Simple waves of the two dimensional compressible full euler equations

Acta Mathematica Scientia 2015,35B(4):855–875 http://actams.wipm.ac.cn

SIMPLE WAVES OF THE TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS∗ Dedicated to Professor Tai-Ping Liu on the occasion of his 70th birthday

‰)

Yu CHEN (

±Á)

Yi ZHOU (

School of Mathematical Sciences, Fudan University, Shanghai 200433, China E-mail : [email protected]; [email protected] Abstract In this paper, we establish the existence of four families of simple wave solution for two dimensional compressible full Euler system in the self-similar plane. For the 2 × 2 quasilinear non-reducible hyperbolic system, there not necessarily exists any simple wave solution. We prove the result that there are simple wave solutions for this 4 × 4 nonreducible hyperbolic system, its simple wave flow is covered by four straight characteristics λ0 = λ1 , λ2 , λ3 and the solutions keep constants along these lines. We also investigate the existence of simple wave solution for the isentropic relativistic hydrodynamic system in the self-similar plane. Key words

simple waves; Euler equations; characteristic decomposition

2010 MR Subject Classification

1

35C06; 76D05; 35L72

Introduction We are interested in the two-dimensional compressible full Euler systems  ρt + (ρu)x + (ρv)y = 0,      (ρu)t + (ρu2 + p)x + (ρuv)y = 0, 2

 (ρv)t + (ρuv)x + (ρv + p)y = 0,     (ρE)t + (ρuE + up)x + (ρvE + vp)y = 0,

(1.1) (1.2) (1.3) (1.4)

where ρ stands for the density, (u, v) represents the two dimensional velocity, S denotes the entropy, the pressure is denoted by p(ρ, S) = A(S)ργ , where γ > 1 is the gas constant (see [3]), 2 2 and E = u +v + e denotes the total energy, e is the internal energy, as usual. 2 A basic feature of the Euler system is that, even for smooth initial data, the solution of the Cauchy problem may develop discontinuities in finite time[13]. It is well known that Riemann invariants are useful in the construction of solutions for hyperbolic system, for example, the construction of D’Alembert formula for the wave equation, and the proof of development of singularities [24]. For the system of two dimensional steady isentropic irrotational Euler system. Asignificant property is: A non-constant state of flow adjacent to a constant state is always a ∗ Received

February 4, 2015.

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simple wave by the existence of Riemann invariants [3]. Its simple wave is a type of flow whose flow region is covered by a one parametric family of straight characteristics, along each of which (u, v) and consequently (p, ρ, c) remain constant [3]. In 1990, T. Zhang and Y.X. Zheng[27] considered the two-dimensional Riemann problem for compressible Euler equations. They constructed the boundaries of interaction of four planar waves coming from infinity case by case. Each boundary consists of characteristics shocks and sonic curves. Based on the contents of one-dimensional gas dynamics systems, two-dimensional Riemann problem of scalar conservation[28] and reflection of shocks[5][6]. They formulated a set of conjectures of the wave patterns in the domains of interaction. For the two dimensional pseudo-steady isentropic irrotational Euler system, that governed by 2 × 2 self-similar system, whose coefficient matrix depends on ξ and η as well as (u, v). It turns out that, one is unable to find explicit forms of the Rimann invariants, so the treatment in [9] breaks down. In [16], ˘ c and Keyfitz explored simple wave further and generwith self-similar transformation, Cani´ alized Courant and Friedirichs’ theorem [3] by allowing the coefficients of a 2 × 2 system to depend on the independent variables (t, x) linearly as well as the dependent variables (u, v). In [18], J.Q. Li, T. Zhang and Y.X. Zheng, used the characteristic decomposition technique to study the quasilinear hyperbolic system. They clarified the simple waves for pseudo-steady isentropic irrotational Euler equation and proved that for two dimensional self-similar flow, a non-constant state adjacent to a region of constant state is always a simple wave. Furthermore, they extented irrotational results to the adiabatic full Euler systems, and proved a very important result: Adjacent to a constant state in the self-similar plane of the adiabatic Euler system is a simple wave, in which the physical variables (u, v, c, p, ρ) are constant along a family of wave characteristics which are straight lines. The characteristic decomposition as a powerful tool for building patches of global smooth solutions was first revealed for the pressure-gradient system by Z.H. Dai and Y.X. Zheng [10] and then used extensively for studying other systems to studied the interaction of rarefection waves [2, 7, 19–21]. In this paper, we study the compressible full Euler system (1.1)–(1.4) in the self-similar (ξ, η) = ( xt , yt ) plane as following:  (u − ξ)ρξ + (v − η)ρη + ρuξ + ρvη = 0, (1.5)      ρ(u − ξ)uξ + ρ(v − η)uη + pξ = 0, (1.6)  ρ(u − ξ)vξ + ρ(v − η)vη + pη = 0,     (u − ξ)pξ + (v − η)pη + γpuξ + γpvη = 0.

Setting U = u − ξ, V = v − η, this system is transformed to the following form  U ρξ + V ρη + ρuξ + ρvη = 0,      ρU uξ + ρV uη + pξ = 0,  ρU vξ + ρV vη + pη = 0,     U pξ + V pη + γpuξ + γpvη = 0.

(1.7)

(1.8)

(1.9) (1.10) (1.11) (1.12)

Solutions of the system (1.9)–(1.12) is called pseudo-steady or self-similar. Moreover, (1.9)–(1.12) can be rewritten as Wξ + A(W, ξ, η)Wη = 0,

(1.13)

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where W = (ρ, p, u, v)T and A(W, ξ, η) is a 4×4 variable coefficient matrix. There not necessarily exists any simple wave solution for 2×2 non-reducible system, thus Y.B. Hu and W.C. Sheng [11] by using the characteristic decomposition established a nice sufficient condition for the existence of characteristic decomposition to the general 2×2 non-reducible quasilinear hyperbolic system. For full non-reducible pseudo-steady Euler system, through generalized characteristic analysis, we obtain the eigenvalues: p U V ± c2 (U 2 + V 2 − c2 ) V λ0 = λ1 = , λ2,3 = , (1.14) U U 2 − c2 which depend on the all variables (U, V, ρ, p) and c is the sound speed. Through computation, they satisfy Burgers equation as discovered by J.Q. Li, T. Zhang and Y.X. Zheng[18]: ∂U λi + λi ∂V λi = 0

(i = 0, 1, 2, 3).

(1.15)

With the important equation (1.15), we prove there are simple wave solutions for the compressible full Euler system and get the following result: Theorem 1.1 For the compressible full Euler system of the self-similar form (1.9)–(1.12), if U + V 2 > c2 , then there are four families of simple wave solution in the plane (ξ, η), and simple wave flow region is covered by straight characteristics λ0 = λ1 , λ2 , λ3 , moreover all physical variables (ρ, p, u, v, c) keep constants along these lines. 2

For proving our main result, we introduce: ωi = Li (W, ξ, η)Wη

(i = 0, 1, 2, 3),

(1.16)

where W = (ρ, p, u, v)T and Li (W, ξ, η), (i = 0, 1, 2, 3) denotes the i-th left eigenvector of the system (1.13). Moreover we assume that Li (W, ξ, η)Rj (W, ξ, η) = δij

(i, j = 0, 1, 2, 3),

(1.17)

where δij stands for the Kronecker’s symbol. By (1.17), it is easy to see Wη =

3 X

ωk Rk (W, ξ, η).

