数理解析研究所講究録
第 1782 巻 2012 年 74-84

74

On the steady supersonic flow past a curved cone
Yongqian Zhang*

School of Mathematical Sciences, Fudan University, Shanghai, China

Problem and the Main Result

1

We are concerned with the problem of the steady potential supersonic flow past a Lipschitz
curved cone. The flow and the cone have axis-symmetry. The surface of the cone is formedby the rotation of the graph $\{y=b(x), x>0\}$ , where the function $b(x)$ is a Lipschitz function

satisfying the following (see Fig. 1):(Al) $b(x)<0$ for $x>0$ and$b(x)=b_{0}x$ ,$x\in[0, t_{0}]$for some constants $b_{0}<0$ and $t_{0}>0$ ; moreover$b_{+}’(x)= \lim_{tarrow x+0}\frac{b(t)-b(x)}{t-x}\in BV([0, +\infty))$and$b_{+}’(x)$,equals to some negative constant for $x>t_{*}$ for some $t_{*}>t_{0}$ .

With the coordinates$x$and , the equations of the flow can be written as
$y$

(1.1)

$-\underline{\mu}$$(\rho u)_{x}+(\rho v)_{y}$

$=$

’
$y$

$v_{x}-u_{y}$

$=$

$0$

(1.2)

,

with the Bernoulli equation:$\frac{u^{2}+v^{2}}{2}+\frac{c^{2}}{\gamma-1}=\frac{u_{\infty}^{2}}{2}+\frac{c_{\infty}^{2}}{\gamma-1}$

are components of the flow velocity in the direction of the axis of the cone(or in

Here

$u$

the

-direction) and in the

$x$

flow;

$\rho$

and

$v$

.

$y$

-direction respectively;

is the density of the flow and$A>0;c_{\infty}=\sqrt{\gamma A\rho_{\infty}^{\gamma-1}}$

and$\rho_{\infty}$$c$$(u_{\infty}, 0)$

the sonic speed with

$c=\sqrt{\gamma A\rho^{\gamma-1}}$

for some constantis the density of the incoming flow. Moreover we assume that

(A2) The velocity of the incoming flow is supersonic, i.e.,’Email: yongqianz@fudan.edu.cn

is the velocity of the incoming$u_{\infty}>c_{\infty}$.

75

Figure 1: Supersonic conical flow

The original problem is reduced to the problem of finding the global weak solution to (1.1-1.2)
in$\Omega$with$(u, v)|_{x\leq 0}=(u_{\infty}, 0)$$(u, v)\cdot\vec{n}|_{\partial\Omega}=0$

where $\Omega=\{(x, y)|x>0, y<b(x)\}$ and$b$$\vec{n}=\vec{n}(x, y)$

(1.3)

,

,

(1.4)

is the normal to$\partial\Omega$at differential points of.
This problem has been studied by many authors, for instance, Courant, Friedrichs, Shuxing

Chen, Zhouping, Xin, Huicheng Yin and W. Lien, Tai-Ping Liu etc. We generalize Lien and$Lius$ work as follows (see also [24]).

Theorem 1.1. Suppose thut the conditions A(l) and A(2) are satisfied, und that $1<\gamma<3$ and$-b_{*}<b_{0}<0$ .
$(1.4)$ has

If T.

$Vb’(\cdot)$

is sufficiently small and$u_{\infty}\iota s$

sufficiently large, then problem $(l.l)-$a global solution $U(x, y)$ with bounded total vari ation. The solution contains a l-shock

front, which is a small perturbation of $y=s_{0}x$ , and the solution in between the shock front andsurface of the cone is a small perturbation of the self-similar solution of the problem. Heredenote the slope of the corresponding shock front when the location of the surface of the cone is$s_{0}$

given by $y=b_{0}x$ , and$b_{*}=( \frac{1}{2}(-1+\sqrt{\frac{\gamma+7}{\gamma-1}}))^{1/2}$

I will talk about this result and briefly recall some related results and explain the proof of

the above theorem.

76

Figure 2: Supersonic flow

Some Related Results

2

In this section we recall some results on the following aspects:

1 Steady supersonic flow past a wedge;
2 Steady Supersonic Flow past a cone (previous results).

2.1

Steady supersonic flow past a wedge

We assume the lower part of the wedge is given by the graph of the function $y=b(x)$ , see Fig.3.

