Tải bản đầy đủ - 0 (trang)
1 The (generalized) Korteweg-de Vries pquation (KdVE)

1 The (generalized) Korteweg-de Vries pquation (KdVE)

Tải bản đầy đủ - 0trang

THE

1.1.

(GENERALIZED)

is

KORTEWEG-DE 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

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 ). Hne-3 (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 Hn-solution) 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

well-posedness

considered in this book is the

Theorem 1.1.5 Let

any

integer

into

> 2

n

and

f (u)

uo E

sense

of the

global

for

a

periodic Hn-solution

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

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(n-1)]2

X

qn(U7

...

(n-2))

)U X

dx,

n

=:

2,3,4,...,

0

where Cn

periodic

are

constants and qn

Hn-solution

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

Hn-solutions.

an

any uo

c --4

is,

problem (1. 1-4),

(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(21-k) u 2

(x)dx

+

d2m-kU)

(dX2m-k

f

dx. 0

dx

<

THE

1.1.

Let

(GENERALIZED)

take

us

(1.1.4),(1.1.5) by

arbitrary

an

the iteration

Wnt + Wnxxx +

KORTEWEG-DE 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(Wn-1)W(n-1)xi

==

nx

(KDVE)

C S.

=

t >

0,

n

(1.1-6)

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

19Wn-1

f(Wn-1) -51- dx

<

-,E

-00

a2Wn

I

YX2

2

) (,94 )2

Wn

+

9X4

dx+

-00

+C1(jjUn-1jjP2+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

(1-1.8)

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 JWn-1 112(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(Wn-1)W(n-1)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)

al-2

f( Wn-1)W(n-j)x]

ax 1-2

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.

X2m-kW(,)

nx

101Wn

dx,

aXI

Pi

I

ci

=

X

2m-17(Wn_l )Wndx

-00

00

and

00

Pi

=

ciI X2m7(Wn-1)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

d-i

[C3(c, K)W2

6

+

K

W2

xx] dx

+ -C2"

=

=-('Od-I

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)

KORTEWEG-DE 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)

<

+

n-1

gn2]dx,

gn

=

Wn

Wn-1,

-

-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 (1-1-4),(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)

(1-14),(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 (1-1.4), (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

(1.1-13)

(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.

KORTEWEG-DE 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),

right-hand

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

right-hand

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

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

IX12m-1

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)

KORTEWEG-DE 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 2m-1(U(n))2

dx

=

I 1XI 2m-1(U(n))2

E +E ) I jXj2m-1(U(n))2

(k=-00

dx +

2

+

(JUjL2(k,k+1)

X

-1

1-n

U

2dX

X

k

r

.5 C111U112 + C"(r)2

2m-1

X

2

1-n

r

1

+

k=-00

f

2m-1

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 12m-1

1),

2 dt

I

(1.1.22)

I

2

wdx=-

00

2m

x

wf (w)wxdx

+ 2m

I

2m-1

X

W2dx

X

-c

third terms in the

the H61der's

right-hand

wwxxdx-

I

X2m WW (4)dx.

X

-00

00

1.1.94.1.11,

2m-1

X

00

00

_M

I

-00

-00

00

Due to Lemmas

and

-2, -3,

follows.

00

2m

x

=

expression

00

1 d

and

22m-1X2M-1 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 1.1.9-1.1.12

of the

ability

T*

whose

S-solution

and cannot be continued

[0,T*)

point

t

T*.

=

lim

t

corresponding

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.

for all t > 0, solv-

Suppose

of the

the existence of

problem (1. 1.4),(1.1.5),

be continued onto the half-interval

can

on an

S understood in the

global,

let uo E S.

arbitrary right half-neighborhood

due to the above-indicated

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 1.1.9-1.1.12 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 (1-1.3) 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

KORTEWEG-DE 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

n-oo

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

Tài liệu bạn tìm kiếm đã sẵn sàng tải về

1 The (generalized) Korteweg-de Vries pquation (KdVE)

Tải bản đầy đủ ngay(0 tr)

×