(1.18)

k=0

With (1.16) and (1.18), we know ωi (i = 0, 1, 2, 3) and W (ξ, η) can be determined by each other. Then we can prove there are simple wave solutions of the full compressible Euler system, only if (ω0 , ω1 , ω2 , ω3 ) can determine a simple wave solution corresponding to the following specitial initial data: I : ω00 = ω10 = ω20 = 0, II :

ω00

III :

ω20

=

ω10

ω30

=

=

ω30

= 0.

= 0,

ω30 6= 0,

(1.19)

ω20

(1.20)

6= 0,

(1.21)

We also consider the existence of simple wave solution of two-dimensional isentropic relativistic

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hydrodynamic system in the self-similar plane (ξ, η) = ( xt , yt ),  ∂u b ∂p ∂u ∂u ∂p   + 2u +u +v +b = 0,   ∂t c ∂t ∂x ∂y ∂x     ∂v b ∂p ∂v ∂v ∂p     ∂t + c2 v ∂t + u ∂x + v ∂y + b ∂y = 0,        a3 ρ ∂u ∂v ∂ρ a3 ρ ∂u ∂v ∂u ∂v     c2 u ∂t + v ∂y + a ∂t + c2 u u ∂x + v ∂x + v u ∂y + v ∂y           ∂ρ ∂ρ ∂u ∂v   + +a u +v = 0, + ρa ∂x ∂y ∂x ∂y

(1.22) (1.23)

(1.24)

1

where ρ is the density, (u, v) is the velocity p = p(ρ) is the pressure and c = (p(ρ)) 2 is the local sound speed. Moreover we set b=

c2 − u 2 , ǫc2 + p2

h=

ǫc2 + p2 , ρc2

1 a= q . 2 2 1 − u c+v 2

(1.25)

Using the same approach, we can get Theorem 1.2 For the two-dimensional isentropic relativistic hydrodynamic self-similar system, we set U = u − ξ, V = v − η, if U 2 + V 2 > c2 . Then there are simple solutions in the self-similar (ξ, η) plane, its simple wave flow region is covered by the characteristics and (ρ, u, v) remain constants along these lines. The paper is organized as follows: In Section 2, we will describe the self-similar system for the compressible full Euler system and compute the eigenvalues as well as the corresponding eigenvectors. In Section 3, we present the concept and property of simple wave of pesudosteady compressible full Euler system. In Section 4, we prove there are simple wave solutions for compressible full Euler system, whose simple wave region is covered by characteristics under three different cases of initial data. In Section 5, we discuss isentropic relativistic hydrodynamics system in the self-similar plane.

2

Systems of Pseudo-steady Compressible Full Euler System

In this section, we will consider pseudo-steady compressible full Euler system (1.9)–(1.12), and compute its eigenvalues and corresponding eigenvectors by using generalized characteristic analysis. For pseudo-steady flow, the pseudo-stream line is defined as a continous line drawn through the pseudo-steady flow so that it has the direction of the pseudo-flow velocity (U, V ) at every point. By computation, we have  x˙  x˙ x 1 U ξ˙ = = − 2 = (u − ξ) = , (2.1) t t t t t  y˙  y˙ y 1 V = − 2 = (v − ξ) = , (2.2) η˙ = t t t t t where a dot (.) denotes the differentiation with respect to the time t in Lagrange’s coordinate. From (2.1) and (2.2), we have that in the (ξ, η) plane particles move in the direction of the pseudo-flow velocity. We can easily derive ∂s (pρ−γ ) = 0,

(2.3)

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where ∂s := (u − ξ)∂ξ + (v − η)∂η . Thus entropy S = pρ−γ is constant along the pseudo-flow lines. In the following we rewrite system (1.9)–(1.12) into the matrix form:       U 0 ρ 0 ρ V 0 0 ρ ρ        0 1 ρU 0   p  0 0 ρV   0       p  (2.4)    +     = 0.  0 0 0 ρU  u  0 1 0 ρV  u       0 U γp 0 v 0 V 0 γp v ξ

η

Let

and

 U 0 ρ   0 1 ρU  A0 (U, V, ρ, p) =  0 0 0  0 U γp 

V

0

0

  0 0 ρV  A1 (U, V, ρ, p) =  0 1 0  0 V 0

0



 0   , ρU   0 ρ



 0   . ρV   γp

The eigenvalues of system (2.4) are determined by |λA0 − A1 | = 0, that is,   |λA0 − A1 | = ρ2 (λU − V )2 (U 2 − c2 )λ2 − 2U V λ + V 2 − c2 = 0,

where c is the speed of sound and c2 = Therefore

(2.5)

γp ρ .

λ0 = λ1 =

V , U

(2.6)

and

p c2 (U 2 + V 2 − c2 ) λ2,3 = λ± = . (2.7) U 2 − c2 Thus system (2.4) is hyperbolic if and only if the flow is pseudo-supersonic, i.e., U 2 + V 2 > c2 . Furthermore, we can get the generalized left eigenvectors    2  l = c , 0, 0, −1 , (2.8)  0          l1 = 0, U, V, 0 , (2.9)    2 2  (2.10)  l2 = 0, c λ2 , −c , V − λ2 U ,         l3 = 0, c2 λ3 , −c2 , V − λ3 U . (2.11) UV ±

In order to get the characteristic form of the system (2.4), multiplying the generalized left eigenvectors li (ρ, p, U, V ) (i = 0, 1, 2, 3) on the left to system (2.4), we get     ρ ρ     p  p      li (ρ, p, U, V )A0 (U, V, ρ, p)   + li (ρ, p, U, V )A1 (U, V, ρ, p)   = 0. (2.12) u u     v

ξ

v

η

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So we get the characteristic form of the self-similar system     ρ ρ     p  p    li (ρ, p, U, V )A0 (U, V, ρ, p)   + λi (U, V, ρ, p)      u u    v v ξ

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

η

    = 0,   

(2.13)

where λi (U, V, ρ, p) (i = 0, 1, 2, 3) is the eigenvalue of the system. Therefore we get the left eigenvectors of the equivalent system Wξ + A(U, V, ρ, p)Wη = 0,

(2.14)

where W = (ρ, p, u, v)T and since U = u − ξ, V = v − η, so the coefficient matrix A(U, V, ρ, η) = A(W, ξ, η) is a 4 × 4 matrix, which depends on ξ and η as well as W ,   V ρV ρ2 V γρp  U U (ρU 2 − γp) γp − ρU 2 U (ρU 2 − γp)     −ρU V γpρ −γpρU  0    γp − ρU 2 γp − ρU 2 γp − ρU 2   A(W, ξ, η) =  ,   V −ρU V γp 0  2 2 2   γp − ρU γp − ρU γp − ρU     1 V 0 0 ρU U that is,    2  L = c U, −U, 0, 0 , 0         2  L = 0, U, ρU , ρU V ,  1   p  − c2 (U 2 + V 2 − c2 )  2 2  L = 0, , c V, −c U ,  2   ρ   p      c2 (U 2 + V 2 − c2 ) 2   L3 = 0, , c V, −c2 U . ρ

