AXISYMMETRIC SOLUTIONS OF A TWO-DIMENSIONAL NONLINEAR WAVE SYSTEM WITH A TWO-CONSTANT EQUATION OF STATE

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1 Eectronic Journa of Differentia Equations, Vo. 07 (07), No. 56, pp. 8. ISSN: URL: or AXISYMMETRIC SOLUTIONS OF A TWO-DIMENSIONAL NONLINEAR WAVE SYSTEM WITH A TWO-CONSTANT EQUATION OF STATE GUODONG WANG, YANBO HU, HUAYONG LIU Abstract. We study a specia cass of Riemann probem with axisymmetry for two-dimensiona noninear wave equations with the equation of state p = A ρ γ + A ρ γ, A i < 0, 3 < γ i < (i =, ). The main difficuty ies in that the equations can not be directy reduced to an autonomous system of ordinary differentia equations. To sove it, we use the axisymmetry and sefsimiarity assumptions to reduce the equations to a decouped system which incudes three components of soution. By soving the decouped system, we obtain the structures of the corresponding soutions and their existence.. Introduction This artice concerns a specia cass of Riemann probem to the two-dimensiona noninear wave system ρ t + (ρu) x + (ρv) y = 0, with the equation of state (ρu) t + p x = 0, (ρv) t + p y = 0, (.) p(ρ) = A ρ γ + A ρ γ, (.) where the variabes (u, v), ρ represent the veocity and the density respectivey, and A i < 0, 3 < γ i < (i =, ). System (.) can be obtained either by starting with the isentropic gas dynamics equations and negecting the quadratic terms in the veocity or by writing the noninear wave equation as a first-order system [, ]. If the equation of state is taken as the foowing form p = Aρ γ (.3) with γ < 0 and A < 0, (.3) is caed as the generaized Chapygin gas. For the case γ = and A =, (.3) was introduced by Chapygin[4], Tsien and von arman [, 4] as a suitabe mathematica approximation for cacuating the ifting force on a wing of an airpane in aerodynamics. Sen and Scherrer generaized this mode to aow for the case γ <. The transient Chapygin gas mode provides a 00 Mathematics Subject Cassification. 35L65, 35J70, 35R35. ey words and phrases. Noninear wave system; generaized Chapygin gas; axisymmetry; decouped system. c 07 Texas State University. Submitted February 7, 07. Pubished June 8, 07.

2 G. WANG, Y. HU, H. LIU EJDE-07/56 possibe mechanism to aow for a currenty acceerating universe without a future horizon and can be taken to be a mode for dark energy aone []. The initia data for a genera Riemann probem are constant aong radia direction from the origin and piecewise constant as a function of ange. There are a set of conjectures offered by Zhang and Zheng in [6] for the soutions to the twodimensiona Riemann probem. Foowing Zhang and Zheng s work, many efforts have been made to prove these conjectures for the compressibe Euer equations or some reduced systems in the past twenty years [9, 9]. Unfortunatey, unti now, none of them has been competey proved due to the compicated structure of soutions. For reated resuts, we can consut the survey [0] and references cited therein. The study of two-dimensiona Riemann probem (or shock refection) have been devoted for system (.) with genera poytropic gas [, 6,,, 4, 5, 6, 7, 8, 3]. Čanié, eyfitz, im studied the Mach stem of shock soution by soving a free boundary probem []. Tesda, Sanders and eyfitz presented numerica soutions for the noninear wave system which is used to describe the Mach refection of weak shock waves [3]. In particuar, im and Lee studied the configuration that the transonic shock interacted with the sonic circe and obtained a goba transonic soution to this configuration. im estabished a goba transonic soution to the interaction of a transonic shock with a rarefaction wave[4, 5, 6, 7, 8]. Furthermore, Chen et a. [6] estabished a goba theory of existence and optima reguarity for a shock diffraction probem. Hu ang Wang [, ] studied the semihyperboic patches of soutions to the two-dimensiona noninear wave system for Chapygin gases and constructed a goba cassica soution to the interaction of two arbitrary panar rarefaction waves. For a specia cass of Riemann probem with axisymmetry for the two-dimensiona isentropic Euer equations with the equation of state p = Aρ γ, γ >, Zhang and Zheng constructed rigorousy a three-parameter famiy of sef-simiar, gobay bounded, and continuous weak soutions for a positive time to the Euer equations [7, 8, 30]. Under the assumption that v = 0, Hu [0] constructed a famiy of sef-simiar and goba bounded weak soutions to the two-dimensiona isentropic Euer equations with the equation of state (.) and A i > 0, γ i > (i =, ) for axisymmetry initia data. As for the reated works, we refer the reader to [5, 7, 8, 9]. As a continuation of the paper [0], we wi consider the system with the equation of state ρ t + (ρu) x + (ρv) y = 0, (ρu) t + p x = 0, ϕ t + (ϕu) x + (ϕv) y = 0, (ρv) t + p y = 0, (.4) p = p + p = A ρ γ + A ϕ γ, (.5) for axisymmetry initia data, where p = A ρ γ, p = A ϕ γ, A i < 0, 3 < γ i < (i =, ), ρ 0 and ϕ 0. It is easy to see that, if ϕ = ρ, γ = γ and A = A, then system (.4) reduces to the system in [5]. The purpose of the present paper is to investigate the two-dimensiona Riemann probem with axisymmetry to the noninear wave system (.) and (.). We wi construct rigorousy a famiy of sefsimiar and goba soutions for a positive time to (.4) with (.5). The difference

