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EXISTENCE OF SHOCK-WAVE SOLUTIONS FOR SOME NON-CONSERVATIVE SYSTEMS 3 1 ANUPAMPALCHOUDHURY 0 2 Abstract. Inthisarticle,weprovetheexistenceofshockcurvesfortwosys- n temsinnonconservativeformusingtheproductdefinedbyDalMaso,Lefloch, a Muratin[1]. Oneofthesystemsisstrictlyhyperbolicwithgenuinelynonlin- J ear characteristic fields while the other one has repeated eigenvalues with an 0 incompletesetofrighteigenvectors (parabolicdegeneracy). Theshockcurves 3 forthetwosystemsarecomparedbaseduponaparameterk tendingto0. ] P A 1. Introduction . h This article is devoted to the study of the first order non-conservative systems t a ut+uux−σx =0, m (1.1) σt+uσx =0 [ and 1 ut+uux−σx =0, v (1.2) 6 σt+uσx−k2ux =0. 6 in the domain Ω={(x,t):−∞<x<∞, t>0}. Here k is a positive constant. 1 In order to implement the approach using the family of paths (we henceforth refer 7 tothemasDLMpaths)forstudyinganonconservativesystem(see[1],[4]),themost . 1 importantsteptobeginwithistoprovetheexistenceofapathusingwhichwecan 0 address the issue of solving the Riemann problem. In this article, we address this 3 issue in regardto the systems (1.1) and (1.2). 1 : The system (1.2) is strictly hyperbolic with the eigenvalues given by λ1(u,σ) = v u − k, λ (u,σ) = u + k with the corresponding right eigenvectors E (u,σ) = i 2 1 X 1 1 , E (u,σ) = . This system arises in elastodynamics and the Rie- r k 2 −k a (cid:18) (cid:19) (cid:18) (cid:19) mann problem for the system was solved using Volpert’s product in [2]. Thesystem(1.1)onthecontraryfailstobehyperbolic(parabolicdegeneracy)since it has repeated eigenvalues λ (u,σ) = u = λ (u,σ) and an incomplete set of right 1 2 eigenvectors. Such systems even when in conservative form are tough to analyse due to absenceoffull setofeigenvectors. In [5]a classofsuchsystemsinconserva- tive form had been studied and it was shown that singular concentrations tend to develop in one of the dependent variables. Motivated by this, existence of singular solutions to (1.1) had been studied in [3] using the method of weak asymptotics. But a seemingly more interesting question in this context is to examine the existence of a DLM path so as to be able to find a shock-wave solution to (1.1) for Riemann type initial data. It has been shown in [3] that the Riemann problem for 2010 Mathematics Subject Classification. 35L65,35L67. Key words and phrases. Hyperbolicsystemsofconservationlaws,Non-conservativesystems. 1 2 ANUPAMPALCHOUDHURY (1.1) cannot be solved using Volpert’s product. Another interesting question would be to explore the connection between the solutions of (1.1) and (1.2) as k tends to 0. We recall that a DLM path φ is a locally Lipschitz map φ :[0,1]×R2×R2 →R2 satisfying the following properties: 1. φ(0;vL,vR)=vL and φ(1;vL,vR)=vR for any vL and vR in R2, 2. φ(t;v,v)=v, for any v in R2 and t∈[0,1], 3. For every bounded set Ω of R2, there exists k ≥1 such that |φt(t;vL,vR)−φt(t;wL,wR)|≤k|(vL−wL)−(vR−wR)|, for every vL,vR,wL,wR in Ω and for almost every t∈[0,1]. The paths φ=(φ ,φ ) and φ˜=(φ˜ ,φ˜ ) given by 1 2 1 2 uL+2t(uR−uL), t∈[0,1], φ(t;vL,vR):φ1(t;uL,uR)=(uR, t∈[21,1]. 