Furthermore, by the assumption Li (W, ξ, η)Rj (W, ξ, η) = δij (i, j = 0, 1, 2, 3) we can corresponging right eigenvectors:   1 T   R = , 0, 0, 0 , 0   c2 U    T  1 V    R1 = 0, 0, , ,   ρ(U 2 + V 2 ) ρU (U 2 + V 2 )      −ρ −ρ   p R2 = , p ,   2 c2 (U 2 + V 2 − c2 ) 2 c2 (U 2 + V 2 − c2 )  2c  p p c2 U + V c2 (U 2 + V 2 − c2 ) c2 V − U c2 (U 2 + V 2 − c2 ) T   p p , ,    2c2 (U 2 + V 2 ) c2 (U 2 + V 2 − c2 ) 2c2 (U 2 + V 2 ) c2 (U 2 + V 2 − c2 )      ρ ρ   R3 = p , p ,   2 2 2 2 2 2 2  2c c (U + V − c ) 2 c (U + V 2 − c2 )   p p    −c2 U + V c2 (U 2 + V 2 − c2 ) −c2 V − U c2 (U 2 + V 2 − c2 ) T   p p , .  2c2 (U 2 + V 2 ) c2 (U 2 + V 2 − c2 ) 2c2 (U 2 + V 2 ) c2 (U 2 + V 2 − c2 )

(2.15) (2.16) (2.17) (2.18) get the

(2.19) (2.20)

(2.21)

(2.22)

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To prove eigenvalue λi (i = 0, 1, 2, 3) keeps constant along its own characteristic line. Recalling the eigenfunction of λ2,3 : (U 2 − c2 )λ2 − 2U V λ + (V 2 − c2 ) = 0,

(2.23)

and regarding that λ depends on the quantities (U, V ) independently, we can easily find λU =

λ(U λ − V ) , λ(c2 − U 2 ) + U V

λV = −

Uλ − V . λ(c2 − U 2 ) + U V

(2.24)

Through the simple computation, we find eigenvalue statisfies: ∂U λi + λi ∂V λi = 0

(i = 0, 1, 2, 3).

Then we compute the eigenvalue along the same characteristic as ∂ ∂λi ∂λi ∂  ∂λi ∂λi + λi λi (U, V, ρ, p) = ∂ξ U + λi ∂η U + ∂ξ V + λi ∂η V ∂ξ ∂η ∂U ∂U ∂V ∂V ∂λi ∂λi ∂λi ∂λi + ∂ξ ρ + λi ∂η ρ + ∂ξ p + λi ∂η p ∂ρ ∂ρ ∂p ∂p ∂λi ∂λi ∂λi = (uξ + λi uη ) + (vξ + λi vη ) + ∂ξ ρ ∂U ∂V ∂ρ ∂λi ∂λi ∂λi +λi ∂η ρ + ∂ξ p + λi ∂η p (i = 0, 1, 2, 3). ∂ρ ∂p ∂p

(2.25)

(2.26)

If λ = λ0 = VU , we can compute it directly and show that it satisfies the (2.25). It implies that the gradient of λi just depends on the differentiation of solutions. Thus if we can prove the solution keeps constant along the simple wave, then the characteristic is a straight line.

3

Simple Waves for Pseudo-steady Compressible Full Euler System

In this section we will introduce the concept and property of the simple wave of pseudosteady compressible full Euler system. Simple waves were systematically studied for hyperbolic systems in two independent variables[24]. From the definition of the simple-wave, we know that, a solution is called a simple wave if it depends only on one single parameter[12]. For the non-reducible hyperbolic systems (2.14), as definition, the simple waves are defined as a special family of solutions of the form W = W (s(ξ, η)),

(3.1)

where s(ξ, η) is a scalar function. Substituting (3.1) into (2.14) yields (sξ + A(W, ξ, η)sη )Ws′ = 0,

(3.2)

−s

which implies that, as long as sη 6= 0, sηξ is an eigenvalue of A(W, ξ, η), the level curves s(ξ, η) = C are characteristic lines, and Ws′ parallels to the associated right eigenvector. On the other hand, since the eigenvalues λi (i = 0, 1, 2, 3) satisfies the characteristic function, where λ0 = λ1 are double eigenvalues. λ0 =

v−η , u−ξ

(3.3)

and [(u − ξ)2 − c2 )λ2 − 2(u − ξ)(v − η)λ + (v − η)2 − c2 ] = 0.

(3.4)

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Noting that W = (ρ, p, u, v)T are constants along the i-th characteristic line, as long as it is an i-th simple wave solution. Differentiating them along their own characteristic lines as dealt in [27], we obtain: dλ2,3 dλ2,3 −2[(η − v) − λ2,3 (ξ − u)](ξ − u) = 2λ2,3 c2 ; (3.5) d2,3 ξ d2,3 ξ i.e.,

p dλ2,3 ± c2 [(ξ − u)2 + (η − v)2 − c2 ] = 0, d2,3 ξ

(3.6)

dλ2,3 = 0, d2,3 ξ

(3.7)

therefore

∂ ∂ where ddi ξ = ∂ξ + λi ∂η . This concludes the property of simple wave: In the self-similar (ξ, η) plane, each simple wave is associated with one characteristic field, say, defined by λk (k = 0, 1, 2, 3), and spans a domain in which characteristics of k-kind are straight along which the solution is constant. According to the property of simple wave,

Ws′ //R(W, ξ, η).

(3.8)

Wη = Ws′ sη ,

(3.9)

Since where sη is scalar function. Thus if (ω0 , ω1 , ω2 , ω3 ) is determined by a i-th simple wave solution of the system, then ωi = Li (W, ξ, η)Wη = δij sη (i = 0, 1, 2, 3). (3.10) Using the characteristic decomposition. We can prove in next section that (ω0 , ω1 , ω2 , ω2 ), which satisfies the system tranformed from system (2.14), determines a λi -simple wave solution of compressible full Euler system with Wη //Ri (W, ξ, η) and Wη =

3 X

(i = 0, 1, 2, 3),

ωk Rk (W, ξ, η).

(3.11)

(3.12)

k=0

By (3.11), we can obtain ∂W ∂W + λi = 0. ∂ξ ∂η

(3.13)

Therefore solution W = (ρ, p, u, v) keep constants along the λi -th characteristics. Hence we find a family of one-parameter simple wave solution, whose flow region covered by straight characteristics, along which W = (ρ, p, u, v) remain constants.

4

Proof of Theorem 1

In this section we will use the decomposition of characteristics to prove there are simple wave solutions of system (2.14) with three different initial cases. Denoting ωi = Li (W, ξ, η)Wη , Li Wη , (4.1)

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where Li (W, ξ, η) denotes the i-th left eigenvector which not only depends on independent variables (ρ, p, u, v), but also depends on dependent variables (ξ, η). By (1.17), it is easy to see that Wη =

3 X

ωk Rk (W, ξ, η) =

k=0

3 X

ωk Rk .

