Chinese Annals of Mathematics, Series B
Chin. Ann. Math. 34B(4), 2013, 557–574 DOI: 10.1007/s11401-013-0780-1
c The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2013
Nonexistence of a Globally Stable Supersonic Conic Shock Wave for the Steady Supersonic Isothermal Euler Flow∗ Yuchen LI1
Gang XU2
Abstract In this paper, for the full Euler system of the isothermal gas, we show that a globally stable supersonic conic shock wave solution does not exist when a uniform supersonic incoming flow hits an infinitely long and curved sharp conic body. Keywords Supersonic flow, Conic shock, Full Euler system, Isothermal gas, Nonexistence 2000 MR Subject Classification 35L70, 35L65, 35L67, 76N15
1 Introduction and Main Results When a uniform supersonic incoming flow (0, 0, q0 ) with constant density ρ0 > 0 and constant pressure P0 hits a circular cone x21 + x22 = b0 x3 along the axis x3 -direction (see Figure 1 below), there will appear a global supersonic conic shock x21 + x22 = s0 x3 (s0 > b0 ) attached at the tip of the cone (see [7]), where b0 is less than a critical value b∗ (b∗ is determined by the parameters of the incoming flow). For the 3-D potential equation, under the different assumptions on the Mach number of the supersonic incoming flow and the appropriate vertex angle of the conic body, the global existence and stability of a conic shock have been established by many authors in [6], [8–9] and [23]. For the full Euler system, under certain restrictions on the perturbed conic bodies, the local supersonic conic shocks have been studied in [4–5]. In addition, Lien and Liu [15] applied the Glimm’s scheme to obtain the global existence and the long-distance asymptotic behavior of a weak solution to the full Euler system in the symmetric conic case under the suitable conditions on the Mach number, the sharp vertex angle and the shock strength. On the other hand, closely related to our topic on the conic shocks, there are many works on the supersonic oblique shock problem for the supersonic flow past a 2-D curved sharp wedge, where the existence of a weak solution or the local existence of a supersonic oblique shock to the 2-D steady full Euler system has been established (see [2–3], [14], [18] and the references therein). Recently, when the surface of the cone is perturbed, Xu and Yin [25] have shown that there can not exists a globally stable multidimensional supersonic conic shock in the polytropic case. The purpose of this paper is to show that such a global shock wave does not exist in the isothermal case either. Manuscript received February 8, 2012. Revised August 1, 2012. of Mathematics & IMS, Nanjing University, Nanjing 210093, China. E-mail:
[email protected] 2 Corresponding author. Department of Mathematics, Jiangsu University, Zhengjiang 212013, Jiangsu China. E-mail:
[email protected] ∗ Project supported by the National Natural Science Foundation of China (Nos. 11025105, 10931007, 11101190), the Doctorial Program Foundation of Ministry of Education of China (No. 20090091110005) and the Natural Science Fundamental Research Project of Jiangsu Colleges (No. 10KLB110002). 1 Department
Y. C. Li and G. Xu
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Figure 1
The so-called isothermal gas means that the equation of the state is given by P = Aρ, where A = RT , T (temperature) and R are positive constants. From this, one has another equation of the state e(ρ, S) = A ln ρ + T S. ∂ρ P (ρ, S) ≡ A stands for the sound speed. The internal energy In addition, c(ρ, S) = function e(ρ, S) = A ln ρ + T S is different from the polytropic case completely. For the reader’s convenience, we give the derivation in the appendix. The steady full Euler system is described as ⎧ 3 ⎪ ⎪ ⎪ ∂j (ρuj ) = 0, ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ 3 ⎨ ∂j (ρui uj ) + ∂i P = 0, i = 1, 2, 3, (1.1) ⎪ ⎪ j=1 ⎪ ⎪ ⎪ 3 ⎪ ⎪ 1 2 ⎪ ⎪ ρ|u| ρe + ∂ + P uj = 0, j ⎪ ⎩ 2 j=1 where u = (u1 , u2 , u3 ), ρ, P, e and S stand for the velocity, density, pressure, internal energy and specific entropy respectively. Moreover, the pressure function P = P (ρ, S) and the internal energy function e = e(ρ, S) are smooth in their arguments, which satisfy ∂ρ P (ρ, S) > 0 and ∂S e(ρ, S) > 0 for ρ > 0. Suppose that the supersonic incoming flow with the state (ρ0 , 0, 0, q0 , P0 ) hits the perturbed conic body along x3 -direction, whose surface equation is denoted by r = b(x3 ), where r = x21 + x22 , b(x3 ) = b0 x3 + εϕ(x3 ), ε > 0 is sufficiently small, ϕ(x3 ) ≡ 0 and ϕ(x3 ) ∈ C0∞ (0, l) with some fixed positive number l > 0. In addition, b0 > 0 is suitably small such that the resulting supersonic shock will be attached at the vertex of the conic body. By the symmetric property of the perturbed conic surface, it is convenient to introduce the following cylindrical coordinates (x3 , r) to study our problem
r = x21 + x22 , x3 = x3 . (1.2)
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For the isothermal gas and the axisymmetric solution to (1.1), i.e., x1 x2 (ρ(x), u1 (x), u2 (x), u3 (x), P (x)) ≡ ρ(x3 , r), U (x3 , r) , U (x3 , r) , u3 (x3 , r), P (x3 , r) , r r (1.1) can be simplified as ⎧ ⎪ ⎨∂r (rρU ) + ∂3 (rρu3 ) = 0, ∂r (rρU 2 ) + ∂3 (rρU u3 ) + r∂r P = 0, ⎪ ⎩ ∂r (rρU u3 ) + ∂3 (rρu23 ) + r∂3 P = 0
(1.3)
and 1 2 1 (U + u23 ) + A ln ρ + T S = q02 + A ln ρ0 + T S0 ≡ C0 , 2 2
(1.4)
where (1.4) is the Bernoulli’s law. Suppose that the flow field behind the supersonic shock r = χ(x3 ) with χ(0) = 0 is denoted + by (ρ+ (x3 , r), U + (x3 , r), u+ 3 (x3 , r), P (x3 , r)). Then, in the domain Ω+ = {(x3 , r) : x3 > + 0, b(x3 ) < r < χ(x3 )}, (ρ+ , U + , u3 , P + ) satisfies ⎧ ∂r (rρ+ U + ) + ∂3 (rρ+ u+ ⎪ 3 ) = 0, ⎪ ⎪ ⎪ + + 2 + + + ⎪ ⎨∂r (rρ (U ) ) + ∂3 (rρ U u3 ) + r∂r P + = 0, + + 2 + (1.5) ∂r (rρ+ U + u+ = 0, 3 ) + ∂3 (rρ (u3 ) ) + r∂3 P ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ((U + )2 + (u+ )2 ) + A ln ρ+ + T S + = C 0 3 2 2 with (U + )2 + (u+ 3 ) > A. On the shock r = χ(x3 ), the Rankine-Hugoniot conditions imply ⎧ ⎪ ⎨[ρU ] − χ (x3 )[ρu3 ] = 0, 2 [P + ρU ] − χ (x3 )[ρU u3 ] = 0, ⎪ ⎩ [ρU u3 ] − χ (x3 )[P + ρu23 ] = 0.
