Non-isentropic multi-transonic solutions of Euler-Poisson system

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.1 (1-18)

Available online at www.sciencedirect.com

ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Non-isentropic multi-transonic solutions of Euler-Poisson system Ben Duan, Yan Zhou ∗ School of Mathematics, Dalian University of Technology, Dalian, People’s Republic of China Received 30 December 2018; revised 22 August 2019; accepted 15 November 2019

Abstract In this paper, we consider the boundary value problem for the one-dimensional non-isentropic EulerPoisson system. The physical subsonic inflow and outflow conditions are prescribed at boundaries. We prove the existence of multi-transonic solutions, i.e., the existence of subsonic-sonic-supersonic-shocksubsonic flows in fact nozzles. Furthermore, the monotonicity between the location of the transonic shock and the density at the exit of the nozzle will be elucidated. © 2019 Elsevier Inc. All rights reserved.

MSC: 35L65; 35R35; 76H05 Keywords: Euler-Poisson system; Sonic; Shock waves

1. Introduction Euler-Poisson system models several physical flows including the propagation of electrons in submicron semiconductor devices and plasma, and the biological transport of ions for channel proteins. The time-dependent equations for the one-dimensional non-isentropic Euler-Poisson flow are * Corresponding author.

E-mail addresses: [email protected] (B. Duan), [email protected] (Y. Zhou). https://doi.org/10.1016/j.jde.2019.11.073 0022-0396/© 2019 Elsevier Inc. All rights reserved.

JID:YJDEQ AID:10150 /FLA

2

[m1+; v1.304; Prn:25/11/2019; 12:28] P.2 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

⎧ ρt + (ρu)x = 0, ⎪ ⎪ ⎪ ⎨ (ρu)t + (ρu2 + p(ρ, S))x = ρE, ⎪ (ρE)t + (ρuE + pu)x = ρuE, ⎪ ⎪ ⎩ Ex = ρ − b,

(1.1)

where the pressure p and the energy density E are given by p = eS ρ γ ,

E=

|u|2 p + , 2 (γ − 1)ρ

(1.2)

γ > 1 is the adiabatic exponent. In the hydrodynamical model of semiconductor devices or plasma, u, ρ and S represent the average particle velocity, electron density and the specific entropy, respectively. E is the electric field, which is generated by the Coulomb force of particles. b > 0 stands for the density of fixed, positively charged background ions. The biological model describes the transport of ions between the extracellular side and the cytoplasmic side of the membranes. In this case, ρ, ρu and E are the ion concentration, the ions translational mass, and the electric field, respectively. In this paper, we consider multi-transonic shock problem of the following steady EulerPoisson system ⎧ (ρu)x = 0, ⎪ ⎪ ⎪ ⎨(ρu2 + p(ρ, S)) = ρE, x ⎪ (ρuE + pu)x = ρuE, ⎪ ⎪ ⎩ Ex = ρ − b.

(1.3)

Assuming that p satisfies p(0) = 0, p  (ρ) > 0, p  (ρ) > 0, for ρ > 0, p(+∞) = +∞.

(1.4)

For given fixed L, length of the nozzle, the boundary conditions for (1.3) in an interval 0 ≤ x ≤ L are given as follows (ρ, u, S, E)(0) = (ρ0 , u0 , S0 , E0 ),

ρ(L) = ρe .

(1.5)

 Introduce the sound speed c = pρ (ρ, S) from gas dynamics. If u < c, the flow is subsonic and the flow is supersonic if u > c. For Euler flows, in [5], R. Courant and K.O. Friedrichs explained that if the pressure at the exit of the nozzle lies in a certain interval, then subsonic inflow will accelerate to the sonic at the throat of the de Laval nozzle, becomes supersonic at the divergent part and then the flow becomes subsonic behind a shock front. Such phenomena have been observed in experiments, however, there have been only a few theoretical analysis for the transonic shock flows. Some significant works for quasi-one-dimensional system were obtained by T.P. Liu [12,13]. Z.P. Xin and H.C. Yin proved the existence and stability of transonic shocks in two dimensional and three dimensional nozzles with some additional restrictions ([15,16]). G.Q. Chen, M. Feldman ([4]) and F. Xie, C.P. Wang ([14]) investigated the transonic shock wave of steady potential compressible fluids in an infinite cylindrical nozzle. In [2], M. Bae, G.Q. Chen and M. Feldman

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.3 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

