On the local well posedness of a Benjamin Ono Boussinesq system

(1)

BENJAMIN-ONO-BOUSSINESQ SYSTEM

RUYING XUE

Received 30 March 2005 and in revised form 25 September 2005

Consider a Benjamin-Ono-Boussinesq systemηt+ux+auxxx+ (uη)x=0,ut+ηx+uux+

cηxxx−duxxt=0, wherea,c, anddare constants satisfyinga=c >0,d >0 ora <0,c <0,

d >0. We prove that this system is locally well posed in Sobolev spaceHs₍_R₎_×_Hs+1₍_R_), withs >1/4.

1. Introduction and main results

We consider the Cauchy problem for a Benjamin-Ono-Boussinesq system:

ηt+ux+auxxx+ (uη)x=0, t >0,x∈R,

ut+ηx+uux+cηxxx−duxxt=0, t >0,x∈R,

η|t=0= f(x), u|t=0=g(x),

(1.1)

wherea,c, anddare constants satisfying

a=c >0, d >0 or a <0, c <0, d >0. (1.2)

The system is called a Benjamin-Ono-Boussinesq system because it can be reduced to a pair of equations whose linearization uncouples to a pair of linear Benjamin-Ono equa-tions.

Equations of type (1.1) are a class of essential model equations appearing in physics and fluid mechanics. To describe two-dimensional irrotational flows of an inviscid liquid in a uniform rectangular channel, Boussinesq in 1871 derived from the Euler equation the classical Boussinesq system:

ηt+ux+ (uη)x=0, t >0,x∈R,

ut+ηx+uux+₃1ηxxt=0, t >0,x∈R.

(1.3)

In [1], Bona et al. derived by considering first-order approximations to the Euler equation the following alternative (a four-parameter Boussinesq system) to the classical Boussinesq

(2)

system:

ηt+ux+auxxx+ (uη)x−bηxxt=0,

ut+ηx+uux+cηxxx−duxxt=0, (1.4)

where the constants obey the relations

a+b=1 2

θ2₋1 3

, c+d=1 2

1−θ2_≥_0, _a₊_b₊_c₊_d₌1

3, (1.5)

withθ∈[0, 1]. The system (1.1) is one of the four-parameter systems associated with b=0. Whenb=0, Bona et al. in [1] determined exactly that the four-parameter systems are linearly well posed if and only ifa,c, anddsatisfy the relation (1.2). The local well-posedness of the nonlinear system (1.1) is considered in [2]. They prove that the system (1.1) associated with (1.2) is locally well posed in the Sobolev spaceHs₍_R₎_×_Hs+1₍_R_), withs≥1. In this work, we will give some local well-posedness for the Cauchy problem (1.1) in the Sobolev spaceHs₍_R₎_×_Hs+1₍_R_{), with}_{s >}_{1/4, by using the so-called}_Lp₋_Lq

smoothing eﬀect of the Strichartz type.

Denote byJthe Fourier multiplier with symbol (1 +ξ2₎1/2_{, and denote by}_Ᏼ_{the usual} Hilbert transform. Our result is the following.

Theorem 1.1. Fix s >1/4. Then, for every(f(x),g(x))∈Hs₍_R₎_×_Hs+1₍_R₎_{, there exist} T >0depending only onf(x)Hs+g(x)_Hs+1 and a unique solution of (1.1) on the time

interval[0,T]satisfying

J−1_Ᏼ_η,u_∈_C_[0,T], _L2₍_R₎_×_L2₍_R₎_,

J−1_Ᏼ_η

x,ux∈C[0,T], L∞(R)×L∞(R).

(1.6)

Moreover, for any R >0, there exists T depending on R such that the nonlinear map

(f(x),g(x))→(η,u)is continuous from the ball of radiusRofHs₍_R₎_×_Hs+1₍_R₎_to_C([0,_T_]; Hs₍_R₎_×_Hs+1₍_R₎₎_.

In the sequel, we say that the pair (p,q)∈R2_{is admissible if it satisfies} 2

p+ 1 q=

1

2, p >4. (1.7)

We denote byJ the Fourier multiplier with symbol (1 +ξ2₎1/2_{, denote by} _Ᏼ_{the usual} Hilbert transform, and denote bym(D) the Fourier multiplier associated with symbol m(ξ). We also denote the dyadic integers 2k_,_k_≥_{0, by} _λ_or_{µ. Whenever a summation}

overλorµappears, it means that we sum over the dyadic integers. The notationAB (resp.,AB) means that there exists a harmless positive constantCsuch thatA≤CB (resp.,A≥CB). We denote byLTpX(resp.,LIpX) the space ofX-valued measurable and

p-integrable functions defined on [0,T] (resp., I), equipped with the natural norm. We also use the notation(u1,u2,. . .,uk)X= u1X+···+ukX.

(3)

2. Some estimates

In this section, we give some smoothing eﬀects for (1.1). These estimate will be the main ingredient in the proof of local well-posedness of the Cauchy problem (1.1). Consider the following linear system:

ηt+ux+auxxx=F(x,t), t >0,x∈R,

ut+ηx+cηxxx−duxxt=G(x,t), t >0,x∈R. (2.1)

Let

σ(ξ)= aξ2−1

cξ2₋₁ dξ2_{+ 1}

1/2

, h(ξ)= aξ2−1

dξ2_{+ 1} cξ2₋₁

1/2

. (2.2)

Consider the change of variables

η=h(D)(v+w), u=v−w, η=h−1_(D)η₌_v₊_w, _(2.3)

whereh(D) (resp.,h−1_{(D)) is the Fourier multiplier with the symbol}_{h(ξ) (resp.,}_h−1_(ξ)). Then we have

vt+σ(D)∂xv=1

2h

−1_(D)F₊1 2

1 +dD2−1_G,

wt−σ(D)∂xw=1₂h−1(D)F−1₂

1 +dD2−1_G.