Meanwhile, the Lax’s geometrical entropy condition (see [22]) holds ⎧ + + λ (U + (x3 , χ(x3 )), u+ ⎪ 3 (x3 , χ(x3 ))) < χ (x3 ) < λ4 (U (x3 , χ(x3 )), u3 (x3 , χ(x3 ))), ⎪ ⎨ 3 √ A ⎪ ⎪ < χ (x3 ), ⎩ 2 q0 − A where λ1,4 (U, u3 ) =
U u3 ∓
(1.6)
(1.7)
√ A U 2 + u23 − A u23 − A
and λ2 (U, u3 ) ≡ λ3 (U, u3 ) =
U . u3
Due to the fixed wall condition, we have on the conic surface r = b(x3 ), U + = b (x3 )u+ 3. The main result in our paper is the following theorem.
(1.8)
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Theorem 1.1 (Nonexistence of a Globally Stable Supersonic Conic Shock) Under the assumptions above, for the isothermal gas and suitably small b0 , there exists an ε0 > 0 such that for + ε < ε0 , the problem (1.5) with (1.6)–(1.8) has no global solution (ρ+ (x), U + (x), u+ 3 (x), P (x); χ(x3 )) in Ω+ which admits the following properties: (i) χ(x3 ) ∈ C 2 [0, ∞) and χ (x3 ) − s0 L∞ [0,∞) ≤ Cε. + 1 ∞ (ii) (ρ+ , U + , u+ 3 , P ) ∈ C (Ω+ \ (0, 0)) ∩ L (Ω+ ) and r + + + + , u ρ, U 3 , P ) ≤ Cε, (ρ , U , u3 , P )(x3 , r) − ( x3 C 1 (Ω+ )
r r r where ρ xr3 , U ,u 3 , P stands for the extension of the self-similar downstream su x3r xr3 x3r r 3 x3 , P x3 behind the shock r = s0 x3 , which is formed for the personic state ρ x3 , U x3 , u supersonic incoming flow (ρ0 , 0, 0, q0 , P0 ) past the cone {x : x21 + x22 < b0 x3 }. With respect to
r r r r
r r much more detailed information on ρ x3 , U 3 x3 , P x3 and ρ xr3 , U 3 x3 , x3 , u x3 , u
r , one can see Lemmas 2.1–2.2 and Remark 2.1 in §2. P x3
Remark 1.1 Due to b(x3 ) − b0 x3 = εϕ(x3 ) ∈ C0∞ (0, l), then it follows from the property of the finite propagation speed of hyperbolic systems that there exists a neighborhood B(0, δ0 ) of the origin O = (0, 0) such that the solution + +, u + ρ+ , U + (ρ+ , U + , u+ 3 , P ; χ(x3 )) ≡ ( 3 , P ; s0 x3 )
in the region Ω+ ∩ B(0, δ0 ). On the other hand, by the local existence and stability of a multidimensional shock wave in [16–17] or the Appendix of [11], we know that (i) and (ii) of Theorem 1.1 are reasonable in any bounded sub-domain of Ω+ . Particularly, in the potential equation case, (i) and (ii) of Theorem 1.1 have been shown in [6], [8–9] and [23] under some different restrictions on the supersonic incoming flow and the perturbed conic body. Remark 1.2 Our result illustrates the essential differences between the 3-D full Euler system and the potential equation when one studies the global supersonic conic shock problem, which are induced by the rotation rot u ≡ (∂2 u3 − ∂3 u2 , ∂3 u1 − ∂1 u3 , ∂1 u2 − ∂2 u1 ). In the latter case, rot u ≡ 0 holds, and the existence and stability of a global conic shock have been established by us in [6] and [8–9]. In order to prove Theorem 1.1, at first we show that the supersonic shock curve must be straight under the assumptions of Theorem 1.1. Subsequently, we r derive that the
down
further r r , u , P by stream supersonic solution must be the background solution ρ xr3 , U 3 x3 x3 x3 use of the uniqueness result of the solutions to the hyperbolic equations and the perturbed conic surface is just r = b0 x3 . However, this is obviously a contradiction to the property of b(x3 ) ≡ b0 x3 . From this, we complete the proof of Theorem 1.1. Our paper is organized as follows. In Section 2, for the full Euler system, we will show that there exists an attached supersonic conic shock r = s0 x3 when a uniform supersonic incoming flow (ρ0 , 0, 0, q0 , P0 ) hits the sharp cone r = b0 x3 . In Section 3, under the assumptions of Theorem 1.1, it is shown that a globally stable supersonic shock curve must be straight. In Section 4, by use of the standard theory on the second order quasilinear hyperbolic equations, we prove that the downstream supersonic solution is the same as the background solution and further derive that the perturbed conic surface r = b(x3 ) is r = b0 x3 . By this contradiction, Theorem 1.1 can be shown.