3

studied the regularity of shock reflection for potential flow and then the transonic shock problems for potential flow in a multidimensional divergent nozzle with an arbitrary smooth cross-section were investigated by M. Bae and M. Feldman in [3]. Then for two dimensional full Euler system, S.X. Chen in [6] studied the transonic shock problem in straight ducts firstly. Other interesting results of transonic shocks for Euler system were investigated in [7], [9] and [17]. For Euler-Poisson system, there has electric field in the nozzles, which is quiet different from Euler system. In [1], U.M. Asher, P.A. Markowich and P. Pietra considered the boundary value problem for isentropic case with linear pressure function and the special subsonic boundary conditions ρ(0) = ρ(L). Then I. Gamba in [8] constructed a transonic solution may contain transonic shocks by using the method of vanishing viscosity, however, the solutions as the limit of vanishing viscosity may contain boundary layer. General cases were considered by Z.P. Xin and T. Luo [10], in one-dimensional Euler-Poisson system, they investigated transonic shock solutions to the boundary value problem case by case. Later, in [11], T. Luo, J. Rauch, C.J. Xie and Z.P. Xin established the structural stability and dynamical stability of the transonic shocks, provided that the electric field is positive at the shock location. All above results study transonic flows with two sides, supersonic upstream and subsonic downstream. In this paper, this is the first study of the multi-transonic Euler-Poisson flows. And for multi-transonic solutions we obtain similar properties, such as monotonicity and structure stability, as the results in [11]. The rest of this paper is organized as follows. First we investigate the detail behavior of the subsonic-sonic-supersonic-shock-subsonic solutions for Euler-Poisson system in section 2. And section 3 is devoted to the monotonicity between the shock position and the density at the exit and the unique existence of the multi-transonic solution. 2. The flow pattern of the multi-transonic solutions In this section, we will study the system (1.3) with the boundary conditions (1.5). Here, we investigate situation with restriction on the background charge that b < ρs , which is the second case in [10]. It follows from the first equation in (1.3) that ρu = J , where the mass flux J is a constant. Moreover, there is a unique solution ρ = ρs for the equation pρ (ρ, S)ρ 2 = J 2 ,

(2.1)

pρ (ρ, S)ρ 2 < J 2 , i.e., ρ < ρs ,

(2.2)

which is the sonic state. In this case, if

the flow is supersonic. Then the flow is subsonic, if pρ (ρ, S)ρ 2 > J 2 , i.e., ρ > ρs .

(2.3)

Then the system (1.3) is equivalent to ⎧ p(ρ,S) S ⎪ ⎪ ⎨ ργ = e , 2

J ( 2ρ 2 +

γ eS γ −1 )x γ −1 ρ

⎪ ⎪ ⎩E = ρ − b. x

= E,

(2.4)

JID:YJDEQ AID:10150 /FLA

4

[m1+; v1.304; Prn:25/11/2019; 12:28] P.4 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

When the solutions of (1.3) have no singularity, the entropy S is a constant along the stream line. Therefore, the smooth solutions of (1.3) can be analyzed in (ρ, E)-phase plane. The trajectories in (ρ, E)-phase plane satisfy the following equation  d

 1 2 E − H (ρ, S) = 0, 2

(2.5)

where   ρ−b J2 S γ −1 Hρ (ρ, S) = − 2 . γe ρ ρ ρ

(2.6)

The trajectory passing through the point (ρ0 , E0 ) with ρ0 > 0 is given by 1 2 E − 2

ρ

1 Hτ (τ, S)dτ = E02 . 2

(2.7)

ρ0

The critical trajectory is defined by the trajectory passing through the point (ρs , 0) and satisfying the equation 1 2 E − 2

ρ Hτ (τ, S)dτ = 0.

(2.8)

ρs

The critical trajectory can be divided into two branches: a supersonic branch and a subsonic branch. The subsonic branch is for ρ > ρs and the supersonic branch is for ρm ≤ ρ < ρs , where ρm is determined by ρm Hτ (τ, S)dτ = 0,

0 < ρ m < ρs .

(2.9)

ρs

In this paper, we focus on the boundary value problem (1.3) and (1.5) with the flow pattern as subsonic-sonic-supersonic-shock-subsonic solutions. Therefore, the initial value (ρ0 , E0 ; S0 ) is given on the lower subsonic branch of the critical trajectory, i.e., 1 2 E − 2 0

ρ0 Hτ (τ, S0 )dτ = 0, ρ0 > ρs and E0 < 0.

(2.10)

ρs

Suppose the location of the shock is at point a ∈ (0, L), then we have the following RankineHugoniot conditions

  ⎧

J2 J2 ⎪ ) = p(ρ, S) + (a p(ρ, S) + − ⎪ ρ  ρ (a+ ), ⎨

2 γp(ρ,S) γp(ρ,S) J2 J + (γ −1)ρ (a− ) = 2ρ 2 + (γ −1)ρ (a+ ), 2ρ 2 ⎪ ⎪ ⎩ E(a− ) = E(a+ ),

(2.11)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.5 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

5

and the Lax’s entropy condition ρ(a+ ) > ρ(a− ).

(2.12)

ρ(a+ ) > ρs > ρ(a− ).

(2.13)

Moreover, the shock is transonic if

The definition of subsonic-sonic-supersonic-shock-subsonic solutions for the boundary value problem (1.3) and (1.5) is given as follows Definition 2.1. The subsonic-sonic-supersonic-shock-subsonic solutions to the boundary value problem (1.3) and (1.5) have the following form ⎧ (ρI , EI ; SI ), 0 ≤ x < x∗ , ⎪ ⎪ ⎪ ⎨(ρ , 0; S ), x = x , s s ∗ (ρ, E; S) = ∗ ⎪ , E ; S ), x (ρ − − − ∗
(2.14)

The solution (ρ, E; S) is piecewise smooth and the Rankine-Hugoniot shock jump conditions are satisfied at the shock location x = x ∗ as the following    J2 J2 ∗ p(ρ− , S− ) + (x ) = p(ρ+ , S+ ) + (x ∗ ), ρ− ρ+

γp(ρ− , S− ) J2 γp(ρ+ , S+ ) J2 ∗ + + (x ) = (x ∗ ), 2 2 (γ − 1)ρ− (γ − 1)ρ+ 2ρ− 2ρ+ 

(2.15)

E− (x ∗ ) =E+ (x ∗ ). Furthermore, ρ > ρs ,

for x ∈ [0, x∗ ),

ρ = ρs ,

for x = x∗ ,

ρ < ρs ,

for x ∈ (x∗ , x ∗ ),

ρ > ρs ,

for x ∈ (x ∗ , L],

and (ρ, E; S) satisfies the system (1.3) on the interval (0, x ∗ ) and (x ∗ , L), respectively. Moreover, (ρ, E; S) satisfies the boundary conditions (1.5). The flow pattern of multi-transonic solutions can be described in the following phase plane. In Fig. 1, from A to B, the flow is smooth transonic on [0, x ∗ ). Acrossing the shock at x = x ∗ , the trajectory jumps from B to C. Accordingly, S+ is the entropy after the shock and the phase plane also changes. And for new sonic state ρ = ρs+ , the subsonic trajectory of the flow starts from C.

