a 2D selfsimilar nonlinear wave system.Global classical solutions to these Goursat problems are obtained by the method of characteristics.The solutions constructed in the paper may be used as building blocks of solutions of 2D Riemann problems.

Keywords Nonlinear wave system,Rarefaction wave,Shock-rarefaction composite wave,Wave interaction,Characteristic decomposition

1 Introduction

The 2D nonlinear wave system takes the form

whereρrepresents the density,(u,v) represents the velocity,andp=p(ρ) is the pressure.This system is derived from the compressible Euler system by neglecting the quadratic terms in the velocity,or by writing the nonlinear wave system as a first order system(see [5]for more details).This system is similar to the pressure gradient system which is also derived from the compressible Euler system (see [1,38]).

The global existence of solution to the Cauchy problem for multi-dimensional nonlinear hyperbolic systems of conservation laws is still a complicated open problem.Thus it has been profitable to consider some special problems,such as 2D Riemann problems,which refer to Cauchy problems with special initial data that are constant along each ray from the origin.Recently,several types of 2D Riemann problems for the compressible Euler system and system(1.1) have been studied by many researchers,see [3,6,8–11,18,20,37,39]for shock reflection problems; [7]for shock diffraction problem; [4,15]for supersonic flows around a convex wedge;[19]for the interaction of transonic shock and rarefaction wave; and [2,14,16–17,21,27–29,31,36]for the interactions of rarefaction waves.

Figure1 Initial data of the expansion of a wedge of gas to vacuum.

In this paper,we consider the system (1.1) with the following initial data:

where (m,n) = (ρu,ρv) is the momentum,ρ0>0,andθ∈0,π2(see Figure1).Here,the momentum in vacuum is not specified.This problem describes the expansion of a wedge of gas at rest into vacuum.It also plays an important role in 2D Riemann problems,since it catches several important types of wave interactions.

2D Riemann problems allow us to consider the so-called self-similar solutions,that are the solutions which depend only on the self-similar variablesξ=xtandη=yt.Then by self-similar transformation,system (1.1) can be changed into the form

which is called the 2D self-similar nonlinear wave system.The greatest feature of the system(1.3)is that its type is a priori unknown,and the type is determined by the local Mach numberrepresents the speed of sound.The system (1.3) is hyperbolic if and only ifM >1,and elliptic-hyperbolic if and only ifM <1.

In this paper,we consider a nonconvex equation of statep=p(ρ) which is assumed to satisfy:

Nonconvex equations of state frequently appear in van der Waals gases(see[22–25]).We divide the discussions into the following two cases:0< ρ0≤ρcandρ0> ρc.Let us briefly describe the results of the paper.

Figure2 Interaction of rarefaction waves.

If 0< ρ0≤ρcthen the gas away from the sharp corner of the wedge expands to vacuum as two symmetrical planar rarefaction wavesR1andR2.As illustrated in Figure2(right),the rarefaction wavesR1andR2meet at some pointP,then interaction starts.ThroughPdraw aC−(C+,resp.) cross characteristic curvel−(l+,resp.) inR1(R2,resp.).Then,by solving a standard Goursat problem (SGP for short) for the 2D self-similar nonlinear wave system (1.3)withl+andl−as the characteristic boundaries (see the SGP (1.3),(3.2) in Subsection 3.2),we can construct the solution in a region Ω bounded by characteristic curvesl+,l−,and a level curveρ= 0,where the two rarefaction waves interact.The main result about this interaction is stated as Theorem 3.1 where we obtain the existence of global classical solution to the SGP(1.3),(3.2).