3 EJDE-07/56 AXISYMMETRIC SOLUTIONS OF THE D NWS 3 with this paper [0] is that we do not assume v = 0 here, since we may decoupe the first three equations from the fourth equation in (.4) by the axisymmetry and sef-simiarity assumptions. Moreover, we find that the shock soutions appear in the case u 0 > 0, instead of u 0 < 0. And then, we obtain the detaied structures of soutions as we as their existence for system (.4) with (.5). If ϕ = ρ at time zero in the soutions obtained from system (.4), we may obtain the existence of soutions and their structures for the specia cass of Riemann probem to system (.) with (.). The rest of this paper is organized as foows. In section, we present some preiminaries, give axisymmetry induction, intermediate fied equations and the Rankine-Hugoniot reations. The goba soutions for two cases u 0 0 and u 0 > 0 are constructed in sections 3 and 4, respectivey. The concusion is deivered in Section 5.. Preiminaries We impose axisymmetry to the system (.4). That is that the soutions (ρ, u, v, ϕ) have the property ( u(t, r, θ) v(t, r, θ) ρ(t, r, θ) = ρ(t, r, 0), ϕ(t, r, θ) = ϕ(t, r, 0), ) ( cos θ sin θ = sin θ cos θ ) ( u(t, r, 0) v(t, r, 0) ), (.) for a t 0, θ R and r > 0, where (r, θ) are the poar coordinates of the (x, y)-pane. Substituting (.) into (.4) for the smooth soutions reduces ρ t + (ρu) r + ρu r = 0, (ρu) t + (p + p ) r = 0, ϕ t + (ϕu) r + ϕu r = 0, (ρv) t = 0, (.) where u = u(t, r, 0) and v = v(t, r, 0) are the radia and pure rotationa veocities, respectivey. We require the Riemann initia data as foows ( ρ(0, r, θ), u(0, r, θ), v(0, r, θ), ϕ(0, r, θ) ) = ( ρ0 (θ), u 0 (θ), v 0 (θ), ϕ 0 (θ) ), which impies, by the axisymmetry condition (.), that our data are imited to ρ(0, r, θ) = ρ 0, ϕ(0, r, θ) = ϕ 0, u(0, r, θ) = u 0 cos θ v 0 sin θ, v(0, r, θ) = u 0 sin θ + v 0 cos θ, (.3) where ρ 0 > 0, ϕ 0 > 0, u 0 and v 0 are arbitrary constants.

4 4 G. WANG, Y. HU, H. LIU EJDE-07/56 Since the probem (.) is invariant under sef-simiar transformations, we ook for sef-simiar soutions (ρ, u, v, ϕ)(ξ)(ξ = r/t). Thus, we obtain from (.) that and ρu ρ ξ = p + ϕ, ξ u ξ = u(p + ϕ ξu) ξ(p + ϕ ξ ), ϕ ξ = (γ )ϕu p + ϕ, ξ uv v ξ = p + ϕ, ξ (.4) im ξ + (ρ, u, ϕ, v) = (ρ 0, u 0, γ p (ϕ 0 )/ρ 0, v 0 ), (.5) where ϕ = γ p (ϕ)/ρ. It is easy to see that the first three equations of (.4) do not invove the component v. Hence, we consider a subsystem for the initia pure radia fow ρu ρ ξ = p + ϕ, ξ and u ξ = u(p + ϕ ξu) ξ(p + ϕ ξ ), ϕ ξ = (γ )ϕu p + ϕ ξ, (.6) im ξ + (ρ, u, ϕ) = (ρ 0, u 0, γ p (ϕ 0 )/ρ 0 ). (.7) In this paper, by (.6) and (.7), we wi construct goba soutions to the probem (.4) and (.5) for any ρ 0 > 0, ϕ 0 > 0 and u 0 R, v 0 R... Far-fied soutions and intermediate fied equations. Let s = /ξ, then (.6) and (.7) become and dρ ds = ρu s (p + ϕ), du ds = u(sp + sϕ u) s (p + ϕ), dϕ ds = (γ )ϕu s (p + ϕ), (.8) (ρ, u, ϕ) s=0 = (ρ 0, u 0, γ p (ϕ 0 )/ρ 0 ). (.9) We can see that (.8) is we-posed and has a unique oca soution for any initia datum with ρ 0 > 0 and ϕ 0 > 0. By introducing the variabes I = su, = s p (ρ), H = s ϕ, (.0)