JID:YJDEQ AID:10150 /FLA

6

[m1+; v1.304; Prn:25/11/2019; 12:28] P.6 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

Fig. 1. Flow pattern of the multi-transonic solutions.

3. The existence of the multi-transonic flow pattern In this section, the existence of multi-transonic solutions and the monotonicity between the shock location and the density at the exit will be proved. When (ρ1 , E1 ; S) and (ρ2 , E2 ; S) are on the same trajectory, let l((ρ1 , E1 ), (ρ2 , E2 ); S) be the length in x for the trajectory of (1.3) to travel from the state (ρ1 , E1 ; S) to the state (ρ2 , E2 ; S). If E doesn’t change sign,

l((ρ1 , E1 ), (ρ2 , E2 ); S) =

ρ2 p (ρ, S) − ρ

J2 ρ2

ρE(ρ)

dρ,

(3.1)

ρ1

if E is strictly increasing or decreasing, E2 l((ρ1 , E1 ), (ρ2 , E2 ); S) = E1

dE . ρ(E, ρ1 ) − b

(3.2)

In this paper, we study the case of the shock front appears at the upper supersonic branch of the

 1 −1 γ +1 critical trajectory. For given positive constants J , L, S0 , γ > 1, let ρmin = γ2γ ρs < ρs satisfying 1 E(ρmin )2 − 2

ρ min

Hτ (τ, S0 )dτ = 0. ρs

Then, let umax =

J ρmin ,

it holds that

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.7 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

2 Mmax

where M =

u c

u2 J2 = 2 max = = c (ρmin ) γ eS0 ρ γ +1 min



ρs ρmin

γ +1

=

2γ , γ −1

7

(3.3)

is the mach number. Define ρ min

L=

pτ (τ, S0 ) −

J2 τ2

τ E(τ )

dτ,

(3.4)

ρs

and 1 B = {(ρ, E) : ρ > ρs , E < 0, E 2 − 2

ρ Hτ (τ, S0 )dτ = 0, max{0, L − L} < L0 < L}, ρs

where 0 L0 = E(ρ)

dt . τ (t, ρ) − b 2

Remark 3.1. At the sonic state, E(ρs ) and pρs (ρs , S0 ) − Jρ 2 are both equal to zero, hence the defs inition of L is a singular integral. But L is well-defined which can be guaranteed by Remark 3.4. We need the following lemmas to prove the existence of multi-transonic shocks and the monotonicity between the shock location and the density at the exit. Lemma 3.1. For given positive constants J , L, S0 , γ > 1 and (ρ0 , E0 ) ∈ B, there exists x∗ ≤ L, such that the flow is subsonic on [0, x∗ ). Proof. For fixed (ρ0 , E0 ) ∈ B, we define 0 x∗ (ρ0 ) = E0 (ρ0 )

dE . ρ(E, ρ0 ) − b

(3.5)

For each τ ∈ (ρs , ρ0 ), let E(τ )

x(τ ) := l((ρ0 , E0 ), (τ, E(τ )); S0 ) = E0

dt , ρ(t, ρ0 ) − b

(3.6)

where E(τ ) is given by 1 2 E (ρ0 ) − 2 0

ρ0 τ

1 Ht (t, S0 )dt = E 2 (τ ). 2

(3.7)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.8 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

8

Therefore, it holds that E(τ )E  (τ ) = Hτ (τ, S0 ),

(3.8)

x(ρs ) = x∗ (ρ0 ).

(3.9)

and

Differentiating (3.6) with respect to τ gives   E  (τ ) J2 Hτ (τ, S0 ) 1 x (τ ) = = = pτ (τ, S0 ) − 2 . τ − b (τ − b)E(τ ) τ E(τ ) τ 

(3.10)

Then, 0 < ρs < τ and E0 < E(τ ) < 0 imply x  (τ ) < 0.

(3.11)

This completes the proof of the lemma. 2 Remark 3.2. The Lemma 3.1 implies that the flow is subsonic on [0, x∗ ), i.e., for any x ∈ [0, x ∗ ), ρ(x) ∈ (ρs , ρ0 ] and u(x) ∈ [u0 , c(ρs )). For convenience, let 0 L1 := x∗ (ρ0 ) = E0 (ρ0 )

dE , ρ(E, ρ0 ) − b

L1 ∈ (0, L),

(3.12)

ρl ∈ (ρmin , ρs ).

