On the Cauchy problem for the generalized shallow water wave equation

J. Differential Equations 245 (2008) 1838–1852 www.elsevier.com/locate/jde

On the Cauchy problem for the generalized shallow water wave equation Lixin Tian a,∗ , Guilong Gui a , Yue Liu b a Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China b Department of Mathematics, University of Texas, Arlington, TX 76019, USA

Received 22 April 2007; revised 13 June 2008 Available online 25 July 2008

Abstract In this paper we mainly study the Cauchy problem for the generalized shallow water wave equation in the Sobolev space H s of lower order s. Using the crucial bilinear estimates in the Fourier transform restriction spaces related to the shallow water wave equation, we establish local well-posedness in H s with any s  − 11 16 . © 2008 Elsevier Inc. All rights reserved. MSC: 35Q53; 35A07 Keywords: Cauchy problem; Local well-posedness; Generalized shallow water wave equation

1. Introduction A number of basic equations in the study of nonlinear waves takes the form [1]   ut + f (u) x + Lu = 0,

(1.1)

where f (u) is a function of u and L is a linear operator with constant coefficients. When f (u) = 3 2 ∂3 2 u , L = ∂x 3 , (1.2) becomes the Korteweg–de Vries equation. In [2], J.L. Bona and R.S. Smith considered the following fifth-order shallow water equation   ut + αuxxxxx + βuxxx + γ ux + μ∂x u2 = 0, (1.2) * Corresponding author.

E-mail addresses: [email protected] (L. Tian), [email protected] (G. Gui), [email protected] (Y. Liu). 0022-0396/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2008.07.006

L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852 5

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3

∂ ∂ ∂ which is given as f (u) = μu2 , L = α ∂x 5 + β ∂x 3 + γ ∂x in (1.1), where α = 0, β and γ are real constants and μ is complex constant. The model described in (1.2) arises in the study of water waves with surface tension in which the Bond number takes on the critical value, where the Bond number represents a dimensionless magnitude of surface tension in the shallow water regime. In [17], by using the Fourier restriction norm method [3,4,7], the local well-posedness of the Cauchy problem for the above equation is established for low regularity data in Sobolev spaces H s (s  − 38 ). In [5,6,13], the study of the third-order K(m, n) equations was generalized by including into the equation higher-order nonlinear dispersive terms, like for example, the fifth-order K(m, n, p) equations of the form

      ut + β1 um x + β2 un xxx + β3 up xxxxx = 0, where m, n > 1, p  1, and β1 , β2 , β3 are constants. In particular, we observe K(2, 2, 1) equation     ut + u2 x ± u2 xxx + (u)xxxxx = 0. When we choose the sign “−” before u2 , the above equation becomes    ut + uxxxxx + ∂x 1 − ∂x2 u2 = 0, where the term ∂x (1 − ∂x2 )(u2 ) gives the dispersive and convection effects. Seeking to understand the role of nonlinear dispersive and nonlinear convection effects in K(2, 2, 1), we introduce and study the Cauchy problem for the new fifth-order generalized shallow water equation: 

 1   ut + uxxxxx + ∂x 1 − ∂x2 2 u2 = 0, u(0, x) = u0 (x),

Let ξ = x − ct, and c =

1 16 ,

x ∈ R, t > 0.

(1.3)

then we have the travelling wave solutions of (1.3) as follows 1

1

1

1

u(t, x) = c1 e 2 (x− 16 t) + c2 e− 2 (x− 16 t) + c3 , where c1 , c2 , c3 are constants. In fact, we can use the method of the Fourier restriction norm to consider the well-posedness of Cauchy problem for (1.3), since the equation has a stronger smoothing effect stemmed from the fifth-order space derivative. This effect enables us to handle the nonlinear terms. Eq. (1.3) can be written as the integral equation t u(t) = U (t)u0 −

 1   U (t − t  )∂x 1 − ∂x2 2 u2 (t  ) dt  ,

0

where U (t)u0 (x) = St ∗ u0 (x) and St (·) is defined by the following oscillatory integral

(1.4)

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L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

+∞ St (x) = c eixξ +itφ(ξ ) dξ

(1.5)