5 EJDE-07/56 AXISYMMETRIC SOLUTIONS OF THE D NWS 5 system (.8) can be put into the form s di ds = I( + I H ) H, s d ds s dh ds = ( γ I H ) H, = H( γ I H ) H. Introducing a new parameter τ, it is easy to get from (.) that (.) di dτ = I( + I H ), d dτ = ( γ I H ), (.) dh dτ = H( γ I H ). and ds dτ = s( H ). (.3) Corresponding to the initia data (.0), we wi ook for soutions of (.) and (.3) with the initia condition (I,, H) s(u 0, p (ρ 0), γ p (ϕ 0 )/ρ 0 ), as s 0 +. (.4).. Rankine-Hugoniot reation. In sef-simiar coordinates (ξ, η) = (x/t, y/t), system (.4) can be rewritten as ξρ ξ ηρ η + (ρu) ξ + (ρv) η = 0, ξ(ρu) ξ η(ρu) η + (p + p ) ξ = 0, ξϕ ξ ηϕ η + (ϕu) ξ + (ϕv) η = 0, ξ(ρv) ξ η(ρv) η + (p + p ) η = 0. Let a discontinuous curve be given η = η(ξ) and the sope of the curve be denoted by σ = η (ξ), then the Rankine-Hugoniot reation is σ = [(η v)ρ], [ξρu p p ]σ = [ηρu], [(u ξ)ϕ]σ = [(η v)ϕ], [ξρv]σ = [ηρv p p ], (.5) where σ = η (ξ) and [h] = h r h, h, h r are the imit states on the eft- and righthand sides of the discontinuity curve η = η(ξ), respectivey. When axisymmetry is given, the discontinuity curve has an infinite sope at the ξ-axis, that is σ = in (.5). So we can obtain ξ[ρ] = [ρu], ξ[ρu] = [p + p ], ξ[ϕ] = [ϕu], [ρv] = 0,

6 6 G. WANG, Y. HU, H. LIU EJDE-07/56 which yieds the sip ine ξ = 0, p + p = p r + p r, and the forward or backward discontinuity ( [p + p ] ) /, ξ = ± [ρ] (u r ξ)ρ r = (u ξ)ρ, (u r ξ)ϕ r = (u ξ)ϕ, ρ r v r = ρ v. Recaing ξ = /s and (.0), the system (.6) can be transformed to γ r ( I r )H γ γ ( I r ) r = ( I ) γ r = γ ( γ γ r γ, γ (γ ) γ r = ( I )H γ γ ) ( + H γ r ρ r v r = ρ v. γ (γ ), γ r H γ ), (.6) (.7) 3. Soutions for u 0 0 In this section, we construct goba bounded continuous soutions to probem (.6) and (.7) with initia negative radia veocity. Here we ony consider the case γ γ, the resut for case γ = γ, which corresponds to the generaized Chapygin gas, can be found in [5]. If u 0 = 0, we get from (.8) that ρ = ρ 0, ϕ = γ p (ϕ 0 )/ρ 0 and u = 0 is a trivia soution. So we ony consider the case u 0 < 0 in the present section and divide this into two cases: 3 < γ < γ < and 3 < γ < γ <. We introduce A := γ I H, B := + I H, Then (.) and (.3) become C := γ I H, D := H. di dτ = IB, d dτ = C, ds dτ = D. dh dτ = HA, 3.. Case one: 3 < γ < γ <. We denote the set { Ω : I < 0, > 0, H > 0; B > 0 for < I /γ, } (3.) A > 0 for /γ I < 0, see Figure. It is not difficut to show that the far-fieds soutions starting at s = 0 + enter the region Ω and do not eave Ω as s increases. Notice that s > 0 is increasing

7 EJDE-07/56 AXISYMMETRIC SOLUTIONS OF THE D NWS 7 A=0 C=0 Q E I B=0 /γ /γ H Q Figure. Integra curves for u 0 < 0 in Ω. function of τ in Ω by (3) and one can find that the stationary points of (3) in the cosure Ω are (H, I, ) = (, 0, 0), (H, I, ) = (0, 0, ), (H, I, ) = (0,, 0), Q = (0, γ, +γ γ ), Q = ( +γ γ, γ, 0), and the points on the edge E : + H =, I = 0, > 0, H > 0. Therefore, there is no stationary point in the open region Ω and a the stationary points are on the boundary of Ω. Lemma 3.. Soutions inside Ω do not eave Ω from its sides (excuding possiby edge or corners) as s increases. Proof. It is easy to see from (3) that the sides of Ω in the surfaces H = 0, I = 0 or = 0 are invariant regions. We need ony to verify that no soution eaves Ω from the foowing surfaces à and B à := { (H, I, ) : A = 0, H > 0, > 0, /γ < I < 0 }, B := { (H, I, ) : B = 0, H > 0, > 0, < I < /γ }. We easiy obtain that in the coordinate order (H, I, ) the outward norma of surface B is given by n eb = ( 4H,, 4 ). Noting that A < 0 and C < 0 and cacuating the inner product of the norma n eb with the tangent vector of an integra curve of (3) on the surface B yieds ( dh n eb dτ, di dτ, d ) = 4H A IB + 4 C < 0, dτ