Here without confusion, we use the same notations as that in the case of the conical flow. In
the case of potential flow, the governing equations are$(\rho u)_{x}+(\rho v)_{y}$

$=$

$0$

,

(2.1)

$v_{x}-u_{y}$

$=$

$0$

,

(2.2)

Bernoulli equations:$\frac{u^{2}+v^{2}}{2}+\frac{c^{2}}{\gamma-1}=\frac{u_{\infty}^{2}}{2}+\frac{c_{\infty}^{2}}{\gamma-1}$,

The first equation means the conservation of mass, while the second one means that the flow isirrotational. Here

$c=\sqrt{\gamma A\rho^{\gamma-1}}$

for some$A>0;c_{\infty}=\sqrt{\gamma A\rho_{\infty}^{\gamma-1}};(u, v)$

satisfying the following conditions:$(u, v)|_{x\leq 0}=(u_{\infty}, 0)$,$(u, v)\cdot\vec{n}|_{\partial\Omega}=0$,

is the velocity at $(x, y)$77
where $\Omega=\{(x, y)|x>0, y<b(x)\}$ and$b$$\vec{n}=\vec{n}(x, y)$is the normal to$\partial\Omega$. here we assume again that the incoming flow is supersonic, that is,

at differential points of$u_{\infty}>c_{\infty}$.

Shock Front
Figure 3: $q_{0}=(u_{\infty}, 0)$ : supersonic.

$q_{1}$: supersonic or subsonic.

For the straight wedge with the attack angle less than a critical value, Courant and Friedrichs
gave in their famous book [13] the solution containing one straight shock issuing from the vertex
by making use of the shock polar. The state behind shock is the constant state$q_{1}$, see Fig.3.

The shock polar is given by the Rakine-Hugoniot equations as shown in shown in the Fig.

4, the line $Oq_{\sup}$ has the same slop as the boundary of the wedge. Therefore, for the wedge
having an attack angle less than the critical value, there are two solutions, one of which is calledsupersonic-supersonic shock with $q_{1}=q_{\sup}$ , while the another is called supersonic-subsonic shockwith $q_{1}=q_{sub}$ . Many authors consider the perturbation of these two solutions.

Figure 4: Shock polar(1) Case: The perturbation problem of supersonic-supersonic shock solution.

78
The local piecewise smooth solutions containing a curved supersonic shock issuing from thevertex were obtained by Chaohao Gu for potential flow in [17], and by T. T. Li for the full Euler
equations in [18], and by D. G. Schaffer in [22]. In the above works, the flow is assumed to besupersonic. Therefore, the equations are of hyperbolic type, and the characteristic method areused to construct the solutions. In addition, with the sharp estimates on the decay of solutions,Chen constructed the global piecewise solutions containing reflecting shocks for the concave andthe convex wedges with piecewise smooth wedge in [6],[7] and [8], Yin constructed the global
solutions containing one shock for the smooth wedge in [25]. In a different approach, Shuxing
Chen constructed the piecewise smooth solution with a smooth shock surface for 3-d case in [5].
For the case of the wedge with Lipschitz boundary, besides the leading shock issuing fromthe vertex of the wedge, there may be many shocks in between the leading shock front andthe boundary of the wedge. Also the leading shock may disappear. Therefore it is hard touse the characteristic method, we use instead the Glimm difference method to construct the
global weak solutions. For the potential flow, Zhang established the global weak solutions forthe wedge with opening angle less than a critical value and small total variation of the tangent
along the boundary in [27] and [28]. The asymptotic behaviour was also studied in [29]. For thecompressible flows governed by the full Euler equations, in [1] we proved the following:

Theorem 2.1. Suppose thut arctan $b’(O)|$ is less than a criticul value and $TV(b’)$ is sufficiently
small, und thut the incoming flow is supersonic. Then (2.1-2.3) has a global solution $(u, v,p)$ .
Moreover, $\lim_{xarrow+\infty}(u, v,p)(x, y)=(u_{+},$ $v_{+},p+I$ uniformly in$(u_{+}, v+,p_{+})$$y$for some constant supersonic state, and the state $(u+, v_{+},p_{+})$ can be determined by the shock polar through$U_{\infty}$

andthe relation: $v+/u+= \lim_{xarrow+\infty}b’(x)$ .

In the above works, we use the modified Glimm scheme to construct the approximate solutions. To deal with the interactions of the waves at the boundary, we introduce a new functional
containing angles only to control the increasing strengthes of the reflecting waves, and use thefact that some weighted linear functional over the strengthes is decreasing at the boundary. In
addition, the key point in the proof of this theorem is to show the following$|K_{1}||K_{2}|<1$

which implies the decay of the strengthes after the reflections at both boundary and the leadingandare the coefficients of the reflecting waves at the boundary andshock front. Here$K_{1}$$K_{2}$

the leading shock front. The$L^{1}$-stability for solutions under the small BV perturbations of theincoming flow was given by Guiqiang Chen and Tianhong Li in [2].
(2) Case: The perturbation problems of the supersonic-subsonic shock. In thiscase, the shock is also called the transonic shock. For the wedge which is the small perturbationof the straight one, the existence and stability of the transonic shock have been studied by S.