(3.13)

and define ρl by ρl L − L1 =

pτ (τ, S0 ) −

J2 τ2

τ E(τ )

dτ,

ρs

Lemma 3.2. For given positive constants J , L, S0 , γ > 1 and (ρ0 , E0 ) ∈ B, there exists x ∗ (ρ− ) ∈ (L1 , L) such that the flow is supersonic on (L1 , x ∗ ), where ρ− ∈ [ρl , ρs ). Furthermore, for any supersonic state (ρ− , E− ; S− ), there exists a unique subsonic state (ρ+ , E+ ; S+ ) such that the Rankine-Hugoniot conditions hold at x = x ∗ . Proof. First, define ∗

x (ρ− ) := L1 +

ρ− p (ρ, S ) − ρ 0 ρE(ρ) ρs

Then, for each τ ∈ [ρ− , ρs ), let

J2 ρ2

dρ,

ρ− ∈ [ρl , ρs ).

(3.14)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.9 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

τ p (ρ, S ) − ρ 0

x(τ ) = L1 +

J2 ρ2

ρE(ρ)

dρ.

9

(3.15)

ρs

Clearly, x(ρ− ) = x ∗ (ρ− ).

(3.16)

Differentiating (3.15) with respect to τ gives 

x (τ ) =

pτ (τ, S0 ) − τ E(τ )

J2 τ2

.

(3.17)

Therefore, x  (τ ) < 0,

(3.18)

by noting that 0 < ρ− < τ < ρs and E(τ ) > 0. Hence, the flow is supersonic on (L1 , x ∗ ). The Rankine-Hugoniot conditions acrossing the shock at x = x ∗ can be written as ⎧ (ρ− u− )(x ∗ ) = (ρ+ u+ )(x ∗ ), ⎪ ⎪ ⎪ ⎪ γ γ ⎪ S 2 ∗ S 2 ∗ ⎪ ⎨(e − ρ− + ρ− u− )(x ) = (e + ρ+ + ρ+ u+ )(x ),     γ −1 γ −1 S γ eS− ρ− 1 2 ∗ ) = 1 |u2 | + γ e + ρ+ ⎪ |u | + (x (x ∗ ), ⎪ 2 − γ −1 2 + γ −1 ⎪ ⎪ ⎪ ⎪ ⎩ E− (x ∗ ) = E+ (x ∗ ). Substituting u =

J ρ

(3.19)

into (3.19) gets

  ⎧

J2 J2 ∗ ∗ ⎪ ⎨ p(ρ− , S− ) + ρ− (x ) = p(ρ+ , S+ ) + ρ+ (x ),     2 2 ⎪ − ,S− ) + ,S+ ) ⎩ J 2 + γp(ρ (x ∗ ) = J 2 + γp(ρ (x ∗ ). (γ −1)ρ− (γ −1)ρ+ 2ρ−

(3.20)

2ρ+

Combining (3.20) and the entropy conditions ρ+ (x ∗ ) > ρ− (x ∗ ),

S+ (x ∗ ) > S− (x ∗ ),

(3.21)

implies u+ (x ∗ ) =

γ −1 2γ γ −1 2γ p− (x ∗ ) = u− (x ∗ ) + u− (x ∗ ) + p− (x ∗ ), (3.22) γ +1 (γ + 1)ρ− u− γ +1 (γ + 1)J 2 (γ + 1)M− (γ + 1)u2− J ρ− (x ∗ ) = ρ (x ∗ ), = 2 ∗ 2 2 − u+ (x ) 2c− + (γ − 1)u− 2 + (γ − 1)M−   2 (x ∗ ) u2+ (x ∗ ) 2 + (γ − 1)M− ρs+ (x ∗ ) 2 2 ∗ M+ (x ) = 2 = = . 2 (x ∗ ) − (γ − 1) ρ+ c+ (x ∗ ) 2γ M−

ρ+ (x ∗ ) =

(3.23) (3.24)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.10 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

10

Furthermore, implicit function theorem gives S+ (x ∗ ) = S+ (ρ− , u− , S− )(x ∗ ),

(3.25)

which together with the entropy conditions implies there exists unique (ρ+, E+ ; S+ ) such that the R-H conditions hold at x = x ∗ . Then, differentiating (3.19)2 and (3.19)3 with respect to x ∗ gives γ −1 γ eS− ρ− (x ∗ ) −

J2 2 (x ∗ ) ρ−



J2 dρ− (x ∗ ) dρ+ (x ∗ ) S+ γ −1 ∗ = γ e ρ (x ) − + 2 (x ∗ ) dx ∗ dx ∗ ρ+ + eS+ ρ+ (x ∗ ) γ

dS+ (x ∗ ) , dx ∗

(3.26)

and γ −2 γ eS− ρ− (x ∗ ) −

J2

3 (x ∗ ) ρ−



J2 dρ− (x ∗ ) dρ+ (x ∗ ) S+ γ −2 ∗ = γ e ρ (x ) − + 3 (x ∗ ) dx ∗ dx ∗ ρ+ γ −1

+

γ eS+ ρ+ (x ∗ ) dS+ (x ∗ ) , γ −1 dx ∗

(3.27)

together with dρ− (x ∗ ) ρ− (x ∗ )E− (x ∗ ) = γ −1 ∗ dx ∗ γ eS− (x ) ρ− (x ∗ ) −

J2 2 (x ∗ ) ρ−

,

(3.28)

imply dρ+ (x ∗ ) γρ− (x ∗ )E− (x ∗ ) − (γ − 1)ρ+ (x ∗ )E+ (x ∗ ) = , 2 γ −1 dx ∗ γ eS+ ρ+ (x ∗ ) − 2J