8 8 G. WANG, Y. HU, H. LIU EJDE-07/56 which impies that no soution eaves Ω from the surface B as s increases. An outward norma of the surface à is given by n ea = ( γ H,, ) in the order (H, I, ). We simiary compute the inner product of the norma n ea to obtain with the tangent vector of the integra curve on the surface n ea ( dh n ea dτ, di dτ, d ) dτ = H A + γ IB + C < 0, because B < 0, C < 0 and γ < 0 on the surface Ã. Hence, no soution eaves Ω from the surface Ã. Thus, we compete the proof. Now, we study the oca structure of integra curves at the stationary points of (3). We first notice that no integra curve from inside Ω goes to (H, I, ) = (0, 0, ) or Q, since the component H, by (3), is an increasing function of τ in Ω for I ( γ, 0 ). Setting Ĥ = H + γ γ, Î = I γ, ˆ =, and inearizing system (3) at point Q yieds dî dτ = Î 4 + γ Ĥ, γ γ γ dĥ dτ = γ We compute the eigenvaues to get d ˆ dτ = γ γ ˆ, γ + γ γ Î + γ γ Ĥ. λ 0 Q = γ γ γ < 0, λ Q = + γ γ < 0, λ + Q =, (3.) which indicate that the stationary point Q is hyperboic. Simiary, we obtain that the stationary point Q is aso hyperboic. For each integra curve ending at Q, we find that s + and thus ξ 0 + as τ + due to D = γ γ > 0. Hence, this famiy of integra curves (caed the transitiona soutions) yieds smooth soutions of probem (.6) and (.7) in the entire region ξ (0, + ). Now, we inearize system (3) at stationary points on curve E {(H, I, ) = (, 0, 0)}. Setting for 0 < α, one can obtain Ĥ = H α, Î = I, ˆ = α dî dτ = Î, d ˆ dτ = γ α Î ( α ) ˆ α α Ĥ, dĥ dτ = γ αî α α ˆ α Ĥ. (3.3)

9 EJDE-07/56 AXISYMMETRIC SOLUTIONS OF THE D NWS 9 which has three eigenvaues λ =, λ = and λ 3 = 0. So, soutions of (3) near E {(H, I, ) = (, 0, 0)} approach E {(H, I, ) = (, 0, 0)} exponentiay as τ +. From equation (3), we find ( s ) τ ( n = H ) dτ, s 0 τ 0 where s 0 = s(τ 0 ) > 0. Hence, s approaches a finite number as τ + since H approaches zero exponentiay. We inearize system (3) at point (H, I, ) = (0,, 0) and get dî dτ = Î, d ˆ dτ = + γ ˆ, dĥ dτ = + γ Î, which has three eigenvaues λ =, λ = +γ < 0 and λ 3 = +γ < 0. Simiary, we concude that the parameter s approaches a finite number as τ +. We now depict the integra curves inside Ω in Figure. It is usefu to observe that there exists a stabe manifod of system (3) at the point Q (see [3, 3] for detais) which contains the transitiona curve in H = 0, the transitiona integra curve in = 0 and the heterocinic orbit from the point Q. There are three kinds of integra curves reative to this manifod. The first kind consists of integra curves that are beow the manifod and go to the stationary point (0,, 0). Each of the second king is right on the manifod and goes to the stationary point Q. The third kind consists of integra curves that are above the manifod and go to E {(, 0, 0)}. None of the integra curves from inside Ω goes to point (H, I, ) = (0, 0, ) or Q. We construct goba soutions for (.) when u 0 < 0 and 3 < γ < γ <. For each integra curve ( which ends at point (H, I, ) = (0,, 0), there exists a ( ) soution (/ρ, u, ϕ) = ) γ A γs, I/s, H /s defined for s (0, s ] for some positive number s < +, and then continue the soution by /ρ = 0, ϕ = 0 in s [s, + ). We here do not need to specify the function u in the above state since system (3) has /ρ or ϕ as a factor in every term. For each integra curve ending at a point on E {(, 0, 0)}, there exists a soution defined for s (0, s ] for some positive s < +. We next continue the soution by the constant state (/ρ, u, ϕ) = (/ρ(s ), 0, ϕ(s )) in s [s, + ). 3.. Case two: 3 < γ < γ <. It can be verified that a far-fied soutions of system (.) with u 0 < 0, ρ 0 > 0 and ϕ 0 > 0 enter the domain Ω in s > 0 where Ω R 3 be the set of points (H, I, ) satisfying { I < 0, > 0, H > 0; B > 0 for < I /γ, Ω : } (3.4) A > 0 for /γ I < 0, see Figure. In this case, system (3) has simiar stationary points with the case γ < γ. Anaogousy, there is no stationary point in the interior of Ω. So, we can get the foowing emma.

