On the existence of shock waves of small intensity

ON THE EXISTENCE OF SIIOCK WAVES OF SMALL INTENSITY (0

SUSHCHESTVOVANII MALOI

PMM

Vo1.26,

No.3,

1962,

18.

G.

UDARNYKH VOLN

INTENSIVNOSTI) PP.

511-519

LIUBARSKII

(Kharkbv) (Received

As is

well

known from the

hydrodynamics, transition

the

algebraic

velocity

to

the

f 00. Let

form

it

a given

is

concerning

transitional

solution

the

Lll.

by Gel’fand

shock

wave in magnetohydrodynamics

Germain he also

and Liubimov

of

the

approach

Part

to

was proposed systems equations Processes

of

coefficients the

of

problem

by Godunov nonlinear

of

E5.61.

of

of of

of

the

hydro-

of

magnetohydromust have

u_ and u+, when

conditions the

case

under

plane

are

the paper

treats

Particular

cases

of

such which

consider

slow

proven

that

from zero.

The present

Be shall

of the

transitional

for

a fast Under more

[41 have

different

of

of

assumption

of

was formu-

flow.

existence

the

existence

existence

of

the

of

equations

and Liubimov

existence

the problem.

wave.

wave moving

processes,

nonlinear

of ordinary and magnetohydrodynamics are taken into account.

1. Statement

every

as transitional.

[3j

waves

of shock

equations

dissipative

dissipation

equations.

a steady

established

shock the

discontinuity

system

in the

assumptions

and slow

discontinu-

[21 has proven

Kulikovskii fast

know which

to

not

do not.

system

shock

of

a shock

important

the

restrictive existence

well-defined

represents

sufficiency

lated

wave.

at the

certain known that

solutions

in the

and magneto-

quantities

also

Ut) , which tend to

these the

is

particular

account

us designate

in ordinary

actually

corresponds

u = u(x -

The problem the

it

relations

u+ there

into

waves

a wave must satisfy

means that

takes

1962)

hydrodynamic

wave and which

u_ -

U. This

of

these

a shock

which

of

12,

shock of

applications

quantities

dynamics, solutions x -

with

of

Furthermore,

us assume that

dynamic with

front

satisfying

represent

Let

the

relationships.

Khen dealing ities

theory

discontinuities

through

discontinuity

February

only

the a

A new

solutions

dissipative systems the

are the

dissipative

discontinuities

C.Ia.

762

of

sufficiently

small

discontinuity natural

magnitude.

closely

cities

of

The velocity of the phase velocities

equals

to distinguish

For example,

Liuharskii

between

ordinary

in magnetohydrodynamics

slow

and fast

is

a singular

velocity.

of

an ordinary

phase

In the case

of

A precise velocity

waves,

whereas of

be given

systems

the

of such a system. It is

phase

velocities

definition

will

dissipative

and singular

ordinary

magneto-sound

transition V of the

velocities.

are

the velo-

the Ufven

velocity

a dissipative

system

and

below. following

statement

will

be

proven:

To any discontinuous into

account

a transitional i.e. of a discontinuous velocity

Ii of

one of

the

ciently

solution

of

solution

transition

ordinary

of

a system,

is of

phase

formulated

there

processes, an exact stable

system,

with

if

regard

the discontinuity

velocities

without

corresponds

and (cl

(a)

to

is

taking

a shock

wave,

the discontinuity

attenuation,

sufficiently

(b)

close

the discontinuity

is

the to

suffi-

small.

If fast

solution

the dissipative

the

foregoing

shock

waves

is

established

of

dissipation. ‘lhe proof

Theorem

conditions of

based

fulfilled,

small

on the

the

intensity

regardless

immediately,

is

are

sufficiently

of

following

the

existence

of

slow

and

in magnetohydrodynamics value

of

the

coefficients

theorem.

