1 The (generalized) Kortewegde Vries pquation (KdVE)
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THE
1.1.
(GENERALIZED)
addition, if u(., t)
is
KORTEWEGDE VRIES
H'solution of the problem
a
(1. 1. 1), (1. 1. 2),
00
I 2 u'(x, t)
El(u(.,t))=
and
f 7(s)ds,
F(u)
0
t G
R,
i.
the
e.
junctionals E0
and
El
place
we
we
initial data. Instead of it
with respect to the
shall
exploit this
solutions in the
case
2 and
Chapter
TI, T2
=
independent of
Cauchy problem
certain result for the
for the standard KdVE with
x
problem
f (u)
=
u;
definition of
following
periodic.
are
Hpn,,(A) for
G
uo
u,
We call
0.
a
We introduce the
4.
when the initial data
f (u)
for the
consider
spatial variable
result in
Definition 1. 1.4 Let
>
determined and
conservation laws.
are
with
periodic
are
0
A result similar to Theorem 1.1.3 takes
periodic
F(u(x, t)) dx,
U
f f (s)ds
=

x
00
U
7(u)
quantities
I
and
00
where
then the
11
00
u'(x,t)dx
Eo (u (, t))
EQUATION (KDVE)
function u(, t)
A
some
0 and inte
>
C((Ti, T2); Hln,,,(A)) n
C1 ((_ T1 T2 ). Hne3 (A)) a solution of the problem (1.1.1),(1.1.2) periodic in x with
P
the period A > 0 (Or simply a periodic Hnsolution) if u(., 0)
uo(.) in the space
t
1.
holds
the
G
in
and,
for
sense
(Ti, T2), equality (1. 1)
any
Hpn,,,(A)
of the space
3
Hpn,,r
(A) after the substitution of the function u in it.
ger
n
,
7
>
a
E
r,
=
As
onto
earlier,
it is correct to
The result
the
on
wellposedness
considered in this book is the
Theorem 1.1.5 Let
any
integer
into
about
> 2
n
and
f (u)
uo E
sense
of the
global
for
a
periodic Hnsolution
solution
(defined for all t E R).
problem (1.1.1),(1.1.2) in the periodic case
=
u
so
Hpn,,(A)
that
deal with the standard KdVE. Then
we
there exists
This solution
that
continuation of
a
a
following.
of the problem
in the
speak
wider interval of time and about
a
a
for
unique global periodic H'solution
continuously depends
any T > 0 the map uo
i
)
C((T, T); Hpn,,,,(A)) n C'(( T, T); Hpn,,,3 (A)).