2 (1.3) φ2(t;σL,σR)=σL+t(σR−σL), t∈[0,1]. and  φ˜(t;vL,vR):φφ˜˜21((tt;;σuLL,,σuRR))==((σσuuLLLR,,++tt(2∈∈2t(t[[u0−21R,,1121−)]],(uσLR)−, tσ∈L)[,0,t21∈],[1,1] (1.4) 2 where vL = (uL,σL), vR = (uR,σR) ∈ R2, satisfy the above conditions and are thus examples of DLM paths. Given a system vt+A(v)vx =0, v(x,t)∈R2, x∈R, t>0 in non-conservative form with Riemann type initial data: v(x,0)=vL if x<0, v(x,0)=vR if x>0, andaDLMpathφ,itwasshownin[1]thatashockwavesolutionexistswithspeed sifandonlyifvL,vR andssatisfythefollowingRankine-Hugoniot(R-H)condition: 1 {−sI+A(φ(t;vL,vR))}φt(t;vL,vR)dt=0, (1.5) Z0 whereI denotes the identity matrix. We remarkthatthe matrix Afor the systems u −1 u −1 (1.1) and (1.2) are given by A(u,σ) = and A(u,σ) = 0 u −k2 u (cid:18) (cid:19) (cid:18) (cid:19) respectively. In Section 2, we derive a condition for a DLM path to satisfy the R-H condition for the systems (1.1) and (1.2). In Section 3, we show that the curves φ and φ˜ mentioned above satisfy the R-H condition and find the shock curves for the solutions of the Riemann problem for (1.1) and (1.2). SHOCK-WAVES FOR SOME NON-CONSERVATIVE SYSTEMS 3 2. Condition on a DLM path and the initial data to satisfy the Rankine-Hugoniot condition In this section, we derive a condition on a DLM path φ=(φ ,φ ) in order that 1 2 it might satisfy the R-H condition (1.5) for the systems (1.1) and (1.2) for given Riemann type initial data vL =(uL,σL), x<0, v(x,0)=(u(x,0),σ(x,0))= (2.1) (vR =(uR,σR), x>0. Lax’sadmissibilityconditionappliedtothesystems(1.1)and(1.2)impliesthatfor a shock-wave type solution we must have uL >uR. Theorem 2.1. Given initial states vL = (uL,σL) and vR = (uR,σR) and a DLM path φ=(φ ,φ ), the R-H condition (1.5) for the system (1.1) is satisfied if 1 2 1 [u2]−[σ] φ1(t;uL,uR)(φ2)t(t;σL,σR)dt=[σ]. 2 [u] , (2.2) Z0 where [u] = (uR −uL) denotes the jump in u and s = [u22[]u−][σ] is the speed of the shock. u −1 Proof. We recall that for the system (1.1) we have A(u,σ)= . Substi- 0 u (cid:18) (cid:19) tuting this in (1.5), we obtain 1 −s+φ1 −1 (φ1)t =0. 0 0 −s+φ1 (φ2)t (cid:18) (cid:19)(cid:18) (cid:19) Thus we have the relations R 1 −s(φ1)t+φ1(φ1)t−(φ2)t dt=0, (2.3) Z0 and 1 −s(φ2)t+φ1(φ2)t dt=0. (2.4) Z0 Now (2.3) on simplification (using the fact φ1(0) = uL, φ1(1) = uR, φ2(0) = σL, φ2(1)=σR) gives s= [u22[]u−][σ]. Substituting this in (2.4) we obtain 1 [u2]−[σ] φ1(t;uL,uR)(φ2)t(t;σL,σR)dt=[σ]. 2 [u] . Z0 (cid:3) Proceeding similarly as in the above theorem, we can prove the corresponding result for the system (1.2). Theorem 2.2. Given initial states vL = (uL,σL) and vR = (uR,σR) and a DLM path φ=(φ ,φ ), the R-H condition (1.5) for the system (1.2) is satisfied if 1 2 1 [u2]−[σ] φ1(t;uL,uR)(φ2)t(t;σL,σR)dt=[σ]. 2 [u] +k2[u], (2.5) Z0 where [u] = (uR −uL) denotes the jump in u and s = [u22[]u−][σ] is the speed of the shock. 4 ANUPAMPALCHOUDHURY Proof. Asimilar calculationasin the proofofTheorem2.1with the corresponding u −1 matrix A(u,σ)= for the system (1.2) gives the result. (cid:3) −k2 u (cid:18) (cid:19) 3. Existence of DLM paths satisfying the Rankine-Hugoniot condition In this section, we prove the existence of DLM paths satisfying the conditions (2.2) and (2.5) for the systems (1.1) and (1.2) respectively. In particular, we show thatgivenaleftstatevL =(uL,σL)thepathsφandφ˜mentionedintheintroduction give rise to shock curves passing through vL. Thus for any state vR = (uR,σR) (with uR < uL) lying on these curves, we can solve the Riemann problem using a shock wave with speed s as given in the previous section. Theorem 3.1. Given a left state vL =(uL,σL), the DLM path φ defined in (1.3) gives a shock-wave solution for the system (1.1) with the right states vR =(uR,σR) on the shock curve 1 S1 :σ =σL− (u−uL)2, u<uL. (3.1) 4 Proof. Theresultfollowsfromastraightforwardcalculationbysubstitutingφ and 1 φ from (1.3) in (2.2). (cid:3) 2 Similar to the