10 0 G. WANG, Y. HU, H. LIU EJDE-07/56 C=0 A=0 E I Q B=0 /γ /γ Q H Figure. Integra curves for u 0 < 0 in Ω. Lemma 3.. Soutions inside Ω do not eave Ω from its sides (excuding possiby edge or corners) as s increases. Proof. It is easy to see from (3) that the sides of Ω in the surfaces H = 0, I = 0 or = 0 are invariant regions. We need ony to verify that no soution eaves Ω from the foowing surfaces B and C B := {(H, I, ) : B = 0, H > 0, > 0, < I < }, γ C := { (H, I, ) : C = 0, H > 0, > 0, } < I < 0. γ Anaogous to the proof of Lemma 3., we can obtain that no soution eaves Ω from the surface B as s increases. An outward norma of the surface C is n ec = ( γ H,, ) in the order (H, I, ). We directy compute on the surface C ( dh n ec dτ, di dτ, d ) = H A + γ IB + C < 0, dτ since A < 0, B < 0 and γ < 0 on the surface C. Hence, no soution eaves Ω from the surface C. These above indicate that the Lemma is true. Simiar to the case γ < γ, no integra curve inside Ω goes to points (H, I, ) = (, 0, 0) or Q, since the function is an increasing function of τ in Ω for I (0, /γ ) by (3). For the oca structure of soutions at E {(0, 0, )} and the point (H, I, ) = (0,, 0), we may see the reated resuts for the case γ < γ and here

11 EJDE-07/56 AXISYMMETRIC SOLUTIONS OF THE D NWS omit the detais. We next consider the oca structure of soutions at the stationary point Q. Setting and inearizing system (3) gives which three eigenvectors are Ĥ = H, Î = I γ, ˆ = + γ γ, dî dτ = + γ d ˆ dτ = γ Î 4 + γ ˆ, γ γ γ + γ Î + γ ˆ, γ γ dĥ dτ = γ γ γ Ĥ, λ Q = γ γ γ, λ ± Q = ( + γ ) ± ( + γ )(5γ 7) 4γ. In this case, we easiy see that λ Q < 0, λ Q < 0 and λ + Q > 0, which indicates that the stationary point Q is hyperboic. The hyperboicity of the stationary point Q can be estabished in a simiar way. The integra curves in Ω in Figure can be depicted as foows. There aso exists a stabe manifod of system (3) for 3 < γ < γ < at the point Q which contains the transitiona integra curve in H = 0, the transitiona integra curve in = 0 and the heterocinic orbit from the point Q. There are three kinds of integra curves reative to this stabe manifod. The first kind consists of integra curves that are beow the manifod and go to the stationary point (0,, 0). Each of the second kind is right on the manifod and goes to the stationary point Q. The third kind consists of integra curves that are above the manifod and go to E {(0, 0, )}. None of the integra curves from inside Ω goes to (, 0, 0) or Q. Now the construction of the goba soutions of system (3) for u 0 < 0 and 3 < γ < γ <. For each integra curve ending at (H, I, ) = (0,, 0), we continue the soution by vacuum state /ρ = 0, ϕ = 0 in s [s 3, + ) for some positive s 3 < +. We aso do not specify the function u in the vacuum. For each soution curve ending on E {(0, 0, )}, we continue the soution by the constat state (/ρ(s 4), 0, ϕ(s 4)) in s [s 4, + ) for some positive finite number s 4. The integra curves ending at point Q (that is the second kind) are aready defined for a s > 0 since τ + and the right hand side of equation (.3)) does not vanish at Q. Furthermore, from the ast equation of (.4), we have ( s u ) v(s) = v 0 exp 0 H ds, (3.5) which is a finite number for u < 0 and H > 0. In summary, we have constructed goba bounded continuous soutions to system (.4) and (.5) in the case u 0 0.

12 G. WANG, Y. HU, H. LIU EJDE-07/56 4. Soutions for u 0 > 0 In this section, we construct goba soutions for probem (.6) and (.7) with initia positive pure radia veocity for the case γ < γ. For the other case γ > γ, we can obtain the same resuts, simiary. It is not difficut to prove that the integra curves of system (3) starting at the origin with u 0 > 0, ρ 0 > 0 and ϕ 0 > 0 enter the region Ω 3 := Ω 3 Ω 3, where Ω 3 := {(H, I, ) : H > 0, I > 0, > 0, B > 0, 0 < I < }, Ω 3 := {(H, I, ) : H > 0, I > 0, > 0, D > 0, I > }, see Figure 3. Denote Ω 3 := {H > 0, I > 0, > 0, D < 0, B < 0}. Obviousy, according to (3), the variabe s is increasing in region Ω 3. We concude, however, that the integra curves in Ω 3 do not aways ie in it. I E B=0 O C=0 H Figure 3. Integra curves for u 0 > 0 in the case γ < γ. Lemma 4.. Each integra curve of system (3) from the origin passes through the surface D = 0 or B = 0 at a finite point (τ, H, I, ) and enters the domain Ω 3, and then crosses the surface C = 0 at a finite τ from C > 0 to C < 0. Proof. To demonstrate the first part of the Lemma, we divide it into two parts. () If the integra curves from the origin do not enter Ω 3. It is easy to obtain from B > 0 that ( + H ) < + I, from which, we find that C = γ I ( + H ) ( + H ) γ + I.