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2 The Analysis on the Self-similar Background Solution Suppose that there is a uniform supersonic flow (ρ0 , 0, 0, q0 , P0 ) which hits the circular cone r = b0 x3 along the x3 -direction. Then as described in [7], there exists a critical value b∗ such that there will appear an attached supersonic conic shock r = s0 x3 (s0 > b0 ) for b0 < b∗ . Moreover the solution to (1.5) has such a form: ρ+ (x) = ρ(s),
u+ 1 (x) = U (s)
x1 , r
u+ 2 (x) = U(s)
x2 , r
u+ 3 (s) 3 (x) = u
(s), u P + (x) = P(s) with s = xr3 . With respect to the existence of ( ρ(s), U 3 (s), P(s); s0 ), Sections 154–156 of [7] have given the outline of the proof procedure. However, for the reader’s convenience and use later on, we will give a detailed proof and further establish some precise (s), u properties on the background supersonic solution ( ρ(s), U 3 (s), P (s)). (s), u By use of (1.5) and a direct computation, we obtain that ( ρ(s), U 3 (s), P(s)) satisfies ⎧ ) (s ρU u3 − U ⎪ ⎪ ⎪ρ (s) = − , ⎪ ⎪ 2 2) ⎪ s((1 + s )A − (s u3 − U) ⎪ ⎪ ⎪ ⎪ AU ⎪ ⎪ ⎪ , ⎨U (s) = − 2) s((1 + s2 )A − (s u3 − U) (2.1) ⎪ AU ⎪ ⎪ , u 3 (s) = ⎪ ⎪ )2 ⎪ (1 + s2 )A − (s u3 − U ⎪ ⎪ ⎪ ⎪ (s ⎪ ρAU u3 − U) ⎪ ⎪ ⎩P (s) = − 2 2) s((1 + s )A − (s u3 − U) 2 (s) + u for b0 ≤ s ≤ s0 , where U 23 (s) > A. Moreover, the Bernoulli’s law holds 1 2 ≡ C0 . (U (s) + u 23 (s)) + A ln ρ(s) + T S(s) 2
(2.2)
(s), u For the notational convenience, we will drop “∼” below in the solution ( ρ(s), U 3 (s), P (s)). According to Lemma 2.2 below, we know that the denominator (1 + s2 )A − (su3 − U )2 > 0 in (2.1) holds for b0 ≤ s ≤ s0 . This means that the system (2.1) is meaningful. On the shock surface r = s0 x3 , due to the Rankine-Hugoniot conditions and the Lax’s geometrical entropy condition, one has ⎧ ⎪ ⎨[ρU ] − s0 [ρu3 ] = 0, (2.3) [P + ρU 2 ] − s0 [ρU u3 ] = 0, ⎪ ⎩ 2 [ρU u3 ] − s0 [P + ρu3 ] = 0 and ⎧ λ (U (s0 ), u3 (s0 )) < s0 < λ4 (U (s0 ), u3 (s0 )), ⎪ ⎪ ⎨ 3 √ A ⎪ ⎪ < s0 . ⎩ 2 q0 − A
(2.4)
Additionally, U (b0 ) = b0 u3 (b0 ).
(2.5)
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With respect to the nonlinear system (2.1)–(2.2) with the free boundary value conditions (2.3)–(2.4) and the fixed boundary value condition (2.5), we have the following result. Lemma 2.1 For suitably small b0 > 0, we can conclude that (2.1)–(2.5) has a smooth solution √(ρ(s), U (s), u3 (s), P (s)) and admits a determined shock position s = s0 such that u3 (s) > A holds true for b0 ≤ s ≤ s0 . Proof Set ρ+ = For convenience, let
lim ρ(s), U+ =
s→s0 −0 σ = 1s , and
lim U (s), u3+ =
s→s0 −0
lim u3 (s) and P+ =
s→s0 −0
then (2.1) can be rewritten as
⎧ ρU (u3 − σU ) ⎪ ⎪ ρ (σ) = , ⎪ ⎪ (1 + σ 2 )A − (u3 − σU )2 ⎪ ⎪ ⎪ ⎪ ⎪ σAU ⎪ ⎪ ⎨U (σ) = (1 + σ 2 )A − (u − σU )2 , 3 AU ⎪ ⎪ ⎪ , u3 (σ) = − ⎪ ⎪ (1 + σ 2 )A − (u3 − σU )2 ⎪ ⎪ ⎪ ⎪ ρAU (u3 − σU ) ⎪ ⎪ ⎩P (σ) = (1 + σ 2 )A − (u3 − σU )2 for σ0 ≤ σ ≤ We have
1 b0 ,
where σ0 =
lim P (s).
s→s0 −0
(2.6)
1 s0 .
P (σ) = Aρ(σ),
(2.7)
and then by the fourth equation in (1.5) and (2.7), we know that ρ(σ), P (σ) can be expressed as the smooth functions of (U (σ), u3 (σ)) respectively. Namely, ρ(σ) = ρ(U (σ), u3 (σ)),
P (σ) = P (U (σ), u3 (σ)).
