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International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN (P): 2249-6955; ISSN (E): 2249-8060 Vol. 8, Issue 4, Aug 2018, 1-6 © TJPRC Pvt. Ltd.

ILL–POSEDNESS FOR THE CAUCHY PROBLEM FOR A FAMILY OF TWO–DIMENSIONAL BENJAMIN–ONO EQUATION JULIO LIZARAZO Department of Mathematics, Universidad Nacional de Colombia, Bogota D.C., Colombia ABSTRACT In this paper going to be showing for all + + + ( , , ) = ( , ), where

= ,

∈ ℝ, when

( , )∈ℝ ,

is the Hilbert transform in the first variable, and

< 2, the ill-posedness for the Cauchy problem 1.

≥ , ≤

(1) ≤ .

KEYWORDS: Phrases & Math Subject Classification: Ill-posedness, Bidimensional, Benjamin-Ono Equation, Picard Scheme, 35A01, 35A02, 35A22, 35A23 & 35B30

1. INTRODUCTION Is known that The Picard iterative scheme cannot be used to prove the local well-posedness (LWP) for the initial value problem (IVP) with the Benjamin-Ono equation 2. +ℋ + (0, ) = " .

= 0,

,! ∈ ℝ

Original Article

Received: May 27, 2018; Accepted: Jun 17, 2018; Published: Jul 14, 2018; Paper Id.: IJMCARAUG20181

(2)

Also, when the equation has a general nonlinear part, in [8] using the Picard scheme the authors proved the LWP for the IVP 3 +ℋ + (0, ) = " .

$

= 0,

,! ∈ ℝ

(3)

Using the ideas exposed in [8] Kenig-Martel and Robiano in [7] proved the LWP for + %& + (0, ) = " .

$

= 0, , ! ∈ ℝ with1 ≤ , ≤ 2

(4)

It is very interesting that cannot be used the same argument for the bidimensional extension 1, because in

this paper gonna be show a bounded sequence of functions in -. over cannot be applied the Picard Scheme.

In [2] shows LWP using energy estimates for 1 when 0 ≤ , ≤ 1 and prove the ill–posedness for all / ∈

ℝ. For LWP of 1 the reader can see [3] and [11]

2. ILLPOSEDNESS Using the ideas in [10], and [2] here prove that the IVP 1 cannot be solved by a Picard iterative scheme based on the Duhamel’s formula. In other words, if

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Julio Lizarazo

(!) = 0& (!)1 − 3" 0& (! − !′)5 (!′)

(!′)67!′

(5)

That is the integral equation corresponding to the IVP 1 with initial datum 1, in wich (0& (!)1)∧ (9, :): = < =>

?|A|B ACDEF(A)G H I

1J(9, :),

(6)

the following assertion holds. Theorem 1 Let , ∈ 51,2), / ∈ ℝ and K > 0. Then, there does not exist a space MN continuously embedded in

O(50, K6; - D (ℝQ )) such that there exist O > 0 with R|0& (!)1|RS ≤ OR|1|RU V(ℝH )

(7)

T

and WX3" 0& (! − !′)( (!′)

(!′) 7!′)XW

ST

Q

≤ OR| |RS ,

∈ MN

T

(8)

Let us observe that the conditions 7 and 8 are necessary to apply the Picard iterative method scheme in the integral equation 5 Proof. Reasoning as in [10] can be supposed there exist a space MN such that 7 and 8 holds. Let us define in 8

: = 0& (!)1. Then

WX3" 0& (! − ! Y )5(0& (! Y )1)(0& (! Y )1 )6 7! Y XW

ST

Q

≤ OR|0& (!)1|RS

T

In this way, using 7 and the fact that MN °O(50, K6; - D (ℝQ )), can be concluded for each ! ∈ 50, K6 WX3" 0& (! − !′)5(0& (!′)1)(0& (!′)1 )6 7!′XW

U V (ℝH )

Q

≾ R|0& (!)1|RU V(ℝH)

(9)

Will show 9 is not true, exhibiting a bounded sequence {1] }]∈ℕ ∈ - D (ℝQ ), such that for ! ∈ 5K/2, K6, lim WX3" 0& (! − ! Y )5(0& (! Y )1] )(0& (!′)(1] ) )6 7! Y XW

]→.