2. ‘Ike system m-1

+ . . . + a,$$-

amy

@o(y-$)

=

E 2

y'""pk (y, y', . . . , y!m-l), E; t?)-:-

k=l +

Ei’

-

i

&?Pl

bikEk

zz

fi

(y,

(y)

y’,

-t

.

. .

EFi

, y(m-1),

(y,

E;

g;

e)

(1.1)

. . , m)

(i=2,3,.

6)

k=z gi

=

fi

has a transitional fies

the

(y>

-+

&Fi

solution

conditions

(Y7

~(x,

E;

(i=m+l,.

E)

E),


y( -00,

E) = 0, y( +a, m-l roots of tie polynomial tZmum+ ‘lN_ 1v simple, if the determinant jbikjZRL is not functions

F; (i = 1, . . . , n), fi m - 1) are analytic with respect value Vie 3 I.

of

the parameter

pr0o.f

of

this

Dissipative

E is

and

is

ideal

equal

= 2, 3,

to

their

:;iven

E))~“,

E) = 1 + 0(s2), + . . . + nlv are

(i

sufficiently

theorem

. . , n)

to

zero,

which

if

all

the

. . . , n) and vk (fi = 1,

argrments

satis-

if all the real and

and if

the

...,

absolute

small. in

systems.

tlie appendix. The

systems

of

equations

of

Shock

waves

of

intensity

small

763

ordinary and magnetohydrodynamicsmay be written schematically in onedimensional form as follows:

ai (u) +

-g +

Ui

(U) +

&

&bj (U) =

-& bi (u)

=

G&(U)

bi (u) = 0

G&i(U)

(i = 1, . . . , kl)

ZZ Ci

(U)

Ez ci (u)

(i=k,+l,.

(i=k%+l,.

. . , k2)

(2.1)

. . , n)

where u = {uj),n is a set of quantities which describe.the condition of the fluid (pressure, temperature, velocity components, magnetic field and others), ai( hi(u) and ci(u) are certain analytical functions, -' > 0 are coefficients of dissipation, namely viscosity, magnetic 'i viscosity, thermal conductivity. The system (2.1) admits.of solutions of the form u = u" which describe constant homogeneous flows. Any vector may be substituted for u", provided it satisfies the system of equations yi(uo) = ?(i = k, + 1, . . . . n). Let the system (2.1) be called a dissipating system if all of its solutions are as~ptotically stable with respect to small disturbances of the initial condition. If system (2.1) is dissipative, then it is easily seen that all the roots as(k) ( -m < k < a) of equation D (id, ik, u”) 5 det 1iodij (u”) Uij (u”) = 2

ikbij (u”) _and

Gi$ij (u”) 1=, 0

SO on.

(2.2)

j Iu--u*

are located in the upper half-plane. *In ordinary and ma~netol~ydrodyn~ic fluids all free vibrations vanish with time. Therefore the systems of differential equations by which they are defined are dissipative. If in one of the equations (2.1) we assume ui = w or ai = 0, then that equation assumes the following form $i (U) = 09

+

Ui (U) +

2

bi (U) = 0

or

&

bi (u) = 0

Assuming that a part of the coefficients ai vanish, while the rest tend to infinity, we will have an ideal system. Consequently in this manner there are vn _ - k separate ideal systems associated with the system (2.1). Tle ideal system in which all oj = m (oi = 0) we shall call the lowest (the highest) system.

764

Liubarskii

G. Ia.

We shall

call

two systems

neighboring

systems

if

they

differ

by only

one equation. If

we assume

that

rest

equal

will

be substituted

kOaij(uo) have of

-

the

a part

infinity, ikbij(uo).

following

are

roots

the

In this

form:

the corresponding

system

of

then a part by the lines

case

os = Vsk,

ideal

of

coefficients

of the yij(uo) the

solutions

where

system.

u. vanishes

and that

lines in the determinant and a part by the lines of

the

(2.2)

Equation

(2.2)

will

Vs - Vs(uo) are phase velocities

The phase

velocities

of

the

lowest

the equation Vaij (u”) + bij (u”) I

I-

A (V) z

dct

j

(Z..?)