−∞

with φ(ξ ) = ξ 5 . About the application of harmonic analysis in partial difference equation see [8–12,14–16]. About shallow water wave see [18–28]. The goal of the present paper is to study the Cauchy problem of (1.3) with the positive dispersion β > 0. The main result is that (1.3) is locally well-posed for data in the Sobolev space H s with s  − 11 16 . The essential element of the method is the new Fourier transform restriction spaces that are strongly related to the symbol of the linear equation [4]. Combining the crucial analysis for χ with the Strichartz-type inequalities will enable us to extend the bilinear estimates for the nonlinearity to the Bourgain function spaces associated with the index s for s  − 11 16 . Before formulating our main achievements, notation is introduced and some preliminary mathematical points are brought to the fore. We denote the Sobolev spaces X b (R2 ) the completion of the Schwarz space S(R2 ) equipped with the norm   uXb (R2 ) = τ b u(τ, ˆ ξ )L2

,

(τ,ξ )

where  · = (1 + | · |). By Hxs we denote the Sobolev space which measures the regularity with respect to the x variable equipped with the norm   uHxs = ξ s u(·, ˆ ξ )L2 (·,ξ ) .

(1.6)

In the same vein, we define the space Bsb equipped with the norm   b uBsb =  τ + φ(ξ ) ξ s u(τ, ˆ ξ )L2 (τ,ξ ) .

(1.7)

Clearly, B0b = B b . Let I ⊂ R be an interval. Then we define the space Bsb (I ) equipped with the norm

uBsb (I ) = inf ωBsb , ω(t, ·) = u(t, ·) on I . ω∈Bsb

(1.8)

Note that for b > 12 , one-dimensional Sobolev embedding injects Bsb into C(R, Hxs (R)). 2. Preliminary estimates Lemma 2.1. If u0 ∈ L1 (R), then u(t) = U (t)u0 ∈ L∞ x (R) and   U (t)u0 

L∞ x

1

 C|t|− 5 u0 L1 .

(2.1)

Remark. By the interpolation between (2.1) and the following identity   U (t)u0 

L2x

 u0 L2x .

(2.2)

L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

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One can readily obtain   U (t)u0 

p

Lx

− 15 (1− p2 )

 C|t|

u0 

p

Lx

where

,

1 1 = 1. + p p

(2.3)

For convenience, in what follows, we denote ρ (τ, ξ ) = F

f (τ, ξ ) . (1 + |τ − ξ 5 |)ρ

(2.4)

Lemma 2.2 (Global smoothing effects). (See [9].) Assume that the phase function φ is not linear and φ(ξ ) = (R(ξ ))α , where R(ξ ) is a rational function and α ∈ R. Define Wγ for γ  0 and (t, x) ∈ R2 by  Wγ (t)u0 (x) =

γ /2 ei(xξ +tφ(ξ )) φ  (ξ ) uˆ 0 (ξ ) dξ.

(2.5)

R

Then, (1) for any ϑ ∈ [0, 1],   Wϑ/2 (t)u0 

q p Lt Lx

 ∞ (2) −∞

 Cu0 L2 ,

4 sup W0 (t)u0 (x) dx

where (q, p) =

1



4

 Cφ

t

R

 2 4 , , ϑ 1−ϑ

1  12 2 φ  (ξ ) 2 uˆ 0 (ξ ) φ  (ξ ) dξ .

(2.6)

(2.7)

Lemma 2.3. The group {U (t)}+∞ −∞ satisfies   U (t)ϕ 

12 L12 t Lx

 CϕL2 ,

 2  D U (t)ϕ  ∞ 2  Cϕ 2 , L x L L t

x

 − 14  Dx U (t)ϕ 

L4x L∞ t

(2.8) (2.9)

 CϕL2 ,

(2.10)

 CϕL2 .

(2.11)

and   21 Dx U (t)ϕ 

L6t L6x

Proof. Firstly we shall prove (2.8). More generally, we come to obtain   U (t)ϕ  p q  Cϕ 2 , L L L t

x

(2.12)

where p  2, q  2, q2 = 15 (1 − p2 ). When p = 12, q = 12, (2.12) is corresponding to (2.8). In fact, by duality, we need to bound

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L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

     U (t)f (t, ·) dt    p

q

for f ∈ Lt Lx , f 

p

q

Lt Lx

= 1, where

1 p

+

1 p

Squaring (2.13) and using the fact that f 

= 1 and p

q

Lt Lx

(2.13) L2x

1 q

+

1 q

= 1.