(3.29)

dS+ (x ∗ ) (γ − 1)(ρ+ (x ∗ ) − ρ− (x ∗ ))E− (x ∗ ) . = γ dx ∗ eS+ ρ+ (x ∗ )

(3.30)

ρ+

(x ∗ )

and

Then, the entropy condition ρ+ (x ∗ ) > ρ− (x ∗ ) leads to dS+ (x ∗ ) > 0. dx ∗ 2 (x ∗ ) ≤ M 2 Combining M− max =

2γ γ −1

and (3.23) gives

ρ+ (x ∗ ) γ < , ∗ ρ− (x ) γ − 1

for

x ∗ ∈ (L1 , L),

(3.31)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.11 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–••• γ −1

which together with γ eS+ ρ+ (x ∗ ) −

J2 2 (x ∗ ) ρ+

11

> 0 implies

dρ+ (x ∗ ) > 0. dx ∗

(3.32)

Furthermore, the following computation gives that the state (ρ+, E+ ; S+ ) is subsonic, p+ (x ∗ ) γp+ (x ∗ )u+ (x ∗ ) 2 ∗ (x ) − = u + ρ+ (x ∗ ) J   ∗ p+ (x ) = u+ (x ∗ ) u+ (x ∗ ) − γ J

2 (x ∗ ) c− ∗ ∗ = u+ (x )u− (x ) − 1 < 0. u2− (x ∗ )

u+ 2 (x ∗ ) − c+ 2 (x ∗ ) = u+ 2 (x ∗ ) − γ

Then, by using the same argument in Lemma 3.1, the flow is subsonic on (x ∗ , L].

(3.33)

2

Remark 3.3. There is no singularity on [0, x ∗ ), and the entropy S ≡ S0 is a constant on [0, x ∗ ). When shock front appearing at x ∗ , S and the phase plane will change acrossing the shock front. Before shock front appearing, the trajectory satisfies 1 2 E − 2

ρ

1 Hτ (τ, S0 )dτ = E02 , 2

(3.34)

ρ0

then after shock, the trajectory satisfies 1 2 E − 2

ρ

ρ+ (x ∗ )

1 2 ∗ Hτ (τ, S+ (x ∗ ))dτ = E+ (x ). 2

(3.35)

Remark 3.4. (ρ, E; S) is a smooth solution on [0, x ∗ ). (ρs , 0; S0 ) is the only possible singular point, therefore it suffices to show that the solution is regular at x = L1 . For any (ρ, E; S0 ) on the critical trajectory, we have 1 2 E − H (ρ, S0 ) = 0 2 together with ρx =

ρE , c2 −u2

(3.36)

implies √ √ ρ 3 2 H (ρ, S0 ) ρx = − S , |e 0 γρ γ +1 − J 2 |

Set F (ρ) = eS0 γρ γ +1 − J 2 . It can be proved that

for ρ = ρs .

(3.37)

JID:YJDEQ AID:10150 /FLA

12

[m1+; v1.304; Prn:25/11/2019; 12:28] P.12 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

F (ρs ) = 0,

F (k) (ρs ) = 0,

for any k ≥ 1.

(3.38)

By Taylor expansion, F (ρ) can be written as F (ρ) = (ρ − ρs )(kj =1 G(j ) (ρs )(ρ − ρs )j −1 + Rk (ρ)(ρ − ρs )k−1 )

(3.39)

with G(j ) (ρs ) =

F (j ) (ρs ) , j!

lim Rk (ρ) = 0.

ρ→ρs

Let F (ρ) = (ρ − ρs )f (ρ),

(3.40)

where f (ρ) is infinitely many differentiable at ρ = ρs , and f (k) (ρs ) = 0 for all k = 0, 1, 2, . . .. Note that sgnF (ρ) = sgn(ρ − ρs ),

(3.41)

and this implies f (ρ) > 0 for all ρ > 0. Therefore, √ 3√ 2ρ H (ρ, S0 ) ρx = − . f (ρ)|ρ − ρs |

(3.42)

√ H (ρ, S0 ) h(ρ) := , |ρ − ρs |

(3.43)

Set

and the following limit can be derived lim h2 (ρ) = lim

ρ→ρs

ρ→ρs

= lim

ρ→ρs

Hρ (ρ, S0 ) H (ρ, S0 ) = lim (ρ − ρs )2 ρ→ρs 2(ρ − ρs ) (ρ − b)f (ρ) (ρs − b)f (ρs ) = . 2ρ 3 2ρs3

(3.44)

Combining (3.42) and (3.44) implies 3√ ρs2 ρs − b . lim ρx = − √ ρ→ρs f (ρs )

(3.45)

Then √ 3√ 2ρ H (ρ, S0 ) ρ3 ρx = − =− b(ρ), f (ρ) |ρ − ρs | f (ρ)

for ρ = ρs ,

(3.46)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.13 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

where b(ρ) = sgn(ρ − ρs )

√

ρ

2H (ρ,S0 ) ρ−ρs

(k+1)

=

= k 

−E ρ−ρs .

13

For each k ≥ 1, we have √

C(k, j )(

j =0

2ρ 3 (j ) (k−j ) (ρ), ) b f (ρ)

(3.47)

where C(k, j ) = (k−jk!)!j ! . Direct computation implies that the existence of limρ→ρs b(j ) (ρ) for each j = 1, 2, . . ., therefore ρ is infinitely many differentiable at ρ = ρs , so is E, i.e., (ρ, E; S0 ) is smooth on [0, x ∗ ). Lemma 3.3. For any ρ+ (x ∗ ) determined in Lemma 3.2, there exists a unique ρ (x ∗ ) on ρ-axis satisfying the following equation 1 2 ∗ E (x ) − 2 +

and

d ρ (x ∗ ) dx ∗

∗ ρ+ (x )

Hτ (τ, S+ (x ∗ ))dτ = 0,

(3.48)

ρ (x ∗ )

> 0.