Ifρ0> ρcthen the gas away from the sharp corner of the wedge expands to vacuum as two symmetrical planar shock-rarefaction composite wavesS1∪R1andS2∪R2,whereSandRrepresent shock and rarefaction wave,respectively.Here,the shock-rarefaction composite waves consist of a rarefaction shock from the front side state (ρ0,0,0) to an intermediate state with the densityρ∗which is defined so thatfollowed by a rarefaction wave from the intermediate state to the vacuum (see Figure3).As illustrated in Figure3(right),these two composite waves meet at some pointP,then interaction starts.ThroughPdraw aC−(C+,resp.) cross characteristic curvel−(l+,resp.) inR1(R2,resp.).Then,by solving a discontinuous Goursat problem (DGP for short) for system (1.3) withl+andl−as the characteristic boundaries (see the DGP (1.3),(4.1) in Subsection 4.2),we can construct the solution in a region Ω bounded by characteristic curvesl+,l−,and a level curveρ=0,where the two composite waves interact.Here,the discontinuous Goursat problem means that the boundary data is discontinuous atP.The main result about this interaction is stated as Theorem 4.1 where we obtain the existence of global piecewise smooth solution to the DGP(1.3),(4.1).

In [16–17,21],the authors considered rarefaction wave interactions for the nonlinear wave system for polytropic gasesp=ργ.They used the idea of Dai and Zhang [14]to convert the 2D self-similar nonlinear wave system (1.3) into the following second order equation:

Figure3 Interaction of shock-rarefaction composite waves.

and obtained global solutions of rarefaction wave interactions by solving some standard Goursat problems for(1.5).However,for the general equation of statep=p(ρ),if we still use this way to study wave interactions then the process will become complicated.Motivated by the results of Zheng et al.[12,29–31]in investigating rarefaction wave interactions of the compressible Euler equations,we derive some characteristic equations and characteristic decompositions of the 2D self-similar nonlinear wave system (1.3).These characteristic equations and characteristic decompositions will be extensively used to establish the a prioriC1norm estimates of solutions.Using these a prioriC1norm estimates,we construct the global solutions of the Goursat problems.Since the main purposes of the paper is the wave interactions,we do not consider the flow after the interactions,i.e.,we do not construct a global solution to the 2D Riemann problem(1.1)–(1.2).But,the wave structures constructed in this paper may be used as building blocks of solutions of 2D Riemann problems.

The rest of the paper is organized as follows.Section 2 is mainly concerned with the 2D selfsimilar nonlinear wave system (1.3).The concepts ofC±characteristic directions,Mach angle,andC±characteristic anglesαandβare presented in Subsection 2.1.A group of characteristic equations in terms of the variablesα,β,andρare derived in Subsection 2.2.These equations will be extensively used to control the hyperbolicity of the system (1.3) and to establish the uniform a prioriC0norm estimates of solutions.Characteristic decompositions for (1.3) are derived in Subsection 2.3.These decompositions will be used to establish the uniform a priori gradient estimates of solutions.Section 3 is devoted to study the interaction of the rarefaction waves.Section 4 is devoted to study the interaction of the shock-rarefaction composite waves.

2 2D Self-Similar Nonlinear Wave System

2.1 Characteristics

The eigenvalues of (1.3) are determined by

which yields the eigenvalues

whereq2=ξ2+η2.So,if and only ifq2> c2(supersonic) system (1.3) is hyperbolic and has two families of wave characteristicsC±defined as the integral curves ofand a family of stream linesC0defined as the integral curves of

Figure4 Characteristic directions and characteristic angles.

See Figure4.The direction of the wave characteristics is defined as the tangent direction that forms an acute angleAwith the vector (-ξ,-η).By simple computation,we see that theC+characteristic direction forms with the direction(-ξ,-η)the angleAfromC+to (-ξ,-η)in the clockwise direction,and theC−characteristic direction forms with the direction(-ξ,-η)the angleAfromC−to (-ξ,-η) in the counterclockwise direction.By computation,we have

The angleAis called the Mach angle.

Following [13]and [31],we use the concept of characteristic angles.TheC+(C−) characteristic angle is defined as the counterclockwise angle from the positiveξ-axis to theC+(C−)characteristic direction.We denote byαandβtheC+andC−characteristic angle,respectively.Letσbe the counterclockwise angle from the positiveξ-axis to the direction (-ξ,-η).Then,we have

and

The first equation of (1.3) can be written as

by the last two equations of (1.3).