= 1, it follows from the unitary property of

the linear group that         U (s − t)f (t, ·) dt U (s − t)f (t, ·), f (s, ·) ds dt  C   

p q

,

(2.14)

Ls Lx

where ·,· is defined as the inner product in L2x . Substitution of (2.3) in (2.14) gives the bound      − 15 (1− p2 )     f (t) q  dt  C  |s − t| (2.15)   Cf  p q   C, Lx

p

Ls

Lt Lx

where use has been made of the Hardy–Littlewood–Sobolev inequality. This proves (2.12). Secondly, a simple computation shows that for ξ = 0, φ  (ξ ) = 5ξ 4 > 0. So φ is invertible. Thus we have  U (t)ϕ =

 e

ixξ itφ(ξ )

e

ϕ(ξ ˆ ) dξ =

 1 −1  = Ft−1 eixφ ϕˆ φ −1  . φ

1  −1 eixφ eitφ ϕˆ φ −1  dφ φ (2.16)

Making change of the variable ξ = φ −1 yields 

2       U (t)ϕ 2 2 = F −1 eixφ −1 ϕˆ φ −1 1   t Lt  φ L2t   2   2 1 1 ˆ )  = ϕˆ φ −1  2 dφ = ϕ(ξ φ  (ξ ) dξ |φ | |φ (ξ )|2   2 1 2 1 ˆ ) ˆ )  dξ = ϕ(ξ  ϕ(ξ dξ. |φ (ξ )| |ξ |4

(2.17)

The above inequality then implies (2.9). Attention is now turned to prove (2.10). From Lemma 2.6, one can see that   U (t)ϕ 2 4

Lx L∞ t

 C

1  2 φ  (ξ ) 2 2 1 ϕ(ξ ˆ )  dξ  C ϕ(ξ ˆ ) |ξ | 2 dξ. φ (ξ )

(2.18)

This completes the proof of (2.10). Finally, (2.11) can be obtained by interpolation between (2.9) and (2.10). 2

L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

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Lemma 2.4. Let h ∈ L2 (R2 ) and Hρ be defined by ρ (τ, ξ ) = h(τ, ξ ) . H (1 + |τ |)ρ Then for ρ > 1/2, Hρ L2x L∞  ChL2 L2 . t

(2.19)

τ

ξ

Proof. In view of the Hölder inequality and the Plancherel identity, we can easily obtain (2.19). 2 Lemma 2.5. Let R(t, x) ∈ R2 be defined by ρ (τ, ξ ), ρ (τ, ξ ) = χ(ξ )F R where χ ∈ C0∞ (R), χ(ξ ) = 1 if |ξ |  1 and χ(ξ ) = 0 if |ξ | > 2. Then for ρ > 1/2 we have Rρ L2x L∞  Cf L2 L2 , t

(2.20)

τ

ξ

where Fρ is defined in (2.4). Proof. Since ∞ ∞ Rρ (t, x) =

ei(xξ +tτ ) χ(ξ )

−∞ −∞

∞ ∞ =

f (τ, ξ ) dξ dτ (1 + |τ + φ(ξ )|)ρ

ei(xξ +tλ) e−itφ(ξ ) χ(ξ )

−∞ −∞

f (λ − φ(ξ ), ξ ) dξ dλ, (1 + |λ|)ρ

this in turn implies that ρ (λ, ξ ) = e−itφ(ξ ) χ(ξ ) f (λ − φ(ξ ), ξ ) . R (1 + |λ|)ρ Hence applying Lemma 2.4 to the function Rρ , we obtain (2.20).

2

Lemma 2.6. Let ρ > 38 . Then  83  D x F ρ 

L4x L4t

ρ (τ, ξ ) = Proof. Recall F relation

f (τ,ξ ) (1+|τ +φ(ξ )|)ρ .