u(, t)
on
the initial data
is continuous
from
In addition there exists
a
Hpn,,(A)
sequence
of quantities
A
Eo (u)
A
I u'(x)dx,
Ej(u)
0
I
2
U2(X)
U
X
6
3(X) dx,
0
A
E.,, (u)
1 2 [U(n)12
X
+ CnU
[U(n1)]2
X
qn(U7
...
(n2))
)U X
dx,
n
=:
2,3,4,...,
0
where Cn
periodic
are
constants and qn
Hnsolution
u(.,t) of
are
polynomials,
the
problem (1.1.1),(1.1.2) (with f(u)
such that
for
any
integer
=
n
u)
> 2
and
a
the quanti
EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE
CHAPTERL
12
ties
Our
do not
E,,(u(., t))
Eo(u(.,
conservation laws
proof
depend
for periodic
Wt
of the
f(W)W.'
+
+ WXXX +
1.1.6 Let
be
f()
get
(x, 0)
=
Wo
we
consider the
(1.1.4)
e>O,
(1.1.5)
(x)
the
infinitely differentiable function satisfying
E S and
following
statement which
f()
Then, for
(1.1.3).
be
E
E
to Coo ([0,
take the limit
we
1.1.7 Let
Proposition
estimate
first,
xER, t>O,
0,
=
X
an
(1. 1. 3).
At the second step,
fact,
(4)
6W
Then, for any uo
unique global solution which belongs
we
are
following:
Proposition
a
junctionals Eo,...' E,,
problem (1.1.1),(1.1.2):
W
estimate
the
e.
of Theorem 1.1.3 consists of several steps. At
following regularization
and prove the
t, i.
on
Hnsolutions.
an
any uo
c 4
is,
problem (1. 14),
(0, 1]
n); S) for an arbitrary n 1, 2,3,....
=
+0 in the problem
of course, of
an
(1. 1.4),(1.1.5).
independent
S there exists
a
In
interest.
infinitely differentiable function satisfying
E
the
(1. 1. 5) has
the
unique solution u(.,t)
the
E
00
U C((n, n); S) of the problem (1. 1. 1), (1. 1.2).
n=1
At the third step,
Now
we
using Proposition 1.1.7,
turn to
proving Proposition
Lemma 1.1.8 The system
we
prove Theorem 1.1.3.
1.1.6. We
begin with
the
following:
of seminorms
I
00
P1,0(u)
II (
=
2
2
dx1
)
dx
00
generates the topology in the
00
I
and
1
po,,(u)
x21u2(x)dx
00
1
I
=
0, 1, 2,...
space S.
Proof follows from the relations
00
2
PM
21
1(u)
x
U(X)
(dM )
00
2
dm
dx
u(x)
dxm
,
00
Cl""
[X2,dmu(x) ]
dxm
00
min m;211
:5
dx
E
k=O
jI
.
2
x
2(21k) u 2
(x)dx
+
d2mkU)
(dX2mk
f
dx. 0
dx
<
THE
1.1.
Let
(GENERALIZED)
take
us
(1.1.4),(1.1.5) by
arbitrary
an
the iteration
Wnt + Wnxxx +
KORTEWEGDE VRIES EQUATION
IEW(4)
(0, 1].
E
c
(x, t; c)
= uo
C ([0,
m); S),
w,
(x)
13
problem
procedure
Wn(Xj 0)
where
We construct solutions of the
f(Wn1)W(n1)xi
==
nx
(KDVE)
C S.
=
t >
0,
n
(1.16)
2,3,4,...,
=
U0(X)j
the Fourier
Using
R,
E
X
(1.1.7)
transform,
easily show
one can
that
00
U
Wn E
n
=
2,3,4,....
M=1
into account
Taking
from
get
(1.1.3)
and
00
00
+
(194 )
Wn
dX
2
00

C0
I(
c
92 Wn
0XI
)2
dx
00
00
I (Wn
a4
+
Wn)
OX4
19Wn1
f(Wn1) 51 dx
<
,E
00
a2Wn
I
YX2
2
) (,94 )2
Wn
+
9X4
dx+
00
+C1(jjUn1jjP2+1
+
1)(IW(4) 12 + JWn 12)
to(,E)
>
<
nx
In view of the Gronwell's
C2(f)(1
<

21luol 122
ax,
for all t E
2,3,4,...,
n
,
axl
(11.8)
the existence
t E
[01to]
(1.1.9)
2
2
c(E,
<

I
=
3, 4, 5,
[0, to] and n
2, 3, 4,
By using the
(1. 1.9) and embedding theorems, we get:
2 dt
11 W j 122).
lemma, inequality (1.1.8) immediately implies
19'Wn
a IWn
+
obtain the estimates
us now
I d
I JWn1 112(p+l)
2
+
0 such that
JjWnj 122
estimate
dx
OX4
(X)
Let
we
0"
2
192Wn
I [Wn ( WX ) 21
2
2 dt
=
embedding inequalities,
(1.1.6):
1 d
of to
Sobolev
applying
....
induction in
1, equation (1. 1. 6),
00
2
'9X 21
2
(f(Wn1)W(n1)x+Wnxxx+ EW(4) )dx
nx
<

c
I
9X1+2
122_
00
W(1+2)
nx
and the estimates
Now,
m,
n
=
let
us
(1.1.10)
al2
f( Wn1)W(nj)x]
ax 12
are
dx <
C(c, 1)
proved.
show the existence of tj
=
ti(c)
E
1, 2,3,...
00
0:5t
sup
00
X
2mW 2dx
n
< c,
(m, c)
(O,to(c)]
such that for any
EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE
CHAPTER1.
14
where cl (m,
because the
c)
is
1
case m
can
independent of
constant
positive
a
be treated
(
X
2mW2dx
c
=
n
of the
integrals
are
I
X2m W 2
=
> 2
derive
we
Pi
+
kinds:
following
00
I
Ci
xdx
n
00
Pi
m
00
00
where Pi
that
00
j
2 dt
can assume
by analogy. By integrating by parts,
00
d
We
n.
X2mkW(,)
nx
101Wn
dx,
aXI
Pi
I
ci
=
X
2m17(Wn_l )Wndx
00
00
and
00
Pi
=
ciI X2m7(Wn1)Wn.,dx
00
with
k,
I
0, 1,
=
where 0 < m'
2 and
r
0,
=
1. In view of the
(1.1.9)
and the estimates
< m,
the terms P of the first type with I
2 and
=
inequality x2?n'
r
=
In
:5
I