79
Chen, Z. Xin and Yin, Beixiang Fang and Guiqiang Chen, see for instance [15], [4] and [26]and the references therein. In such a case, the equations become of elliptic type in the domain

between the shock front and the wedge. Therefore some conditions are needed to impose on thevelocity and the pressure. The weighted Holder space or weighted Sobolev space were introduced
to describe the regularities of the solutions near the vertex and the infinity.

It is mentioned that Shiffman and Bers dealt with Subsonic flow past a obstacle, and C.

Morawetz [21], Chen, Dafermos, Dehua Wang and Slemrod used compensated compactness
approach to studied the transonic flow past a obstacle, see [3]. Recently, Elling and Liu [14]studied the unsteady case.

One of open problems is to find the global weak solution for the steady supersonic flow pasta wedge with an opening angle larger than the critical value. In such case, the experiments showthat the shock is detached from the wedge. Formal analysis shows that the equations may beof mixed type. That is to say, it is a transonic shock. New ideas are needed to deal with thisproblem.

2.2

Some results on Steady Supersonic Flow past a cone

For the circular cone, in [13] Courant-Friedrich gave the solutions by making use of the shockpolar and the apple curves, see Fig. 5. Due to their results, there are two solutions for the conewith an opening angle less than a critical value, one of which is of supersonic-supersonic shock

solution. In such case, the equations are of hyperbolic type over the flow field. Many authors
consider the small perturbation of this solution.

$v$

Figure 5: Apple Curve

The local piecewise smooth solution near the vertex is given in [9]. Such solution contains asmooth conical shock front issuing from the vertex. One of global results was given by W. Lien80
and Tai-Ping Liu in [20]. They assume that the cone has small opening angle and has small
total variation of and that the initial strength of the relatively strong shock is sufficiently
weak and that the Mach number of incoming flow is sufficiently large. Then for $1<\gamma<3$ the$b’$boundary value problem (1.1-1.4) has a global solution with bounded total variation. To get

the global weak solution, they used a modified Glimm type difference scheme to construct theapproximate solutions. Different from the standard Glimm scheme, the Riemann solutions andthe self-similar solutions are used as the building block. The smallness of the opening angle andthe strength of the leading shock play an important role in proving the monotonicity of Glimm
type functional.

For smooth function , the global piecewise smooth solution is constructed by Shuxing Chen,
Zhoupin Xin and Huicheng Yin in [11]. In the paper the attack angle is assumed to be small$b$and the Mach number of incoming flow is assumed to be sufficiently large. $1<\gamma<3$ is also
assumed. Then they considered the case that $b-b_{0}$ is small in suitable weighted sobolev norm

. The solution they constructed has aand the derivatives of (up to some order) are decay atsmooth conical shock ffont issuing from the vertex of the cone. In their proof the local piecewise$b$$+\infty$

smooth solution constructed in [9] was used. Then to extend this solution, the weighted Sobolev
space are introduced to describe the regularity of the solutions and the Hardy type inequality
are established for the solution. Yin and his students also made the improvement of this result,see [26] and the reference therein.
In Courant and Friedrichs’ result, the supersonic-subsonic shock solutions were also given forthe circular cone. For the cone that is a smooth perturbation of the straight circular one, Yin
and students and Beixiang Fang and Guiqiang Chen constructed the piecewise smooth solutions,see [26] and [4] and the references therein. Such solutions also have smooth conical shock front
issuing from the vertex of the cone, but the equations are of elliptic type in and between theshock front and the surface of the cone. In addition, some conditions on the velocities are
imposed at $x=+\infty$ to find the solution. For the three dimensional case. self-similar solutions

were given by S. Chen in [10].
An open problem is to determine the solution for the steady supersonic flow past a cone ofgeneral shape.