(2.4)

Rewriteσ(ξ)=(ac/d)1/2_|_ξ_|₊_{γ(ξ) with}

γ(ξ)= d−(ac+ad+dc)ξ2

daξ2₋₁_dξ2_{+ 1}_cξ2₋₁ 1/2_{+ (acd)}1/2_|_ξ_|_dξ2_{+ 1}. (2.5)

LetΓ(D) be the Fourier multiplier with the symboliξγ(ξ), which is a skew-adjoint oper-ator inL2₍_R_{) and is bounded from}_Lp₍_R_{) to}_Lp₍_R_{) for 1}_{< p <}₊_∞_{. Then}

vt+

ac

d 1/2

Ᏼ∂xxv=Γ(D)v+1₂h−1(D)F+1₂

1 +dD2−1_G,

wt−

_ac

d 1/2

Ᏼ∂xxw= −Γ(D)w+1

2h

−1_(D)F₋1 2

1 +dD2−1_G.

(2.6)

(4)

for admissible pair (p,q),

v_Lp

ILqvL∞IL2+

−Γ(D)v+1 2h

−1_(D)F₊1 2

1 +dD2−1_G

L1

IL2

vL∞IL2+vL1IL2+

h−1_(D)F₊_{1 +}_dD2−1_G

L1

IL2

1 +|I|vL∞IL2+h

−1_(D)F

L1

IL2+

1 +dD2−1_G

L1

IL2

,

(2.7)

and similarly

w_Lp ILq

1 +|I|wL∞IL2+h

−1_(D)F

L1

IL2+

1 +dD2−1_G

L1

IL2

. (2.8)

Thus by (2.3),

₍_η,u)

LIpLq

1 +|I|(η,u_L∞

IL2+

_h−1_(D)F(x,t)

L1

IL2

+1 +dD2−1_G(x,t)

L1

IL2

. (2.9)

Consider a standard Littlewood-Paley decomposition:

u=

λ

uλ, η=

λ

ηλ, η=

λ

ηλ, (2.10)

where

uλ=∆λu, ηλ=∆λη, ηλ=∆λη, (2.11)

∆λ are the Fourier multipliers with symbolsφ(ξ/λ) when λ=2k withk≥1, andχ(ξ)

whenλ=1, and where the nonnegative functionsχ∈C0∞(R) andφ∈C0∞(R) satisfy

χ(ξ) +

λ

φ(ξ/λ)=1,

φ(ξ)=       

0, if|ξ| ≤5

8 or|ξ| ≥2, 1, if 1≤ |ξ| ≤5

4.

(2.12)

For a dyadic integerλ, we set

∆λ=

  

∆λ/2+∆λ+∆2λ, ifλ >1, ∆1+∆2, ifλ=1.

(2.13)

Lemma2.1. Let{aλ},{dλ}, and{δλ}be three sequences indexed on positive dyadic integers

(5)

Then

λ

δλ

µ≥λ/8 aµdλ

λ

δ2

λa2λ

1/2

λ

d2

λ

1/2

, (2.14)

and hence by duality,

λ

δ2

λ

µ≥λ/8 aµ

2

λ

δ2

λa2λ. (2.15)

Proof. Note thatκ1δλ≤δ2λ≤κ2δλimpliesδλ/δ2k_λ≤κ−₁k. Using the Cauchy-Schwarz

in-equality, we get

λ

δλ

µ≥λ/8 aµdλ=

λ

δλ

∞

k=−3,2k_λ_≥1

a2k_λd_λ=

∞

k=−3 κ−k

1

λ≥2−k

δ2k_λa₂k_λd_λ

≤ ∞

k=−3 κ−1k

λ≥2−k

δ₂2k_λa2₂k_λ

1/2

λ≥2−k

d2_λ 1/2

λ

δ2_λa2_λ

1/2

λ

d_λ2 1/2

.

(2.16)

Lemma2.2. FixT >0andσ >1/2. Let(η,u)be a smooth solution of the system (1.1). Then for every admissible pair(p,q),

λ

λ2σᏴηλ,uλ

2

LpTLq

1/2

(1 +T)1/ p_{1 +}_Jσ_u L∞TL2

×1 +u,ux,Ᏼη,∂ xᏴη,ηL1

TL∞

λ

λ2/ p+2σᏴηλ,uλ2L∞TL2

1/2 ,

(2.17)

where η=h−1_(D)η_,_η

λ=h−1(D)∆λη, h−1(D)is the Fourier multiplier with the symbol

h−1_(ξ)_{defined in (}_2.2_).

Proof. Let (η,u) be a smooth solution of the system (1.1). Then (ηλ,uλ) satisfies the

fol-lowing system:

ηλ

t+

uλ

x+a

uλ

xxx= −

uh(D)ηλ+

∆λ,uh(D) ηλ

x, t >0,x∈R,

uλ

t+

ηλ

x+c

ηλ

xxx−d

uλ

xxt= −

uuλ

x+

∆λ,u∂x u

(6)

Using (2.9) to the system (2.18) and choosing the intervalI satisfying|I| ≤1 and|I| ≤ 1/λ, we get

_Ᏼ_η_λ_,u_λ

LpILq

ηλ,uλLIpLq

1 +|I|ηλ,uλL∞IL2+

_∂_x_h−1_(D)_uh(D)_η

λ+

∆λ,uh(D) ηλL1

IL2

+1 +dD2−1uuλ

x+

∆λ,u∂x u_L1

IL2

1 +|I|ηλ,uλL∞

IL2+

_uh(D)_η_λ

L1

IL2

+∆λ,uh(D) ηλL1

IL2+

_u uλ

xL1

IL2+

_∆_λ_,u∂_x _u

L1

IL2

1 +|I|ηλ,uλL∞IL2+

_u

L∞IL∞

_h(D)_η_λ

L1

IL2

IL2+uL

∞

IL∞uλ

xL1

IL2+

_∆_λ_,u∂_x _u

L1

IL2.