K
are
is
> 0
a
+ C2 (1611
7n'7 M)
7
get the following for
we
00
X2rn (Wnxx) 2dx + c(c, m)
I
X2mW2dx + C, (c, m),
n
00
00
where K
E,X2m
0:
00
,
(1.1.10),
and
<
sufficiently large
trivial. Consider also the
constant. The estimates for I
case r
I
=
=
I and k
=
0
or
k
=
=
1.
0
or
Then,
I
=
1,
r
=
0
have
we
d
00
00
Pi :
X2mW2 xdx
E
C2 + ICil
d=oo
<
X2m
C2'
d
di
[C3(c, K)W2
6
+
K
W2
xx] dx
+ C2"
=
=('OdI
00
X2m
C21
where the constant K
estimated
by analogy.
>
0 is
For
[C3(c,
6
K )W2 +
n
K
n
+
C22
The terms Pi of other kinds
arbitrarily large.
example,
W2xx] dx
for the terms Pi of the second kind
we
can
be
have
00
Pj :5 C + C
f X2m(W2_,
n
+
W2n)dx.
00
00
So,
we can
choose the constant K > 0
so
large that
the term
e
f
X
2mW 2x.,dx becomes
n
00
00
larger
than the
sum
of all terms of the kind 2z
f
K
X
2mW2x.,dx. Therefore,
n
we
get
00
00
(X,
I d
2 dt
I
X2m W 2dx
n
<

C(c, m)
1 +
00
I
00

X
2m(W2,
n
+
W
2)dx
n
(1.1.12)
THE
1.1.
(GENERALIZED)
The estimate
(1.1.11)
KORTEWEGDE VRIES EQUATION
follows from
fWn}n=1,2,3,...
C([O, ti(r)]; S). Also,
00
I d
2
the compactness of the sequence
the estimate
00
f gndx
Tt
15
(1.1.12).
Inequalities (1.1.9)(1.1.11) immediately yield
in the space
(KDVE)
f [g2
C3 (E)
<
+
n1
gn2]dx,
gn
=
Wn
Wn1,

00
is
implied by equation (1.1.6)
and the estimates
(1.1.9),(1.1.10). Therefore, the sequence fWn}n=1,2,3....
w(x, t; c) in the space Q0, t'(c)]; L2)
where t' E (0, ti] is sufficiently small. Hence, due to its compactness in C([O,
ti(c)]; S),
this sequence converges to w(x, t; c) in the space Q0, t'(c)]; S). Thus, taking the limit
in (1.1.6),(1.1.7) as n
oo, we get the local solvability of the problem (114),(1.1.5)
in the space C([O, t'(c)]; S).
converges to
a
function
+
To show the
solutions
w'(x, t;,E)
I
W
=
W
uniqueness
_
w
2,
W2(X, t; 6)
and
easily
we
of this
solution,
derive from
Q0, T]; S)
equation (1.1.4):
t;
6)]2 dx
<
C(c)
00
where
1
W
a
2
=
W
constant
C(c)
according
above class of the
Now
we
>
to the Gronwell's
depend
be
T > 0.
some
on
t; 6)] 2dx,
E R and t E
x
uniqueness
[0, T]. Therefore,
of
is proved.
estimates, uniform with respect
(1. 1.4), (1. 1. 5) of the class
solution
of
the
a
solution of the
and T > 0 there exists R2
>
of Proposition
t > 0.
Also, for
0 such that
for
to
c
E
an
Then
any C >
w(x, t; 6) E
lw(., t; 6)12 is a nonin0, p E (0, 4), R, > 0
arbitrary infinitely differentiable
condition
these constants C and p, any
and
C
T' E
solution
w(x, t; c)
(114),(1.1.5), satisfying
for
all t E
the condition
for
L 1. 6 be valid and let
function f (), satisfying
(1. 1. 3) with
C([O, T']; S) (where
(0, 1],
C([O, T]; S).
problem (11.4), (1.1.5).
creasing function of the argument
a
Setting
problem (1.1.4),(1.1.5)
problem
a
I [W(X,
and the
lemma,
Lemma 1. 1.9 Let the assumptions
C([O, T]; S)
some
00
0 does not
want to make
solutions of the
with
00
I [W(X,
dt
suppose the existence of two
us
of the class
00
d
let
Jjw(,0;c)jjj
E (0, 1]
arbitrary) of the problem
R1, one has Jjw(,t;'E)jjj < R2
(0, T]
<
E
is
[0, T].
Proof. To prove the first statement of
our
00
I d
2 dt
I
Lemma,
it suffices to observe that
2. (X,
WX.
E) dx
CIO
2
W
(X, t; c)dx
=:
I
e
t;
< 0.
00
Let
problem
us
prove the second statement. For
a
solution
w(x, t; 6)
E
Q[0, T]; S)
by applying embedding theorems and the proved
of the
statement of
CHAPTERI.
16
the
lemma,
get:
we
00
00
I d