(2.8)
In addition, by the second and third equations in (2.6), we find that U (σ) = −σ. u3 (σ) As indicated in Section 155 of [7], it is particularly amenable to treat U as a function of u3 . In this case, one has U (u3 ) = −σ
(2.9)
and u3 (σ) = −
1 U (u
3)
,
U (σ) = −
U (u3 ) . U (u3 )
(2.10)
Substituting (2.10) into the second equation of (2.6) and using (2.7)–(2.8) yield U U (u3 ) = 1 + (U (u3 ))2 −
(u3 + U U (u3 ))2 . A
Assume that the parameter equation of the shock polar is u3 = u3 (t),
U = U (t),
(2.11)
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which describes the relation of u3 and U on the (u3 , U )-plane in terms of the R-H conditions (2.3). From (2.3), using [f g] = [f ]g+ + f− [g], one has ⎧ ⎪ ⎨[ρU ] − s0 [ρu3 ] = 0, (2.12) s0 [U ] + [u3 ] = 0, ⎪ ⎩ ρ0 q0 [u3 ] + A[ρ] = 0. The first and second equations yield ρU 2 − (q0 − u3 )(ρu3 − ρ0 q0 ) = 0.
(2.13)
From the third equation, one has 1 q0 ρ = ρ0 1 + q02 − u3 . A A
(2.14)
Substituting (2.14) into (2.13), the equation of the shock polar for isothermal gas is U2 =
A
(q0 − u3 )2 (q0 u3 − A) A + q02 − q0 u3
< u3 < q0 , which is similar to the (121.04) of [7]. Its picture can be drawn in Figure 2 (see [7, p. 313]). q0
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Next, we discuss the existence of the solution to the equation (2.11), which starts from any point M (u3 (t0 ), U (t0 )) of the shock polar in the supersonic part. In this case, only U > 0, u3 > 0 and σ > 0 are considered. In the light of (2.3), we can look for the related σ, which is denoted by σ0 (t0 ) =
U (t0 ) . q0 − u3 (t0 )
From (2.9), we have Uu 3 (u3 (t0 )) = −σ0 (t0 ). Now we study the following initial value problem AU U (u3 ) = (A − U 2 )(U )2 (u3 ) − 2u3 U U (u3 ) + (A − u23 ), U (u03 ) = U 0 , U (u03 ) = −σ0 ,
(2.15)
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u03
0
where = u3 (t0 ), U = U (t0 ) and σ0 = σ0 (t0 ). From the theory of ODE and U 0 > 0, (2.15) is locally solvable. Next, we assert that (2.15) is actually solved in the first quadrant when σ0 ≤ σ ≤ b10 . Indeed, in the downstream supersonic domain, the right hand side of (2.15) is (A − U 2 )(U )2 (u3 ) − 2u3 U U (u3 ) + (A − u23 ) = σ 2 ((1 + s2 )A − (su3 − U )2 ) > 0. This derives that U (u3 ) > 0 holds. By (2.10), one has u3 (σ) < 0 and U (σ) > 0, which means that the solution curve of (2.15) extends from southeast to northwest and U (u3 ) de creases along solution curve (but |U (u3 )| increases) (see Figure 3 below for convenience). the 1 Since σ ∈ σ0 , b0 , then one can derive that
0
U −U =
u3
u03
Uu 3 (z)dz
u03
= u3
σdz ≤
0
u03
1 u0 dz ≤ 3 . b0 b0
u03 b0
Thus, U 0 ≤ U ≤ +U 0 holds. Together with |Uu 3 | ≤ b10 , it follows from the extension theorem of the solution to the ODE that (2.15) is always solvable in the supersonic domain of the first quadrant. On the other hand, for suitably small b0 , u3 > c always holds (see Lemma 2.2(ii) 1 below). Thus, (2.15) is solved for σ ∈ σ0 , b0 and we keep the flow field to be supersonic in x3 -direction. 6 6
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Next, for fixed t0 , we associate the equation (2.15) with the shock boundary condition and the fixed wall boundary condition in (2.3) and (2.5), respectively. For the requirement of (2.5), we need to look for a point N in (u3 , U )-plane such that the solution curve of (2.15) ends N and fulfills (u3 , U )(1, Uu 3 )|N = 0, which is equivalent to arctan
U π − arctan U (u3 ) = u3 2
at N.
(2.16)
Now we show that there exists a determined point N such that (2.16) holds. When point N moves along the solution curve of (2.16) (correspondingly, σ increases), then arctan uU3 increases and arctan U (u3 ) decreases. At σ = σ0 , from the R-H condition (2.3), we know U0 1 q0 − u03 < k = = . shock u03 σ0 U0
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This derives arctan
U0 U0 U0 π − arctan U (u03 ) = arctan 0 + arctan ≤ , 0 u3 u3 q0 − u03 2
where the equality holds if and only if (u03 , U 0 ) = (q0 , 0). Noting that U U0 ≥ → +∞ u3 u3 and arctan we have
U π → u3 2
as u3 → 0,
− arctan Uu 3 ≥ arctan σ0
holds. This yields arctan
U π − arctan U (u3 ) > u3 2
as u3 → 0.