(2.19)

By the Sobolev embedding,|I| ≤1 and|I| ≤1/λ,

uL∞

IL∞J

σ_u L∞IL2,

_h(D)_η_λ

L1

IL2ηλ

L1

IL2+

_∂_x_η_λ

L1

IL2|I|ηλ

L∞

IL2+|I|ληλ

L∞

IL2ηλ

L∞

IL2,

_∂_x_u_λ

L1

IL2|I||λ|

_u_λ

L∞IL2

_u_λ

L∞IL2.

(2.20)

Then, we deduce from (2.19)

_Ᏼ_η_λ_,u_λ

LIpLq

1 +Jσ_u L∞IL2

Ᏼηλ,uλL∞IL2

IL2+

_∆_λ_,u∂_x _u

L1

IL2.

(2.21)

Partition [0,T]=kIk, where each intervalIkis of size≤min{1/λ, 1}. We can chooseIk

such that their number is bounded by (1 +T)λ. Therefore by (2.21), we obtain

_Ᏼ_η_λ_,u_λ_Lp

TLq(1 +T)

1/ p_λ1/ p_{1 +}_Jσ_u L∞

TL2

×Ᏼηλ,uλL∞TL2+

_∆_λ_,uh(D) _η_λ

L1

TL2+

_∆_λ_,_u∂_x _u

L1

TL2

. (2.22)

To estimate the terms[∆λ,uh(D)]ηλL1

TL2and[∆λ,u∂x]uL1TL2, we introduce the

follow-ing estimates which come from [3, Lemmas 4.2 and 4.3]:

_∆_λ_,v∂_x _w

(7)

Then, forλ≥4,

_∆_λ_,uh(D) _∆_λ_η

L2=

∆λ,u∂x h(D)_iD ∆λη

L2

uxL∞

h(D)_iD _∆_λη

L2 _u_x

L∞∆ληL2,

(2.24)

and forλ=1, 2,

_∆_λ_,uh(D) _∆_λ_η

L2=∆_λ,u h(D)∆_λη_L2

uL∞h(D)∆_λη_L2u_L∞∆_λη_L2,

(2.25)

so we get

_∆_λ_,uh(D) _∆_λ_η

L1

TL2

uL1

TL∞+uxL1TL∞

∆ληL∞TL2. (2.26)

Now we consider the term[∆λ,uh(D)](Id−∆λ)ηL2, that is,∆_λ(uh(D)(Id−∆_λ)η)_L2. Note that the frequencies of order≤λ/8 in the Littlewood-Paley decomposition ofudo not contribute, therefore

_∆_λ

uh(D)Id−∆λ

η_L1

TL2=

∆λ

µ≥λ/8 uµh(D)

Id−∆λ

η

L1

TL2

µ≥λ/8

_∆_λ_h(D) Id−∆λ

η_L1

TL∞

_u_µ

L∞TL2.

(2.27)

Denote

A(ξ)=h(ξ)−

ad c

1/2 |ξ|

= (ad+ac−cd)ξ2−c

caξ2₋₁_cξ2₋₁_dξ2_{+ 1} 1/2_{+ (acd)}1/2_|_ξ_|_cξ2₋₁.

(2.28)

Letψ(ξ)∈C∞(R) be a nonnegative function satisfyingψ(ξ)=1 for|ξ|>2 andψ(ξ)=0 for|ξ| ≤1. It is obvious thatψ(ξ)A(ξ)ξ/|ξ| ∈S−1_{; that is,}_{ψ(ξ)A(ξ)ξ/}_|_ξ_{| ∈}_C∞₍_R₎ satis-fies

_∂α ξ

ψ(ξ)A(ξ)ξ |ξ|

1 +ξ2(−1−α)/2 _(2.29)

for any nonnegative integersα. LetΛγbe the Lipschitz space defined by

Λγ=

f : there exists a positive constantAsuch thatfL∞≤A,∆_λf_L_∞≤Aλ−γ

(8)

Stein [4, Chapter 6, Section 5.3, Proposition 6] shows that ψ(D)A(D)Ᏼis a bounded mapping fromΛ1/2toΛ3/2. We have

_∆_λ_h(D)ψ(D)

Id−∆λη_L1

TL∞

ad

c 1/2

∂xᏴψ(D) +A(D)ψ(D)

Id−∆λ

η_L1

TL∞

ψ(D)∂xId−∆λᏴηL1

TL∞+

_A(D)ψ_(D)_Ᏼ

Id−∆λᏴηL1

TL∞

∂x

Ᏼη_L1

TL∞+

_A(D)ψ(D)_ᏴᏴ_η

L1

TΛ3/2

∂x

Ᏼη_L1

TL∞+

_Ᏼ_η

L1

TΛ1/2∂x

Ᏼη_L1

TL∞+

_Ᏼ_η

L1

TL∞,

_∆_λ_h(D)

Id−ψ(D)Id−∆λ

η_L1

TL∞

_h(D)

Id−ψ(D)η_L1

TL∞η

L1

TL∞,

(2.31)

because ofh(ξ)(1−ψ(ξ))∈C0∞(R). Hence, we get

_∆_λ_h(D) Id−∆λ

η_L1

TL∞

∆λh(D)ψ(D)

Id−∆λ

η_L1

TL∞+

_∆_λ_h(D)

Id−ψ(D)Id−∆λ

η_L1

TL∞

∂x

Ᏼη_L1

TL∞+

_Ᏼ_η

L1

TL∞+η

L1

TL∞.

(2.32)

A combination of (2.26) with (2.27) and (2.32) yields

_∆_λ

uh(D)Id−∆λη_L1

TL2

∂xᏴηL1

TL∞+

_Ᏼ_η

L1

TL∞+η

L1

TL∞

µ≥λ/8 _u_µ

L∞

TL2,

(2.33)

and so

_∆_λ_,uh(D) _η_λ

L1

TL2

uL1

TL∞+uxL1TL∞

∆ληL∞TL2

+∂x(Ᏼη) L1

TL∞+ᏴηL

1

TL∞+ηL1TL∞

µ≥λ/8 _u_µ

L∞TL2.