 Cf L2 L2 . ξ

τ

(2.21)

Changing the variable τ = λ − φ(ξ ), there appears the

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L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

∞ ∞

ei(xξ +tτ )

Fρ (t, x) = −∞ −∞

∞ =

∞ e

ei(xξ −tφ(ξ ))

itλ

−∞

f (τ, ξ ) dξ dτ (1 + |τ + φ(ξ )|)ρ

−∞

f (λ − φ(ξ ), ξ ) dξ dλ. (1 + |λ|)ρ

(2.22)

Therefore, in view of (2.11), Plancherel’s identity and Minkowski’s integral inequality, and taking into account ρ > 12 , we obtain that ∞

 21  D x F ρ 

L6x L6t

C

   f λ − φ(ξ ), ξ 

−∞

L2ξ

1 dλ  Cf L2 L2 . ξ τ (1 + |λ|)ρ

(2.23)

On the other hand, we have F0 L2 L2  Cf L2 L2 . x

t

ξ

(2.24)

τ

Inequality (2.21) then follows from (2.23) interpolated with (2.24). Lemma 2.7. If ρ >

θ 2

2

with θ ∈ [0, 1], then  2θ  D F ρ  x

2

Lx1−θ L2t

 Cf L2 L2 .

(2.25)

τ

ξ

In particular, when θ = 0, Fρ L2 L2  Cf L2 L2 x

t

ξ

for ρ > 0,

τ

and when θ = 1,  2  D F ρ 

2 L∞ x Lt

x

1 for ρ > . 2

 Cf L2 L2 ξ

τ

Proof. The argument in (2.23) and the estimate (2.9) show that for ρ >  2  D Fρ  x

2 L∞ x Lt

 Cf L2 L2 . ξ

Lemma 2.8. Suppose ρ >

1 6(q−2) 2 5q .

(2.26)

τ

Again, applying interpolation (2.26) with (2.24) yields (2.25).

1 2

2

Then for 2  q  12, we have Fρ Lqx Lqt  Cf L2 L2 . ξ

τ

In particular, when q = 2 with ρ  0 Fρ L2 L2  Cf L2 L2 , x

t

ξ

τ

(2.27)

L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

when q = 3 with ρ >

1845

1 5

Fρ L3 L3  Cf L2 L2 , t

x

and when q = 4 with ρ >

τ

ξ

3 10

Fρ L4 L4  Cf L2 L2 . t

x

τ

ξ

1 2

Proof. The argument in (2.23) and the estimate (2.8) show that for ρ > Fρ L12 L12  Cf L2 L2 . t

x

(2.28)

τ

ξ

Thus a suitable interpolation between (2.24) and (2.28) provides the needed inequality.

2

3. Bilinear estimates In this section, we are in the position to state a bilinear estimate which will be the main tool in the proof of local existence for (1.3) (Theorem 4.2 below). Theorem 3.1. If s  − 11 16 then for sufficiently small ε > 0, we have    1 ∂x 1 − ∂ 2 2 (uv) x

− 21 +3ε

Bs

 cu

v

.

(3.1)

σ2 = σ (τ − τ1 , ξ − ξ1 ),

(3.2)

1 +2ε

Bs2

1 +2ε

Bs2

For convenience, we define   1 +2ε s ξ u(τ, ˆ ξ ), fˆ(τ, ξ ) = τ + φ(ξ ) 2   1 +2ε s g(τ, ˆ ξ ) = τ + φ(ξ ) 2 ξ v(τ, ˆ ξ ), σ1 = σ (τ1 , ξ1 ),

σ = σ (τ, ξ ) = τ + φ(ξ ), k(τ, ξ ) =

ξ1

−s ξ

σ1

1 2 +2ε

− ξ1 σ2

−s

1 2 +2ε

=

(ξ1 ξ2 1

)−s 1

σ1 2 +2ε σ2 2 +2ε

(3.3)

,

ξ2 = ξ − ξ1 . Then (3.1) is equivalent to the estimate     − 1 +3ε  1+s σ 2 ˆ |ξ |ξ k(τ, ξ )f (τ1 , ξ1 )g(τ ˆ − τ1 , ξ2 ) dτ1 dξ1    R2

L2 (τ,ξ )

 cf L2 gL2 . Hence by the self-duality of L2 , it is easy to see that (3.4) is equivalent to

(3.4)

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L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

 ˆ ξ ) dτ1 dξ1 dτ dξ k1 (τ, ξ, τ1 , ξ1 )fˆ(τ1 , ξ1 )g(τ ˆ − τ1 , ξ − ξ1 )h(τ, R4

 cf L2 gL2 hL2 ,

(3.5)

where k1 (τ, ξ, τ1 , ξ1 ) =

|ξ |ξ 1+s ξ1 −s ξ2 −s 1

1

1

σ1 2 +2ε σ2 2 +2ε σ 2 −3ε

.