2
Tt
f
EQUATIONS. RESULTS ON EXISTENCE
EVOLUTIONARY
W2 dx
=
f
e
X
00
00
00
d
W2
dx +
X
F(w(x, t; r))dx
Tt
00
6
I
<
00
00
00
00
<
5X rf(w)]wxxxdx
r

W
2,.,dx
X
+
CCc(l
+
d
1+1.+P3
6)
IWXXX12
+
I F(w(x,

dt
c))dx
t;
<
00
00
00
d
f F(w(x,
dt
c))dx
+
C2F
C1, C2
>
0
t;
00
because I + 1 +
3
on
P <
6
C', C"
constants
2, and where constants
from the
> 0
IUIC
<
Since due to condition
depend only
on
C,
R, and
p,
multiplicative inequalities
i
I
6
6
and
:5
C3(U
+
C4IUIPF2+2 1UXI
C'IU12 IUXXX12
(1.1.3) F(w)
JU, 12
2
+
a
i
3
3
CIIIU12 IUXXX12
!5
JUIp+2)
where p E
(0, 4),
have
we
by
theorems
embedding
00
1
F (w (x, t;
E)) dx
2
P2
IIU.12 + C5,
<
2
(1.1.14)
2
4
00
where the
following inequality
has been used:
JUIp+2
and where
C3, C4, C5 and C6
are
<
1
1+
2
p+2
positive
1
i
1 U 122
C61U12
constants
p+2
depending only
on
Rj,C
Now the second statement of Lemma 1.1.9 follows from the first statement,
and
(1.113)
(1. 1. 14). n
Lemma I.1.10 Let C >
R2
and p.
=
R2 (C7 p, R1, T)
arbitrary
twice
>
0,
0 be the
p E
(0,4), R,
>
corresponding
0 and T > 0 be
constant
continuously differentiable function f ()
sup
from
we
I u 1,, F2 (C, p, f, R1, T)
=
u(=HI: jjujjj:5R2
arbitrary
Lemma L 1. 9.
and let
For
an
set:
sup
I f'(u) I
1U1
and
F3 (C, p, f, R1, T)
=
sup
If" (u) I
lul
(here W
large R3
<
>
oo
0.
in view
Then,
of the embedding of H1
into
C).
there exists R4 > 0 such that
Take
for
any
an
6
arbitrary sufficiently
E
(0, 1],
an
arbitrary
1.1.
KORTEWEGDE VRIES
(GENERALIZED)
THE
EQUATION (KDVE)
infinitely differentiable function f(.), satisfying (1.1.3) with
f, R1, T) :! , R3
and p and such that F2 (C, p,
an
solution
arbitrary
w(x,t;c)
and F3 (C, p,
C([O,T];S) of
E
the above constants C
f, RI, T)
<
R3,
and
for
problem (1.1.4),(1.1.5) (T'
the
(0,T]), obeying the conditions Ijw(,0;c)IIj :5 R,
I I W (') t 6)112 :5 R4 for all t E [0, T'].
17
IIW(',O;'E)112
and
:5 R3,
one
E
has
1
Proof. Take
c
E
and let
(0, 1]
constants R1, sufficiently large R3 C > 0, p E (0, 4), some
infinitely differentiable function f (.) satisfy condition (1. 1. 3) with
arbitrary
an
7
these constants C and p, F2 (C, p,
Lemma 1.1.9 and
inequality (1.1.15),
2
Tt
dx
=
dx

x
00
CIO
00
a2
I
WXX
1 (W(4))2
(9X2
If (w)w.,]dx
00
00
dx
00
I f1l (w)wxwxxdx , I f'(w)wxwx.,dx
2
3


x
00
R3. Using
00
I (U(4))2
c
9X2
<
get
we
2
f (a2W )
6
R3 and F3 (C, p, f, R1, T)
<
00
00
d

f, R1, T)