(2.34)

Similarly,

_∆_λ_,_u∂_x _u

L1

TL2

_u_x

L1

TL∞∆λu

L∞TL2+

_u_x

L1

TL∞

µ≥λ/8 _u_µ

(9)

It follows from (2.22), (2.34), and (2.35) that

_Ᏼ_η_λ_,u_λ_Lp TLq

(1 +T)1/ p_λ1/ p_{1 +}_Jσ_u L∞TL2

×Ᏼηλ,uλL∞TL2+

_∆_λ_,uh(D) _η_λ

L1

TL2+

_∆_λ_,u∂_x _u

L1

TL2

(1 +T)1/ p_λ1/ p_{1 +}_Jσ_u L∞TL2

×

_u,u_x

L1

TL∞∆λᏴη,∆λu

L∞TL2+

_Ᏼ_η_λ_,u_λ

L∞TL2

+Ᏼη,∂xᏴη,η,ux_L1

TL∞

µ≥λ/8 _u_µ

L∞TL2

.

(2.36)

Hence,

λ

λ2σ_Ᏼ_η

λ,uλ2LTpLq

λ

(1 +T)2/ pλ2/ p+2σ_{1 +}_Jσ_u L∞TL2

2

× 

_u,u_x2

L1

TL∞∆λᏴη,∆λu

2

L∞

TL2+

_Ᏼ_η_λ_,u_λ2

L∞

TL2

+Ᏼη,∂ xᏴη,η,u xL1

TL∞

µ≥λ/8 _u_µ

L∞

TL2

2 

(1 +T)2/ p_{1 +}_Jσ_u L∞TL2

2

1 +u,ux,Ᏼη,∂xᏴη,η2_L1

TL∞

×  

λ

λ2/ p+2σ_Ᏼ_η λ,uλ

2

L∞TL2+∆λᏴη,∆λu

L∞TL2

+

λ

λ2/ p+2σ

µ≥λ/8 _u_µ

L∞TL2

2 

(1 +T)2/ p1 +Jσu_L∞

TL2

2

1 +u,ux,Ᏼη,∂xᏴη,η

2

L1

TL∞

×

λ

λ2/ p+2σ_Ᏼ_η

λ,uλ2L∞

TL2,

(2.37)

where we have used the inequality due toLemma 2.1:

λ

λ2/ p+2σ

µ≥λ/8 _u_µ

L∞TL2

2

λ

(10)

and the inequality

λ

λ2/ p+2σ_∆

λᏴη,∆λu_L∞

TL2

λ

λ2/ p+2σ_Ᏼ_η

λ,uλ2_L∞

TL2. (2.39)

Thus we get

λ

λ2σ_Ᏼ_η

λ,uλ2LpTLq(1 +T)

2/ p_{1 +}_Jσ_u L∞

TL2

2

×1 +u,ux,Ᏼη,∂ xᏴη,η

2

L1

TL∞

λ

λ2/ p+2σ_Ᏼ_η λ,uλ

2

L∞TL2.

(2.40)

We complete the proof.

Lemma2.3. Let1≤κ1≤κ2, and let{δλ}be the dyadic sequence of positive numbers sat-isfyingκ1δλ≤δ2λ≤κ2δλandλ≤δλ≤λ2for all dyadic integersλ. Then for allτ,t∈I, the smooth solution(η,u)of (1.1) satisfies

λ

δ_λ2ηλ(t),uλ(t)

2

L2

λ

δ_λ2ηλ(τ),uλ(τ)

2

L2exp

2u,ux,Ᏼη,∂ xᏴηL1

IL∞

.

(2.41)

Proof. LetF= −(uη)xandG= −uuxin (2.6), and without loss of generality assumeτ < t.

Multiplying (2.6) by (∆λvλ,∆λwλ) and integrating by parts, we get

1 2

d dt



vλ(t)2L2 _w_λ_(t)2

L2  ₌_Re d

dt    

! vλ!vλdξ

! wλw!λdξ

   

=Re    

− #h−1(D)(uη)x

λ+

1 +dD2−1uux

λ

$ vλdx

− #h−1(D)(uη)x

λ−

1 +dD2−1uux

λ

$ wλdx

   ,

(2.42)

where we denote (!vλ,w!λ)=Ᏺx(vλ,vλ). Using (h−1(D)η,u)=(v+w,v−w)=(η,u), we

get

d dt

vλ(t)2L2+wλ(t)2L2

=2 Re

− h−1_(D)_(uη)

x

ληλdx−

1 +dD2−1_uu

x

λuλdx

,

(11)

and so for the dyadic sequence{δλ}, we get

λ

δ2

ληλ(t),uλ(t)2L2

λ

δ2

λ

vλ(t)

2

L2+w_λ(t) 2 L2 λ δ2 λ

vλ(τ)2_L2+wλ(τ)2_L2 + t τ λ δ2

λh−1(D)

u(σ)η(σ)_x_λη(σ )λdx

dσ + t τ λ δ2 λ

1 +dD2−1_u(σ_)u

x(σ)

λu(σ)λdx

dσ λ δ2

ληλ(τ),uλ(τ)2L2+I+II,

(2.44)

with

I= t

τ λ

δ_λ2h−1(D)(uη)x

ληλdx

dσ,

II= t

τ λ δ2 λ

1 +dD2−1_uu

x

λuλdx

dσ.

(2.45)

The estimate of II. Usingλ≤δλ≤λ2, we have

II

λ t

τ

_δ2

λuλL2δ_λ2J−2

uux

λL2dσ

t

τ

λ

δ2

λuλ

2

L2 1/2

λ

δ2

λJ−2

uux λ 2 L2 1/2

dσ t τ λ δ2

λuλ2_L2 1/2

λ

λ4_J−2_uu

x_λ

2

L2 1/2

dσ

t

τ

λ

δ_λ2uλ2L2 1/2

_uu_x

L2dσ

t

τ

λ

δ_λ2uλ2L2 1/2

uL2u_x_L_∞dσ

t τ _u_x L∞ λ

δ_λ2uλ2L2dσ.