Lemma 3.2. The following inequality holds



max |σ1 |, |σ2 |, |σ |  c ξ ξ1 (ξ − ξ1 ) max |ξ |2 , |ξ1 |2 , |ξ − ξ1 |2 .

(3.6)

Proof.   σ1 + σ2 − σ = −5ξ ξ1 (ξ − ξ1 ) ξ 2 + ξ12 − ξ ξ1 .

2

Remark. Lemma 3.2 implies that one of the following cases occurs:

|σ1 |  c|ξ ||ξ1 ||ξ − ξ1 | max |ξ |2 , |ξ1 |2 , |ξ − ξ1 |2 ,

|σ2 |  c|ξ ||ξ1 ||ξ − ξ1 | max |ξ |2 , |ξ1 |2 , |ξ − ξ1 |2 ,

|σ |  c|ξ ||ξ1 ||ξ − ξ1 | max |ξ |2 , |ξ1 |2 , |ξ − ξ1 |2 . Proof of Theorem 3.1. We denote by J the left integral of (3.5). Case 1. |ξ |  2. We denote by J1 the restriction of J on this region. Case 1.1. |ξ1 |  1. In this case, |ξ2 |  |ξ | + |ξ1 |  3, k1 (τ, ξ, τ1 , ξ1 ) = 

|ξ |ξ 1+s ξ1 −s ξ2 −s 1

1

1

1

1

1

σ1 2 +2ε σ2 2 +2ε σ 2 −3ε c σ1 2 +2ε σ2 2 +2ε σ 2 −3ε

,

so,     ˆ ξ) ˆ − τ1 , ξ − ξ1 )h(τ, fˆ(τ1 , ξ1 )g(τ J11  c dτ1 dξ1 dτ dξ 1 1 1 +2ε +2ε −3ε σ1 2 σ2 2 σ 2 4 R

 cF 1 +2ε L4 L4 G 1 +2ε L4 L4 H 1 −3ε L2 L2 2

x

t

2

 cf L2 gL2 hL2 .

x

t

2

x

t

(3.7) (3.8) (3.9)

L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

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Case 1.2. |ξ1 |  1, |ξ2 |  1. Without loss of generality, we assume |ξ1 |  |ξ2 | and denote by s˜ = −s. In this case, by Lemma 3.2, we consider the three subcases (3.7), (3.8) and (3.9) separately. If (3.7) holds, we have 1

1

1

1

σ1 2 +2ε  c|ξ1 |r1 ( 2 +2ε) |ξ2 |r2 ( 2 +2ε) |ξ | 2 +2ε , where r1 + r2 = 4, 1  r1 , r2  3, so 1

1

|ξ1 |s˜−r1 ( 2 +2ε) |ξ2 |s˜−r2 ( 2 +2ε)

k1 (τ, ξ, τ1 , ξ1 )  c

1

1

σ2 2 +2ε σ 2 −3ε 1

|ξ2 |2˜s −4( 2 +2ε)

c

1

1

σ2 2 +2ε σ 2 −3ε

.

Hence, we can obtain     1 ˆ − τ1 , ξ − ξ1 )|ξ − ξ1 |2˜s −4( 2 +2ε) fˆ(τ1 , ξ1 )g(τ dτ dξ dτ dξ J12  c 1 1 1 1 2 +2ε σ 2 −3ε σ 2 4 R

 3   cF0 L2 L2 Dx8 G 1 +2ε L4 L4 H 1 −3ε L4 L4 x

t

t

x

2

x

2

t

 cf L2 gL2 hL2 , where 2˜s − 4( 12 + 2ε)  38 . If (3.8) holds, similar to the estimate above, we can easily get the estimate J12  cf L2 gL2 hL2 . If (3.9) holds, we have 1

1

1

1

σ 2 −3ε  c|ξ | 2 −3ε |ξ1 |α( 2 −3ε) |ξ2 |2( 2 −3ε) , so 1

k1 (τ, ξ, τ1 , ξ1 )  c

1

|ξ1 |s˜−2( 2 −3ε) |ξ2 |s˜−α( 2 −3ε) 1

1

σ1 2 +2ε σ2 2 +2ε

.