=O llij C"")

When referring velocities

of

Phase system

velocities

(2.1)

of

are

real

velocities

of

system

(2.1)

we mean the phase

system, all

ideal

[81 (see

systems also

associated

with

the dissipative

[91) .

Ordinary phase velocities. Let V = V(u_) be one of the phase

3.

velocities

of

following (a)

system

for

(b)

are

We shall

call

it

an ordinary

velocity

if

the

fulfilled.

V is a simple

root

of

the characteristic

equation

uo = u .-

In one of

V for

(c)

(2.1).

conditions The quantity

(2.3)

from

to;phase

the lowest

the

neighboring

systems

all

phase

velocities

differ*

u” = u_. system

The uppermost

has no phase

velocity

identical

with

V for

u” = u_* (d)

To every

value

of

II from any neighborhood

of

the number

corresponds

a discontinuity, propagating with velocity (u+ - u_)/(LI - I”) tends as II + V, while the ratio

zero for

(U -

V) - 0.

(e) All with

roots

the exception

w of of

equation (kl

In magnetohydrodynamics waves hand,

*

D( -

system

If system (2.1) property (b) .

in this is

vV, -v,

+ 1) which the Alfven

because for them condition fast and slow magneto-sound

uppermost

V there

II and tending to to a finite limit

case

dissipative

is

root

and real,

v = 0.

and entropic waves Cl01 are singular not fulfilled [ll] . On the other

(d) is waves

are

has an infinite then

u_) = 0 are simple

a multiple

the

ordinary

waves.

viscosity property

(a)

Indeed,

the

and thermal is

the

result

of

Shock

conductivity

and zero

PH

electric

a%

o

of

means that

o -zI

one phase

of

the magnetic

such

a system

are

isothennal

phase

velocities

ordinary

sound

velocity

of

the property

task

of

(e)

In the case

of

been

by Godunov

proven

a system

equals

vz,

and the

condition of

thermal

inthe

conductivity

is

velocities

oscillations

may be obtained by replacing

the

sound velocity.

be the checking already

a sound wave in ordinary

are

consider

Snail

magnetohydrodynamics

isothermal

rest

(b)

vanish.

and the phase

proved

we have

,

viscosity

would

is

0

765

In such

coefficient

ideal

by the

The most complicated

intensity

To check

and ordinary

from the

ever,

aT ax

velocity

finite. Condition (c) is fulfilled. neighboring system, in which the infinite,

small

conductivity.

==,

x=’ This

waves

in

of

condition

the work of

hydrodynamics

(e) .!low-

Germain

condition

[21.

(e!

has

151 .

4. nre system which determines the structure of a shock wave. To the stationary hhock wave, propagating with velocity II, there corresponds where of

u( km)

the

a solution of = u*, u’(*m)

system

of

system (2.1) of = 0. Evidently

For the purpose R(u)

of

further

Un (u) +

b(u)}

considerations

=

-

it

is

necessary

of system (4.1). Let us introduce and write Fguation (4.1) in the V(U_)U(U),

-$ {The set

of

coordinates operators

EU

(u) + B (u)} = c(u)

all

vectors,

vanish,

will

of

spectively.

the

in which be denoted

orthogonal

These

the

are

(E =

first

by H,(G,)

projection

operators

onto

given

k,

u -I; (last

denote

by al,

B,

QlCl-- QlCl = 0, From (4.2) ing

the

it

generality

follows of

the

that

ol{

k

to make

the notations following form:

(4.2)

pi_)) n -

kl)

of

the

. Let P, and f-)1 be the the

subspaces

II, and G, re-

by the matrices

aBi

Bi/; (u_) = au

aik (U_>,

we shall

= u(x - Ut), the solution

c(u)

transformations

= b(u)

which

t) is

equations

$ {certain

the form u(n, function u(n)

aci u=u_

and cl.

