Open Journal of Applied Sciences, 2013, 3, 79-83
doi:10.4236/ojapps.2013.31B1016 Published Online April 2013 (http://www.scirp.org/journal/ojapps)
Courant-Friedrichs’ Hypothesis and Stability of the Weak
Shock Wave Satisfying the Lopatinski Condition
Dmitry Tkachev, Alexander Blokhin
Mechanics and Mathematics Department, Novosibirsk State University, Sobolev Institute of Mathematics, Novosibirsk, Russia
Email: tkachev@math.nsc.ru, blokhin@math.nsc.ru
Received 2013
ABSTRACT
We are studying the problem of a stationary supersonic flow of an inviscid non-heat-conducting gas in thermodynami-
cal equilibrium onto a planar infinite wedge. It is known that theoretically this problem has two solutions: the solution
with a strong shock wave (when the velocity behind the front of the shock wave is subsonic) and the solution with a
weak shock wave (when, generally speaking, the velocity behind the front of the shock wave is supersonic). In the pre-
sent paper, the case of a weak shock wave is studied. It is proved that if the Lopatinski condition for the shock wave is
satisfied (in a weak sense), then the corresponding linearized initial boundary-value problem is well-posed, and its clas-
sical solution is found. In this case, unlike the case when the uniform Lopatinski condition holds, additional plane
waves appear. It is shown that for compactly supported initial data the solution of the linearized problem converges in
finite time to the zero solution. Therefore, for the case of a weak shock wave and when the Lopatinski condition holds
in a weak sense these results complete the verification of the well-known Courant-Friedrichs' conjecture that the strong
shock wave solution is unstable whereas the weak shock wave solution is stable.
Keywords: Weak Shock Wave; Asymptotic Stability (in the Sense of Lyapunov)
1. Introduction
It is well-known that the classical problem of a stationary
supersonic flow of an inviscid non-heat-conducting gas
onto an infinite plane wedge (with a sufficiently small
angle σ at the vertex) has two solutions: one of them
corresponds to the case of a strong shock wave, when the
components of the velocity vector behind the
shock satisfies the inequality (0
c is the
sound speed of the gas behind the shock wave), and the
second corresponds to the case of a weak shock wave,
that is [1-3] (see Figure 1).
000
:,Uuv
22 2
00 0
uvc
22
00 0
uvc
2
In Figure 1 the angular coordinates determine
strong and weak shocks respectively. The vector of flow
velocity is parallel to the axis.
Paradoxically, but in practice, in physical or numerical
experiments, only the weak shock wave solution is actu-
ally realized. One possible explanation was suggested by
Courant and Friedrichs [1]. They conjectured that the
solution corresponding to a strong shock wave is unsta-
ble by Lyapunov whereas the weak shock wave solution
is stable.
The Courant-Friedrichs' hypothesis was verified in
[4-7] (but this conclusion was based only on some quali-
tative reasons). A strict mathematical justification (and
this is very important) of this statement for the linearized
problem appeared in recent years in [8-12]. Briefly
speaking, it was shown in [8-12] that in the case of a
strong shock wave (for compactly supported initial data!)
the perturbation arrives to the wedge's vertex as time
increases having the growth r
(0
) or a logarith-
mic growth in space variables, and this causes instability
of the steady-state solution under consideration.
The situation with the weak shock wave is totally dif-
ferent. In this case the perturbation decays with time.
Moreover, this solution is asymptotically stable by
Lyapunov. It is assumed that in the both cases the shock
satisfies the well-known uniform Lopatinski condition
[13].
Figure 1.
Copyright © 2013 SciRes. OJAppS
D. TKACHEV, A. BLOKHIN
80
The present work is a continuation of [10]. We con-
sider the more general case when the Lopatinski condi-
tion on the shock wave is satisfied only in a weak sense
[13], i.e. the uniform Lopatinski condition can be vio-
lated. In particular, this makes finding the classical solu-
tion of this problem more difficult. The key point of the
work is the analysis of an explicit form of the solution
and we extensively use the technique developed in [10].
2. Statement of the Original and the
Auxiliary Problems and Formulation of
Main Results
The linear problem of finding a supersonic stationary gas
flow onto a planar infinite wedge can be stated as follows
[14]. We seek for a solution of the system of acoustic
equations
0,
tx y
AUBUC U
  (1)
in the domain ,0, tantxy x
 that satisfies the fol-
lowing boundary conditions at the shock wave ()
and on the wedge

tan :yx
13 342
3
0, 0,,
tan ;
y
ty
uduuuuF
FF u

 

(2)
21
tn ,auu
(3)
and at it also satisfies the initial data
 
00
0,,( ,),0,.UxyUxyFyFy
(4)
Here 1234 1234
are smooth
perturbations of the components of the velocity, the
pressure and the entropy respectively;
(,,) (, , ,): , , ,Utxy uuuuuuuu
(, )
x
Fty is a
small displacement of the discontinuity front, and