Differentiating the system (2.14) with respect to η, we can get ∂ ∂ Wξ + (A(W, ξ, η)Wη ) = 0. ∂η ∂η Using (4.2), we can get 3 3   X ∂  ∂ X ωk Rk + A(W, ξ, η) ωk Rk = 0, ∂ξ ∂η k=0

(4.2)

k=0

(4.3)

(4.4)

k=0

which equivalent to 3 h  i X ∂  ∂  ωk Rk + A(W, ξ, η)Rk ωk = 0. ∂ξ ∂η

(4.5)

k=0

Noting A(W, ξ, η)Rk = λk (W, ξ, η)Rk , system (4.5) can be written as: 3 h  i X ∂  ∂  ωk Rk + λk (W, ξ, η)Rk ωk = 0, ∂ξ ∂η

(4.6)

k=0

that is, 3 h X ∂ωk

∂ξ

k=0

Rk + ωk

∂Rk ∂λk ∂Rk ∂ωk i + Rk ωk + λk ωk + λk Rk = 0. ∂ξ ∂η ∂η ∂η

(4.7)

Equation (4.7) multiplies the left eigenvector Li (W, ξ, η) to give 3 h X ∂ωk ∂Rk λk ∂Rk ∂ωk i δik + Li ωk + δik ωk + λk Li ωk + λk δik = 0. ∂ξ ∂ξ ∂η ∂η ∂η

(4.8)

k=0

Rearranging terms, we obtain 3

3

k=0

k=0

X ∂Rk X ∂ωi ∂λi ∂Rk ∂ωi + λi =− Li ωk − ωi − λk Li ωk . ∂ξ ∂η ∂ξ ∂η ∂η

(4.9)

In the following we will compute the right hand of equation (4.9) terms by terms. Using the equation Wξ = −A(W, ξ, η)Wη , we can rewrite the first term as Li

3  ∂R ∂W  X ∂Rk k j ωk + Li (Rk,ξ )ωk ωk = Li ∂ξ ∂W ∂ξ j j=0

=−

=−

3 X j=0

  Li ∇W Rk A(W, ξ, η)Rj ωj ωk + Li Rk,ξ ωk

3 X

 Li ∇W Rk λj Rj ωj )ωk + Li Rk,ξ ωk ,

j=0

(4.10)

where Rk,ξ denotes the Rk (W, ξ, η) differentiates with respect to the variable ξ. The second term is 3 X ∂λi ωi = ∇W λi Rj ωj ωi + λi,η ωi , (4.11) ∂η j=0

864

ACTA MATHEMATICA SCIENTIA

where ∇W λi denotes the gradient of λi about the variables (ρ, p, u, v) and λi,η = The third item is λk Li

Vol.35 Ser.B ∂λi (W,ξ,η) . ∂η

3  ∂R ∂W  X ∂Rk k j ωk = λk Li ωk + λk Li Rk,η ωk ∂η ∂Wj ∂η j=0   = λk Li ∇W Rk Wη ωk + λk Li Rk,η ωk

=

3 X j=0

  λk Li ∇W Rk Rj ωj ωk + λk Li Rk,η ωk ,

(4.12)

where Rk,η denotes Rk (W, ξ, η) differentiates with respect to variable η. Substituting (4.10), (4.11) and (4.12) into (4.9), we get the differential equation satisfied by ωi along the i-th characteristic 3 3   X X ∂ωi ∂ωi + λi = Li ∇W Rk λj Rj ωj ωk − Li Rk,ξ ωj ∂ξ ∂η j,k=0

−

k=0

3 X

j,k=0

−

3 X

3  X λk Li ∇W Rk Rj ωj ωk − λk Li Rk,η ωk − λi,η ωi



k=0

∇W λi Rj ωj ωi

j=0

=

3 X

j,k=0

−

  (λj − λk )Li ∇W Rk Rj ωj ωk − λi,η ωi

3 X j=0

∇W λi Rj ωj ωi −

3 X

k=0

  Li Rk,ξ + λk Rk,η ωk .

(4.13)

Then we will provide the suitable initial nonconstant data to the system in three conditions to clarify there are different four kinds simple wave of the compressible full Euler system: Case I We choose the suitable initial data ω00 = ω10 = ω20 = 0 and ω30 6= 0. By (4.13), We can obtain:  ∂ω0 ∂ω0   + λ0 = −L0 (R3,ξ + λ3 R3,η )ω3 , (4.14)   ∂ξ ∂η      ∂ω1 ∂ω1   + λ1 = −L1 (R3,ξ + λ3 R3,η )ω3 , (4.15)  ∂ξ ∂η ∂ω2 ∂ω2    + λ2 = −L2 (R3,ξ + λ3 R3,η )ω3 , (4.16)   ∂ξ ∂ξ     ∂ω3 ∂ω3    + λ3 = −L3 (R3,ξ + λ3 R3,η )ω3 − ∇W λ3 R3 ω3 ω3 − λ3,η ω3 . (4.17) ∂ξ ∂η In the following, we can simplify the system (4.14)–(4.17). Noting right eigenvector Ri = Ri (W, ξ, η) and U = u − ξ, V = v − η, we have ∂Ri,l ∂Ri,l = ∂ξ ∂ρ ∂Ri,l = ∂ρ

∂ρ ∂Ri,l ∂p ∂Ri,l ∂u ∂Ri,l ∂v ∂Ri,l + + + + ∂ξ ∂p ∂ξ ∂u ∂ξ ∂v ∂ξ ∂ξ ∂ρ ∂Ri,l ∂p ∂Ri,l ∂u ∂Ri,l ∂v ∂Ri,l + + + − (l = 1, 2, 3, 4), ∂ξ ∂p ∂ξ ∂U ∂ξ ∂V ∂ξ ∂U

(4.18)

No.4

Y. Chen & Y. Zhou: TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

∂Ri,l ∂Ri,l = ∂η ∂ρ ∂Ri,l = ∂ρ

∂ρ ∂Ri,l ∂p ∂Ri,l + + ∂η ∂p ∂η ∂u ∂ρ ∂Ri,l ∂p ∂Ri,l + + ∂η ∂p ∂η ∂U

∂Ri,l ∂u ∂Ri,l ∂v + + ∂η ∂v ∂η ∂η ∂Ri,l ∂u ∂Ri,l ∂v + − (l = 1, 2, 3, 4). ∂η ∂V ∂η ∂V

Differentiating (1.17) along the k-th characteristic:   d (Li Rj ) = Li,ξ + λk Li,η Rj + Li (Rj,ξ + λk Rj,η ) = 0. dk ξ

865

(4.19)

(4.20)

Therefore −Li (Rj,ξ + λk Rj,η ) = (Li,ξ + λk Li,η )Rj .

(4.21)

Using eigenvaluves we can obtain: λ2 − λ3 =

2

p c2 (U 2 + V 2 − c2 ) . U 2 − c2

We rewrite the left eigenvectors as follow:    2  L = c U, −U, 0, 0 ,  0          L1 = 0, U, ρU 2, ρU V ,    −(U 2 − c2 )(λ − λ )  2 3 2 2  L = 0, , c V, −c U ,  2   2ρ     (U 2 − c2 )(λ − λ )    2 3  , c2 V, −c2 U .  L3 = 0, 2ρ

(4.22)

(4.23) (4.24) (4.25) (4.26)

Noting (2.25), we have:

L0,ξ + λ3 L0,η = (−c2 , 1, 0, 0),   L1,ξ + λ3 L1,η = 0, −1, −2ρU, −ρ(V + λ3 U ) ,  U (λ − λ ) U 2 − c2  2 3 L2,ξ + λ3 L2,η = 0, + (∂U λ2 + λ3 ∂V λ2 ), −c2 λ3 , c2 , ρ 2ρ   U (λ2 − λ3 ) U 2 − c2 − (∂U λ2 + λ3 ∂V λ2 ), −c2 λ3 , c2 . L3,ξ + λ3 L3,η = 0, − ρ 2ρ

(4.27) (4.28) (4.29) (4.30)

By the eigenfunction of λ2,3 : (U 2 − c2 )λ2 − 2U V λ + (V 2 − c2 ) = 0.