Thus, by the continuity and monotone of arctan uU3 − arctan U (u3 ) with respect to u3 , there must exist a unique N such that the arc M N corresponds to the solution to (2.11) together with two boundary values U (u03 ) = U 0 and (u3 , U )(1, Uu 3 )|N = 0. Next, we show that for any sharp body, there exists a unique supersonic shock such that the boundary value problem (2.15) with (2.3) and (2.5) is always solvable. Indeed, when M (u3 (t0 ), U (t0 )) moves on the shock polar in the subsonic domain, it follows from (u3 (t0 ), U (t0 )) ∈ C k and the continuous dependence of the solution on the initial values that (u3 , U )|N is of C k on t0 . This continuous curve (together with the transonic shock part), which is composed of N , is called the apple curve which √ lies above the shock polar (one can see the following Figure 4 drawn in [7], where c∗ = c = A). TIPDL /
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Figure 4
If b0 → 0, by use of Uu 3 |N = − b10 , then we have Uu 3 |N → −∞ and arctan Uu 3 |N → − π2 . Hence, it follows from (2.16) and 0 < u3 ≤ u03 < ∞ that arctan uU3 |N → 0 and U |N → 0. This the radial line U = b0 u3 will intersect with the apple curve implies, when b0 is suitably small, √ in the supersonic part and u3 > A holds. Moreover, by use of (2.16) and the uniqueness theorem of the solution to ODE, we know that the nonlinear mapping from N to M , which is determined by (2.15)–(2.16), is one to one between the apple curve and the shock polar. Namely, the supersonic solution to (2.15) exists uniquely when b0 is suitably small. This, together with (2.8), yields Lemma 2.1.
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Next, we establish some basic properties on (ρ(s), U (s), u3 (s), P (s)) as in [6], which can be used to illustrate that the system (2.1) makes sense and to treat our nonlinear problem (1.5) with (1.6)–(1.8). Lemma 2.2 Set λk (s) = λk (U (s), u3 (s)), k = 1, 4. If b0 is suitably small, then for b0 ≤ s ≤ s0 , the solution (ρ(s), U (s), u3 (s), P (s)) to (2.1)–(2.5) satisfies 2 2 (i) U (s) < 0, √ u3 (s) > 0, ρ (s) < 0 and A(1 + s ) − (su3 (s) − U (s)) > 0; (ii) u3 (s) > A, λ4 (s) > s0 . Proof (i) From (2.12), it follows that [ρU ] − s0 [ρu3 ] = 0, s0 [U ] + [u3 ] = 0.
(2.17)
Then we have ⎧ s0 q0 (ρ+ − ρ0 ) ⎪ ⎪ ⎨U+ = (1 + s2 )ρ , 0 + ⎪ s20 q0 (ρ+ − ρ0 ) ⎪ ⎩u3+ = q0 − . (1 + s20 )ρ+
(2.18)
By the entropy condition (which leads to ρ+ < ρ+ ), we find U+ > 0. Also, from (2.18) and a direct computation, we can derive that s0 u3+ − U+ = Since
s0 u3+ −U+ 1
(1+s20 ) 2
s0 ρ0 q0 > 0. ρ+
(2.19)
is the normal velocity on the shock front, the entropy condition also implies s0 u3+ − U+ (1 +
1 s20 ) 2
<
√ A.
(2.20)
The physical explanation to (2.20) is that across the shock front the normal velocity of the supersonic flow becomes subsonic. By the continuity of ρ(s), U (s) and u3 (s), (2.20) is also valid in s0 − δ0 ≤ s ≤ s0 with small δ0 > 0, and then (2.6) makes sense in this interval. Also, by (2.6), we obtain ρ (s) < 0, U (s) < 0 √ (s) and u3 (s) > 0 in s0 −δ0 ≤ s ≤ s0 . Hence the function A− s0 u3 (s)−U is a decreasing function 1 2 of s in s0 − δ0 ≤ s ≤ s0 . From this and (2.20), we find
(1+s ) 2
A(1 + s2 ) − (su3 (s) − U (s))2 √ s0 u3 (s) − U (s) √ s0 u3 (s) − U (s) = (1 + s2 ) A − A+ 1 1 (1 + s2 ) 2 (1 + s2 ) 2 √ √ s0 u3 (s) − U (s) > 0. > A(1 + b20 ) A − 1 (1 + s2 ) 2
(2.21)
From (2.19), one can derive that the denominator of (2.1) is lower bounded away from zero as long as the solution to (2.1) exists. Therefore, (2.19) holds in the whole interval [b0 , s0 ], and meanwhile the solution to (2.1) exists uniquely by the proof procedure of Lemma 2.1, which satisfies ρ (s) < 0, U (s) < 0, u3 (s) > 0, P (s) < 0.
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(ii) By a direct computation, we have (u3 (s) −
√ A) =
which yields u3 (s) −
AU > 0, A(1 + s2 ) − (su3 − U )2
√ √ A > u3 (b0 ) − A > 0.
Since
∂λ4 ∂λ4 U (s) + u (s), ∂U ∂u3 3 by use of (ii) of Lemma 2.2, we arrive at √ ⎧ ∂λ4 AU 1 ⎪ ⎪ ⎪ = u + > 0, 3 ⎨ ∂U 2 u3 − A U 2 + u23 − A √ √ ⎪ 1 2 AU 2 u3 + Au3 (u23 − A) ∂λ4 ⎪ 2 ⎪ U (u3 + A) + =− 2 < 0, ⎩ ∂u3 (u3 − A)2 U 2 + u23 − A λ4 (s) =
(2.22)
which together with U (s) < 0, u3 (s) > 0 and ρ (s) < 0, yields λ4 (s) < 0 for b0 ≤ s ≤ s0 . Combining with the entropy condition (2.4), we have λ4 (s) ≥ λ4 (s0 ) > s0 . Remark 2.1 Since the denominator of the system (2.1) is positive in [b0 , s0 ], we can extend (s), u the background solution ( ρ(s), U 3 (s), P(s)) of (2.1)–(2.5) to some interval [b0 , s0 + τ0 ] with (s), u ρ(s), U 3 (s), P (s)). τ0 > 0. The related extensions are denoted by (
3 Some Crucial Properties on Globally Stable Supersonic Conic Waves In this section, under the assumptions of Theorem 1.1, we focus on some basic observations on the globally stable supersonic conic wave for the full Euler system. It is noted that it follows from the first equation of (1.5) that one can introduce a stream function ψ(x3 , r) with ψ(0, 0) = 0 such that ∂x3 ψ(x3 , r) = −rρ+ U + and ∂r ψ(x3 , r) = rρ+ u+ 3. + ∂ )S = 0. In addition, by (1.5) and the state equation, we can easily deduce (U + ∂r + u+ 3 x3 This means that S + = constant holds along each stream line (the global existence of the stream line for x3 > 0 is guaranteed by (ii) of Theorem 1.1), which is denoted by a function S. Namely, S can be expressed as S = S(ψ).