00
00
00
6
I (W(4))2
(1.1.16)
dx + I, (w) + 12 (w).
x
00
Let
estimate the terms
us
inequality (1.1.15)
and Lemma
I, (w) :5 F31W
1.1.9,
13 IWxxI2
<
6
where the constant C,
and
we
I2(w) separately.
11(w), applying
For
get
CjF31WX122
depends only
0
>
Il(w)
IWXX12
2
<

CIR 2F
31W
2
=
12,
the constant from the
on
(1.1.17)
2
embedding
in
equality (1. 1. 15).
Let
estimate
us
I2(w).
We have
00
5 d
12 (W)
6dt
CX)
j f (w)wx.,dx
5
+
6
00
1 (f (w) f, (w)
w3 +
x
f"(W) W3 w.,.,)dx+
x
00
00
5
+
c
6
I [2f(w)w
2
x xx
+
fll(w)w'wxxxdx
x
+
4f(w)wxw.,xwxxx]dx.
(1.1.18)
0.
The second term in the
as
from
II(w)
(1.1.16),
righthand
so
that
we
side of this
equality
can
be estimated
completely
have
00
5
6
1 jf(W)f,(W)
3
W
00
where F,
=
sup
IUI
If (u) 1.
x
+
fll(W)W3W. } dx
x
<
C2 (F, F2 + F3) (I wxx 122 + 1)
(1.1.19)
CHAPTERL
18
Due to
EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE
the term from the
embedding theorems,
c
the coefficient
can
6
be estimated
righthand
side of
(1. 1.18)
with
as
00
I (W(4))2
dx +
x
C3 (Fl, F2, F3, R2)
(1.1.20)

co
00
for
Finally,
43(w)
f 7(w)w.,xdx
6
have
we
00
I3(W)
<
C(I
+
where the constant
:5 4 C2 R22(1 + C4R 2)2 +
C4jjWjjP1)jWj2 lWxxl2
C4
0
>
depends only
Lemma 1. 1. 11 Under the
constants from
on
In view of Lemma 1.1.9 and the estimates
(1.1.16)(1.1.21),
1
4
1 WX. 12,
2
(1.1.21)
embedding inequalities.
Lemma 1.1.10 is
assumptions of Theorem L 1. 3 for
integer
any
proved.E]
I > 2 and
T > 0 there exists
c(l, T) > 0 such that for any c E (0, 1] and an arbitrary solution
E
w(x, t; 6)
C([O, T']; S) (here T' E (0, T] is arbitrary) of the problem (1. 1.4), (1. 1.5)
one has
:5 c(l, T) for all t E [0, T'].
9xj
121W 12
Proof. We
proved
case
I
the induction in 1. For I
use
2 the statement of Lemma is
with Lemma 1. 1. 10. Let this statement be valid for I
=
r
+ 1.
Using
00
d
I
=
(Or+lw
2 dt
axr+1
the
integration by parts
dx
c
=
0.0
I(
W
oXr+3
)
dx
I

00
gXr+l
Cl(JJW112)
Lemma 1.1.12 Let the
integer
and
a
m
> 0 be
solution
(1. 1.4), (1. 1. 5)
the
C2(11W112)
+
E
+_1
_5XIWI
dx <
axr+l
2
2
assumptions of Theorem L1.3 be valid and let T
arbitrary. Then,there
w(x, t; 6)
If (W)WX]
00
ar+IW
<
get
we
00
2
ar+
C([O, T'j; S),
following
already
2,..., r. Consider the
embedding theorems,
00
2
)
and
=
exists
c(m)
where T' E
estimate takes
> 0
(0, T]
such that
is
for
any
arbitrary, of
f
the
>
0 and
(0, 1]
E
problem
place:
00
f
x
2mW2 dx
c(m),
<
[0, T'j.
t E
00
Proof. First of
and
integer
r
> 0
all,
we
shall show that for any
such that for
u
E
S
we
00
IX12m1
00
dnU
dXn
=:
1, 2, 3,
...
there exist C
>
0
(X)
2
)
m
have
dx < C
1
JJU112
r
+
IIUI12 +
2
f
00
X2m u2(x) dx
1
(1.1.22)
where I
0
=
1
or
I
=
or
EQUATION (KDVE)
KORTEWEGDE VRIES
(GENERALIZED)
THE
1.1.
1
2 and
x
(0, 1),
n
1
=
or n
2. For this
=
aim,
we use
19
the obvious
estimate
Jkl
Jk+xj
I
<