P,c -EU(U)

investigation,

9 ?Jote the

= c,

relations

PIG, =

+ B(u)) we shall

Cl

= const.

%thout

assume a(~_)

lirnit-

= 0,

C.la.

768

Liubarskii

Rfu_) = 0. Also -

sQla (u) + QII: (u) = 0

(4.3)

and Equation (4.2) then assumes the following form:

& f-

EP,U (u) + P1B (u)}

zxz R

(u) - EQ~G (u) (R = QIR -i- C\

(4.4)

Let US prove that for certain known conditions system (4.4) has a transitional solution. From (4.4) it follows that quantities u_ = u(--co) and u+ = uf +m) satisfy relation Z?(u_) -

sQln (u_) = R (II+) -- E&Q (u+) = 0

(4.5)

5. Discontinuities propagating with velocity close to the ordinary phase velocity. Let us investigate Equation (~4.5)in detail. Let us assume u+ = u_ + ev+ and expand the vector-functions R(u+) and a(~+,)in Taylor series. \?eobtain B,c, + E& (v+f+ . . . = EQ~CI~U++ c2Q1al (v+) + . . .

(5 11

iiere R, is a particular or intrinsic operator. 'Ibisfollows directly = 0. Let from (2.3). Therefore there exists a vector u", such that r'Tlvo us introduce in the investigation also vector us, which satisfies relation I' '1*v* = 0, where R,* is the operator conjugate to R,. Notice, that

crvo==:Prc*$ z P112r# z= 0 According to property (d) the ordinary phase velocity vector v+ tends to a finite limit, when E _ 0. From (5.1) it follows that

The number r_~ is easily found by multiplying Equation (5.1) by U+ and then approaching the limit E - 0; we obtain the result

p (II”, I& (0”)) = (v*,

Q,n,v")

Let,us prove that (v*, ,'l)lnlvO) # 0. According to property (a) of an ordinary phase velocity the number V_ = V(U_) is a simple zero of funcTherelroreA'(V_) # 0. %I the otlier tion A(V) = det 18, - (V - C'_)Q,CX~/. hand

Shock

where Tik are algebraic

waves

of

supplements

small

76’7

intensity

of determinant

det

/RI/.

Since according to (5.3) not all fib vanish and since the operator is particular, the numbers rik may be represented in the form rik = if the vectors v” and v* are suitably normalized. Substituting Vi*VkOl this expression of rik in (5.3) we obtain

R,

A’ (J’_)= -- (u*, Qlulvoj + o Consequently

(a*, R, (q) # 0.

p = (U”>Ql4~7 / (u*, -Rz(0

6. Reduction of system (4.3) to (4.4) to the canonical form. Let us note that dB(u)/dx = &u)du/dx. According to property (c) of an ordinary phase velocity the operator k(u_) - R, has a reverse Therefore for sufficiently small 1~1 and 1~ - u_] Equation operator. (4.2) may be rewritten in the form du

--

dx

= [h(U) -

E;t (u)]-‘c

Introducing .the notations Hz = B1-“N, fine the operator P,, assuming P&J = cp

(u)

(6.1)

and G, = B,-‘G,,

P#

(cpE l~zft

= 0

19,f.

we shall

de-

G2l

To simplify further derivations we shall assume, without limiting the generality, u_ = 0. Let us separate the desired function u(x) into two terms u = z $- 2L’ and substitute

it

(z s Hz. II:E Gzt

in (4.3)

Q&z

-I- Q,B,w = EQ# (EC)-t Q1 I&U - B (U)I

Since 2 E Ez, then R, z E H, and Q,B, z = 0 and Q,B,w = Blw. Therefore w = B1-‘Q, Apply the operator

d;

-z

{EU (u) +-

P, to both parts

= P,B,-‘c,u

-I- P, ([u’(u) -

Blu - R (u)}

of

(6.1).