(3.1)
Next, we derive a crucial relation between S(ψ) and the rotation ∂3 U + − ∂r u+ 3. Substituting the first equation of (1.5) and the expression P + = Aρ+ into the second and the third equations in (1.5) yields, respectively ⎧ + + + + ⎪ ⎨∂r (rρ U ) + ∂3 (rρ u3 ) = 0, + + + + + (3.2) ρ U ∂r U + ρ u3 ∂3 U + + ∂r P + = 0, ⎪ ⎩ + + + + + + + ρ U ∂r u3 + ρ u3 ∂3 u3 + ∂3 P = 0 and
⎧ ⎪ ∂r (rρ+ U + ) + ∂3 (rρ+ u+ 3 ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ A∂r ρ+ + U + ∂r U + + u+ = 0, 3 ∂3 U + ρ+ ⎪ ⎪ ⎪ ⎪ ⎪ A∂3 ρ+ + + ⎪ ⎩U + ∂r u+ + u ∂ u + = 0. 3 3 3 3 ρ+
(3.3)
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In addition, it follows from Bernoulli’s law that + U + ∂r U + + u+ 3 ∂r u3 +
A∂r ρ+ + T S (ψ)∂r ψ = 0. ρ+
This together with the second equation in (3.3) yields S (ψ) =
∂3 U + − ∂r u+ 3 . (ρ+ T )r
(3.4)
Thus, one has 0 lim (∂3 U + − ∂r u+ 3 )(x3 , r(x3 , x3 )) = ∞,
x3 →∞
if S (ψ(x03 , χ(x03 ))) = 0, with x03 > 0,
(3.5)
where r(x3 , x03 ) stands for the stream line starting from the point (x03 , χ(x03 )) at the shock curve, i.e., it is determined by dr(x3 , x03 ) U + (x3 , r(x3 , x03 )), = dx3 u+ 3
r(x3 , x03 )|x3 =x03 = χ(x03 ).
In order to fulfill (i)–(ii) of Theorem 1.1, by (3.4)–(3.5), it is required that (∂3 U + − ∂r u+ 3 )(x3 , χ(x3 )) ≡ 0 should hold for x3 > 0. Furthermore, we have the following lemma. Lemma 3.1 Under the assumptions of Theorem 1.1, for x3 > 0, (∂3 U + −∂r u+ 3 )(x3 , χ(x3 )) ≡ 0 holds if and only if χ (x3 ) ≡ 0. Remark 3.1 By Lemma 3.1, it is easy to know that (∂3 U + − ∂r u+ 3 )(x3 , χ(x3 )) ≡ 0 for x3 > 0 if and only if χ(x3 ) = s0 x3 due to χ(0) = 0 and χ (0) = s0 . Proof of Lemma 3.1 It follows from (1.6) that on r = χ(x3 ), we have ⎧ + + + + ⎪ ⎨ρ U − ρ u3 χ = −ρ0 q0 χ , + + + 2 + + + P + ρ (U ) − χ ρ U u3 = P0 , ⎪ ⎩ + + + 2 2 ρ U u3 − P + χ − ρ+ (u+ 3 ) χ = −P0 χ (x3 ) − ρ0 q0 χ .
(3.6)
Multiplying χ on the two sides of the second equation in (3.6), and then adding the third equation in (3.6), we have + + + + 2 (U + χ + u+ 3 )(ρ U − ρ u3 χ ) = −ρ0 q0 χ .
From this, together with the first equation in (3.6) and the fact χ = 0, we obtain on r = χ(x3 ) that U + χ + u + 3 = q0 .
(3.7)
+ 2 + χ ∂3 U + + U + χ + ∂3 u+ 3 + (χ ) ∂r U + ∂r u3 χ = 0 on r = χ(x3 ).
(3.8)
Taking ∂x3 on (3.7) yields
In addition, by differentiating the second equation of (3.6) with respect to the variable x3 , we arrive at + + + + + ∂3 P + + χ ∂r P + + ∂3 (U + (ρ+ U + − ρ+ u+ 3 χ )) + χ ∂r (U (ρ U − ρ u3 χ )) = 0.
(3.9)
Nonexistence of a Globally Stable Supersonic Conic Shock Wave
569
It follows from the second and third equations of (3.2) and (3.9) that + + + + + 2 + − ρ+ U + ∂r u+ 3 − ρ u3 ∂3 u3 − ρ u3 (χ ) ∂r U + d + ∂3 U + (ρ+ U + − 2ρ+ u+ (ρ+ U + − ρ+ u+ 3χ )+U 3 χ ) = 0. dx3
This together with the first equation in (3.6) yields + + + + + 2 + − ρ+ U + ∂r u+ 3 − ρ u3 ∂3 u3 − ρ u3 (χ ) ∂r U
+ + ∂3 U + (ρ+ U + − 2ρ+ u+ 3 χ ) = ρ0 q0 U χ .
(3.10)
Combining (3.8) with (3.10), we have + + + + + (ρ+ U + − ρ+ u+ 3 χ )(∂3 U − ∂r u3 ) = (ρ0 q0 − ρ u3 )U χ .
