Global well-posedness for the Benjamin equation in low regularity
aa r X i v : . [ m a t h . A P ] O c t Global well-posedness for the Benjaminequation in low regularity ∗ Yongsheng Li and Yifei Wu † Department of Mathematics, South China University of Technology,Guangzhou, Guangdong 510640, P. R. China
Abstract
In this paper we consider the initial value problem of the Benjaminequation ∂ t u + ν H ( ∂ x u ) + µ∂ x u + ∂ x u = 0 , where u : R × [0 , T ] R , and the constants ν, µ ∈ R , µ = 0. We use the I-method to show that it is globally well-posed in Sobolev spaces H s ( R ) for s > − / Keywords:
Benjamin equation, Bourgain space, global well-posedness, I -method MR(2000) Subject Classification:
We consider the initial value problem (IVP) for the Benjamin equation ∂ t u + ν H ( ∂ x u ) + µ∂ x u + ∂ x u = 0 , u : R × [0 , T ] R , (1.1) u ( x,
0) = u ( x ) ∈ H s ( R ) , (1.2) ∗ This work is supported by National Natural Science Foundation of China under grant numbers10471047 and 10771074. † Email: [email protected] (Y. S. Li) and [email protected] (Y. F. Wu) ν, µ ∈ R , µ = 0, H denotes the Hilbert transform defined by H f ( x ) = P . V . π Z f ( x − y ) y dy, i.e. c H f ( ξ ) = − i sgn( ξ ) ˆ f ( ξ ). The hat ˆ denotes the Fourier transform.The equation (1.1) was introduced by Benjamin [2] to describe a class of the interme-diate waves in the stratified fluid. The equation is also applied in other fluids. Recently,Gleeson, Hammerton, Papageorgiou and Vanden-Broeck [11] found a new application ininterfacial electrohydrodynamics, they considered the waves on a layer of finite depthwith the influence of vertical electric fluid and derived a Benjamin equation. The linearpart of (1.1) is formed by combining the linear parts of the Korteweg-de Vries (KdV) andBenjamin-Ono equation together, so (1.1) is often called the Korteweg-de Vries–Benjamin-Ono equation.The Benjamin equation was studied by several authors on the low regularity theories.Linares [19] proved the global well-posedness of IVP of (1.1)-(1.2) in L ( R ); Kozono,Ogawa and Tanisaka [18] showed the local well-posedness in negative index space H s ( R )with s > − / H s ( R ) for s > / s ≥ − / L ( R ).We consider the global well-posedness for (1.1)-(1.2) in H s ( R ) for s < I -method) is introduced by Colliander, Keel, Staffilani,Takaoka and Tao (see [6], [8] for examples) to study the global well-posedness theory inlow regular space. It is mainly dependent on an almost conservation law and the iterationwhich is based on the former and the local existence intervals. If the solution of anequation lacks the scale invariance, unlike the KdV equation ( ν = 0 , µ = 1 in (1.1)), thenthe threshold of the global well-posedness in H s ( R ) is decided by two ingredients: theincrement of the almost conserved quantities and the lifetime in the local theory. One ofthe argument here is to lengthen the lifetime of the local existence by establishing a variant2ocal well-posedness result, which is based on a special bilinear estimate (see Proposition3.2 below). We believe that these techniques are of independent interest and may beuseful for other equations which lacks the solution of scale invariance. Indeed, we havesucceed in applying this argument to establish the global well-posedness results of NLS-KdV system in H s ( R ) × H s ( R ) for s > /
2, which improve the results in [20]. Moreover,in order to establish global well-posedness in H s ( R ) for any s >
34 , it also requires thedevelopment of the techniques in [8], because of the complexity of the linear principleoperator which makes some troubles to give the pointwise estimates on the multipliers(for more detailed explanations, see Section 4). In this purpose, we employ some multiplierdecomposition argument, which is featured by convenient operation. More precisely, wesplit the multiplier ( M , defined in Section 4) into two parts ( ¯ M , ˜ M ), then we remain¯ M , and deduce ˜ M into a higher order cancelation by introducing the next generationmodified energy. Some notations.