& (u)]-1 c (u) - B,-‘qu)

(6.2)

768

Note dz

-

Liubarskii

C. la.

that

clu~Hl,

+ F(u),

= II,-‘c,u

dx

System

(6.2)

Before

B,-lcl~ti

to

F (u) =

(6.3)

proceeding XIV ’

equation

Let

if

el,

el

= v’.

e”

in G,.

is

t[B

let

P,J31-1cl~

8; (u)]-l c (U) -

(~1

-

to

the

initial

that

us prove

Q,R,v” = Q1 (Q,B + cl) that.B1ro

. . . . eh(z

In addition,

EN,

= n-

let

(6.3j

(4.1).

Starting

from

k1)

there

v*= QIBv”

and v” EB~-~H~

(6.4)

= 8,.

be any starting

be a certain

base

starting

in !I,,

base

es+

where 1,

....

11

.?n sp”yei +

x

EQei,

W =

2

in Equations

following

(6.2),

form is

Y’ = s $

coordinates

(6.3)

(6.5)

&“Eiei

km-+1

i=2

If

Bl-lClu}

system

v” E H,.

and

Assume

2 z

the

= B1-lclu

we obtain

follows, e2,

P,

equivalent

further

= 0,

0= Hence

Y2. Iherefore

are

introduced,

a system

of

obtained

P&j + a (a?/ + by)“- -I- s2’p (y, I&F- f * 9 En; E) (I;.(;)

j=2 II‘ Ei’

-

2

PijEj

=

ai.!/

+

big2

$

‘EFi

(_I/,

E,2,

+ . . y En;

(i=

E)

L

3.

. . . . n)

j-2

ii

Let

=

my

+

by2

us represent

+

the characteristic istic

(6.6)

is

polynomial

In part

. . 7 Em;

(i

&)

=

m + 1,

. . , n)

by

D, (Y, E, u_) = det

System

E2,.

&Fi(y,

1v (- FP~u~+ P,B,) - R, $- .sOlnTj

polynomial equivalent is

identical

of to

system

(4.4)

the system to D,(v,

(4.4).

E, u_)

at the point ‘Iberefore except

for

u = u_. its

character-

the multiplier.

we obtain

11, (Y, 0, II_) = conStY dct Assume

I/3ij -VyBij

\zffl

(6.7.

Shock

Q (Y,

o, u_)

=f

waves

of

small

a,,,~ + a,+_.l~m--l + . . . + a,v = VAX(Y)

Let us apply now the operator By the well known Hamilton-Kelly

A(d/dx) to the first of equations theorem [121, we obtain m-1

+

769

intensity

(6.6).

m-1

(6.8)

e2& (y, y’, . . . , y(m-% %21 . . . , En;e)

Let us denote by (A) the system which will be obtained, if in the system (6.6) the first equation is replaced by Equation (6.5). Systems (6.6) and (A) are equivalent in the sense that every transitional solution of one is the transitional solution of the other. This follows from the fact that among the solutions of equation A,fd/dx)cp = 0 only the trivial solution cp f 0 is bounded on the entire x-axis. 7. Existence of shock waves. It will be shown that system (A) all the conis a particular case of system (1.1) and that it satisfies ditions of ‘lbeorem 1.1. for sufficiently small Condition (a) of the Theorem 1.1 is satisfied condition (ef pertainvalues of absolute magnitude of e because of the ing to the ordinary phase velocity, since u1 (Y, 0, u_) == (-Y)

D (-

vv_,- y, U-J

To check condition (b), use is made of the following property of the dissipating systems. Let v = v,(U) be the root of equation (-v)-&l D (-VU, -v, u_) = 0, which becomes zero for U = V_. If velocity V_ possesses then v0 ‘(V-1 < 0. city*,

the property

(b)

of the ordinary

phase velo-

It is easily seen that the characteristic polynomial of the system (A) has the root V(E) = - +Jal + O(s*) for small E. Therefore, &/cl

’

0.