Letω=mη-nξ.Then by the last two equations of (1.3) we have

The left eigenvectors corresponding to the eigenvaluesλ±areMultiplying

where

2.2 2D self-similar nonlinear wave system with ω ≡0

Ifω≡0,we can introduce a potential functionϕ(ξ,η) such thatϕξ=m,ϕη=n.Hence,from the last two equations of (1.3),we obtain the Bernoulli law

Moreover,system (1.3) can be reduced to

supplemented by (2.10).

From (2.5) we have

From (2.12)–(2.13)we have

From (2.10) we have

Combining this with (2.11) and (2.5),we have

Proposition 2.1(Commutator Relation) We have

ProofSee [30]and we omit the proof.

Proposition 2.2For the variableρ,we have the following characteristic decompositions:

ProofWe apply the commutator relation (2.19) fornand useto obtain

Hence,we have

Applying the commutator relation (2.20) forc,we get

Inserting this into (2.23) and using (2.14)–(2.17) we can get

3 Interaction of Rarefaction Waves

3.1 Planar rarefaction waves

If 0< ρ0≤ρcthen the gas away from the sharp corner of the wedge expands to vacuum as two symmetrical planar rarefaction wavesR1andR2(see Figure2).In the (ξ,η) plane,R1andR2can be represented by

whereand the functionis defined so thatHere,R1andR2are obtained by solving a one-dimensional Riemann problem.Since it is very classical,we omit the details.

3.2 Goursat problem

Referring to Figure2,the rarefaction wavesR1andR2start to interact from the pointThroughPdraw aC−(C+,resp.) cross characteristic curvel−(l+,resp.) inR1(R2,resp.).Using (2.2) and (3.1),we know thatl−andl+can be determined by

whereIn order to construct the solution to the interaction ofR1andR2,we consider system (1.3) with the boundary data

Problem (1.3),(3.2) is a standard Goursat problem (SGP for short).By the definition of characteristic angle,we can set

3.3 Global classical solution to the SGP (1.3),(3.2)

Lemma 3.1(Local Solution) Whenε>0 is sufficiently small,the SGP(1.3),(3.2)admits a uniqueC1solution on a triangle domain Ωεclosed byl−,l+,and a level curveρ=ρ0-ε.Moreover,this solution satisfies

ProofThe local classical solution can be obtained by the classical theory for boundary value problems for quasilinear hyperbolic system (see for example Li and Yu [33]).

By computation,we haveω=0 in the rarefaction wavesR1andR2.Then by(2.8)we have Combining this with (2.7) we have that the solution satisfiesω=0.By (2.15) and (2.17) we have

Combining this with (2.21) we have that the solution satisfiesandConsequently,by (2.14) and (2.16) we haveandrespectively.We then have this lemma.

Lemma 3.2(Hyperbolicity) Assume that the SGP (1.3),(3.2) admits aC1solution onΩεwhere 0<ε<ρ0.Then the solution satisfies

ProofFromandwe have

It is easy to check byin Ωε.Thus,byA= arcsinandp′′>0 asρ<ρ0we haveA>arcsinin Ωε.We then have this lemma.

Lemma 3.3(A prioriC0Norm Estimate) Assume that the SGP (1.3),(3.2) admits aC1solution on Ωεwhere 0< ε < ρ0.Then there exists a positive constant H0independent ofε,such that

ProofThis lemma can be proved by integrating (2.19) alongC±characteristic curves.

Lemma 3.4Assume that the SGP (1.3),(3.2) admits a uniqueC1solution on Ωεwhere 0<ε<ρ0.Then the solution satisfies

where

ProofBy (3.5) we have that∈(-M(ε),0) alongand∈(-M(ε),0) alongwhereBεandDεare the points onl−andl+,respectively,such thatρ(Bε) =ρ(Dε) =ρ0-ε.

LetEbe an arbitrary point in Ωε.Ifthen by the first equation of (2.21) we have

at the pointE.Similarly,if= -M(ε) and∈[-M(ε),0),then by (3.6) and the second equation of(2.21)we haveat the pointE.Therefore,by an argument of continuity we can get (3.8).We then have this lemma.