13 EJDE-07/56 AXISYMMETRIC SOLUTIONS OF THE D NWS 3 From the equation for, we find that d = C 3 dτ which impies that bows up at a finite τ = ˆτ. Combining this and the fact 0 < I <, we concude that the integra curve must pass through the surface B = 0 and D = 0, and then enters the domain Ω 3. () If the integra curves from the origin eave the region Ω 3 and then enter the region Ω 3. Simiar to the above procedure, the soution curves must pass through the surface D = 0 at finite τ. Now, we prove that the above curves must pass through the surface B = 0 at a finite τ, and then enter the domain Ω 3. Suppose, on the contrary, such an integra curve is aways under the surface B = 0. It foows from the I equation that I is increasing for a τ. Using the equation, B > 0 and I >, we have d n = γ I ( + H ) γ I, dτ which means that τm ( γ I ) dτ = +, τ 0 for some positive number τ m, from which one can achieve τm ( + γ τ 0 We arrange the equation in the foowing form τ 0 I + + H ) dτ = +. (4.) d dτ = ( B + γ I + + H ). Integrating the above equaity with respect to τ from τ 0 to τ yieds = 0 exp ( τ Bdτ ) ( τ ( + γ exp I + + H ) ) dτ, (4.) τ 0 where 0 = (τ 0 ). One can aso obtain from the I equation that ( τ ) I = I 0 exp Bdτ, (4.3) where I 0 = I(τ 0 ). Combining (4.) with (4.) and (4.3) gives I = ( τ 0 ( + γ exp I + + H ) ) dτ +, as τ τ m. I 0 τ 0 On the other hand, one can obtain from the assumption B > 0 and the fact I > that < ( + H ) < + I < ( + I), which gives < + I I. The right-hand side of the above inequaity is bounded. So we get a contradiction. Next, we prove the second part of the Lemma, that is that a integra curves from Ω 3 cross the surface C = 0 at a finite τ. One can obtain that a integra curves from Ω 3 in I > 0 must end at E := E {(, 0, 0), (0, 0, )}. By the inearization τ 0

14 4 G. WANG, Y. HU, H. LIU EJDE-07/56 equation (3.3) of (3), we get that the eigenvaues of (3.3) are λ =, λ = and λ 3 = 0, and the corresponding eigenvectors are in turn r = (α, 0, α ), r = (α [ ( + γ γ ) (γ γ )α ],, [( γ ) (γ γ )α ] ) α, r 3 = ( α, 0, α ) for 0 α. Since γ < + γ for 3 < γ < γ <, we compute the inner product of the norma n ec with the vector r on E to obtain n ec r = α [ (γ γ )( α ] ) + γ + ( α ) [ ( γ ) (γ γ )α ] + (γ )/ = (γ γ )α + ( γ )/ > 0, which indicates that each integra curve from Ω 3 wi pass through the surface C = 0 at a finite τ and then go to a stationary point on E aong the direction r. Using (3), we can see that the variabe s(τ) starts to decrease as τ increases in the domain {D < 0}, which means that no continuous soutions exist in the case u 0 > 0. Hence we need to use discontinuous soutions to construct goba soutions. Denote the subscripts r and the front (the side cose to ξ = + ) and behind (the side cose to the origin) states, respectivey. Here we have ignored the backward discontinuous soutions since we wi not need it. The entropy condition requires that ρ r < ρ, or equivaenty, by (.7), r >, I r < I or H r > H. (4.4) Using the R-H reation (.7), we get the foowing emma. Lemma 4.. For any state (H, I, ) in the domain {D > 0} {I > 0}, there exists a state (H r, I r, r ) in the domain {D < 0} {I r > 0} satisfying the R-H reation (.7) and the entropy condition (4.4). Proof. It is easy to see from the third equation of (.7) that Setting we have ) = γ ( γ γ γ r ) ( γ γ r γ } {{} f(, r) + ( H γ γ r r H γ ) ( ) γ γ r. } {{} g(, r) γ x = r and x 0 = γ, f(, r ) := f(x 0, x) = x γ x γ 0. γ x x 0 Since ( x γ /γ ) = x γ > 0, ( x γ /γ ) = (γ )x γ < 0 for 0 < x < x 0 and 3 < γ <, we obtain from the entropy condition (4.4) that < f(, r ) < r. (4.5)