U V ?ℝH I

=∞

Let us take 1 ≡ 1] defined through the Fourier transform as follows ghi

ghi

1J(9, :) = f = H jkg (9, :) + f = H l =D jkH (9, :), l ≫ 1, 0 < f ≪ 1, o > 0, o = o(,)

Where pq = 5f/2, f6 × 50, f s 6

and pQ = 5l, l + f6 × 50, f s 6, f to be precise later, and jt denotes the

characteristic function of the set u. Let us note that Q

R|1|RU V(ℝH )

= 3ℝH (1 + 9 Q + : Q )D R1J(9, :)R 79: Q

= 3k (1 + 9 Q + : Q ) D f =(qCs) 79: + g

+ 3k (1 + 9 Q + : Q )D f =(qCs) l =QD 79:~ H

w ~ 3w/Q

Impact Factor (JCC): 6.2037

wi

3"

]Cw

f =(qCs) 79: + 3]

wi

3"

(10)

l QD f =(qCs) l =QD 79:~O

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Ill–Posedness for the Cauchy Problem for a Family of Two–Dimensional Benjamin–Ono Equation

3

Moreover, is defined x(9, :) = −(|9|& 9 + /yz(9): Q ), {(9, :, 9q , :q ) = x(9q , :q ) + x(9 − 9q , : − :q ) − x(9, :),

and |(9, :, 9q , :q ) = 1J(9, :)1J(9 − 9q , : − :q ). Then it can be easily seen that 3" 0& (! − ! Y )5(0& (! Y )1)(0& (! Y )1 )6 7! Y =

= O 3ℝH < >(

<

AC}G) > ~(A,G)

9 •3ℝQ |(9, :, 9q , :Q )

€ •‚ƒ(„,…,„g ,…g ) =q †(A,G,Ag ,Gg )

79q :q ‡ 79:

Considering the following four sets p>ˆ (9, :)

= ‰(9q , :q ) ∈ ℝQ | (9q , :q ) ∈ p> , (9 − 9q , : − :q ) ∈ pˆ Š, ‹, Œ ∈ {1,2}

(11)

To obtain |(9, :, 9q , :q ) ≠ 0, it is required (9q , :q ) ∈ p>ˆ (9, :) for someone of this four sets. Using the notation

{ ≡ {(9, :, 9q , :q ), holds that 3" 0& (! − ! Y )

5(0& (! Y )1)(0& (! Y )1 )6 7! Y =

= O 3ℝH < >( + 3t

HH (A,G)

+ 3t

<

AC}G) > ~(A,G)

€ •‚ƒ=q

f =(qCs) l =QD

gH ∪tHg (A,G)

9 Ž3t

gg (A,G)

€ •‚ƒ =q

79q :q +

f =(qCs) l =D

f =(qCs)

€ •‚ƒ =q †

79q :q +

79q :q • 79:

Define ‘q ( , ’, !)

‘Q ( , ’, !) ‘— ( , ’, !)

: = 3ℝH

: = 3ℝH : = 3ℝH

“A€ •(”„h•…) € •‚–(„,…) w (ghi)

“A€ •(”„h•…) € •‚–(„,…) w (ghi) ]HV

“A€ •(”„h•…) € •‚–(„,…) w (ghi) ]V

Ž3t

gg (A,G)

Ž3t

HH (A,G)

€ •‚ƒ=q †

€ •‚ƒ=q

Ž3t

gH ∪tHg (A,G)

79q :q • 79: 79q :q • 79:

€ •‚ƒ=q †

79q :q • 79:

(12)

˜3" 0& (! − ! Y )5(0& (! Y )1)(0& (! Y )1 )6 7! Y ™ ( , ’) = = ‘q ( , ’, !) + ‘Q ( , ’, !) + ‘— ( , ’, !)

and will can see that / šš?‘›q I ⊂ 5f, 2f6 × •0,2f s ž / šš?‘›Q I ⊂ 52l, 2l + 2f6 × •0,2f s ž

w / šš?‘›— I ⊂ Žl + , l + 2f• × •0,2f s ž Q

The supports are mutually disjoint. We have that WX3" 0& (! − !′)5(0& (!′)1)(0& (!′)1 )6 7!′XW

U V (ℝH )