Proof. There exists a unique ρ (x ∗ ) such that ( ρ (x ∗ ), 0; S+ (x ∗ )) on the trajectory through ∗ ∗ ∗ (ρ+ (x ), E+ (x ); S+ (x )) and satisfies 1 2 ∗ E (x ) − 2 +

∗ ρ+ (x )

Hτ (τ, S+ (x ∗ ))dτ = 0.

ρ (x ∗ )

The equation (3.48) implies that ρ (x ∗ ) is continuous of ρ+ (x ∗ ). Then, by (3.23), ρ+ (x ∗ ) is ∗ 1 continuous on ρ− (x ), which is C with respect to x ∗ . Therefore, ρ (x ∗ ) is C 1 with respect to x ∗ and we have the following relation d ρ (x ∗ ) d ρ (x ∗ ) dρ+ (x ∗ ) = . · dx ∗ dρ+ (x ∗ ) dx ∗ ∗

∗

ρ (x ) d ρ (x ) By noting (3.32), proving d (x1∗ ) < dx ∗ > 0 is equal to prove that dρ+ (x ∗ ) > 0. Hence, suppose ρ ρ (x2∗ ), we need to prove ρ+ (x1∗ ) < ρ+ (x2∗ ). We argue by contradiction, assume ρ+ (x1∗ ) ≥ ρ+ (x2∗ ) is true. Let  be the set of states which can be connected to the states of supersonic branch of the critical trajectory by transonic shocks. l1 , l2 and l3 are defined as follows

l1 : the trajectory from ( ρ (x1∗ ), 0; S+ (x1∗ )) to (ρ+ (x1∗ ), E+ (x1∗ ); S+ (x1∗ )), l2 : the trajectory from ( ρ (x2∗ ), 0; S+ (x2∗ )) to (ρ+ (x2∗ ), E+ (x2∗ ); S+ (x2∗ )), l3 : the trajectory from ( ρ (x2∗ ), 0; S+ (x1∗ )) to (ρ(x ˆ 2∗ ), E+ (x2∗ ); S+ (x1∗ )).

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.14 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

14

Fig. 2. Contradiction arguments in (ρ, E)-phase plane.

The assumption ρ+ (x1∗ ) ≥ ρ+ (x2∗ ) together with ρ (x1∗ ) < ρ (x2∗ ) implies l1 and l2 have at least one intersection point (ρa , Ea ) in (ρ, E)-phase plane (point F in Fig. 2). Furthermore, by the definition of l2 and l3 , the following equation is satisfied 1 2 ∗ E (x ) − 2 + 2

ρ(x ˆ 2∗ )

ρ + (x2∗ )

τ −b 1 2 ∗ (x2 ) − h(τ, S+ (x1∗ ))dτ = E+ τ 2

∗ ρ+ (x2 )

ρ (x2∗ )

τ −b h(τ, S+ (x2∗ ))dτ, (3.49) τ

then ρ(x ˆ 2∗ )

0= ρ (x2∗ )

τ −b h(τ, S+ (x1∗ ))dτ − τ

∗ ρ+ (x2 )

= ρ (x2∗ )

∗ ρ+ (x2 )

ρ (x2∗ )

τ −b h(τ, S+ (x2∗ ))dτ τ

τ −b ∗ ∗ (γ eS+ (x1 ) τ γ −1 − γ eS+ (x2 ) τ γ −1 )dτ − τ

∗ ρ+ (x2 )

ρ(x ˆ 2∗ )

τ −b h(τ, S+ (x1∗ ))dτ τ

(3.50)

:=J1 + J2 . The assumption ρ+ (x1∗ ) ≥ ρ+ (x2∗ ) and (3.32) imply x1∗ ≥ x2∗ and S+ (x1∗ ) ≥ S+ (x2∗ ). Therefore, J1 ≥ 0 can be obtained by S+ (x1∗ ) ≥ S+ (x2∗ ) and τ > b. Then, ρ(x ˆ 2∗ ) < ρ+ (x2∗ ) ≤ ρ+ (x1∗ ),

(3.51)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.15 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

15

due to h(t, S+ (x1∗ )) > 0. Finally, (3.51) together with ρ (x2∗ ) > ρ (x1∗ ) implies l1 and l3 have at least one intersection point (ρb , Eb ) in (ρ, E)-phase plane (point G in Fig. 2), which is a contradiction to uniqueness of the trajectory in the same phase plane. This completes the proof of the lemma. 2 Lemma 3.4. For given positive constants J , L, S0 , γ > 1 and (ρ0 , E0 ) ∈ B, if there exists a transonic solution satisfying the flow is subsonic on [0, L1 ); the flow is supersonic on (L1 , x ∗ ); the flow is subsonic on (x ∗ , L]. Then it holds that ∂ρ(L) ∂x ∗ < 0. Proof. The existence of the subsonic-sonic-supersonic-shock-subsonic solution can be obtained by the Lemma 3.1 – 3.2. We have the following relation from the fixed L, ρ + (L)

J2 τ2

γ eS+ τ γ −1 −

ρ+ (x ∗ )