Thanks to the first equation of (3.6), one has on r = χ(x3 ), + + 2 (∂3 U + − ∂r u+ 3 )ρ0 q0 χ = ρ (U )
χ . χ
(3.11)
Due to χ (x3 ) = 0 and U + = 0, then we have ∂3 U + − ∂r u+ 3 = 0 ⇔ χ = 0 on r = χ(x3 ).
Thus, we complete the proof of Lemma 3.1. In the next section, based on Lemma 3.1 and Remark 3.1, we will complete the proof of Theorem 1.1.
4 Proof of Theorem 1.1 By Lemma 3.1 and Remark 3.1, we know that the shock curve must be straight, whose slope ∂ U + −∂ u+ is just s0 in terms of Remark 1.1. Together with the expression S (ψ) = 3 (ρ+ T )rr 3 , then one has S (ψ) ≡ 0. This derives that S(ψ) ≡ constant holds behind the shock r = s0 x3 . Therefore, we have P + (x3 , r) = Aρ+ (x3 , r),
∂3 U + − ∂r u+ 3 ≡ 0.
(4.1)
From this, it is easy to know that the supersonic flow field behind the shock can be described by the potential flow equation. We now set ∂r φ = U + and ∂x3 φ = u+ 3 with φ(0, 0) = 0. Then it follows from the second and the third equations of (3.2) and the state equation of isothermal gas that 1 |∇φ|2 + h(ρ+ ) = C0 , 2
(4.2)
where h(ρ+ ) = A ln ρ+ is the specific enthalpy. By use of (4.2) and the implicit function theorem, it is easy to know that the density function ρ+ (x) can be expressed as 1 ρ+ = h−1 C0 − |∇φ|2 ≡ H(∇φ). 2
(4.3)
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Substituting (4.3) into the first equation of (3.2) yields 2 φ + (A − (∂r φ)2 )∂r2 φ + (A − (∂3 φ)2 )∂32 φ − 2∂3 φ∂r φ∂rx 3
A ∂r φ = 0. r
(4.4)
Next, we look for the value of φ on the shock. On r = s0 x3 , we have x3 d (φ(x3 , s0 x3 ))dx3 φ(x3 , s0 x3 ) = dx 3 0 x3 = (u3 (s0 ) + U (s0 )s0 )dx3 0
= (u3 (s0 ) + U (s0 )s0 )x3 ≡ ϕ0 (x3 )
(4.5)
and s 1 0 s0 ), U ( s0 )) ·
, −
∂n φ(x3 , s0 x3 ) ≡ (u3 ( 1 + s20 1 + s20 =
s ) − U ( s0 ) s0 u3 ( 0 ≡ ϕ1 (x3 ). 2 1 + s0
(4.6)
In addition, on the conic surface r = b(x3 ), φ satisfies ∂r φ = b (x3 )∂3 φ.
(4.7)
Denote the domain Ω1 ≡ {(x3 , r) : b(x3 ) < r < s0 x3 , x3 > 0} and Ω2 ≡ {(x3 , r) : b0 x3 < r < s0 x3 , x3 > 0}. Then we have 2 Lemma 4.1 Under the assumptions of Theorem 1.1, the Lip(Ω 1 \ (0,
r0))-regular r
1r)∩ C (Ω r ,u solution φ to (4.4)–(4.7) is the same as the background solution ρ x3 , U 3 x3 , P x3 x3 in the domain Ω1 ∩ Ω2 .
Proof We now take a rotational transformation as follows y1 = cos θ0 x3 − sin θ0 r, y2 = sin θ0 x3 + cos θ0 r,
(4.8)
where θ0 = π2 − arctan s0 . In this case, the shock curve r = s0 x3 is changed into y1 = 0, and (4.4)–(4.7) can be rewritten as ⎧ (∂y1 φ)2 2 2∂y1 φ∂y2 φ 2 (∂y2 φ)2 2 ⎪ ⎪ 1− ∂y1 φ − ∂y1 y2 φ + 1 − ∂y2 φ ⎪ ⎪ ⎪ A A A ⎪ ⎪ ⎪ ⎪ cos θ0 ∂y2 φ − sin θ0 ∂y1 φ ⎪ ⎪ ⎪ ⎨ + cos θ0 y2 − sin θ0 y1 = 0 in Ω1 , (4.9) s0 ) + U ( s0 ) s0 ) sin θ0 y2 ≡ ϕ 1 (y2 ), ⎪φ(0, y2 ) = (u3 ( ⎪ ⎪ ⎪ ∂y1 φ(0, y2 ) = s0 u3 ( s0 ) − U ( s0 ) ≡ ϕ 2 (y2 ), ⎪ ⎪ ⎪ ⎪ ⎪cos θ0 ∂y2 φ − sin θ0 ∂y1 φ = (cos θ0 ∂y1 b + sin θ0 ∂y2 b)(cos θ0 ∂y1 φ + sin θ0 ∂y2 φ) ⎪ ⎪ ⎪ ⎩ on y2 = tan θ0 y1 + sec θ0 b(cos θ0 y1 + sin θ0 y2 ), where φ(y) ≡ φ(cos θ0 y1 + sin θ0 y2 , − sin θ0 y1 + cos θ0 y2 ).