We use A . B or B & A to denote the statement that A ≤ CB for some large constant C which may vary from line to line, and may depend on thecoefficients such as µ, ν and the index s . When it is necessary, we will write the constantsby C , C , · · · to see the dependency relationship. We use A ≪ B , or sometimes A = o ( B )to denote the statement A ≤ C − B , and use A ∼ B to mean A . B . A . The notation a + denotes a + ǫ for any small ǫ , and a − for a − ǫ . h·i = (1 + | · | ) / and D αx = ( − ∂ x ) α/ .We use k f k L px L qt to denote the mixed norm (cid:16) Z k f ( x, · ) k pL q dx (cid:17) p . Moreover, we denote F x to be the Fourier transform corresponding to the variable x . We define the Fourierrestriction operators P l , P l respectively as P l f ( x ) = Z | ξ |≥ l e ixξ ˆ f ( ξ ) dξ, P l f ( x ) = Z | ξ |≤ l e ixξ ˆ f ( ξ ) dξ for any l >
0. Finally, we denote the constant a = 2 max (cid:16) , (cid:12)(cid:12)(cid:12) ν µ (cid:12)(cid:12)(cid:12)(cid:17) , it will be often usedin the analysis.Now we introduce some definitions before presenting our main result.For s, b ∈ R , define the Bourgain space X s,b to be the closure of the Schwartz class3nder the norm k u k X s,b ≡ (cid:18)ZZ h ξ i s h τ − φ ( ξ ) i b | ˆ u ( ξ, τ ) | dξdτ (cid:19) , (1.3)where φ ( ξ ) = − νξ | ξ | + µξ is the phase function of the semigroup generated by the linearBenjamin equation.For an interval Ω, we define X Ω s,b to be the restriction of X s,b on R × Ω with the norm k u k X Ω s,b = inf {k U k X s,b : U | t ∈ Ω = u | t ∈ Ω } . (1.4)When Ω = [ − δ, δ ], we write X Ω s,b as X δs,b . By the limiting argument we see that, for every u ∈ X δs,b , there exists an extension ˜ u ∈ X s,b such that ˜ u = u on Ω and k u k X δs,b = k ˜ u k X s,b (see also [12]).Let s < N ≫ I N,s is defined as [ I N,s u ( ξ ) = m N,s ( ξ )ˆ u ( ξ ) , (1.5)where the multiplier m N,s ( ξ ) is a smooth, monotone function satisfing 0 < m N,s ( ξ ) ≤ m N,s ( ξ ) = ( , | ξ | ≤ N,N − s | ξ | s , | ξ | > N. (1.6)Sometimes we denote I N,s and m N,s as I and m respectively for short if there is noconfusion.It is obvious that the operator I N,s maps H s ( R ) into L ( R ) with equivalent norms forany s <
0. More precisely, there exists some positive constant C such that C − k u k H s ≤ k I N,s u k L ≤ CN − s k u k H s . (1.7)Moreover, I N,s can be extended to a map (still denoted by I N,s ) from X s,b to X ,b whichsatisfies C − k u k X s,b ≤ k I N,s u k X ,b ≤ CN − s k u k X s,b for any s < , b ∈ R .Now we are ready to state our main result.4 heorem 1.1 The IVP (1.1)-(1.2) is globally well-posed in H s ( R ) for s > − . Moreprecisely, for any u ∈ H s ( R ) with s > − and T > , (1.1)-(1.2) has a unique solution u ∈ X Ts, + ⊂ C ([0 , T ] , H s ( R )) , and the solution map u u [ u ] is continuous from H s ( R ) to X Ts, + . The rest of this article is organized as follows. In Section 2, we present some prelimi-nary estimates. In Section 3, we will give a key bilinear estimate and establish the variantlocal well-posedness result. In Section 4, we use the I-method to prove Theorem 1.1.