According

l

This than the

to the condition

of

of a discontinuity

relative

proven in TSI for systems of a more special However, the derived proof is valid in the the form (2.1).

statement is system (2.1). systems

of stability

form case

of

770

C. la.

Liabarskii

to dissipation, the following inequality must be satisfied

i.e. it must be that E < 0. And thus @,a1 < 0 and the condition (b) is satisfied. At the same time it was proved that condition (c) is also satisfied, since on account of (6.7) a, = const det 1Pik i2” It remains to prove that yaa = PO. To do this we notice that the coordinates of vector u+ according to (5.2) equal

g!+-fi-Q(&

Ei

z

t?

(&)

(i = 2, 3,.

. . , n)

On the other hand, a constant vector u = u+ is the solution of the system (A). Substituting it into Equation (6.81, we obtain

Hence, Thus,

it is concluded that y,,*

= p,.

the statement formulated in Section I has been proven.

In particular it has been proven that there exists a structure in slow and fast magnetohyJrodynamic waves of sufficiently small intensity. 8.

(1.1)

We shall Appendix. Auxiliary equation, with the investigation of the auxiliary +3

begin the proof equation

y f Ef (?/) -i- E$ (3) = 0

of Theorem

(8.1)

(0
Define two numbers a_ > 0 and a+ < 0 in such manner that the polynomials P(v) + O* have only real and also simple zeros. In regard to functions f(y) and y(x) we assume the following. (I) closed (2)

Function interval

continuous and differentiable f(y) is defined, p1 < y Q pa, where pI a_ = p2a+.

in a

Let

(3) No matter what the interval is (a, exists a number d(a, p) > 0, such that

a) (p,

< a < 0 < (3 < pz),

there

Shock

(4)

Function to

(5)

Y(X)

certain

is

Let

us denote

intervals

( -m,

Theorem 8.1. has

y( -m),

theorem If

set

and (0,

is

denote

conditions

by yf

functions

satisfy

the

(5)

satisfied,

o(x)

following

in the

inequalities

(1)

to

solution

the

set

of

are

and this all

then

solution

functions

is

V(X),

shows that there into the solutions

Equation unique

which

[T].

satisfy

exists an operator y(x) E S of Equa-

(8.1)

The following Theorem

interval is

- UJ< x < 03 and

k 03.

valid

continuous

Theorem (8.1) conditions (4) and (5). TE which transforms functions y k.Yf tion

axis

when x -

(y > 0)

valid:

in S a transitional

We shall

all

(y)= 1 a,

*

on the whole

which

771

f a-(Y
PI),

are

of

m),

intensity

small

v( +m),

inequalities

by S the 0)

The following

(8.1)

determined

limits

The following

of

(YE&

P)la(y)l

If’(~)I>d(a,

tends

waves

8.2.

theorem There

0 < E < EI,

may be proven:

exists

a number ~1 > 0,

and for

all

I~I, v’2 E Y f

such the

that

for

following

all

E in the

evaluation

valid

(l(fm+$g!Pf~). The following Consequence and 0 < E < EI, 11 Yi” -

consequence 8.1.

If

all

follows the

x I 9 (4 I II9 IIx= sup from Theorem 8.2.

conditions

of

Theorem 8.2

are satisfied

then

Yp’ 11< e”k

(i + &

\\ $3 - $1 11

(k=i,2,...,nz-i)

772

G. la.

Liubarskii

where

Proof

of

determinant

1.1. According to the condition of the theorem the 1bik/ 2m does not vanish. Therefore the system of equations

Theorem

Ei’ -

jj

b, ik = qi ‘2)

(IIwi II
k=8

has a bounded solution stant B such that

ei(x)

along the entire

axis.

(i = 2,3,.

. * ,n)

II5i II
There exists

a con-

Assume maxlfi(Y)l==Cli

and denote y, Y’.