3

Remarks on the Proof of the Main Result

Our solution is still a small perturbation of the solution for circular cone. Therefore, we first. Let
studied the asymptotic expansion of the solution for circular cone. In this case$b\equiv b_{0}x$$y=s_{0}x$be the location of shock front. Let $\sigma=y/x$ . Then equations for Circular conical flow

81
can be reduced to:$(- \sigma^{2}(1-\frac{u^{2}}{c^{2}})^{v}-\frac{u}{c}\tau\sigma)u_{\sigma}+(\frac{uv}{c^{2}}+(1-\frac{v^{2}}{c^{2}})\sigma)v_{\sigma}+v=0$$u_{\sigma}+\sigma v_{\sigma}=0$,,$s_{0}<\sigma<b_{0}$$\rho(us_{0}-v)=\rho_{\infty}u_{\infty}s_{0},$$\sigma=s_{0}$$u+vs_{0}=u_{\infty},$,$ub_{0}=v,$$\sigma=s_{0}$$\sigma=b_{0}$

,

(3.1)

(3.2)
,

(3.3)
(3.4)

;

(3.5)

with
$(u(\sigma), v(\sigma))=(u_{\infty}, 0)$,$\sigma<s_{0}$

.

(3.6)

It can be reduced to
,

(3.7)

$v_{\sigma}= \frac{-vc^{2}}{\sigma((1+\sigma^{2})c^{2}-(v-\sigma u)^{2})}$

,

(3.8)

$\rho_{\sigma}=\frac{\rho vc^{2}(v-\sigma u)}{\sigma((1+\sigma^{2})c^{2}-(v-\sigma u)^{2})}$

.

(3.9)

$u_{\sigma}= \frac{vc^{2}}{(1+\sigma^{2})c^{2}-(v-\sigma u)^{2}}$We have the following.

Lemma 3.1. For $1<\gamma<3$ and $b_{0}\in(-b_{*}, 0)$ und$K”>0$ and $K”’>0$ , independent of

unique solution$(u(\sigma), v(\sigma), s_{0})$$u_{\infty}$$\rho_{\infty}>0$, there exist constants $K^{l}>0$ ,, such that for $u_{\infty}>K’’’$ , the problem (3.1-3.5) has$u$constituted by a supersonic conicul shock front issuing from thevertex. Moreover,(3.10),$\lim_{u_{\infty}arrow+\infty}\frac{u(\sigma)}{u_{\infty}}=\frac{1}{1+b_{0}^{2}}=\cos^{2}\theta_{0}$,(3.11)$\lim_{u_{\infty}arrow+\infty}\frac{v(\sigma)}{u_{\infty}}=\frac{b_{0}}{1+b_{0}^{2}}=\sin\theta_{0}\cos\theta_{0}$(3.12)$s_{0}=b_{0}+O(1)u_{\infty}^{-\frac{2}{\gamma-1}}$$\frac{\rho}{\rho_{\infty}}=\{\frac{\gamma-1}{2}\frac{u_{\infty}^{2}s_{0}^{2}}{c_{\infty}(1+s_{0}^{2})}\}^{\frac{1}{\gamma-1}}(1+O(1)u_{\infty}^{-2})$,(3.13)

and$\lim_{u_{\infty}arrow+\infty}(\frac{u(\sigma)}{c(\sigma)}I^{2}=\frac{2}{(\gamma-1)b_{0}^{2}(1+b_{0}^{2})}>1$$\cos(\theta_{0}\pm\theta_{ma}^{0})>0$

where$\theta_{0}=$arctan$b_{0}$

and$\theta_{ma}^{0}=$$\lim_{u_{\infty}arrow+\infty}$$\theta_{ma};O(1)$,,stands for a bounded quantity as(3.14)(3.15)$u_{\infty}arrow+\infty$.

82
Such expansions can be found for instance in the book of Van Dyke [23]. The proof is givenin [11] and [24] and it plays a crucial role in our analysis. Now we regard as time variable,
then the homogeneous system is strictly hyperbolic. The origin problem can be regarded as a$x$

initial boundary value problem of a hyperbolic system with a singular source term. Then weapproximate the boundary by piecewise hne segments as in [28] and use the modified Glimm

scheme by Lien and Liu [20]. To get the bounds on the total variations of the approximate
solutions, we introduced a weighted Glimm type functional. Besides the terms given by Lienand Liu, we introduced the linear terms to control the moving centers and the strengthes of the

waves produced by the flow around the corner. Then, by making use of the above expansion
for straight circular cone, we establish the estimates on coefficients for the wave-reflections forlarge Mach numbers, which show that the total amount of the waves in between the leading

shock and the cone will vanish. This also implies the decreasing of the Glimm type functionals.
Standard arguments [16] give the main result then.

Figure 6: Glimm Scheme

References[1] G. Chen, Y. Zhang and D. Zhu, Existence and stability of supersonic flows past Lipschitz
wedge, Arch. Rational Mech. Anal. 181(2006), 261-310.