(2.46)

The estimate of I. To estimate the termI, we introduce

B(ξ)=h−1_(ξ₎₋

c ad

1/2

(12)

with

B(ξ)=% (cd−ac−ad)ξ2+c

ad|ξ|aξ2₋₁_dξ2_{+ 1}_cξ2₋₁ 1/2_{+ (acd)}1/2_aξ2₋₁_dξ2_{+ 1}&_|_ξ_|. (2.48)

Forλ >2,

λ>2 δ2

λh−1(D)

(uη)xληλdx

λ>2

δ2

λ

_c

ad 1/2

1 |D|

(uη)x

ληλdx

+

λ>2 δ2

λB(D)

(uη)x

ληλdx

.

(2.49)

Using|φ(ξ/λ)B(ξ)|λ−3_for_{λ >}_{2, we have}

λ>2 δ2

λB(D)

(uη)x

ληλdx

λ>2 δ2

λB(D)

(uη)x_λ_L2ηλ_L2

λ>2

δ_λ2B(D)(uη)xλ

2

L2

1/2

λ>2

δ_λ2ηλ2L2 1/2

λ>2

δ_λ2ηλ2L2 1/2

λ>2

λ−2(uη)x

λ

2

L2 1/2

λ>2 δ2

ληλ

2

L2 1/2

_uη

L2.

(2.50)

By (2.28),

uηL2=u·h(D)η_L2=u·

A(D) + _ad

c 1/2

|D|η_L2

uA(D)η_L2+u

|D|η_L2u_L∞A(D)η_L2+u

Ᏼη_x_L2

uL∞η_L2+u

Ᏼη_H1+uxᏴη_L2

uL∞η_L2+uL∞Ᏼη_H1+ux_L∞Ᏼη_L2

uL∞+u_x_L∞

λ

δ2

ληλ

2

L2 1/2

.

(13)

A combination of (2.50) with (2.51) yields λ>2

δ_λ2B(D)(uη)x

ληλdx

uL∞+u_x_L∞

λ

δ2_ληλ2L2. (2.52)

Using (2.28) again, we get

λ>2 δ_λ2 1

|D|

(uη)xλ·ηλdx

= λ>2

δ_λ2 ᏴuA(D) + _ad

c 1/2

|D|η_λ·ηλdx

λ>2 δ2 λ Ᏼ

u|D|η_λ·ηλdx

+ λ>2 δ2 λ Ᏼ

uA(D)η_λ·ηλdx

.

(2.53)

A(ξ)∈᏿−1_{and the inequality}_λ_≤_δ

λ≤λ2imply

λ>2 δ2 λ Ᏼ

uA(D)η_λ·ηλdx

λ>2

δ2_λᏴuA(D)η_λ2_L2

1/2

λ>2

δ_λ2ηλ2L2 1/2

λ>2

δ_λ2ηλ2L2 1/2

_uA(D)_η

H2

λ>2

δ2_ληλ2L2 1/2

uA(D)η_L2+∂x

u·A(D)η_H1

λ>2 δ2

ληλ

2

L2 1/2

uL∞A(D)η_L2+u_x_L∞A(D)η_H1+u_L∞∂_x(A(D)η)_H1

uL∞+u_x_L∞

λ

δ2

ληλ2L2,

(2.54) λ>2 δ2 λ Ᏼ

u|D|η_λ·ηλdx

=

λ>2

δ_λ2 u∂xᏴη

λ·Ᏼηλdx

≤ λ>2 δ2

λ u∂xᏴηλ·Ᏼηλdx

+ λ>2 δ2 λ

∆λ,u∂x Ᏼη·Ᏼηλdx

λ>2

δ_λ2 Ᏼη_λ2·uxdx

+

λ>2

δ_λ2∆λ,u∂x ᏴηL2·ᏴηλL2

uxL∞

λ>2 δ2

λᏴηλ

2

L2+

λ>2 δ2

λ∆λ,u∂x ᏴηL2·Ᏼη_λ_L2.

(14)

Using (2.23), we obtain

_∆_λ_,u∂_x _Ᏼ_η

L2∆λ,u∂x ∆λᏴηL2+∆λ,u∂x Id−∆λ

Ᏼη_L2, _∆_λ_,u∂_x _∆_λ_Ᏼ_η

L2uxL∞∆λᏴηL2.

(2.56)

Then we get

λ>2

δ_λ2∆λ,u∂x ∆λᏴηL2ᏴηλL2uxL∞

λ>2

δ2_λᏴηλ2L2. (2.57)

Now we estimate the term[∆λ,u∂x](Id−∆λ)ᏴηL2. Letu='_µu_µ. We have

_∆_λ_,u∂_x _Id₋_∆_λ_Ᏼη_L2=∆λ

u∂x

Id−∆λ

Ᏼη_L2

µ≥λ/8 _u_µ_∂_x

Id−∆Ᏼη_L2

µ≥λ/8

_∂_x_Ᏼ_η

L∞uµL2, (2.58)

and then by usingLemma 2.1,

λ>2 δ2

λ∆λ,u∂x Id−∆λ

Ᏼη_L2ᏴηλL2

∂xᏴηL∞

λ>2 δ2

λᏴηλL2

µ≥λ/8 _u_µ

L2

∂xᏴηL∞

λ

δ_λ2Ᏼηλ2L2

1/2

λ

δ_λ2uλ2L2 1/2

∂xᏴηL∞

λ

δ2

ληλ

2

L2+

λ

δ2

λuλ

2

L2

.