(4.31)

We can obtain λ2 λ3 =

c2 − V 2 . c2 − U 2

(4.32)

Recalling the derivatives (λU , λV ), we have 1 c2 − V 2 ∂V λ2 λ3 c2 − U 2 λ2 (U λ2 − V ) 1 c2 − V 2 λ2 U − V − = 2 2 2 2 2 λ2 (c − U ) + U V λ3 c − U λ2 (c − U 2 ) + U V 2(U λ2 − V ) = . c2 − U 2

∂U λ2 + λ3 ∂V λ2 = ∂U λ2 +

(4.33)

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ACTA MATHEMATICA SCIENTIA

Vol.35 Ser.B

Substituting (4.27), (4.28), (4.29) and (4.30) into (4.14), (4.15), (4.16) and (4.17) respectively, we can get:  ∂ω0 ∂ω0   + λ0 = 0, (4.34)   ∂ξ ∂η      ∂ω1 ∂ω1   + λ1 = 0, (4.35)  ∂ξ ∂η ∂ω2 ∂ω2    + λ2 = 0, (4.36)   ∂ξ ∂η     ∂ω3 ∂ω3    + λ3 = F1 (W, ξ, η)ω32 + F2 (W, ξ, η)ω3 . (4.37) ∂ξ ∂η

Where F1 (W, ξ, η), F2 (W, ξ, η) are the functions depend on (W, ξ, η). Therefore we can know that (ω0 , ω1 , ω2 , ω3 ) satisfies the system that transformed from system (1.13) and keeps the initial state along its own characteristic line, hence it determines a family of simple wave solution. Moreover, since (ω0 , ω1 , ω2 , ω3 ) and W (ξ, η) can be determinded by each other, therefore we can have: Wη //R3 (W, ξ, η).

(4.38)

Backing to the compressible full Euler system Wξ + A(W, ξ, η)Wη = 0.

(4.39)

Differentiating solution W (ξ, η) along the λ3 characteristic, we obtain dW = Wξ + λ3 (W, ξ, η)Wη = −A(W, ξ, η)Wη + λ3 (W, ξ, η)Wη = 0. d3 ξ

(4.40)

Therefore W = (ρ(ξ, η), p(ξ, η), u(ξ, η), v(ξ, η)) keep constants value on fixed λ3 -th characteristic. Moreover, by (2.26), we can prove ∂ ∂  + λ3 λ3 (W, ξ, η) = ∂U λ3 (uξ + λ3 uη ) + ∂V λ3 (vξ + λ3 vη ) ∂ξ ∂η ∂λ3 ∂λ3 ∂λ3 ∂λ3 + ∂ξ ρ + λ3 ∂η ρ + ∂ξ + λ3 ∂η p ∂ρ ∂ρ ∂p ∂p = 0. (4.41) Hence, using the characteristic decomposition, we prove that there are simple wave solutions of compressible full Euler system which determined by (ω0 , ω1 , ω2 , ω3 ) and the simple wave flow region is covered by a one parametric family of straight characteristic λ3 , along each of these lines W = (ρ, p, u, v) remain constants. Case II Using the same method we can get the simple wave solution when the initial data satisfies ω20 6= 0, ω00 = ω10 = ω30 = 0. By system (4.13), we have  ∂ω0 ∂ω0   + λ0 = −L0 (R2,ξ + λ2 R2,η )ω2 , (4.42)   ∂ξ ∂η      ∂ω1 ∂ω1   + λ1 = −L1 (R2,ξ + λ2 R2,η )ω2 , (4.43)  ∂ξ ∂η  ∂ω2 ∂ω2   + λ2 = −L2 (R2,ξ + λ2 R2,η )ω2 − ∇W λ2 R2 ω2 ω2 − λ2,η ω2 , (4.44)   ∂ξ ∂ξ     ∂ω3 ∂ω3    + λ3 = −L3 (R2,ξ + λ2 R2,η )ω2 . (4.45) ∂ξ ∂η

No.4

Y. Chen & Y. Zhou: TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

867

Through differentiating the left eigenvectors, we get: L0,ξ + λ2 L0,η = (−c2 , 1, 0, 0),   L1,ξ + λ2 L1,η = 0, −1, −2ρU, −ρ(V + λ2 U ) ,   U (λ − λ ) U 2 − c2 2 3 L2,ξ + λ2 L2,η = 0, − (∂U λ3 + λ2 ∂V λ3 ), −c2 λ2 , c2 , ρ 2ρ   U (λ2 − λ3 ) U 2 − c2 L3,ξ + λ2 L3,η = 0, − + (∂U λ3 + λ2 ∂V λ3 ), −c2 λ2 , c2 . ρ 2ρ Substituting (4.46)–(4.49) into (4.42)–(4.45), we can get:  ∂ω0 ∂ω0   + λ0 = 0,   ∂ξ ∂η      ∂ω1 ∂ω1   + λ1 = 0,  ∂ξ ∂η ∂ω2 ∂ω2    + λ2 = G1 (W, ξ, η)ω22 + G2 (W, ξ, η)ω2 ,   ∂ξ ∂η     ∂ω3 ∂ω3    + λ3 = 0. ∂ξ ∂η

(4.46) (4.47) (4.48) (4.49)

(4.50) (4.51) (4.52) (4.53)

Where G1 (W, ξ, η), G2 (W, ξ, η) are the functions depend on (W, ξ, η). Therefore we have obtained: (ω0 , ω1 , ω2 , ω3 ) keeps the initial state along its own charateristic line and determines the λ2 -th simple wave solution of compressible full Euler system with

Furthermore, we have:

Wη //R2 (W, ξ, η).

(4.54)

∂W ∂W + λ2 = 0. ∂ξ ∂η

(4.55)

It implies that the solution W = (ρ, p, u, c) along the λ2 -th straight characteristic line keep constants. Case III Full Euler system is not strict hyperbolic. For the double eigenvalue λ0 (U, V, ρ, p), right eigenvectors R0 and R1 as the basis span the eigenspace Cλ0 . Moreover, arbitrary vector of Cλ0 can be represented by the linear combination of R0 and R1 . In order to prove the simple wave region is covered by characteristics λ0 = λ1 , we need to prove Wη //R0∗ (W, ξ, η), where R0∗ (W, ξ, η) is arbitrary eigenvetor of eigenspace Cλ0 . Since λ0 = λ1 which is the double eigenvalue, we get there is no item Li (∇W R0 R1 ω0 )ω1 on the right hand of the equation (4.13). It implies that there is no affection to ω2 , ω3 from the multiplying of ω0 and ω1 . Thus in order to prove Wη //R0∗ (W, ξ, η), we just discuss when the initial data satisfies (ω20 = ω30 = 0), (ω2 , ω3 ) can satisfy their own system tranformed from compressible full Euler system. By simple computation, we get from (4.13)      ∂ω2 ∂ω2   + λ2 = −L2 R0,ξ + λ0 R0,η ω0 − L2 R1,ξ + λ0 R1,η ω1 , (4.56)  ∂ξ ∂η      ∂ω3 ∂ω3   + λ3 = −L3 R0,ξ + λ0 R0,η ω0 − L3 R1,ξ + λ0 R1,η ω1 . (4.57) ∂ξ ∂η