0
,00 0,Ft F
(5)
that means that we consider the case of a shock wave
attached to the wedge's vertex. It is also assumed that the
components of the vector of initial data are
compactly supported, i.e.
2
0{( ,)|,0},1,2,3,4.
i
suppuRx yx yi
 
The matrices read as follows:
2
2
22
010
00
( ,,1,1),,
1010
0001
M
M
AdiagMM B








0
;A
0000
0010
,tan
0100
0000
CCC







0
0
,
u
MM
c1
is the Mach number behind of the
shock wave (00 0
,tanuv u
are the components of the
velocity vector of the stationary solution, 0 is the
downstream sound speed), and the physical constants
c
,,d
were described in detail in [14]. These constants
depend on the components of the piecewise constant so-
lution corresponding to the step shock as well as on the
state equation of gas (,)spp
(
is the density of
the gas and s is the entropy) and the equation of the
Hugoniot adiabat.
If the solution of problem (1)-(4) is continuous up to
the boundary 0, tanxyx
0,0:tx
y
, then, in view of (5), it
follows from the boundary conditions (2), (3) that the
following compatibility condition should be fulfilled at
the edge
2
3
(tan)( ,0,0)0,0,dut t


i.e. if 2
1tan 0Dd

, then
(, 0, 0)0,0.Ut t
(6)
Remark 1.
We have formulated the initial boundary-value prob-
lems (1)-(5) for the case when the gas flow in a neighbor-
hood of the wedge with shock wave directed along the
- axis is fixed as the main solution (see Figure 2). Oy
In this paper we consider the case of a weak shock
wave, i.e. the case when
22
00
02
0
1.
uv M
Mcos
c
 (7)
Further we will also assume that the state equation of
gas is such that
22
1
0,, (1)0,ddMM
M

 (8)
The coefficient 10D
and, moreover, 1 (for
example, this is true for the polytropic gas [10]; some
examples of state equations satisfying (8) are considered
in [15]).
0D
Figure 2.
Copyright © 2013 SciRes. OJAppS
D. TKACHEV, A. BLOKHIN 81
Assume that the solution of problems (1)-(5) is not just
continuous but also has second derivatives which are
continuous up to the boundary of the domain. Taking
mixed derivatives we can reduce problem (1)-(5) with
condition (6) to the following initial boundary-value
problem for the component (the pressure). In the
domain
3
u
0, 0,tantxyx
 we seek for a smooth
solution of the wave equation
2
222
12 3
0,ML Lu
y


 




(9)
that satisfies the following boundary conditions at the
shock wave () and on the wedge (
0xtanyx
):
22
12 123
20;{}mLnLL Lu
M
  (10)
30;cossin u
yx





 





(11)

3,0,0 0,ut
(12)
and it also satisfies the initial data for :
0t
300301
(, ),()(, )||
ttt
uuxyu ux

y
0
(13)
(note that the derivative 3 can be found from the
third equation of system (1)).
()|
tt
u
In equations (9), (10) we have used the following no-
tations:
2
1112
1,tan,M
LllL l
ty x


 
 
 
1
,
2
22
1, ,M
Mnm d


.
 
The converse also holds, i.e. for each solution 3 of
problems (9)-(13) we can uniquely find a corresponding
smooth solution
u
(, ,),Utxy (, )
F
ty of problem (1)-(6).
This fact can be proved by the same way as for the case
of a half-plane (see [14]).
0x
Unlike [10], we assume that problem (9)—(13) satis-
fies the Lopatinski condition [13] at the boundary 0x
only in a weak sense, i.e. the uniform Lopatinski condi-
tion (see [14]) or equivalently the condi-
tion
0, 0mn
2
2
0, .
M
d
 (14)
can be violated.
In Figure 3 we shade the domain where the Lopatinski
condition is satisfied (in a weak sense).
This domain is described by the two systems of ine-
qualities
2
2
2
2
1,
1,,
and
.
0
d
M
dd M
M
M
d

 




(15)
The strait line
0
1
dM
 corresponds to the gas
dynamic case, and
and 0
are the densities ahead
and behind of the shock front respectively.
Remark 2.
For the state equations from [15] the point (,)d
lies
in the second quadrant and
2
2.
M
d

We point out that the uniform Lopatinski condition
holds at the boundary tanyx
.
Now we make a convenient change of coordinates by
setting
,taxxyyxn.
 (16)
We shall drop the primes in what follows.
Then problem (9)-(13) takes the following form:
22
2
tan
0,, ,0;
{( )()
()}
Mtx xy
utxy
y
2
 

 
 