(2.59)

It follows from (2.55), (2.57), and (2.59) that

λ>2 δ2

λ

Ᏼu

ad c

1/2 |D|

η

λ·

ηλdx

uxL∞+∂xᏴηL∞

λ

δ2_ληλ2L2+

λ

δ_λ2uλ2L2

.

(2.60)

What remains is to estimate the term|δ2

λ

(

h−1_(D)((uη)

x)λ·ηλdx|, withλ=1 or 2.

(15)

implies∂xηλL2η_L2andχ(D)((uη)_x)_λ·η_λ=((uη)_x)_λ·η_λ, we get

δ2

λ h−1(D)

(uη)x

λ·ηλdx

=δ2

λ h−1(D)∂xχ(D)(uη)λ·ηλdx

h−1_(D)∂

xχ(D)

#

u·h(D)χ(D)η_λ·ηλ+

u·h(D)(Id−χ(D)η_λ·ηλ

$ dx

u·h(D)χ(D)η_λ·ηλL1+

u·∂x

h(D)Id−χ(D)

iD η

λ·

ηλ

L1

uL∞η2_L2+

∂xu·

h(D)Id−χ(D)

iD η

λ·

ηλ

L1

+

u·h(D)

Id−χ(D)

iD η

λ·

∂xηλ

L1

uL∞+u_x_L∞

η2_L2

uL∞+u_xv_L∞

λ

δ_λ2ηλ,uλ2L2.

(2.61)

A combination of (2.52), (2.60), and (2.61) with (2.49) yields

λ

δ2

λ h−1(D)

(uη)x_λ·ηλdx

u,ux,Ᏼη,∂ xᏴη_L∞

λ

δ2

ληλ,uλ2_L2, (2.62)

and so

I

t

τ

_u,u_x_,_Ᏼ_η,∂ _x_Ᏼ_η

L∞

λ

δ_λ2ηλ,uλ2L2dσ. (2.63)

Hence, by (2.44), (2.46), and (2.63)

λ

δ2

λη(t),uλ(t)2_L2

λ

δ2

λη(τ),˜ uλ(τ)2_L2+

t

τ

_u,u_x_,_Ᏼ_η,∂ _x_Ᏼ_η

L∞

λ

δ2

λη,uλ2_L2dσ.

(2.64)

The Gronwall inequality implies

λ

δ2

λη(t),uλ(t)

2

L2

λ

δ2

λη(τ),˜ uλ(τ)

2

L2exp

2u,ux,Ᏼη,∂ xᏴηL1

IL∞

.

(2.65)

(16)

Theorem2.4. FixT >0and1< σ≤2−1/ p. Let(η,u)be a smooth solution of (1.1). Then for every admissible pair(p,q),

_Jσ₍_η,u)

LTpLq(1 +T)

1/ p_{1 +}_Jσ_u L∞TL2

Jσ+1/ pη,u _L∞

TL2

×1 +u,ux,Ᏼη,∂ xᏴη,η2_L1

TL∞

. (2.66)

Proof. If (p,q) is an admissible pair, then both p and q are greater than or equal to two and diﬀerent from infinity. Therefore Minkowski inequality, Littlewood-Paley square function theorem, and Mikhlin-H¨ormander theorem show

_Jσ₍_η,u)2

LTpLq

λ

_Jσ_η

λ,uλ2LTpLq

λ

λ2σ_η

λ,uλ2LTpLq. (2.67)

ByLemma 2.2,

λ

λ2σ_η λ,uλ

2

LTpLq

1 +T)2/ p_{1 +}_Jσ_u L∞TL2

2

1 +u,ux,Ᏼη,∂ xᏴη,η

2

L1

TL∞

2

×

λ

λ2σ+2/ pηλ,uλ2L∞

TL2.

(2.68)

Choosingτ=0,δλ=λσ+1/ p, andI=[0,T] in (2.64), we get

λ

λ2σ+2/ p_η

λ,uλ2_L∞

TL2

λ

λ2σ+2/ p_η

λ(0),uλ(0)

2

L2

+

t

0

_u,u_x_,_Ᏼ_η,∂ _x_Ᏼ_η

L∞

λ

λ2σ+2/ p_η

λ(τ),uλ(τ)

2

L2dτ

1 +u,ux,Ᏼη,∂ xᏴη2L1

TL∞

Jσ+1/ p_η,u L∞

TL2.

(2.69)

Theorem 2.4follows from (2.67), (2.68), and (2.69).

3. The local well-posedness

3.1. Uniqueness. Let (η1,u1) and (η2,u2) be two solutions of the system (1.1). Letηj=

(17)

parts easily yield

1 2

d

dtv1−v2,w1−w2 2

L2

= − Γ(D)v1−v2

·v1−v2

dx+ Γ(D)w1−w2

·w1−w2

dx

−1 2

#

h−1_(D)_u

1η1−u2η2_x $

·η1−η2dx

−1 2

1 1 +dD2

u1∂xu1−u2∂xu2

·u1−u2

dx

η1−η2,u1−u22_L2+III+IV,

(3.1)

with

III= −1 2 ∂xh

−1_(D)_u 1−u2

η1+u2

η1−η2 ·

η1−η2

dx

=1 2

u1−u2

η1·∂xh−1(D)

η1−η2

dx−1

2 ∂xh

−1_(D)_u 2

η1−η2

·η1−η2

dx,

IV= −1 2

1 1 +dD2

u1−u2

∂xu1+u2

∂xu1−∂xu2 ·

u1−u2

dx.