2
<

2,
k
2,3,4,...
k
or
(1.1.23)
1, 2,3,...
multiplicative inequality
and the
a+1
2
dnu
WX_;)
dx
C(r)ju IL2
1 a,a+l)
<
(I
U
X(r) IL2(a,a+l)
(1.1.24)
1U1L2(a,.+1)
+
a
where
a
Due to
=
0, 1,::L2,... is arbitrary,
(1.1.24),
and
(1.1.23)
n
I
=
n
or
get for integer
we
I 1XI 2m1(U(n))2
dx
=
I 1XI 2m1(U(n))2
E +E ) I jXj2m1(U(n))2
(k=00
dx +
2
+
(JUjL2(k,k+1)
X
1
1n
U
2dX
X
k
r
.5 C111U112 + C"(r)2
2m1
X
2
1n
r
1
+
k=00
f
2m1
jkj
k=1
k+1
( 1:
E
I U (r) IL2(k,k+l)) 1
+
2
X
+

k
k+1
1:
(k=oo
C"(r)
dx <
X
k=1
C11 JU 112
arbitrary integer.
C o
2
<
is
k+1
2
X
X
> 2
r
2mnl':
>
r
1
00
2 and
=
2
dx
jjUjj2
+
2M
x
u
(JUjL2(k,k+1)
+
I U X(r) 1L2(k,k+1))!!
<
k
00
C (r)
r
IJUI12
2
1
+
2m
X
U2dx
00
where
k
=
we
have used the trivial
1, 2,3,... and
x
Consider the
G
(k, k
+
inequality Ik 12m1
1),
2 dt
I
(1.1.22)
I
2
wdx=
00
2m
x
wf (w)wxdx
+ 2m
I
2m1
X
W2dx
X
c
third terms in the
the H61der's
righthand
wwxxdx
I
X2m WW (4)dx.
X
00
00
1.1.94.1.11,
2m1
X
00
00
_M
I
00
00
00
Due to Lemmas
and
2, 3,
follows.
00
2m
x
=
expression
00
1 d
and
22m1X2M1 for k
<
side of this
inequality
equality
and
can
00
C1 + C2
I
00
2m
X
W2dX
(1.1.22),
be
the
obviously
first,
second and
estimated
as
20
CHAPTER1.
with
some
EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE
C1, C2
constants
integration by parts. So,
0. The last term
>
can
'X,
1 d
Tt
2
be estimated
by analogy after
an
to the estimate
we come
00
I
2mW2dX
X
<