Lemma 3.5(A priori Gradient Estimate) Assume that the SGP (1.3),(3.2) admits aC1solution on Ωεwhere 0< ε < ρ0.Then there exists a positive constant H1depending onε,such that

ProofBy computation,we get

Then the lemma can be obtained by (2.19) and Lemmas 3.2 and 3.4.

Theorem 3.1The SGP (1.3),(3.2) admits aC1solution on the domain

ProofIt is easy to check by∂±ρ <0 that the level curves ofρare non-characteristic.Using Lemmas 3.3 and 3.5,and the standard extension method of [32],we can prove that for anyε∈(0,ρ0),if the SGP (1.3),(3.2) admits aC1solution on Ωεthen there exists ae >0 which depends onε,such that the solution can be extend to Ωε+e.We then have this theorem.

Remark 3.1Hu and Wang[17,Section 5]studied the level curveρ(ξ,η)=0.They proved that the level curveρ(ξ,η)=0 is not a point but a closed curve.

4 Interaction of Shock-Rarefaction Composite Waves

4.1 Planar shock-rarefaction composite waves

Ifρ0> ρcthen the gas away from the sharp corner of the wedge expands to vacuum as two symmetrical planar shock-rarefaction composite wavesS1∪R1andS2∪R2(see Figure3(right)).Sinceρ0>ρcandp′(0)=0,there exists a 0<ρ∗<ρcsuch that

Then by Rankine-Hugoniot conditions for nonclassical shocks (see [26]) we know thatS1andS2are located atandrespectively.Define

Then,the(ρ,m,n)at the backsides ofS1andS2are(ρ∗,-χsinθ,χcosθ)and(ρ∗,-χsinθ,-χcosθ),respectively.

The rarefaction wavesR1andR2can be represented by

and

4.2 Discontinuous Goursat problem

Referring to Figure3,the rarefaction wavesS1andS2start to interact from the pointThroughPdraw aC−(C+,resp.) cross characteristic curvel−(l+,resp.) inR1(R2,resp.).Similarly,l−andl+can be represented by

In order to construct the solution to the interaction ofS1∪R1andS2∪R2,we consider system(1.3) with the boundary data

Problem (1.3),(4.1) is a discontinuous Goursat problem (DGP for short),since the data atPis discontinuous.

4.3 Centered waves for the system (1.3)

In order to solve the DGP (1.3),(4.1),we fist give the definition of centered waves for the system (1.3).

Definition 4.1(see Figure5) Let Ψ(t) be an angular domain with curved boundaries:

whereA function(ρ,m,n)(ξ,η)is called aresp.centered wave for the system (1.3) withPas the center point if the following properties are satisfied (see [33,pp.188–190]):

(1) (ρ,m,n) can be implicitly determined by the functionsη=g(ξ,ν) and (ρ,m,n)(ξ,η) =defined on a rectangular domainT(t):={(ξ,ν) |ξP-t≤ξ≤ξP,η1′(ξP) ≤ν≤η2′(ξP)}.Moreover,gandbelong toC1(T(t)),and for any (ξ,ν)∈T(t){ξ=ξP} there holdsgν(ξ,ν)<0.

(2) The function (ρ,m,n)(ξ,η) defined above satisfies (1.3) on Ψ(t){(ξP,0)}.

(3) For any fixedν∈[η′1(ξP),η′2(ξP)],η=g(ξ,ν) gives theC−(C+,resp.) characteristic line passing throughPwith the slopeνatP,i.e.,

(4)ν=η′1(ξP) andν=η′2(ξP) correspond toη=η1(ξ) andη=η2(ξ),respectively.

Figure5 A C+ centered wave,where νi =η′i(ξP) (i=1,2).

4.4 Principal parts of the C± centered waves

We first consider the principal part of theC−centered wave.From the transformationξ=ξandη=g(ξ,ν),we have the relations

Thus,in the (ξ,ν)-plane (1.3) can be written in the form

From (4.3) we have that for theC−centered wave,

Therefore,using (4.5) and lettingξ→ξP,we have that the principal part of theC−centered waves satisfies

since

Lemma 4.1Consider the initial value problem

There exists aρm> ρcwhereρmdepends onθ,such that ifρc< ρ0< ρmthen there exists a-tanθ <ν∗<0 such that the solution of (4.8) satisfies

ProofBy integration,we have

So,whenχis not large there exists -tanθ < ν∗<0 such that the solution of (4.8) satisfiesWe then have this lemma.