(3.2)

It is obvious that

|IV|∂xu1,∂xu2_L∞u1−u2 2

L2. (3.3)

By (2.28), we have

h(ξ)= _ad

c 1/2

|ξ|+|ξ|+ 1 A(ξ)

1 +|ξ|. (3.4)

Moreover,A(ξ)/(1 +|ξ|)∈H1₍_R_{) and}_{A(ξ) sgn(ξ)/(1 +}_|_ξ_|₎_∈_H1₍_R_{) decay like}_|_ξ_|−2_at infinity, and so they define someL∞-multipliers. Thus,

_η₁

L∞=

ad_c 1/2∂x+ A(D)

1 +|D|∂x−A(D)Ᏼ

1 +|D|

Ᏼη1

L∞

Ᏼη1_L∞+∂xᏴη1_L∞,

(3.5)

and so

u1−u2

η1·∂xh−1(D)η1−η2

dx

η1_L∞η1−η2,u1−u2 2

L2

#Ᏼη1_L∞+∂xᏴη1_L∞ $

(η1−η2,u1−u2 2

L2.

(18)

To bound the term(∂xh−1(D)(u2(η1−η2))·(η1−η2)dx, we use (2.28) again and get

∂xh−1(D)

u2

η1−η2

·η1−η2

dx

= ∂xh−1(D)

u2

_ad

c 1/2

|D|+A(D)η1−η2·η1−η2dx

= )

c ac

1/2

Ᏼ+∂xB(D)

* u2

_ad

c 1/2

|D|η1−η2

·η1−η2

dx

+ ∂xh−1(D)u2A(D)η1−η2 ·η1−η2dx

u2∂xᏴ

η1−η2 ·Ᏼ

η1−η2

dx

+ ∂xB(D)Ᏼu2∂xᏴη1−η2 ·η1−η2dx +u2A(D)

η1−η2_L2η1−η2_L2

∂xu2

Ᏼη1−η2 2

dx+ ∂2

xB(D)Ᏼ

u2Ᏼ

η1−η2 ·

η1−η2

dx

+ ∂xB(D)Ᏼ

∂xu2Ᏼ

η1−η2 ·

η1−η2

dx+u2_L∞η1−η22_L2

#∂xu2_L∞+u2_L∞ $

η1−η2 2

L2.

(3.7)

It follows from (3.1)–(3.7) that

d

dtv1−v2,w1−w2 2

L2 #

1 +Ᏼη1,∂xᏴη1,u2,∂xu2_L∞ $

η1−η2,u1−u2 2

L2. (3.8)

By the Gronwall lemma,

η1−η2,u1−u2

(t)_L2

η1(0)−η2(0),u1(0)−u2(0)_L2exp

1 +Ᏼη1,∂xᏴη1,u2,∂xu2_L1

TL∞

, (3.9)

which clearly implies the uniqueness.

3.2. Existence. Without loss of generality, we assume 1/4< s <1. Let (η,u) be a smooth solution of the system (1.1). Settingσ=s+ 3/4,δλ=λs+1, andI=[0,T] inLemma 2.3,

we deduce that

_Js+1_η(t), _u(t)2

L∞TL2

_Js+1_η(0), _u(0)2

L2exp

2u,ux,Ᏼη,∂xᏴηL1

TL∞

(19)

If (p,q) is an admissible pair, thenσ+ 1/ p < σ+ 1/4=s+ 1<2. Therefore usingTheorem 2.4and (3.10), we get that for every admissible pair (p,q) and everyT >0,

_Jσ₍_η,u)

LTpLq(1 +T)

1/ p_{1 +}_Jσ_u L∞

TL2

1 +u,ux,Ᏼη,∂xᏴη,η2L1

TL∞

×Js+1_η(0), _u(0)

L2exp

2u,ux,Ᏼη,∂xᏴη,ηL1

TL∞

.

(3.11)

Using the Sobolev embedding in the spatial variable together with the H¨older inequality in time variable, we can choose an admissible pair (p,q) such that

_u,u_x_,_Ᏼ_η,_∂_x_Ᏼ_η,_η

L1

TL∞T

1−1/ p_Jσ_η,u

LTpLq. (3.12)

A combination of (3.12) with (3.11) yields

_u,u_x_,_Ᏼ_η,∂_˜ _x_Ᏼ_{η, ˜}_˜ _η

L1

TL∞

T1−1/ p_{(1 +}_T)1/ p_{1 +}_Jσ_u L∞TL2

1 +u,ux,Ᏼη,∂ xᏴη,η

2

L1

TL∞

×Js+1η(0), u(0)_L2exp

2u,ux,Ᏼη,∂ xᏴη,ηL1

TL∞

.

(3.13)

Choosingδλ=λs+1,τ=0, andt=Tin (2.64), we deduce

_Js+1_η(t), _u(t)2

L∞TL2

_Js+1_η(0), _u(0)2

L2

+Js+1η(t), u(t)_L∞

TL2

_u,u_x_,_Ᏼ_η,_∂_x_Ᏼ_η

L1

TL∞.

(3.14)

Then there exists a positive constantC0so small that

_Js+1_η(t), _u(t)

L∞TL2

1 1−C0

_Js+1_η(0), _u(0)

L2 fort∈[0,T] (3.15)

providing that (u,ux,Ᏼη,∂ xᏴη,η)L1

TL∞ ≤C0. Let H(T)= (u,ux,Ᏼη,∂ xᏴη)L1TL∞.

Equations (3.13) and (3.15) imply that

H(T)

1 + 1 1−C0

_Js+1_η(0),u(0)

L2

Js+1_η(0),u(0)

L2

×T1−1/ p_{(1 +}_T)1/ p_{1 +}_H2_(T)_exp_2H(T)

(3.16)

(20)

allT∈[0,TR],

1 + 1

1−C0 R

RT1−1/ p_{(1 +}_T₎1/ p_{1 +}_C2 0

exp2C0

<C0

2 . (3.17)

Then for(η(0),u(0))Hs+1≤R, we deduce from (3.16) that, for allT∈[0,T_R],

H(T)≤C0impliesH(T)≤1

2C0. (3.18)

Note thatH(0)=0. A straightforward continuity argument shows thatH(T)≤C0/2 for allT≤TR(η(0),u(0))−H1s+1. Using (3.10), we obtain that if (η,u) is a smooth solution of the system (1.1), then it satisfies