15 EJDE-07/56 AXISYMMETRIC SOLUTIONS OF THE D NWS 5 Now, we caim that In fact, from (.7), we have H = H < g(, r ) < H r. (4.6) ( where 0 < y = ( r / ) /(γ ) <. Thus which gives ) (γ ) γ Hr := y (γ ) Hr, r g(, r ) = y (γ ) H r y γ γ y, = y (γ ) H r < g(, r ) < H r. Combining (4.5) and (4.5) and (4.6) gives + H < < r + H r, (4.7) from which it foows that (H r, I r, r ) {D < 0} {I r > 0}. From the R-H reation and the entropy condition, we ony have three equations for four variabes (H(I ), I, (I )) and (H r, I r, r ), so we need one more condition to find a unique shock wave transition. Here H = H(I ), = (I ) are determined by the integra curve of system (3). According to (3), the direction of τ is opposite to that of s in the domain {D < 0}. Therefore, shock transition from a state (H, I, ) in D > 0 to a state (H r, I r, r ) on the pane I = 0 is the ony one eading to a goba soution, since the integra curve on the pane I = 0 yieds stationary constant ρ and ϕ soutions to system (.4). Thus, we require that I r = 0. (4.8) Lemma 4.3. For any integra curve in the quadrant {H > 0, I > 0, > 0} of the autonomous system (3), there exists a soution (H r, I r, r, H, I, ) with I > 0 satisfying the R-H reation (.7), the entropy condition (4.4) and the condition (4.8). Proof. We estabish this emma by the method of continuity. Since each integra curve of system (3) in the quadrant {H > 0, I > 0, > 0} crosses the surface D = 0 at a finite τ by Lemma 4., then if we take the cross point as (H, I, ), the soution to the R-H reation (.7) woud be the same point, i.e., (H, I, ) = (H r, I r, r ). Now moving from this point down the integra curve, we notice by the R-H reation (.7) that I r wi become minus when (H, I, ) is near the origin. In fact, if not, we then have r + Hr > by (4.7). Thus, one has by the R-H reation (.7) ( I )( γ + H γ γ (γ ) ) = ( I r )( ( γ r + H γ r γ r + H γ r γ (γ ) r γ (γ ) r ) ) < c for some positive constant c, which contradicts to the fact (H, I, ) (0, 0, 0). Hence, we have I r > 0 when the point (H, I, ) near the origin. By continuity, we must have a soution (H r, I r, r, H, I, ) with I > 0 for equations (.7) and (4.8) and the entropy condition (4.4).

16 6 G. WANG, Y. HU, H. LIU EJDE-07/56 From (3) and (3), we get a subsystem on the pane I = 0, that reads which means that d dτ = ( H ), dh dτ = H( H ), ds dτ = s( H ), d ds = s, dh ds = H s, which impy that s and H s are constant. Then a point (H, I, ) = (H r, 0, r ) on the pane I = 0 gives a constant soution to system (.8) u = 0, ρ = ( A γ ) γ (ξ r ) γ, ϕ = (ξ H r ), (0 < ξ < ξ ), where ξ = /s is the radia coordinate ξ of the shock ocation. We are now ready to construct goba soutions for (.8) in the case u 0 > 0. For any integra curve of (3), there exists a soution (H, I,, H r, I r, r ) satisfying Lemma 4.3 such that (H, I, ) on the integra curve and (H r, I r, r ) on the pane I = 0. For this integra curve from (H, I, ) = (0, 0, 0) to (H, I, ) = (H, I, ), we have a soution (/ρ, u, ϕ) = ( ( /( A γ s )) γ, I/s, H /s ) defined for s (0, s ] for some positive number s < + such that (H, I, ) = (H(s ), I(s ), (s )). The soution for s [s, + ) is specified as (/ρ, u, ϕ) = ((r /( A γ s )) γ, 0, Hr /s ). We use a shock wave to connect the two states (/ρ, u, ϕ) = ((r /( A γ s )) γ, 0, Hr /s ) and (/ρ, u, ϕ) = (( /( A γ s )) γ, I /s, H /s ) at s = s. Here s (i.e., the shock ocation ξ ) is determined by the R-H reation (.7), the entropy condition (4.4), the compatibiity condition (4.8) and the autonomous system (3). From the ast equation in (.7), we have the v component of the soution is v = ρ v ρ. 5. Concusions We first summarize the resuts for system (.4). Theorem 5.. For any datum (ρ 0, u 0, v 0, ϕ 0 ) with ρ 0 > 0 and ϕ 0 > 0, there exists a goba soution (ρ, u, v, ϕ) to the initia-vaue probem (.4) and (.3). Simiar to [0], we obtain from the system (.8) for ρ > 0, ϕ > 0 that dϕ ϕ = (γ ) dρ ρ or dϕ ϕ = dρ ρ, which means that our soutions satisfy ϕ/ρ = ϕ 0 /ρ 0 in the smooth case. If we restrict ϕ 0 = ρ 0 at initia time, then we have ϕ = ρ in the smooth case. Moreover, the constant states continued the smooth parts of soutions for (.4) are aso the constant soutions to system (.). One can easiy check that the shock soution of