According to the right eigenvectors, we have:  1 T R0,ξ = 2 2 , 0, 0, 0 , c U

(4.58)

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ACTA MATHEMATICA SCIENTIA

R0,η = (0, 0, 0, 0)T , T  −3V −2U , , R1,ξ = 0, 0, 2 2 2 2 2 2 ρ(U + V ) ρU (U + V )  −2V U 3 − U V 2 T R1,η = 0, 0, , . ρ(U 2 + V 2 )2 ρU 2 (U 2 + V 2 )2

Vol.35 Ser.B

(4.59) (4.60) (4.61)

Therefore  1 T , 0, 0, 0 , 2 2 c U  T −2 −2V = 0, 0, , . ρU (U 2 + V 2 ) ρU 2 (U 2 + V 2 )

R0,ξ + λ0 R0,η =

(4.62)

R1,ξ + λ0 R1,η

(4.63)

Then  ∂ω2 ∂ω2   + λ2 = 0,  ∂ξ ∂η ∂ω3  ∂ω3   + λ3 = 0. ∂ξ ∂η

(4.64) (4.65)

It is implies that for arbitrary eigenvector R0∗ (W, ξ, η) belongs to Cλ0 , we can obtain Wη //R0∗ (W, ξ, η). Finally, we also have: ∂W ∂W + λ0 = 0. ∂ξ ∂η

(4.66)

Then there is a simple wave solution corresponding to the eigenvalue λ0 = λ1 . In conclusion, we get simple wave solutions of the compressilbe full Euler system under three different cases of initial data. The simple wave flow region corresponding to each case is covered by straight characteristics λ0 = λ1 , λ2 and λ3 respectively.

5

Relativistic Hydrodynamics System

Relativistic Hydrodynamisch (RHD) plays an essential role in many fields of modern physics, e.g., astrophysics. Relativistic flows appear in numerous astrophysical phenomena from stellar to galactic scales, e.g., active galactic nuclei, super-luminal, etc. The relativistic description of fluid dynamics should be taken into account if the local velocity of the flow is close to the light speed in vacuum or the local internal energy density is comparable with the local rest-mass density of the fluid. It should also be used whenever matter is influenced by large gravitational potentials, where the Einstein field theory of gravity has to be considered ([1, 8, 14, 26]).

No.4

Y. Chen & Y. Zhou: TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

5.1

869

Systems of Self-similar Flows

Consider the two-dimensional isentropic Relativistic Hydrodynamics System (1.22)–(1.24), which can be rewritten as the self-similar form:   ∂u ∂u u  ∂p ∂ρ b ∂p ∂ρ  + (v − η) + b 1 − 2ξ − 2 uη = 0,  (u − ξ)   ∂ξ ∂η c ∂ρ ∂ξ c ∂ρ ∂η       ∂v ∂v b ∂p ∂ρ v ∂p ∂ρ     (u − ξ) ∂ξ + (v − η) ∂η − c2 vξ ∂ρ ∂ξ + b 1 − c2 η ∂ρ ∂η = 0,   (5.1) a3 ρ c2 ∂u a3 ρ ∂u a3 ρ ∂v    u(u − ξ) + + u(v − η) + v(u − ξ)   c2 a2 ∂ξ c2 ∂η c2 ∂ξ       3 2  a ρ c ∂ρ ∂ρ ∂v   + 2 v(v − η) + 2 + a(u − ξ) + a(v − η) = 0. c a ∂η ∂ξ ∂η Setting U = (u − ξ) and V = (v − η) as we did in the full Euler system, the system (5.1) can be rewritten into the form:   ∂u u  ∂p ∂ρ b ∂p ∂ρ ∂u  + V + b 1 − ξ − 2 uη = 0, (5.2) U   2  ∂ξ ∂η c ∂ρ ∂ξ c ∂ρ ∂η      ∂v ∂v b ∂p ∂ρ v  ∂p ∂ρ   U + V − vξ + b 1 − η = 0, (5.3)   ∂ξ ∂η c2 ∂ρ ∂ξ c2 ∂ρ ∂η   c2 ∂u a3 ρ ∂u a3 ρ ∂v a3 ρ    2 uU + 2 + 2 uV + 2 vU   c a ∂ξ c ∂η c ∂ξ   (5.4)    3 2   a ρ c ∂v ∂ρ ∂ρ   + 2 vV + 2 + aU + aV = 0. c a ∂η ∂ξ ∂η 5.2

Generalized Characteristic Analysis

System (5.2)–(5.4) can be written into the following matrix form:     u u        A0 (ρ, U, V ) v  + A1 (ρ, U, V ) v   = 0, ρ ρ η

ξ

where 

   A0 (ρ, U, V ) =    

U

0

0

U

ρa3  c2  uU + 2 2 c a

ργ 3 vU c2

and 

   A1 (ρ, U, V ) =    

V 0 ρa3 uV c2

(5.5)

 u  ∂p  b 1− 2ξ c ∂ρ   b ∂p  , − 2 vξ c ∂ρ    U

b ∂p  uη c2 ∂ρ    v  ∂p  . V b 1 − 2η c ∂ρ    ρa3  c2  vV + 2 V c2 a 0

−

Through the simple computation, we get the the eigenfunction of system (5.5), that is |λA0 − A1 | = (λU − V )(h2 λ2 + h3 λ + h4 ) = 0,

(5.6)

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where ρab ∂p ∂p ρba3 ∂p 2 v ξU + 2 uξ − ρav + U 2, c4 ∂ρ c ∂ρ ∂ρ
ρba3 ∂p 2 ρa3 b ∂p 2 ρa3 b ∂p h3 = 4 u ξ − (u2 + v 2 )η U − 4 v ξV + 2 U (v − u) c ∂ρ c ∂ρ c ∂ρ ρab ∂p − 2 (vξ + uη) − 2U V, c ∂ρ ρa3 b ∂p 2 ρa3 b ∂p ∂p h4 = − 4 (u ξ − (u2 + v 2 )η)V − 2 V (v − u) − ρab c ∂ρ c ∂ρ ∂ρ ρab ∂p + 2 vη + V 2 . c ∂ρ

h2 =

Then we have λ0 =

V , U

λ1,2 = λ± =

−h3 ±

p h23 − 4h2 h4 . 2h2

(5.7)

(5.8)

(5.9)

(5.10)

Thus system (5.5) is hyperbolic if and only if h23 − 4h2 h4 > 0. We can get their generalized left eigenvectors:      v  bV  u  b Vb   vξ + b 1 − η , 1 − ξ + uη, 0 , (5.11) l = 0   U c2 c2 U c2 c2         λ+ c2 ρa3 ρa3 (5.12) , − 2 v(λ+ U − V ) − ρa, λ+ U − V , l+ = − 2 u(λ+ U − V ) + 2  c a c          ρa3 λ c2 ρa3   l− = − 2 u(λ− U − V ) + +2 , − 2 v(λ− U − V ) − ρa, λ− U − V . (5.13) c a c