As it’s well-known, the corresponding linear equation of (1.1) ∂ t u + ν H ( ∂ x u ) + µ∂ x u = 0 , x, t ∈ R (2.1)generates a unitary group { S ( t ) } t ∈ R on L ( R ) such that u = S ( t ) u solves (2.1)-(1.2). Itis also defined explicitly by spatial Fourier transform as \ S ( t ) u ( ξ ) , e itφ ( ξ ) b u ( ξ ) . The first part of estimates in this section are some standard Strichartz estimates concern-ing this group. We remark that some Fourier restriction operators shall be used in theseestimates because of the presence of the nontrivial zero points of the phase function φ ( ξ ),which is different from the KdV equation. See [13] for details. Lemma 2.1
For u ∈ L ( R ) , k D x S ( t ) P a u k L ∞ x L t . k u k L , (2.2) (cid:13)(cid:13)(cid:13) D − x S ( t ) P a u (cid:13)(cid:13)(cid:13) L x L ∞ t . k u k L , (2.3) k D αx S ( t ) P a u k L px L qt . k u k L , (2.4) k S ( t ) u k L x L t . k u k L , (2.5) where p = 15 (1 − α ) , q = 110 (4 α + 1) , for any α ∈ [ − , . roof. See [13] for the proof of (2.2), (2.3) and (2.5) (see also [15]). (2.4) follows byinterpolation between (2.2) and (2.3). Lemma 2.2
Let α, p, q be as in Lemma 2.1. For F ∈ X , + , k D αx P a F k L px L qt . k F k X ,
12 + . (2.6) Proof.
We shall omit the details here, since the argument is well-known (see [16]). By interpolating between (2.6) and the following equality k F k L xt = k F k X , , (2.7)we can generalize (2.6) as below. Lemma 2.3
For any θ ∈ [0 , , α ∈ [ − θ , θ ] and F ∈ X , θ + , we have k D αx P a F k L px L qt . k F k X , θ , (2.8) where p = 12 − α − θ , q = 12 + 25 α − θ . Similarly, combining (2.5) with (2.7), we have
Lemma 2.4
For ρ ≥ q − q , q ∈ [2 , and F ∈ X ,ρ + , we have k F k L qxt . k F k X ,ρ + . (2.9)At the end of this part, we introduce an operator which first appeared in [12] (a similarargument was used in [5]). Define the bilinear Fourier integral operator I s ( u, v ) by \ I s ( u, v )( ξ, τ ) = Z ⋆ | φ ′ ( ξ ) − φ ′ ( ξ ) | s ˆ u ( ξ , τ )ˆ v ( ξ , τ ) , (2.10)where Z ⋆ = Z ξ ξ ξ, τ + τ = τ dξ dτ . Now we give some estimates on this operator. Lemma 2.5
Let I be defined by (2.10), then for any u, v ∈ X , + , (cid:13)(cid:13)(cid:13) I ( u, v ) (cid:13)(cid:13)(cid:13) L xt . k u k X ,
12 + k v k X ,
12 + . (2.11)6 roof. We use the argument in [5] to prove the result. By the definition (2.10) and theduality, the left-hand side of (2.11) is equal tosup k h k L ≤ Z | φ ′ ( ξ ) − φ ′ ( ξ ) | ˆ h ( ξ + ξ , τ + τ )ˆ u ( ξ , τ )ˆ v ( ξ , τ ) dξ dξ dτ dτ . (2.12)First, we change variables by setting τ = λ + φ ( ξ ) , τ = λ + φ ( ξ ) , then (2.12) is changed intosup k h k L ≤ Z | φ ′ ( ξ ) − φ ′ ( ξ ) | ˆ h ( ξ + ξ , λ + λ + φ ( ξ ) + φ ( ξ )) · ˆ u ( ξ , λ + φ ( ξ ))ˆ v ( ξ , λ + φ ( ξ )) dλ dλ dξ dξ . (2.13)We change