f
C3 + C4
X2m W2 dx.
_C0
the statement of Lemma 1. 1.12 is
Thus,
Lemmas I.1.1 and 1.1.91.1.12
of the
ability
T*
whose
Ssolution
and cannot be continued
[0,T*)
point
t
T*.
=
lim
t
corresponding
uniqueness has already been proved,
of time
the
immediately imply
problem (1.1.4),(1.1.5). Indeed,
0 such that the
>
proved. El
w(x, t; E)
T*O
w(x, t; c)
the
Cauchy problem
for
6
0, i.
contradiction.
for all t > 0, solv
Suppose
of the
the existence of
problem (1. 1.4),(1.1.5),
be continued onto the halfinterval
can
on an
S understood in the
global,
let uo E S.
arbitrary right halfneighborhood
due to the aboveindicated
Then,
ul E
=
the
sense
results, there
of the space S.
exists
of
limit
a
Thus, considering
equation (1. 1.4) with the initial data w(x, T*; E)
ul (x), we
the
local
of
this
interval
of
get
time [T*, T* + 6) with some
solvability
problem on an
>
e.
Let
get
we
us now
the existence of
obtained
a
prove
u(x, t)
the limit
by taking
So, Proposition
Proposition
solution
a
=
C([O, T); S)
as
for the
belonging
of the
C([O, T); S) for
problem (1.1.4),(1.1.5). The
Now,
Thus, Proposition
turn to
existence and
1.1.7 is
proving Theorem
can
can
be
proved,
be
> 0
can
(1. 1.4),(1.1.5).
any T > 0
to S for any fixed t in the domain t < 0
above construction.
problem
+0 in the problem
of this solution of the class
way
proved.0
1.1.7. Due to Lemmas 1.1.91.1.12 for any T
E
as c +
1. 1.6 is
The
proved
uniqueness
uniqueness
in the
of
be
a
same
solution
proved by analogy with the
too.E1
1.1.3. Let
us take an arbitrary twice continuously differentiable function f () satisfying the estimate (11.3) and let If,,(')jn=1,2,3....
be a sequence of infinitely differentiable functions
satisfying the estimate (1.1.3) with
the
same
for any
we
constants C and p and
I, m
f Un0 1 n=1,2,3....
1,2,3,....
C S be
For each
n + oo.
of the
=
n
a
=
Let
us
sequence
1, 2, 3,
problem
...
converging
also take
C2((_M, M)
x
( 1, 1); R)
2
and T
uo E H
>
0 and let
f (.)
arbitrary
in
converging to uo weakly in H 2 and strongly in H' as
by Un (X, t) E C (( T, T); S) we denote the solution
taken with
f
=
fn and
uo
=
Un.
0
It is clear that the
JR2(CIP) JjUnjjj,T)jn=1,2,3 where the function R2 > 0 is given by Lemma
bounded and let R2
sup R2 (C7 Pi Un 1, T) > 0 Let also R3
Sup I I Un
0 1 12
sequence
....
1. 1. 9, is
to
,
=

n
Then, clearly W3
E
(0, oo).
We set R4
n
=
R4(R3)
where the function R4
R4(R3)
> 0
1.1.
is
Lemma I.1.10.
given by
For t E
[T,O)
x 
and t
R2
!5
these estimates
and
JjUn(') t)112
can
be obtained
W4)
5
we
21
1.1.10,
(T,T).
t E
(1.1.25)
by the simple change
equation (I.1.1). Therefore,
t in
+
EQUATION (KDVE)
due to Lemmas 1.1.9 and
Then,
jju,,(,t)jjj
x
KORTEWEGDE VRIES
(GENERALIZED)
THE
have for t
>
of variables
0
00
0"
I d
(Un
2 dt
Um) 2dx

=
(Un

Um)(fn(Un)Unx


fm(um)um.,)dx
00
00
I J(Un

Um)[f(Un)(Unx
Umx)

Umx(f(Un)
+

f(Um))+
0.0
+(fn(Un)
f(Un))Unx
+
Umx(f(Um)

fm(um)jjdx
00
f (Un
C(T)
um)2dx

+ an,m)
00
where an,m
>
+0
as
n,
m

+oo and
by analogy
convergence of the sequence
fUn}n=1,2,3....
(1.1.25),
Due to the estimates
u(, t)
Indeed, let
us
take
E H
2
and
arbitrary
an
t E
for t
in the space
0. These estimates
<
C([T, TI; L2)
to
JjU(',t)jj2
[T, T].
(1. 1.25),
Due to
yield
some
the
u(x, t).
(1.1.26)
the sequence
2
fUn(*) t)}n=1,2,3.... is weakly compact in H hence it contains a weakly converging
subsequence (without the loss of the generality we accept that it is the sequence
JUn(') t)}n=1,2,3_.). Therefore,
,
u(, t)
E H
2
I JU('7 t) 112:5
and
liM illf
I JUn('7 t) 112
<
W4
noo
and the
properties (1.1.26)
The
for
following
proved.
are
statement
can
be
proved by analogy.
Lemma 1.1.13 For any T
>
Lemma 1. 1. 14
strongly
any t E
[T, T]
If Un0
uo
as n
oo.
0
Un(',t)
+
in H 2
u(.,t)
in
as n + oo
as n +
oo, then
I JU(*) t) Un(* t) 112

i
Proof. Due to Lemma 1. 1. 13 and the above arguments un (,
oo
weakly
in H 2 for any t E R.
Further,
we
have from
C((T,T);H').
t)
u(, t) as
(1. 1.16),(1.1.18) with c
+
=
1
2
JUnz,&i t) 122
2
3
1 Unxx ('1 0) 122
6
0
00
fIf( Un( 3))Un
n
I
9)Unxx( 9) +
i
+
n
0
0