[2] G. Chen and T. Li, Well-posedness for two-dimensional steady supersonic Euler flows pasta Lipchitz wedge, J. Differential Equations 244 (2008), 1521-1550.
[3] G. Chen, C. Dafermos, M. Slemrod and D. Wang, On two-dimensional sonic-subsonic flow,
Commun. Math. Phys. 271 (2007), 635-647.

83[4] G. Chen and B. Fang, Stability of transonic shock-fronts in the three-dimensional conial

steady potentiul flow past a purterbed cone, Discrete Contin. Dyn. Syst. 23 (1-2) (2009),

85-114.
[5] S. Chen, Existence of locul solution to supersonic flow past a three-dimensionul wing, Advancein Appl. Math. 13(2) (1992), 273-304.
[6] S. Chen, Supersonic flow past a concave wedge, Science in China (Ser. A)10(27) (1997),

903-910.
[7] S. Chen, Asymptotic behaviour of supersonic flow past a convex combined wedge, China Ann.
of Math. $19B:3(1998),$ 255-264.

[8] S. Chen, Globul existence of supersonic flow past a curved convex wedge, J. Partial Diff. Eqs
11 (1998), 73-82.

[9] S.X.Chen&D.N.Li: Supersonic Flow Past a Symmetrically Curved Cone, Indiana Univ.

Math. Jour., 49(4)(2000), 1411-1435.
[10] S. Chen, Existence of stationary supersonic flow past a pointed body, Arch. Rational Mech.

Anal. 149 (2) (2001), 141-181.
[11] S.Chen, Z. Xin and H. Yin., Globul shock waves for the supersonic flow past a perturbed

cone, Commun. Math. Phys. 228(2002), 47-84.

[12] S. Chen and B. Fang, Stability of transonic shocks in supersonic flow past a wedge, J.

Differential Equations, 233 (2007), 105-135.
[13] R.. Courant and K. O.

$\mathbb{R}iderich$, Supersonic flow and shock waves, Wiley Interscience, New

York, 1948.

[14] V. Elling and T. Liu, Supersonic flow onto a solid wedge, Comm. Pure Appl. Math. 61(10) (2008), 1347-1448.
[15] B. Fang, Stability of transonic shocks for the full Euler system, Math. Meth. Appl. Sci. 29(2006), 1-26.
[16] J. Glimm, Solution in the large for nonlinear systems of conservation laws, Comm. Pure

Appl. Math. 18(1965), 695-715.
[17] C. Gu, A method for solving the supersonic flow past a curved wedge, Fudan J.(Natur.
Sci.) $7(1962),$ 11-14.

[18] T. Li, On a free boundary problem, Chinese Ann. of Math.(1980), 351-358.

84[19] T. Li and W. Yu, Boundary Value Problems for Quasilineur Hyperbolic Systems,
University (Mathematics Series V), Duke University, Durham, 1985.

Duke[20] W. Lien and T. P. Liu, Nonlinear stability of a self-similur 3-dimensionul gas flow, Commun.

Math. Phys. 204 (1999), 525-549.
[21] C. S. Morawetz, On a weuk solution for transonic flow problem, Comm. Pure Appl. Math.
38(1985), 797-818.

[22] D. G. Schaffer, Supersonic flow past a nearly straight wedge, Duke Math. J. 43(1976),

637-670.
[23] M. Van Dyke, Perturbation Method in Fluid Mechanics. Applied Mathatics and Mechanics,

Vol. 8 Academic Press, New York-Lodon 1964.
[24] Z. Wang and Y. Zhang, Steady supersonic flow past a curved cone, J. Differential Equations

247 (2009), 1817-1850.
[25] H. Yin, Globul existence of a shock for the supersonic flow past a curved wedge, Acta Math.

Sin. (English Ser.) bf 22 (5) (2006), 1425-1432.
[26] G. Xu and H. Yin, Globul multidimensional tmnsonic conic shock wave for the perturbed
supersonic flow past a cone, SIAM J. Math. Anal. 41 (2009), 178-218.

[27] Y. Zhang, Global existence of steady supersonic potential flow past a curved wedge withpiecewise smooth boundary, SIAM J. Math. Anal. 31(1)(1999), 166-183.
[28] Y. Zhang, Steady supersonic flow past an almost straight wedge with large vertex angle, J.

Differ. Eqs. 192 (1) (2003), 1-46.
[29] Y. Zhang, Asymptotic behaviour of steady supersonic flow past a wedge, Chinese Ann.

Matth. Ser. B 26(3) (2005), 379-392.