(17)
2
2
tan tan
1tan0, 0;
{( )[( )
()]()}
d
tytxty
ux
xyy
M








(18)
sin cos0,0;()uy
yx



 (19)
,0,0 0;ut (20)
00 01
(, ),(, )||
ttt
uuxyu uxy

 (21)
Figure 3.
Copyright © 2013 SciRes. OJAppS
D. TKACHEV, A. BLOKHIN
82
(we also drop the subs the unknown function
roblems (1)-(5) and (17)-(21) are equivalent, it
is
he following property
ch
cript 3 by
3
u).
Since p
sufficient to state our main results for the solution
(, ,)utxy of problems (17)-(21).
ition, we also assume that tIn add
aracterizing the behavior of the solution as ,tx
holds: there exist parameters 0
s
and 0
p such
function 00
(, ,)
st px
ee utxy
 isoundeas ,tx
for each fixe
that the
bd
d 0y, i.e.
00
(, ,),
,,0is fix
()
st px
utxy Oe
tx y
 
ed
(22)
Let us introduce the notations
.
2
01,M 0tan
B
0
(1B),
2
1tanDd
 ,
22tanDM
,0
0
1
1
B
LB
and assume that the problem's parameters are linked by
the relation
21
21
||
.
DD L
DD
(23)
The following results hold.
ata for problem (17)-(21) are compactly
su
Theorem 1.
If the initial d
pported (i.e. 2
01
,)
s
uppu uR
and inequality (23)
holds, then the cla the problem exists, is
unique and can be defined by formula
ssical solution of
0
00
()
,
11
0
22 0
2
(,0)
22
0
2
22
8
(, ,)
(, )*(()()tan),/
2
8
,(
*(()tan),
8
/
2
/
,
()
()
(
)
(
Bxy
xy
BB
t
xy
t
t
utxy B
yx
M
M
gtEty Bx
xyBdlt
M
M
Etyxxy d
M

 

 
 

 
 

 

 

 
 
 
 


0
1
2
22
,2(,)
(()( )tan),/
2
tan
,; 2,2
)
(
)()
OQM P
xydd u
M
Ety x
,
x
yddxyx y
 

 
 

 
 
x
 
 
(24)
the first integral is over the line , and the next
two integrals are over the ab
0
yBx
 
scissa axis y0

; in the
last two integrals over the qua we
have the following coordinates: t
0),
drangle
he point
0POQM
(,Qx y

the point 0()
,
22
()
Bx y
xy
P 
 
,
and the point 0(, )
M
xy

olution of t
1(, )uxy
,
; the function is the
fundamental she operatorn (14);
and are the initial dre the
es
(, ,)Etxy
of equatio
ata (whe
0(, )uxy
coordinat
x
y are expressed throuiables gh the var
'' ''
,
x
y); the functions (, ),
g
tx
 (, ),x
 lt (, )
f
tx
are
known, in particular, (, )
f
tx
 is determinllows: ed as fo
(, )Ntx 1
,
0
(,)(,),
px
stn
n
f
txLHps


 (2)
where

5
(, )Ntx
is a certain integer number, (, ) 0Ntx
.
Rem
We do not write down here formulae defining th
functions
ark 3.
e
, (, )
f
tx

(, )
g
tx
, (, )ltx
and (,)
n
H
ps
because of their awkwardness.
Theorem 2.
For problem (17)-(21) the trace
,)*
)

0
0
( ,0)
0
2
00
2
0
2
(, )*
*(()tan),/
2
,8tan
(()( )tan),
2
,2(,)
(()( )tan),/
2
(
)( )
(
)
(
xy
t
t
OQM P
OQM P
ft
y
M
Etyx
xydu u
M
Ety x
xydd u
t
M
Ety x


 

 

 
 

 
 
 
 
 
 
 
 
''
''
0
2
(,)(, ,)||
x
xy
ftxutxy
at the shock wave is a superposition of a fier of
cylindrical and plane waves. Namely, repr
nite numb
esentation
'' ''
0,
(,0,)(,).|
n
(, )
1
p
xs txy
n
uLty Hps

Nty
takes place.
Theorem 3.
If
y
K
(Kis a compact subset of the real half-axis),
then there exists *(,)tK
( is the support of the
initial data 0
u and chat (, )0fty
1
u) suh t
when
*(,),ttK
y
K
.
3. Conclusions
on the linearized level that the solution
w
when one shockont the Li condi
e, i.e., the uniform Lopat-
ins violated.
ported initial data any
So,
1) we prove
ith a weak shock is asymptotically stable (by Lyapunov)
in the case th fropatinsk-
tions is satisfied in a weak sens
ski condition i
2) Moreover, for compactly sup
Copyright © 2013 SciRes. OJAppS
D. TKACHEV, A. BLOKHIN
Copyright © 2013 SciRes. OJAppS
83
solution of the linearized initial boundarye problem
becomes stationary for a finite time.
sta-
ble whereas the weak shock wave solution is stable
(when the Lopatinski condition holds at least a weak
sense).
4.
an
th iaucation
No. 14.B37.2.0355)
also grateful to A.V. Yegitov for the
ing the manuscript.
ctures on Basic Gas Dynamics,”
es of Flow Roun
valu
Our result justifies the classical Courant-Friedrichs'
hypothesis that the strong shock wave solution is un
in
Acknowledgments
This research was supported by the Russian Foundation
for Basic Research (grts No. 10-01-302-a and 11-08-
00286-a) and e Russn Ministry of Ed (grant
1.
The authors are
assistance in prepar
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