In what follows we shall confine ourselves to the case ofρc<ρ0<ρm.

4.5 Global piecewise smooth solution to the SGP (1.3),(4.1)

Centered wave problems for general first order quasilinear hyperbolic systems were first proposed and studied by Li and Yu [33–35].They obtained local centered wave solutions with small amplitude (see [33,Theorem 7.1,p.210]).Zhou [40–41]obtained local centered wave solutions with large amplitude for general first order quasilinear hyperbolic systems.In what follows,we shall use the result of Zhou [40].

Lemma 4.2(Local Solution) There exists a sufficiently smallε >0,such that the SGP(1.3),(4.1) admits a solution on a triangle domain ∆closed byl+,l−,and the straight lineξ=ξP-ε.Moreover,the solution satisfies

ProofAccording to Lemma 4.1,the local existence of solution to the DGP (1.3),(4.1)can be obtained by Zhou [40,Theorem 2.1].The solution contains aC+centered wave ∆+closed byl+,ξ=ξP-ε,and aC+characteristic curve passing throughPwith the slopeν∗atP,and aC−centered wave ∆−closed byl−,ξ=ξP-ε,and aC+characteristic curve passing throughPwith the slope -ν∗atP.The principal part of theC−centered wave is

where -tanθ≤ν≤ν∗.The principal part of theC+centered wave is

By a method which is similar to that of Lemma 3.3,we haveω≡0 in ∆.

By (2.15) and (2.17) we have

By computation,we obtain

in ∆−.we have(ξP,ν)<0.Thus,by (4.11) we have that ifεis sufficiently small then∂+ρ <0 in ∆−.From (2.21) and (4.10) we have that the solution satisfies0 in ∆−.Using (2.14) and (2.16),we also haveandin ∆−.Usingα|l−=π+θ,β|l+=π-θ,and=π+arctanν(-tanθ < ν < ν∗<0),we further haveα < π+θandβ > π-θin ∆−.By symmetry we have0,0,α<π+θ,andβ >π-θin ∆+.

TheC+characteristic curve passing throughPwith the slope -ν∗atPintersects with the straight lineξ=ξP-εat a pointG; theC−characteristic curve passing throughPwith the slopeν∗atPintersects with the straight lineξ=ξP-εat a pointF.Usingandwe can get0,α<π+θ,andβ >π-θin ∆0:=∆(∆−∪∆+).

We then have this lemma.

We are now ready to construct a global solution to the DGP (1.3),(4.1).See Figure6.TheC+characteristic curve throughFintersects withl−at a pointE; theC−characteristic curve throughGintersects withl+at a pointH.By solving a SGP for the system (1.3) withandl−as the characteristic boundaries,we can find a solution in a curved quadrilateral domain closed byl−,andwhereis aC−characteristic curve passing throughFandis a level curveρ(ξ,η)=0.Similarly,by solving a SGP for the system(1.3)withandl+as the characteristic boundaries we can find a solution in a curved quadrilateral domain closed byl+,andwhereis aC+characteristic curve passing throughGandis a level curveρ(ξ,η)=0.In the end,by solving a SGP for the system (1.3)withandas the characteristic boundaries,we can find a solution in a triangle domain closed byandwhereis a level curveρ(ξ,η) = 0.The existence of global classical solutions to these SGPs can be obtained by the same method as in Section 3,since0 are satisfied on theC±characteristic boundaries.We omit the details.Therefore,we have the following theorem.

Theorem 4.1The DGP(1.3),(4.1)admits a piecewise smooth solution on a region Ω closed byl+,l−,and a level curveρ(ξ,η)=0.

where -ν∗≤ν≤tanθ.(Lemma 4.1 implies that

Figure6 Global piecewise smooth solution to the SGP (1.3),(4.1).

AcknowledgementThe authors are grateful to the referee for his (or her)careful reading of the original manuscript and giving helpful suggestions and comments.