above theorem for the system (1.2) we have the following result. Theorem 3.2. Given a left state vL =(uL,σL), the DLM path φ defined in (1.3) gives a shock-wave solution for the system (1.2) with the right states vR =(uR,σR) on the shock curves 1 S1 :σ =σL− ((u−uL)2− (u−uL)4+64k2(u−uL)2), u<uL, (3.2) 8 p 1 S2 :σ =σL− ((u−uL)2+ (u−uL)4+64k2(u−uL)2), u<uL, (3.3) 8 p Proof. Substituting φ andφ from(1.3)in(2.5)weobtainthe followingquadratic 1 2 equation in [σ]: 4[σ]2+(uR−uL)2[σ]−4k2(uR−uL)2 =0. TheaboveequationwhensolvedgivesustherequiredexpressionsforS andS . (cid:3) 1 2 Remark 3.3. As k tends to 0, the shock curve S defined in (3.3) for the system 2 (1.2) tends to the shock curve for the system (1.1) given by (3.1), while the other shock curve S degenerates. 1 Next we state the analogous results for the DLM path φ˜defined in (1.4). Theorem 3.4. Given a left state vL =(uL,σL), the DLM path φ˜ defined in (1.4) gives a shock-wave solution for the system (1.1) with the right states vR =(uR,σR) on the shock curve 1 S1 :σ =σL− (u−uL)2, u<uL. (3.4) 2 Proof. Theresultfollowsfromastraightforwardcalculationbysubstitutingφ˜ and 1 φ˜ from (1.4) in (2.2). (cid:3) 2 SHOCK-WAVES FOR SOME NON-CONSERVATIVE SYSTEMS 5 Theorem 3.5. Given a left state vL =(uL,σL), the DLM path φ˜ defined in (1.4) gives a shock-wave solution for the system (1.2) with the right states vR =(uR,σR) on the shock curves 1 S1 :σ =σL− ((u−uL)2− (u−uL)4+16k2(u−uL)2), u<uL, (3.5) 4 1 p S2 :σ =σL− ((u−uL)2+ (u−uL)4+16k2(u−uL)2), u<uL. (3.6) 4 Proof. Substituting φ˜ andφ˜ fromp(1.4)in(2.5)weobtainthe followingquadratic 1 2 equation in [σ]: 2[σ]2+(uR−uL)2[σ]−2k2(uR−uL)2 =0. TheaboveequationwhensolvedgivesustherequiredexpressionsforS andS . (cid:3) 1 2 Remark 3.6. As k tends to 0, the shock curve S defined in (3.6) for the system 2 (1.2) tends to the shock curve for the system (1.1) given by (3.4), while the other shock curve S degenerates. 1 4. Conclusion Thus we have shown the existence of shock-wavesolutions for the systems (1.1) and (1.2) using DLM paths. This result is of more significance in the case of (1.1) due to the parabolic degeneracy it exhibits. We also remark that a rarefaction wave solution for the system (1.1) with Riemann type initial data is not possible. ThereforetosolvetheRiemannproblemforarbitraryinitialdatawemighthaveto consider the class of singular solutions. We refer to [3] for a discussion on singular solutions for the system (1.1). We have also shown the existence of more than one DLM paths which give rise to shock-wave solutions. As reflected here, different choices of DLM paths in general give rise to different solutions. References [1] G.Dal Maso,P.G.Lefloch,F.Murat, Definition and weak stability of non-conservative prod- ucts,J. Math. Pures Appl.74(1995), 483-548. [2] K.T.Joseph,P.L.Sachdev, Exact solutions for some non-conservative hyperbolic systems,Internat. J. Non-LinearMech.38(2003), 1377-1386. [3] A.PalChoudhury,Singularsolutionsfora2X2systeminnonconservativeformwithincomplete setofeigenvectors,arXiv:1212.0664[math.AP](2012). [4] A.I.Volpert,ThespaceBVandquasilinearequations,Math. USSR Sb. 2(1967), 225-267. [5] Y.Zheng, Systems of conservation laws with incomplete sets of eigenvectors everywhere. Ad- vancesinnonlinearpartialdifferentialequationsandrelatedareas(Beijing,1997),World Sci. Publ., RiverEdge, NJ.(1998),399-426. AnupamPalChoudhury TIFR Centre for Applicable Mathematics, Sharadanagar, Chikkabommasandra, Yela- hankaNew Town,Bangalore560065,India E-mail address: [email protected]