Nonexistence of a Globally Stable Supersonic Conic Shock Wave
571
Meanwhile, the domain Ω1 is changed into 1 : {(y1 , y2 ) : 0 < y1 < cot θ0 y2 − csc θ0 b(cos θ0 y1 + sin θ0 y2 )}. Ω
r r 3 , r) be the potential function of U 0) = 0. Under the Let φ(x 3 x3 with φ(0, x3 , u 3 , r) is changed into the function φ(y). Then φ(y) transformation (4.8), we suppose that φ(x satisfies ⎧ 2 2 ⎪ + 1 − (∂y2 φ) ∂ 2 φ ⎪ 1 − (∂y1 φ) ∂ 2 φ − 2∂y1 φ∂y2 φ ∂ 2 φ ⎪ y1 y1 y2 y2 ⎪ ⎪ A A A ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2, ⎨ + cos θ0 ∂y2 φ − sin θ0 ∂y1 φ = 0 in Ω cos θ0 y2 − sin θ0 y1 (4.10) ⎪ y2 ) = ϕ ⎪ φ(0, 1 (y2 ), ⎪ ⎪ ⎪ ⎪ 2 (y2 ), ⎪ ⎪∂y1 φ(0, y2 ) = ϕ ⎪ ⎪ ⎪ ⎩(1 − b0 tan θ0 )∂y2 φ = (b0 + tan θ0 )∂y1 φ on y2 = b0 + tan θ0 y1 , 1 − b0 tan θ0 where
φ(y) ≡ φ(cos θ0 y1 + sin θ0 y2 , − sin θ0 y1 + cos θ0 y2 ).
Meanwhile, the domain Ω2 is changed into 2 ≡ (y1 , y2 ) : 0 < y1 < 1 − b0 tan θ0 y2 . Ω b0 + tan θ0 in Ω 2. 1 ∩ Ω Next, we prove φ = φ Set w = φ − φ, and then it follows from (4.9)–(4.10) that w satisfies ⎧ 3 2 ⎪ ⎪ 2 ⎪ 1 ∩ Ω 2, a (y)∂ w + bi (y)∂yi w = 0 in Ω ⎪ ij y y i j ⎨ i,j=1
⎪ w(0, y2 ) = 0, ⎪ ⎪ ⎪ ⎩ ∂y1 w(0, y2 ) = 0,
i=1
(4.11)
where a11 = 1 −
0
1
D12 dθ, A
a12 = a21 = −
0
1
D1 D2 dθ, A
a22 = 1 −
1
0
D22 dθ A
≡ θφ + (1 − θ)φ, θ ∈ [0, 1] and Di = ∂y D. In addition, bi (y) (i = 1, 2) are with D(θ, φ, φ) i continuous functions in Ω1 ∩ Ω2 . Noting that 2∂ φ∂ y φ 2 2 2 (∂y2 φ) 4 (∂y1 φ) y1 2 2 + (∂y φ) 2 − A) > 0 1− = ((∂y1 φ) −4 1− 2 A A A A and
2=− A − (∂y1 φ)
1 2 − A)) > 0 s0 + (U (( u23 − A)s20 − 2 u3 U 1 + s20
due to Lemma 2.2(iii), together with (ii) of Theorem 1.1, we have a11 > 0,
a12 < 0,
2. 1 ∩ Ω Δ = 4(a212 − a11 a22 ) > 0 in Ω
(4.12)
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This means that (4.11) is strictly hyperbolic with respect to y1 -direction. By the standard result on the existence and uniqueness of solutions to the hyperbolic equations (see [13]), we 2 . Thus, we complete the proof of Lemma 4.1. 1 ∩ Ω know that w ≡ 0 in Ω Based on Lemma 4.1, we now show the following conclusion. From this, Theorem 1.1 will be easily derived. Theorem 4.1 Under the assumptions of Theorem 1.1, Ω1 = Ω2 holds. Moreover, φ(x3 , r) ≡ 3 , r). φ(x Proof Set Γb ≡ {(x3 , r) : r = b(x3 ), x3 > 0}, L ≡ {(x3 , r) : r = b0 x3 , x3 > 0}. Due to b(x3 ) = b0 x3 + εϕ(x3 ) with ϕ(x3 ) ∈ C0∞ (0, l), then we can conclude that there exist at least two intersection points P1 and P2 in Γb ∩ L such that the arc P1 P2 lies above L or under L. Without loss of generality, we assume that the arc P1 P2 lies above L. In addition, for P1 1 convenience, the coordinate of P1 is denoted by (xP 3 , r ) (see Figure 5 below).
S
STY
ΓC -
1
1 0
Y Figure 5
Consider the following stream line starting from P1 ⎧ ⎪ ⎨ dr = ∂r φ(x3 , r) , 3 , r) dx3 ∂3 φ(x ⎪ ⎩ P1 r(x3 ) = rP1 .
(4.13)
Due to Lemma 4.1, we know that there exist two different stream lines r = b0 x3 and r = b(x3 ) between P1 and P2 for the system (1.5). However, this is contradictory to the uniqueness of C 1 solution to the ordinary differential equation (4.13). Thus, we must have Ω1 = Ω2 . Consequently, the proof of Theorem 4.1 is completed. Based on Theorem 4.1, we now start to prove Theorem 1.1. Proof of Theorem 1.1 By use of Theorem 4.1, Ω1 ≡ Ω2 holds. However, the boundary r = b(x3 ) is obviously different from the boundary r = b0 x3 . Thus, under the assumptions of Theorem 1.1, a globally stable supersonic shock solution does not exist.
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573
5 Appendix Lemma 5.1 For the isothermal gas, the internal energy function e(ρ, S) = A ln ρ + T S holds. Proof From the first law of thermodynamics de = T dS − pdτ , one has eS = T, eτ = −p = −Aρ. Then by integrating the second equation, we have e = −RT ln τ + C(S) = A ln ρ + C(S). From the first equation, we have
C (S) = T.
So C(S) = T S + C. Therefore e = A ln ρ + T S + C. The constant C, which is determined by the initial data of ρ and S, can be eliminated by scaling on ρ, and then on p, i.e., letting ρ = kρ , where k is determined by C. So Lemma 5.1 is proved. Acknowledgements The authors would like to express their gratitude to Professor Yin Huicheng of Nanjing University, and the anonymous referees for their valuable suggestions and comments.
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