τ E+ (τ, x ∗ )

dτ = L − x ∗ ,

(3.52)

where x ∗ is defined by (3.14). Differentiating (3.52) with respect to x ∗ yields −1 =

h(ρ+ (L), S+ ) ∂ρ+ (L) h(ρ+ (x ∗ ), S+ ) dρ+ (x ∗ ) − ρ+ (L)E+ (L) ∂x ∗ ρ+ (x ∗ )E+ (x ∗ ) dx ∗ ρ + (L)

+ ρ+ (x ∗ )

γ eS+ τ γ −1 dS+ dτ − τ E+ (τ, x ∗ ) dx ∗

where h(τ, S) = γ eS τ γ −1 −

ρ + (L)

ρ+ (x ∗ )

J2 , τ2

γ eS+ τ γ −1 −

J2 τ2

2 (τ, x ∗ ) τ E+

∂E+ (τ, x ∗ ) dτ, ∂x ∗

(3.53)

and E+ (τ, x ∗ ) satisfies the following equation

1 2 E (τ, x ∗ ) − 2 +

τ Ht (t, S+ )dt = 0,

(3.54)

ρ (x ∗ )

where ρ (x ∗ ) is given by (3.48). Differentiating (3.54) with respect to x ∗ gives ∂E+ (τ, x ∗ ) ρ (x ∗ ), S+ ) d 1 ρ (x ∗ ) ( ρ (x ∗ ) − b)h( = − ∗ ∗ ∗ ∂x E+ (τ, x ) ρ (x ) dx ∗

τ t − b S+ γ −1 dS+ + dt · ∗ . γe t t dx ρ (x ∗ )

Then substituting (3.55) into (3.53) gives ∂ρ+ (L) ρ+ (L)E+ (L) = ∗ ∂x h(ρ+ (L), S+ )



h(ρ+ (x ∗ ), S+ ) dρ+ (x ∗ ) −1 ρ+ (x ∗ )E+ (x ∗ ) dx ∗



(3.55)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.16 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

16

ρ+ (L)E+ (L) − h(ρ+ (L), S+ )

−

ρ+ (L)E+ (L) h(ρ+ (L), S+ )

ρ + (L) ρ+ (x ∗ ) ρ + (L) ρ+ (x ∗ )

h(τ, S+ )h( ρ (x ∗ ), S+ ) ρ ρ (x ∗ ) (x ∗ ) − b d dτ 3 (τ, x ∗ ) ρ (x ∗ ) dx ∗ τ E+ ⎛ ⎜ ⎝

γ eS+ τ γ −1 τ E+

(τ, x ∗ )

−

τ

h(τ, S+ ) 3 (τ, x ∗ ) τ E+

ρ (x ∗ )

⎞ t − b S+ γ −1 ⎟ dt ⎠ dτ γe t t

dS+ dx ∗ := I1 + I2 + I3 .

·

(3.56)

Direct computation yields  ρ− (x ∗ ) − 1 < 0, ρ+ (x ∗ )

(3.57)

h(τ, S+ )h( ρ (x ∗ ), S+ ) ρ ρ (x ∗ ) (x ∗ ) − b d dτ < 0. 3 (τ, x ∗ ) ρ (x ∗ ) dx ∗ τ E+

(3.58)

γρ+ (L)E+ (L) I1 = h(ρ+ (L), S+ )



and according to Lemma 3.3, ρ+ (L)E+ (L) I2 = − h(ρ+ (L), S+ )

ρ + (L)

ρ+

(x ∗ )

Next,

I3 = −

·

ρ+ (L)E+ (L) h(ρ+ (L), S+ )

ρ + (L) ρ+ (x ∗ )

⎛ ⎜ ⎝

γ eS+ τ γ −1 τ E+

(τ, x ∗ )

−

τ

h(τ, S+ ) 3 (τ, x ∗ ) τ E+

⎞ (t

− b)γ eS+ t γ −1

ρ (x ∗ )

t

⎟ dt ⎠ dτ

dS+ (x ∗ ) dx ∗

(3.59)

ρ+ (L)E+ (L) =− h(ρ+ (L), S+ )

ρ + (L) ρ+ (x ∗ )

N (τ ) 3 (τ, x ∗ ) τ E+

dτ ·

dS+ (x ∗ ) , dx ∗

where S+ γ −1

N(τ ) = γ e τ

2 E+ (τ, x ∗ ) − h(τ, S+ )

τ

ρ (x ∗ ) S+ γ −1

τ

= 2γ e τ

ρ (x ∗ )

τ = ρ (x ∗ )

(t − b)γ eS+ t γ −1 dt t

(t − b)h(t, S+ ) dt − h(τ, S+ ) t

τ

ρ (x ∗ )

(t − b)γ eS+ t γ −1 dt t

γ eS+ (t − b) (γ eS+ t γ +1 τ γ +1 − 2J 2 τ γ +1 + J 2 t γ +1 )dt, τ 2t 3

t ∈ ( ρ (x ∗ ), τ ).

(3.60)

JID:YJDEQ AID:10150 /FLA

[m1+; v1.304; Prn:25/11/2019; 12:28] P.17 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

17

Set f (t, τ ) = γ eS+ t γ +1 τ γ +1 − 2J 2 τ γ +1 + J 2 t γ +1 ,

t ∈ ( ρ (x ∗ ), τ ).