17 EJDE-07/56 AXISYMMETRIC SOLUTIONS OF THE D NWS 7 system (.4) is aso a shock soution to system (.) with (.) if the initia data satisfies ϕ 0 = ρ 0. Therefore, we have the foowing theorem. Theorem 5.. For any datum (ρ 0, u 0, v 0 ) with ρ 0 > 0, there exists a goba soution (ρ, u, v) to the initia-vaue probem (.) and (.3) when p(ρ) = A ρ γ + A ρ γ for any constants A i < 0, 3 < γ i < (i =, ). Acknowedgements. G. D. Wang was supported by the Anhui Provincia Natura Science Foundation (508085M A08). Y. B. Hu was supported by the Nationa Science Foundation of China (308 and 57088), Zhejiang Provincia Natura Science Foundation (LY7A0009) and China Postdoctora Science Foundation (05M58086). References [] S. Čanié, B. L. eyfitz, E. H. im; Mixed hyperboic-eiptic systems in sef-simiar fows, Bo. Soc. Brasi. Mat. 3 (00), -3. [] S. Čanié, B. L. eyfitz, E. H. im; Free boundary probems for noninear wave systems: Mach stems for interacting shocks, SIAM J. Math. Ana. 37 (006), [3] J. Carr; Appications of centre manifod theory, Springer-Verag, New York, 98. [4] S. Chapygin; On gas jets, Sci. Mem. Moscow Univ. Math. Phys. (904), -. [5] G.-Q. Chen, J. Gimm; Goba soutions to the compressibe Euer equations with geometrica structure, Comm. Math. Phys. 80 (996), [6] G.-Q. Chen, X. Deng, W. Xiang; Shock Diffraction by Convex Cornered Wedges for the Noninear Wave System, Arch. Ration. Mech. Ana. (04), 6-. [7] L. Guo, W. Sheng; Axisymmetric soutions of the Chapygin gas for initia negative radia veocity, J. Shanghai University 4 (00), [8] L. Guo, W. Sheng; Axisymmetric soutions of the Euer system for the Chapygin gas (in Chinese), China Ann. Math. 3A (0), [9] Y. Hu; Axisymmetric soutions of the pressure-gradient system, J. Math. Phys. 53 (0) [0] Y. Hu; Axisymmetric soutions of the two-dimensiona Euer equations with a two-constant equation of state, Noninear Ana. Rea Word App. 5 (04), [] Y. Hu, G. Wang; Semi-hyperboic patches of soutions to the two-dimensiona noninear wave system for Chapygin gases, J. Differentia Equations 57(5) (04), [] Y. Hu, G. Wang; The interaction of rarefaction waves of a two-dimensiona noninear wave system, Noninear Ana. Rea Word App. (05), -5. [3] A. eey; The stabe, center stabe, center, center unstabe and unstabe manifods, J. Differentia Equations 3 (967), [4] E. H. im; A goba subsonic soution to an interaction transonic shock for the sef-simiar noninear wave equation, J. Differentia Equations 48 (00), [5] E. H. im; An Interaction of a Rarefaction Wave and a Transonic Shock for the Sef-Simiar Two-Dimensiona Noninear Wave System, Comm. Partia Differentia Equations 37 (0) [6] E. H. im, C.-M. Lee; Numerica soutions to shock refection and shock interaction probems for the sef-simiar transonic two-dimensiona noninear wave system, J. Comput. Sci. 4 (03), [7] E. H. im, C.-M. Lee; Transonic shock refection probems for the sef-simiar twodimensiona noninear wave system, Noninear Ana. 79 (03), [8] E. H. im; Transonic shock and rarefaction wave interactions of two-dimensiona Riemann probems for the sef-simiar noninear wave system, Bu. Braz. Math. Soc. 47 (06), [9] J. Li, T. Zhang, S. Yang; The two-dimensiona Riemann probem in gas dynamics, Longman, New York, 998. [0] J. Li, W. Sheng, T. Zhang, Y. Zheng; Two-dimensiona Riemann probems: from scaar conservation aws to compressibe Euer equations, Acta Math. Sci. 9 (009), [] A. A. Sen, R. J. Scherrer; Generaizing the generaized Chapygin gas, Phys. Rev. D 7(6) (005),

18 8 G. WANG, Y. HU, H. LIU EJDE-07/56 [] H. S. Tsien; Two dimensiona subsonic fow of compressibe fuids, J. Aeron. Sci. 6 (939), [3] A. M. Tesda, R. Sanders, B. L. eyfitz; The tripe point paradox for the noninear wave system, SIAM J. App. Math. 67 (006) [4] T. von ármán; Compressibiity effects in aerodynamics, J. Aeron. Sci. 8 (94) [5] G. Wang, Y. Hu; Axisymmetric soutions of the two-dimensiona noninear wave system with negative pressure, Submitted, 07. [6] T. Zhang, Y. Zheng; Conjecture on the structure of soutions of the Riemann probem for two-dimensiona gas dynamics systems, SIAM J. Math. Ana. (990), [7] T. Zhang, Y. Zheng; Exact spira soutions of the two-dimensiona Euer equations, Discrete Contin. Dyn. Syst. 3 (997), [8] T. Zhang, Y. Zheng; Axisymmetric soutions of the Euer equations for poytropic gases, Arch. Ration. Mech. Ana. 4 (998), [9] Y. Zheng; Systems of conservation aws: two dimensiona Riemann probems, Birkhauser, Boston, 00. [30] Y. Zheng; Absorption of characteristics by sonic curves of the two-dimensiona Euer equations, Discrete Contin. Dyn. Syst. 3 (009), Guodong Wang (corresponding author) Schoo of Mathematics and Physics, Anhui Jianzhu University of China, Hefei 3060, China E-mai address: gdwang@63.com Yanbo Hu Department of Mathematics, Hangzhou Norma University of China, Hangzhou 30036, China E-mai address: yanbo.hu@hotmai.com Huayong Liu Schoo of Mathematics and Physics, Anhui Jianzhu University of China, Hefei 3060, China E-mai address: iuhy@ahjzu.edu.cn