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

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

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THE



1.1.



(GENERALIZED)



addition, if u(., t)



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



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 ). 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



about



> 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



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(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



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



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



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 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



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