≥ R|‘> (⋅,⋅, !)|RU V(ℝH )

(13)

Making the change of variables 9Q : = 9 − 9q and :Q : = : − :q and taking into account that {(9, :, 9Q , :Q ) =

{(9, :, 9Q , :Q ), it is easy to see that 3t

Hg (A,G)

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€ •‚ƒ(„,…,„g ,…g ) =q †(A,G,Ag ,Gg )

79q :q = 3t

gH (A,G)

€ •‚ƒ(„,…,„H ,…H ) =q †(A,G,AH ,GH )

79Q :Q

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Julio Lizarazo

Therefore, ‘— ( , ’, !) = =

Q“ 3 w (ghi) ]V ℝH

9< >(

<

AC}G) > ~(A,G)

Ž3t

gH (A,G)

€ •‚ƒ =q †

79q :q • 79:

and, Q

R|‘— (⋅,⋅, !)|RU V?ℝHI = = 3D

¤ ¡¡?¢ £I

(1 + 9 Q + : Q ) D

“A H

w H(ghi) ]HV

X3t

gH (A,G)

€ •‚ƒ =q †

Q

79q :q X 79:

taking account the support of ‘›— , i.e. (9, :) ∈ Žl + , l + 2f• × •0,2f s ž w Q

Q

R|‘— (⋅,⋅, !)|RU V?ℝHI ~

“]HV ]H

3

¤ w H(ghi) ]HV D ¡¡?¢ £I

€ •‚ƒ=q

X3t

gH (A,G)

Q

79q :q X 79:

take f such a way that fl & = l =¥ , with 0 < ¦ ≪ 1, it follows that

{(9, :, 9q , :q )

= x(9q , :q ) + x(9 − 9q , : − :q ) − x(9, :) = −9q&Cq − (9 − 9q )&Cq + 9 &Cq + 2:q (: − :q ) = −9q&Cq − 59 &Cq − (, + 1)9 & 9q + +§(9q )6 + 9 &Cq + 2:q (: − :q )

=

=

¨where lim

~fl & ,

Taking into account that R|‘— (⋅,⋅, !)|RU V?ℝHI

«(Ag )

= 0¬

Ag →" Ag &Cq & (, + 1)9 9q − 9q − §(9q ) + 2:q (: − :q ) «(A ) 9q Ž(, + 1)9 & − 9q& − g • + 2:q (: − :q ) A g

q=-®¯(°)

∥ O! Q

°

]H

~± for 0 < ± ≪ 1, then if 5K/2, K6, follows: 3

¤ w H(ghi) D ¡¡?¢ £I

“ H ]H

3

¤ w H(ghi) D ¡¡?¢ £I H H “N ]

3 ¤ w H(ghi) D ¡¡?¢ £I “N H ]H

X3t

X³< 3t

gH (A,G)

f l Q

gH (A,G)

Q&

— f Q l Q& ?fqCs I w H(ghi) QCQ& (l =&=¥ )(—Cs)

=l

€ •‚ƒ =q

q=-®¯( †)

X3t

gH (A,G)

Q

79q :q X 79: Q

79q :q X 79: Q

79q :q X 79:.

= l QCQ& f QC(qCs) =

= l Q=(qCs)&=(—Cs)¥ → ∞, if , <

As we can see, if o → 0, then , would be near to 2

Q

qCs

Theorem 2 Fix / ∈ ℝ. Then there does not exist a K > 0, such that 1 admits a unique local solution defined on

50, K6 and such that the flow map data–solution 1 → (!), ! ∈ 50, K6

for 1 is O Q differentiable at zero from - D (ℝQ ) to - D (ℝQ ) Proof. Consider the Cauchy problem

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Ill–Posedness for the Cauchy Problem for a Family of Two–Dimensional Benjamin–Ono Equation

+ %& + ℋ }} + ( , ’, 0) = ±1( , ’), Suppose that

°(

5

= 0, ± ≪ 1, 1 ∈ - D (ℝQ )

(14)

, ’, !) is a local solution of 14 and that the flow map is O Q at the origin from - D (ℝQ ) to - D (ℝQ ).