variables again as follows. Let( η, ω ) = T ( ξ , ξ ) , (2.14)where η = T ( ξ , ξ ) = ξ + ξ ,ω = T ( ξ , ξ ) = λ + λ + φ ( ξ ) + φ ( ξ ) . Then the Jacobian J of this transform satisfies | J | = | φ ′ ( ξ ) − φ ′ ( ξ ) | . Define H ( η, ω, λ , λ ) = ˆ u ˆ v ◦ T − ( η, ω, λ , λ ) , then, by eliminating | J | with | φ ′ ( ξ ) − φ ′ ( ξ ) | , (2.13) has a bound ofsup k h k L ≤ Z ˆ h ( η, ω ) · H ( η, ω, λ , λ ) | J | dηdωdλ dλ . (2.15)Further, by H¨older’s inequality we have(2.15) ≤ sup k h k L ≤ (cid:13)(cid:13) ˆ h (cid:13)(cid:13) L ηω · Z (cid:16) Z | H ( η, ω, λ , λ ) | | J | dηω (cid:17) dλ dλ . Z k ˆ u ( ξ , λ + φ ( ξ )) k L ξ dλ · Z k ˆ v ( ξ , λ + φ ( ξ )) k L ξ dλ . k u k X ,
12 + k v k X ,
12 + , When s = 0, by (2.9) we have k uv k L xt ≤ k u k L xt k v k L . k u k X ,
12 + k v k X ,
16 + . (2.16)Interpolation between (2.11) and (2.16), we have Corollary 2.6
Let I s be defined by (2.10), then for any s ∈ [0 ,
12 ] , ˜ b ≥
16 + 23 s , k I s ( u, v ) k L xt . k u k X ,
12 + k v k X , ˜ b + . (2.17)It’s easy to verify that Lemma 2.5 and Corollary 2.6 still hold if one replaces theoperator I s by the one (still denoted by I s ) defined as \ I s ( u, v )( ξ , τ ) = Z ⋆ | φ ′ ( ξ ) − φ ′ ( ξ ) | ˆ u ( ξ, τ )ˆ v ( ξ , τ ) . (2.18)We now continue to present some estimates of the group { S ( t ) } in X s,b . We denote ψ ( t ) to be an even smooth characteristic function of the interval [ − , Lemma 2.7 ([16]) Let δ ∈ (0 , , s ∈ R , then the following estimates hold: (i) k u k C t ( H sx ; R ) . k u k X s,b , ∀ b ∈ ( 12 , , u ∈ X s,b ; (ii) k ψ ( t ) S ( t ) u k X s,b . k u k H s , ∀ b ∈ ( 12 , , u ∈ H s ( R ) ; (iii) (cid:13)(cid:13)(cid:13)(cid:13) ψ ( t ) Z t S ( t − s ) F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) X s,b . k F k X s,b − , ∀ b ∈ ( 12 , , F ∈ X s,b − ; (iv) k ψ ( t/δ ) f k X s,b ′ . δ b − b ′ k f k X s,b , for ≤ b ′ ≤ b < . Remark.
See [5] for the proof of Lemma 2.7 (iv) when b = b ′ . Corollary 2.8
Let b ∈ [0 ,
12 ) , δ ∈ (0 , , then k u k X δs,b . δ ( − b ) − k u k X δs, . (2.19) Proof.
Let ˜ u be the extension of u ∈ X δs, defined after Definition 1.1. By Lemma 2.7(iv), k u k X δs,b ≤ k ψ ( t/δ )˜ u k X s,b . δ ( − b ) − k ˜ u k X s, = δ ( − b ) − k u k X δs, . This completes the proof of the corollary. A Bilinear Estimate and the Local Well-posedness
In this section, we will establish a variant local well-posedness result as follows.
Proposition 3.1
Let s > − / , then IVP (1.1)-(1.2) is locally well-posed in H s ( R ) .Moreover, the solution exists on the interval [0 , δ ] with the lifetime δ ∼ k I N,s u k − − L (3.1) when N ≥ N for some large number N such that N − +0 · k I N ,s u k L ∼ , (3.2) further, the solution satisfies the estimate k I N,s u k X δ ,
12 + . k I N,s u k L . (3.3) Remark.