Therefore, one can see that  1 1 ˆ ξ) ˆ − τ1 , ξ − ξ1 )|ξ − ξ1 |s˜−2( 2 −3ε) h(τ, fˆ(τ1 , ξ1 )|ξ1 |s˜−2( 2 −3ε) g(τ J12  c dτ dξ dτ dξ 1 1 1 1 2 +2ε σ2 2 +2ε σ 1 4 R

 3  cDx8 F 1

2 +2ε

 

L4x L4t

 83 Dx G 1

2 +2ε

 

L4x L4t

 cf L2 gL2 hL2 , where s˜ − 2( 12 − 3ε) = s˜ − 1 + 6ε  38 .

H0 L2 L2 x

t

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L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

Case 2. |ξ |  2. We denote by J2 the restriction of J on this region. Case 2.1. |ξ1 |  1. In this case, we have |ξ − ξ1 |  1 and |ξ | ∼ |ξ − ξ1 | ∼ ξ ∼ ξ − ξ1 , so |ξ2 |2

k1 (τ, ξ, τ1 , ξ1 ) 

σ1

1 2 +2ε

1

1

σ2 2 +2ε σ 2 −3ε

.

Therefore, we obtain     ˆ ξ) ˆ − τ1 , ξ − ξ1 )|ξ − ξ1 |2 h(τ, fˆ(τ1 , ξ1 )g(τ J21  c dτ1 dξ1 dτ dξ 1 1 1 σ1 2 +2ε σ2 2 +2ε σ 2 −3ε R4  2  D G 1   cF 1 +2ε L2x L∞ +2ε L∞ L2 H 1 −2ε L2 L2 t 2

t

x

2

x

2

t

 cf L2 gL2 hL2 . Case 2.2. |ξ2 |  1. This statement is similar to Case 2.1. Case 2.3. |ξ1 |  1 and |ξ2 |  1. In this case, we denote by J23 the restriction of J2 on this region. From Lemma 3.2, we must consider the three cases (3.7), (3.8) and (3.9) separately. If (3.7) holds, without loss of generality, we assume |ξ1 | < |ξ2 |, then k1 (τ, ξ, τ1 , ξ1 ) 

c|ξ |2−˜s |ξ1 |s˜ |ξ2 |s˜ σ1

1 2 +2ε

1

1

σ2 2 +2ε σ 2 −3ε 1



1

1

1

σ2 2 +2ε σ 2 −3ε 3



1

c|ξ |2−˜s −3( 2 +2ε) |ξ1 |s˜−( 2 +2ε) |ξ2 |s˜−( 2 +2ε) 1

c|ξ |2−˜s −( 2 +6ε) |ξ2 |2˜s −2( 2 +2ε) 1

1

σ2 2 +2ε σ 2 −3ε

Hence, we have

.

 3 1 |ξ |2−˜s −( 2 +6ε) |ξ2 |2˜s −2( 2 +2ε) fˆgˆ hˆ J23  c dτ1 dξ1 dτ dξ 1 1 +2ε −3ε σ2 2 σ 2 4 R

 3   cF0 L2 L2 D 8 G 1 +2ε L4 L4 H 1 −2ε L4 L4 x

t

2

 cf L2 gL2 hL2 , where 2˜s − 2( 12 + 2ε)  38 .

x

t

2

x

t

L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

1849

If (3.8) holds, we can deal with this statement similar to the one above. If (3.9) holds, then 1

k1 (τ, ξ, τ1 , ξ1 )  c

1

1

|ξ |2−˜s −(3−20ε)( 2 −3ε) |ξ1 |s˜−(1+10ε)( 2 −3ε) |ξ2 |s˜−(1+10ε)( 2 −3ε) 1

1

σ1 2 +2ε σ2 2 +2ε

.

Hence, we have  3   3  J23  cDx8 F 1 +2ε L4 L4 Dx8 G 1 +2ε L4 L4 H0 L2 L2 x

2

t

x

t

x

2

 cf L2 gL2 hL2 .

t

2

4. Local well-posedness Let ψ be a cut-off function such that ψ ∈ C0∞ (R),

supp ψ ⊂ [−2, 2],

ψ = 1 over the interval [−1, 1].