Then µ ¶ X µ°H °·"

= −2 3" 0& (! − ! Y )5(0& (! Y )1)(0& (! Y )1 )6 7! Y

The assumption of O Q regularity yields WX3" 0& (! − !′)5(0& (!′)1)(0& (!′)1 )6 7!′XW

U V (ℝH )

Q

≾ R|0& (!)1|RU V(ℝH)

(15)

But the above estimate is 9, which has been shown to fail.

3. CONCLUSIONS •

Can’t be posible to use the Picard Scheme to prove the local well posedness for the initial value problem (1)

The map datum-solution is not at the origin, this result is known as ill-posedness

REFERENCES 1.

Jaime Angulo, Márcia Scialom, and Carlos Banquet. The regularized Benjamin-Ono and BBM equations: well-posedness and nonlinear stability. J. Differential Equations, 250(11):4011–4036, 2011.

2.

Jiménez J. Mejía J. Bustamante, E. The cauchy problem for a family of two-dimensional fractional benjamin-ono. Preprint, 2017. Arxiv, math. AP, 1710.08380v2.

3.

Omar Duque. Sobre una versión bidimensional de la ecuación Benjamin-Ono generalizada. PhD thesis, Universidad Nacional de Colombia, 2014.

4.

Amin Esfahani and Ademir Pastor. Ill-posedness results for the (generalized) Benjamin-Ono-Zakharov-Kuznetsov equation. Proc. Amer. Math. Soc., 139(3):943–956, 2011.

5.

El-Giar, O. S. A. M. A., And A. M. El-Feky. "Two Dimensional Full Multigrid Method For Solving Stokes Equation In A NonStaggered Grid Using Fas Scheme."]

6.

Germán Fonseca and Gustavo Ponce. The IVP for the Benjamin-Ono equation in weighted Sobolev spaces. J. Funct. Anal., 260(2):436–459, 2011.

7.

Germán Fonseca, Guilermo Rodríguez-Blanco, and Wilson Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Preprint, 2013. Arxiv, math.AP, 1304.6454.

8.

Jaipal, Rakesh C., And Vn Kala. "A Numerical And Analytical Approach For Solving Nonlinear Water Hammer Equations."

9.

2 C. E. Kenig, Y. Martel, and L. Robbiano. Local well-posedness and blow-up in the energy space for a class of L critical dispersion generalized Benjamin-Ono equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 28(6):853–887, 2011.

10. Khambra, Krishna, and Nirmal Yadav. "Effect of Tightness factors on dimensional properties of knitted fabric." Man-Made Textiles in India 52.2 (2009). 11. Carlos E. Kenig, Gustavo Ponce, and Luis Vega. On the generalized Benjamin-Ono equation. Trans. Amer. Math. Soc., 342(1):155–172, 1994. www.tjprc.org

editor@tjprc.org


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Julio Lizarazo 12. Carlos E. Kenig, Gustavo Ponce, and Luis Vega. On the ill-posedness of some canonical dispersive equations. Duke Math. J., 106(3):617–633, 2001. 13. L. Molinet, J. C. Saut, and N. Tzvetkov. Ill-posedness issues for the Benjamin-Ono and related equations. SIAM J. Math. Anal., 33(4):982–988 (electronic), 2001. 14. Mohammed, Zerf, and Bengoua Ali. "Effect Dimensional of Delimiters on Implement Ation of Speed, Balance and the Agility in Dribbling among Soccer (Under 15 Year)." International Journal of Educational Science and Research (IJESR) (2015): 6772. 15. Fabián Sánchez Salazar. El problema de Cauchy asociado a una ecuación del tipo rBO-ZK. PhD thesis, Universidad Nacional de Colombia, Junio 2015.

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ILL–POSEDNESS FOR THE CAUCHY PROBLEM FOR A FAMILY OF TWO–DIMENSIONAL BENJAMIN–ONO EQUATION  

In this paper going to be showing for all s∈R, when 1≤α<2, the ill-posedness for the Cauchy problem 1. {■(&u_t+D_x^α u_x+H_x u_yy+uu_x=0,...

ILL–POSEDNESS FOR THE CAUCHY PROBLEM FOR A FAMILY OF TWO–DIMENSIONAL BENJAMIN–ONO EQUATION  

In this paper going to be showing for all s∈R, when 1≤α<2, the ill-posedness for the Cauchy problem 1. {■(&u_t+D_x^α u_x+H_x u_yy+uu_x=0,...

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