In order to get the characteristic form of the system (5.5), multiplying the left eigenvectors we can obtain:      u u          = 0, (5.14) li (ρ, U, V )A0 (ρ, U, V ) v  + λi (ρ, U, V )  v     ρ ρ η

ξ

where λi (ρ, U, V ) is the eigenvalue. So we get the left eigenvectors:   bV    v  u  bU 2 2  L = vξ + bU 1 − η , bV 1 − ξ + uη, b (c − uξ − vη) ,  0  2  c2 c2 c2  c  L+ = − ρaV, −ρaU, M1 (λ1 U − V ) − M2 λ1 + ρabvξ ,        L = − ρaV, −ρaU, M (λ U − V ) − M λ + ρabvξ , − 1 2 2 2

(5.15) (5.16) (5.17)

where

M1 = −bρa3 u(1 −

u ρa3 2 ξ) + v ξ + U, c2 c2

u ξ). c2 satisfies the characteristic equation M2 = ρabc2 (1 −

Recalling that λ±

(5.18) (5.19)

h2 λ2 + h3 λ + h4 = 0.

(5.20)

Regarding that λ depends on the quantities (U, V ) independently, we can easily find λU = −

2 ∂h2 3 λ ∂h ∂U + λ ∂U , 2λh2 + h3

λV = −

∂h4 3 λ ∂h ∂V + ∂V . 2λh2 + h3

(5.21)

No.4

Y. Chen & Y. Zhou: TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

871

Computing directly, we find ρba3 ∂p 2 v ξ + 2U, c4 ∂ρ ρba3 ∂p 2 ρba3 ∂p ∂U h3 = 4 (u ξ − (u2 + v 2 )η) + 2 (v − u) − 2V, c ∂ρ c ∂ρ ∂U h4 = 0, ∂U h2 =

∂V h2 = 0,

(5.22) (5.23) (5.24) (5.25)

3

ρa b ∂p 2 v ξ − 2U, c4 ∂ρ ρa3 b ∂p 2 ρa3 b ∂p ∂V h4 = − 4 (u ξ − (u2 + v 2 )η) − 2 (v − u) + 2V. c ∂ρ c ∂ρ ∂V h3 = −

(5.26) (5.27)

Thus combing (5.22)–(5.27), we can also get the important equation: ∂U λi + λi ∂V λi = 0,

(i = 0, 1, 2),

(5.28)

and if λ = λ0 , then we can compute it directly to show that it satisfies the equation (5.28). Then   ∂ ∂ + λi λi (ρ, U, V ) ∂ξ ∂η = ∂U λi (uξ + λi uη ) + ∂V λi (vξ + λi vη ) + ∂ρ λi (ρξ + λi ρη ) (i = 0, 1, 2). (5.29) 5.3

Proof of Theorem 2

In this section, we will proof there are simple wave solutions to the isentropic Relativistic Hydrodynamic system with the same method. We rewrite the system (5.2)–(5.4) as following: Qξ + A(U, V )Qη = 0, where Q = (u, v, ρ)T and A(Q, ξ, η) =

A−1 0 A1

(5.30)

is a 3 × 3 matrix depends on (Q, ξ, η). Setting

ωi = Li (Q, ξ, η)Qη , Li Qη ,

(5.31)

where Li (Q, ξ, η) denotes the i-th left eigenvector which not only depends on independent variables (ρ, u, v) but also dependent variables (ξ, η). Furthermore we assume Li (Q, ξ, η)Rj (Q, ξ, η) = δij

(i, j = 0, 1, 2),

(5.32)

where δij stands for the Kronecker’s symbol. Thus with (5.32) we can get the corresponding right eigenvector Ri (Q, ξ, η) of the λi (Q, ξ, η), and it is easy to see that Qη =

2 X

ωk Rk (Q, ξ, η) =

k=0

2 X

ωk Rk .

(5.33)

k=0

Differentiating the equation (5.30) we can get ∂ ∂ Qξ + (A(Q, ξ, η)Qη ) = 0. ∂η ∂η Using the similar procedures as before, we can obtain 2 2   X X ∂ωi ∂ωi + λi = Li ∇Q Rk λj Rj ωj ωk − Li Rk,ξ ωk ∂ξ ∂η j,k=0

k=0

(5.34)

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−

2 X

j,k=0

−

2 X

Vol.35 Ser.B

2   X λk Li ∇Q Rk Rj ωj ωk − λk Li Rk,η ωk k=0

∇Q λi Rj ωj ωi − λi,η ωi

j=0

=

2 X

j,k=0

−

2     X (λj − λk )Li ∇Q Rk Rj ωj ωk − Li Rk,ξ + λk Rk,η ωk

2 X

k=0

∇Q λi Rj ωj ωi − λi,η ωi .

(5.35)

j=0

According to the property of the simple wave, if the flow region covered by the simple wave Q(s(ξ, η)) which depends on one single parameter then Q′s //Ri (Q, ξ, η),

(5.36)

Qη = Q′s sη ,

(5.37)

moreover where sη is a scaler function. Thus ωi = Li Qη = δij sη

(i = 0, 1, 2).

(5.38)

Here we only consider the case of suitable initial data satisfies ω00 6= 0, ω10 = ω20 = 0. By (5.35) we can obtain:  ∂ω0 ∂ω0  + λ0 = −L0 (R0,ξ + λ0 R0,η )ω0 − ∇Q λ0 R0 ω0 ω0 − λ0,η ω0 , (5.39)    ∂ξ ∂η    ∂ω ∂ω1 1 + λ1 = −L1 (R0,ξ + λ0 R0,η )ω0 , (5.40)  ∂ξ ∂η       ∂ω2 + λ2 ∂ω2 = −L2 (R0,ξ + λ0 R0,η )ω0 , (5.41) ∂ξ ∂η

denotes λ0 (Q, ξ, η) differentiates with respect to the variable η. where λ0,η = ∂λ0 (Q,ξ,η) ∂η ∇Q λ0 denotes the gradients of λ0 (Q, ξ, η) about the independent variables (ρ, u, v). According the left eigenvectors we get Right eigenvectors as follow:   T   R0 = R01 , R02 , R03 , (5.42)      T R1 = R11 , R12 , R13 , (5.43)     T    R = R ,R ,R , (5.44) 2 21 22 23

where

R01 = R02 =

−c2 U , c2 bV 2 (1 − cu2 ξ) − c2 bU 2 (1 − cv2 η) − bU V vξ + bU V uη

(5.45)

c2 V −

(5.46)

c2 bV 2 (1

−

u c2 ξ)

−

c2 bU 2 (1

v c2 η)

− bU V vξ + bU V uη

,

R03 = 0, −b(c2 − uξ − bη)2 ρaU c2 − c2 bV (1 − cu2 ξ)S1 − bU uηS1 R11 = , H(Q, ξ, η)(S1 − S2 )

(5.47) (5.48)