We truncate (1.4) the integral equivalent formulation of (1.2) with the data u(0, x) = u0 (x) as follows t u(t) = ψ(t)U (t)u0 − 2ψ(t/T )

U (t − t  )u(t  )∂x u(t  ) dt  .

(4.1)

0

Since the solutions of (4.1) will correspond local solutions of (1.6) in the time interval [−T , T ], 1

+2ε

we shall apply a fixed point argument to solve (4.1) in Bs2 for sufficiently small ε. To solve problem (4.1), we need to estimate the two terms in the right-hand side of (4.1). Firstly, we have the following linear estimates. Lemma 4.1 (Linear estimates). Assume s  − 11 16 and T ∈ (0, 1]. Then the following estimates hold for sufficiently small ε > 0:   ψ(t/T )U (t)u0  1  c0 T −2ε u0 H s , (4.2) +2ε   ψ(t/T )h

Bs2

1 +2ε

Bs2

 CT −2ε h

  t       ψ(t/T ) U (t − t )w(t ) dt   

1 +3ε Bs2

0

1 +2ε

Bs2

(4.3)

,

 CT −3ε w

,

(4.4)

.

(4.5)

− 12 +3ε

Bs

and   t     ψ(t/T ) U (t − t  )w(t  ) dt     0

1 +2ε Bs2

 CT ε w

Now we are in the position to prove our main theorem.

− 21 +3ε

Bs

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L. Tian et al. / J. Differential Equations 245 (2008) 1838–1852

s Theorem 4.2. Let s  − 11 16 and u0 ∈ H . Then for sufficiently small ε > 0 there exist a constant T > 0 and a unique local solution u(t) of (1.3) satisfying u ∈ C([−T , T ]; H s ) ∩ 1

Bs2

+2ε

([−T , T ]) with u(0) = u0 .

b Proof. Let u0 (x) ∈ H s (R) with s  − 11 16 . For u ∈ Bs we define a mapping

t Φu0 (u) = Φ(u) = ψ(t)U (t)u0 − ψ(t/δ)

 1   U (t − t  )∂x 1 − ∂x2 2 u2 (t  ) dt  .

0

We shall prove that Φ(·) defines a contraction mapping on

B = u ∈ Bsb : uBsb  2c0 u0 H s s with b = 12 + 2ε and s  − 11 16 , provided that δ is small enough and depends only on u0 H . In fact, (4.2) yields ψ(t)U (t)u0 ∈ B which guarantees that B is not empty. Moreover, if u ∈ B, then, when T = δ, in view of relations (4.2), (4.5) and Theorem 3.1, we have

     1   Φ(u) b  c0 u0 H s + Cδ ε ∂x 1 − ∂ 2 2 u2  x B s

 c0 u0 H s + Cδ ε u2

− 21 +3ε

1 +2ε

Bs2

Bs

 c0 u0 H s + 4c02 Cδ ε u0 2H s . Thus, if we fix δ such that 4c0 Cδ ε u0 H s < 1/2, then Φ(B) ⊆ B. On the other hand, if u, v ∈ B with the same data u0 (x), then by using (4.5) again, and combining Theorem 3.1, we have   Φ(u) − Φ(v)

1 +2ε Bs2

  = Φu0 (u) − Φu0 (v) 

1 +2ε

Bs2

 1   C ε δ ∂x 1 − ∂x2 2 (u + v)(u − v)  − 1 +3ε 2 Bs 2

1  Cδ ε u + v 1 +2ε u − v 1 +2ε 2 Bs2 Bs2  4c0 Cδ ε u0 H s u − v

1 +2ε

Bs2

1  u − v 1 +2ε 2 Bs2

and Φ is a contraction mapping in B. Hence, we can apply the contraction mapping principle for small T which completes the proof of Theorem 4.2. 2 Acknowledgments Research was supported by the National Nature Science Foundation of China (No. 10771088) and Nature Science Foundation of Jiangsu (No. 2007098) and Outstanding Personnel Program in Six Fields of Jiangsu (No. 6-A-029).

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