(3.61)

According to entropy condition S+ > S− > ln 2S− , we have eS+ > 2eS− . Then, γ +1

ρs

=

J2 2J 2 γ +1 > S = 2ρs+ . S γe − γe +

(3.62)

Direct computation gives ft (t, τ ) = (γ + 1)γ eS+ t γ τ γ +1 + (γ + 1)J 2 t γ > 0,

(3.63)

and β γ +1 =

γ +1

1+

2ρs+ γ +1

ρ γ +1 < 2ρs+ ,

(3.64)

s+

τ

where β satisfying f (β, τ ) ≡ 0. Therefore, we have 1

f (β, τ ) < f (2 γ +1 ρs+ , τ ) < f (ρs , τ ) < f ( ρ (x ∗ ), τ ),

(3.65)

by noting that ρs < ρ (x ∗ ) and (3.63). Then, f (t, τ ) > 0,

for

t ∈ ( ρ (x ∗ ), τ ),

which together with (3.31), (3.59) and (3.60) implies I3 < 0. Therefore, ∂ρ+ (L) = I1 + I2 + I3 < 0. ∂x ∗

(3.66)

This completes the proof of the lemma. 2 Finally, the following theorem can be obtained from Lemma 3.1 – 3.4. Theorem 3.1. For given positive constants J , L, S0 and γ > 1, there exists a non-empty parameter set B. For any (ρ0 , E0 ) ∈ B, there exists an interval I = (ρ, ρ) and when ρe ∈ I , the boundary value problem (1.3) and (1.5) exists a unique solution (ρ, E; S) on [0, L] in the form of (2.14). Furthermore, the solution satisfies the Rankine-Hugoniot condition (2.15) and entropy condition (3.21) at x = x ∗ . Proof. In Lemma 3.4, we have proved the monotonicity between the shock location x ∗ and the end density ρ(L). Then set ρ = ρL (L) and ρ = ρL1 (L), there exists a unique shock location x ∗ such that ρx ∗ (L) = ρe ∈ (ρ, ρ). Furthermore, the proofs in Lemma 3.1 – 3.3 imply the boundary value problem (1.3) and (1.5) exists a unique piecewise smooth solution in the form of (2.14), which satisfies R-H conditions (2.15) at x = x ∗ and the entropy condition (3.21). 2

JID:YJDEQ AID:10150 /FLA

18

[m1+; v1.304; Prn:25/11/2019; 12:28] P.18 (1-18)

B. Duan, Y. Zhou / J. Differential Equations ••• (••••) •••–•••

Remark 3.5. It’s worth to mention that, the subsonic-sonic-supersonic-shock-subsonic solution can be divided into two transonic parts, one part is smooth transonic, the other is transonic shock. Acknowledgment The research of authors is partially supported by NSFC No. 11871133, No. 11671412, the Fundamental Research Funds for the Central Universities grant DUT18RC(3)000 and the Highlevel innovative and entrepreneurial talents support plan in Dalian grant 2017RQ041. References [1] U.M. Asher, P.A. Markowich, P. Pietra, C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Methods Appl. Sci. (1991) 347–376. [2] M. Bae, G.Q. Chen, M. Feldman, Regularity of solutions to regular shock reflection for potential flow, Invent. Math. 175 (2009) 505–543. [3] M. Bae, M. Feldman, Transonic shocks in multidimensional divergent nozzles, Arch. Ration. Mech. Anal. 201 (2011) 777–840. [4] G.Q. Chen, M. Feldman, Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185–242. [5] R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers Inc., New York, 1948. [6] S.X. Chen, Stability of transonic shock fronts in two-dimensional Euler systems, Trans. Am. Math. Soc. 357 (2005) 287–308. [7] S.X. Chen, Z.P. Xin, H.C. Yin, Global shock waves for the supersonic flow past a perturbed cone, Commun. Math. Phys. 228 (2) (2002) 47–84. [8] I.M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Commun. Partial Differ. Equ. 17 (1992) 553–577. [9] J. Li, Z.P. Xin, H.C. Yin, On transonic shocks in a nozzle with variable end pressures, Commun. Math. Phys. 291 (2009) 111–150. [10] T. Luo, Z.P. Xin, Transonic shock solutions for a system of Euler-Poisson equations, Commun. Math. Sci. 10 (2012) 419–462. [11] T. Luo, J. Rauch, C.J. Xie, Z.P. Xin, Stability of transonic shock solutions for one-dimensional Euler-Poisson equations, Arch. Ration. Mech. Anal. 202 (2011) 787–827. [12] T.P. Liu, Transonic gas flow in a variable area duct, Arch. Ration. Mech. Anal. 80 (1982) 1–18. [13] T.P. Liu, Nonlinear stability and instability of transonic gas flow through a nozzle, Commun. Math. Phys. 83 (1982) 243–260. [14] F. Xie, C.P. Wang, Transonic shock wave in an infinite nozzle asymptotically converging to a cylinder, J. Differ. Equ. 242 (2007) 86–120. [15] Z.P. Xin, H.C. Yin, Transonic shock in a nozzle I: two dimensional case, Commun. Pure Appl. Math. 58 (2005) 999–1050. [16] Z.P. Xin, H.C. Yin, The transonic shock in a nozzle, 2-D and 3-D complete Euler systems, J. Differ. Equ. 245 (2008) 1014–1085. [17] Z.P. Xin, W. Yan, H.C. Yin, Transonic shock problem for the Euler system in a nozzle, Arch. Ration. Mech. Anal. 194 (2009) 1–47.