No.4

Y. Chen & Y. Zhou: TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

b(c2 − uξ − vη)2 ρaV c2 + c2 bU (1 − cv2 η)S1 + bV vξS1 , H(Q, ξ, η)(S1 − S2 ) 1 =− , S1 − S2 b(c2 − uξ − vη)2 ρaU c2 + c2 bV (1 − cu2 ξ)S2 + bU uηS2 , = H(Q, ξ, η)(S1 − S2 ) b(c2 − uξ − vη)2 ρaV c2 + c2 bU (1 − cv2 η)S2 + bV vξS2 = , H(Q, ξ, η)(S1 − S2 ) 1 = , S1 − S2

873

R12 =

(5.49)

R13

(5.50)

R21 R22 R23

(5.51) (5.52) (5.53)

where v u ξ) − c2 bU 2 (1 − 2 η)ρa − bU V ρavξ + bU V ρauη, c2 c S1 = M1 (λ1 U − V ) − M2 λ1 + ρabvξ, H(Q, ξ, η) = c2 bV 2 ρa(1 −

S2 = M1 (λ2 U − V ) − M2 λ2 + ρabvξ. Thus we get the derivation as we did before  ∂S1  L1,ξ = 0, ρa, , ∂ξ  ∂S2  L2,ξ = 0, ρa, , ∂ξ

(5.54) (5.55) (5.56)

 ∂S1  L1,η = ρa, 0, , ∂η  ∂S2  L2,η = ρa, 0, . ∂η

(5.57) (5.58)

We can get:  ∂S1 ∂S1  L1,ξ + λ1 L1,η = λ1 ρa, ρa, + λ1 , ∂ξ ∂η  ∂S2 ∂S2  L2,ξ + λ1 L2,η = λ1 ρa, ρa, + λ1 . ∂ξ ∂η Then we can obtain   L1,ξ + λ1 L1,η R1 = 

 L2,ξ + λ1 L2,η R1 =

(5.59) (5.60)

−c2 V ρa + c2 ρaV = 0, c2 bV 2 (1 − cu2 ξ) − c2 bU 2 (1 − cv2 η) + bU V uη − bU V vξ

(5.61)

−c2 V ρa + c2 ρaV = 0. − c2 bU 2 (1 − cv2 η) + bU V uη − bU V vξ

(5.62)

c2 bV 2 (1

−

u c2 ξ)

Substituting (5.61), (5.62) into (5.39), (5.40) and (5.41) respectively, we can get:  ∂ω0 ∂ω0 2    ∂ξ + λ0 ∂η = P1 (Q, ξ, η)ω0 + P2 (Q, ξ, η)ω0 ,     ∂ω ∂ω1 1 + λ1 = 0,  ∂ξ ∂η       ∂ω2 + λ2 ∂ω2 = 0. ∂ξ ∂η

(5.63) (5.64) (5.65)

Where P1 (Q, ξ, η), P2 (Q, ξ, η) are the functions depend on (Q, ξ, η). Therefore we can know that (ω0 , ω1 , ω2 ) satisfies the system transformed from the pseudo-steady isentropic Hydrodynamics system and it keeps the intial state along its own characteristic line. Moreover (ω0 , ω1 , ω2 ) and Q(ξ, η) can be determined by each other. Thus under the initial condition, we can have Qη //R0 (Q, ξ, η).

(5.66)

874

ACTA MATHEMATICA SCIENTIA

Vol.35 Ser.B

Backing to the pseudo-steady isentropic hydrodynamics system Qξ + A(Q, ξ, η)Qη = 0.

(5.67)

Differentiating the solution Q(ξ, η) along the λ0 -th characteristic and using the pseudo-steady isentropic hydrodynamic equation, we obtain dQ = Qξ + λ0 (ρ, U, V )Qη = −A(Q, ξ, η)Qη + λ0 (ρ, U, V )Qη = 0. d0 ξ

(5.68)

Therefore Q = (ρ(ξ, η), u(ξ, η), v(ξ, η)) keep constants value on fixed λ0 -th characteristic. Moreover according (5.29), we can prove d ∂c2 λ0 = ∂U λ0 (uξ + λ0 uη ) + ∂V λ0 (vξ + λ0 vη ) + ∂ρ λ0 (ρξ + λ0 ρη ) = 0. d0 ξ ∂ρ

(5.69)

Hence, using the characteristic decomposition, we get the simple wave solutions of the pseudosteady isentropic hydrodynamic system and its simple wave flow region is covered by a one parametric family of straight characteristic λ0 , along each of which Q = (u, v, ρ) remain constants. Remark We use the same method as above can prove when the initial data satisfies 0 ω1 6= 0, ω00 = ω20 = 0 and ω00 = ω10 = 0, ω20 6= 0 respectively, (ω0 , ω1 , ω2 ) also determines simple wave solution of isentropic hydrodynamic system in each case. Furthermore, we find the simple wave flow region is covered by straight characteristics λ1 , λ2 respectively. References [1] Anile A M, Choquet-Bruhat Y. Relativistic Fluid Dynamics. Berlin, Heidelverg: Springer-Verlag, 1989 [2] Bang S. Interaction of four rarefection waves of the pressure gradient system. J Differ Equ, 2009, 246: 453–481 [3] Courant R, Friedrichs K O. Supersonic Flow and Shock Waves. New York: Interscience, 1948 [4] Chang T, Hsiao L. The Riemann problem and interaction of waves in gas dynamics//Pitman Monographs and Surveys in Pure and Applied Mathematics, 41. Harlow: Longman Scientific Technical, 1989 [5] Chang T, Chen G Q, Yang S L. On the 2-D Riemann problem for the compressible Euler equations, I, Interaction of shock waves and rarefaction waves. Disc Cont Dyn Syst, 1995, 1: 555–584 [6] Chang T, Chen G Q, Yang S L. On the 2-D Riemann problem for the compressible Euler equations, II, Interaction of contact discontinuities. Disc Cont Dyn Syst, 2000, 6: 419–430 [7] Chen X, Zheng Y X. The direct approach to the interaction of rarefaction waves of the two-dimensional Euler equations. Indian J Math, 2010, 59: 231–256 [8] Chiu H H. Relativistic gas dynamics. Phys Fluids, 1973, 16: 825–881 [9] Dafermos C. Hyperbolic Conservation Laws in Continuum Physics. Heidelberg: Springer, 2000 [10] Dai Z H, Zheng Y X. Existence of a global smooth solution for a degenerate Goursat problem of gas dynamics. Arch Ration Mech Anal, 2000, 155: 277–298 [11] Hu Y B, Sheng W C. Characteristic decomposition of the 2 × 2 quasilinear strictly hyperbolic systems. Appl Math Lett, 2012, 25: 262–267 [12] John F. Partial Differential Equations. New York: Springer-Verlag, 1982 [13] John F. Formation of singularities in one-dimensional nonlinear wave propagation. Commun. Pure Appl. Math, 1974, 27: 377–405 [14] K¨ onigl A. Relativistic gasdynamics in two dimensions. Phys Fluids, 1980, 23(6): 1083–1090 [15] Kurganov A, Tadmor E. Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. Num Meth Part Diff Eqs, 2002, 18: 584–608 ˘ [16] Keyfitz B L, Cani´ c S. Quasi-one-dimensional Riemann problems and their role in self-similar two dimensional problems. Arch Rat Mech Anal, 1998, 144: 233–258 [17] Li J Q. On the two-dimensional gas expansion for compressible Euler equations. SIAM J Appl Math, 2001, 62: 831–852

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Y. Chen & Y. Zhou: TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

875

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