_u,u_x_,_Ᏼ_η,∂ _x_Ᏼ_η

L1

TL∞≤C ∀T≤TR, (3.19)

ηλ,uλL∞THs+1η(0),u(0)

Hs+1 ∀T≤T_R. (3.20)

The bounds (3.19) and (3.20) enable us to perform a standard compactness argument. More precisely, consider that the smooth sequence{fn(x),gn(x)}satisfies( ˜fn,gn)Hs+1 ≤Rfor some positive constantR, which converges to (f(x),g(x)) inHs₍_R₎_×_Hs+1₍_R_), where we denote by ˜fn=h−1(D)fn. Let {ηn,un} be the solution of the system (1.1)

with data (fn(x),gn(x)) which exists globally in time due to [2, Theorem 3.5]. We will

prove that{ηn,un}converges and the limit object is a solution of the system (1.1) with

data (f(x),g(x)). Indeed, (3.20) implies that{ηn,un}converges in weak∗-topology of

L∞([0,TR] :Hs×Hs+1) to some limit (η,u). Using (3.9), we deduce that {ηn,un}

con-verges strongly to (η,u) in L∞([0,TR] :H−1×L2), and therefore (unηn)x and un(un)x

converge to (uη)x anduux, respectively, in a distributional sense. This proves that the

limit (η,u) satisfies the system (1.1) in a distributional sense. The map [0,TR] t→

(η(t),u(t))∈Hs₍_R₎_×_Hs+1₍_R_{) is weakly continuous.}_{Lemma 2.3}_{, with}_δ

λ=λs+1, implies

that the map [0,TR] t→ η(t)Hs+u(t)_Hs+1 is continuous because exp (C(u,u_x, Ᏼη,∂ xᏴη)L1

IL∞) tends to one asτtends totifI=[τ,t]. Hence [0,TR] t→(η(t),u(t))∈

Hs₍_R₎_×_Hs+1₍_R_{) is continuous.}

3.3. Continuous dependence on the data. We present a proof of continuous dependence

on the data based onLemma 2.3.

Lemma3.1. Fixs∈[0, 1). Suppose that(νn_,γn₎_→₍_ν_,_γ)_in_Hs+1_{. Then there exists a}

se-quence{δλ}of positive numbers which satisfies2s+1δλ≤δ2λ≤4δλ,λ≤δλ≤λ2, andδλ/λs+1

→ ∞such thatsup_n'λδλ2(∆λνn,∆λγn)2L2<+∞.

Proof. Forλ=2j_{, set}_an

j =λ2(s+1)(νλn,γnλ)2L2,a_j=λ2(s+1)(ν_λ,γ_λ)2_L2. The assumptions imply that{an_j}j∈N→ {aj}j∈Nin1(N). Then for allk∈N, there existsNksuch that

Nk≥k, sup n

∞

j=Nk

an

(21)

For a fixed j∈N, there exists a uniquek∈Nsuch thatNk−1≤j < Nk. We setµj=2k(1−s)

andδλ=λs+1µjforλ=2j. Obviously 2s+1δλ≤δ2λ≤4δλandδλ/λs+1→+∞, and

sup

n

λ

δ2

λ∆λνn,∆λγn2L2≤ ∞

j=1 µjanj≤

∞

k=0

Nk+1

j=Nk

µjanj

≤ ∞

k=0 2k(1−s)

Nk+1

j=Nk

anj≤

∞

k=0

2k(1−s)2−2k<+∞.

(3.22)

Let{(ηn_,un₎_}_{be a sequence of solutions in}_C([0,_T_];Hs−1_×_Hs_{) with (η}n_(0),_un₍₀₎₎_→

(η(0),u(0)) inHs_×_Hs+1_{. As in the proof of the existence of solutions, we have}

ηn_(t),_un_(t)_−→_η(t), _u(t) _in_C_[0,T_];L2_×_L2_. _(3.23)

Using Lemmas3.1and2.3, we deduce

sup

n 0sup≤t≤T

λ

δ2

λ

ηn

λ(t),unλ(t)

2

L2+ηλ(t),uλ(t)

2

L2

<+∞. (3.24)

SetηΛ='λ≤Ληλ,uΛ=

'

λ≤Λuλ. Fix>0, there exists by (3.24) aΛsuch that for every

t∈[0,T],

sup

n

ηn_Λ(t)−ηn(t),un_Λ(t)−un(t)_Hs+1+ηΛ(t)−η(t),u Λ(t)−u(t)_Hs+1

< 2. (3.25)

By (3.23), there existsn0such that forn≥n0and 0≤t≤T, ηn

Λ(t)−ηΛ(t),unΛ(t)−uΛ(t)Hs+1≤(2Λ)s+1ηn_Λ(t)−ηΛ(t),un_Λ(t)−uΛ(t)_L2< 2. (3.26)

Therefore, we get forn≥n0and 0≤t≤Tthat ηn

Λ(t)−η(t), unΛ(t)−u(t)L∞THs+1

≤ηn_Λ(t)−ηΛ(t),uΛn(t)−uΛ(t)L∞THs+1

+ηn_Λ(t)−ηn_(t),_un

Λ(t)−un(t)Hs+1 +ηΛ(t)−η(t),u Λ(t)−u(t)L∞

THs+1<.

(3.27)

We complete the proof of the continuous dependence on the data.

Acknowledgment

(22)

References

[1] J. L. Bona, M. Chen, and J.-C. Saut,Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci.12

(2002), no. 4, 283–318.

[2] , Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory, Nonlinearity17(2004), no. 3, 925–952.

[3] C. Gallo,Korteweg-de Vries and Benjamin-Ono equations on Zhidkov spaces, Adv. Diﬀerential Equations10(2005), no. 3, 277–308.

[4] E. M. Stein,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, New Jersey, 1993.

Ruying Xue: Department of Mathematics, Zhejiang University, Xixi Campus, Hangzhou 310028, Zhejiang, China