Y’:

(---Libya)

(i =

2,

by D the region in (a f II)- I)-dimensional (m-1) aa.9 Y ; &, &’ ..*, Cn, determined

--1
jy’“)j
that vector-function

(i=2.. {Y(X),

space of variables by the inequalities

,.,n;

k=i

I.e.1 m-

belongs

to the

I) Functions y(x) and ei(x) ( i = 2.3, . . . , A) are continuous whole x-axis and tend toward a certain limit for x -, k w;

on the

set

It may be asserted S(D), if:

I&I<(B+i)&

3, . . * , n)

2) Function y(r) has continuous derivatives, of the (a - 1) order, all of which tend to zero 3) for <,ce.

every value x(--m < x
4) Function

xy(x)

y(x),

C(X))

including the derivative for x - f a; y’(x),

. . . . y +1)(x),

is not negative.

for comparison with every vector function {y, Let us associate the vector function A{y. 4) = fyx, Ex), which satisfies from S(D) relations: following P

@ the

ii

Shock

fi”’

-

t~aves

of

773

intensity

snail

5 bij Ej” = upi(x)

(i = 2,3,

. . . , m)

j=2 (i=m+l,...,n)

Ef” = 4%(4 where m-l 9 (~1 =

-

2

yfK) ‘Pk (y, y’, . . . , yim--lf,

82, . . . , En; e) -

k=l

From Theorem 8.1 it follows that these relations, indeed, determine operator A. The Theorem 8.2 and its corollary 8.1 show that for sufficiently small E > 0 the operator A does not invalidate the vector functions in the region S(D) and that it is compressible relative to the norm

IIY* EII = slc”p[I Y W

I + $ “il

I Ytk) (4

k=l

with the coefficient

of compressibility

I+

5

I Ei WI}

i=?&

p < 1.

Hence it follows that there exists a vector function (y(x), C(X)) {y, 5). This function serves as the transitional such that A{y,
BIBLIOGRAPHY 1.

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2.

Germain, P., dynaaique

Contribution des

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a

la

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Q.N.T.R.A.

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en

Ragneto-

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3.

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4.

Liubimov. G. A., Struktura ~agnito~i(iro(iinamicheskoi udarnoi volny v gaze s anizotropnoi provodiPiost’ iu (Structure of magnetohydro dynamic shock waves in a gas with anisotropic conductivity). PMM Vol. 25, No. 2, 1961.

7-l<

5.

G.Ia.

S. K. , Interesnyi

Godunov,

ing class No. 6.

of

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sistemy

continuity SSSR

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Whilham,

Vol.

136,

No.

of

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Vol.

139,

razryva

a quasilinear

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smearing

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1961.

2,

Vol.

SSSR

140,

No.

L.D.

to

12, NO. 1.

and Lifshits,

11. Polovin,

R.V.,

of

Udarnye

Gantmakher,

F. R. , Teoriia

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Conm.

Pure

1959.

E.M.,

volny

1961.

magnetohydrodynamics.

Media).

v magnitnoi UFN Vol.

matri

(On the

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Continuous

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6,

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application

Vol.

Math.

Nauk

Some comments

with

(Electrodynamics

12.

razmazyvaniia

0 strukture udarnykh voln Liubarskii,. G. Ia., shock waves). PMM Vol. 25, No. 6, 1961.

10. Landau,

(An Interest-

Nauk

Postroenie perekhodnykh reshenii nelineinykh (Structure of transitional solutions of nonlinear equa-

G. R.,

structure

9.

sistem Akad.

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solution

t ions) . Dokl . Akad.

Appl.

Dokl.

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a.

kvazilineinykh

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S. K. , 0 needinstvennosti

kvazilineinoi

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quasilinear

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Nauk

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structure

sploshnykh

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gidrodinamike

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1,

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of

waves

1960.

Gostekhizdat,

1954.

Translated

by

J.R.K.