The condition (3.2) is reasonable by taking N & k u k s + H s , since k Iu k L ≤ CN − s k u k H s .Compared with the standard local well-posedness result, this proposition is establishedfor adapting to the I-method. It gives the estimates on the lifetime and the solution underthe X δs, + - norm, with the operator I N,s . The proposition is based on the following bilinearestimates.
Proposition 3.2
Let s ∈ ( − , , b = 12 + , δ ∈ (0 , , N ≫ , then for any u, v ∈ X δs,b , k ψ ( t/δ ) ∂ x I (˜ u ˜ v ) k X ,b − . ( δ − + N − + ) k Iu k X δ ,b k Iv k X δ ,b , (3.4) where I = I N,s , ˜ u and ˜ v are the extensions of u | t ∈ [ − δ,δ ] and v | t ∈ [ − δ,δ ] such that k Iu k X δ ,b = k I ˜ u k X ,b and k Iv k X δ ,b = k I ˜ v k X ,b . Remark.
As we described in Section 1, the global well-posedness result shall be effectedby the estimate (3.1) on the lifetime. By the equivalence (1.7), we have from Proposition3.1 that δ − ∼ k Iu k − L . On the other hand, a similar local well-posedness result also can9e achieved by a standard process. In fact, we can obtain the bilinear estimates (betterthan what obtained in [17]) that k ∂ x ( uv ) k X s,b − . k u k X s,b ′ k u k X s,b ′ for any s ∈ ( − , , b ∈ ( 12 ,
34 + 13 s ] , b ′ ∈ ( 12 ,
1] (we omit the proofs here). Then by ageneral result (see [9]), we have k ∂ x I ( uv ) k X ,b − . k Iu k X ,b ′ k Iu k X ,b ′ under the same assumptions. Thus we can establish the local well-posedness similar toProposition 3.1 but replacing the lifetime estimate by δ b − b ′ ∼ k Iu k − L . However, since b − b ′ <
14 + 13 s , it is much weaker than (3.1).Now we turn to an arithmetic fact which is often used below, before the proof of themain results of this section. We note that | ( τ − φ ( ξ )) − ( τ − φ ( ξ )) − ( τ − φ ( ξ )) | & | ξ || ξ || ξ | , (3.5)where ξ = ξ + ξ and max {| ξ | , | ξ | , | ξ |} ≥ a . In fact, we may assume that | ξ | ≥ | ξ | bysymmetry, then ξ · ξ ≥
0. We only consider the case: ξ, ξ ≥ , ξ ≤ τ − φ ( ξ )) − ( τ − φ ( ξ )) − ( τ − φ ( ξ )) = φ ( ξ ) + φ ( ξ ) − φ ( ξ )= − ν ( ξ − ξ − ξ ) − µξξ ξ = ξξ (2 ν − µξ ) , since | ξ | ≥ a , we have (3.5). According to (3.5), one of the following three cases alwaysoccurs:( a ) | τ − φ ( ξ ) | & | ξ || ξ || ξ | ; ( b ) | τ − φ ( ξ ) | & | ξ || ξ || ξ | ; ( c ) | τ − φ ( ξ ) | & | ξ || ξ || ξ | . (3.6) Proof of Proposition 3.2.
By duality and Plancherel’s identity, it suffices to show that ZZ ξm ( ξ ) b h δ ( ξ, τ ) c ˜ u ˜ v ( ξ, τ ) dξdτ . K k h k X , − b k I ˜ u k X ,b k I ˜ v k X ,b ≡ K · RHS (3.7)10or any h ∈ X , − b . Here we denote h δ ( x, t ) = ψ ( t/δ ) h ( x, t ) and K = δ − + N − + forshort.We writeˆ f ( ξ, τ ) = c I ˜ u ( ξ, τ ) = m ( ξ ) b ˜ u ( ξ, τ ) , ˆ g ( ξ, τ ) = c I ˜ v ( ξ, τ ) = m ( ξ ) b ˜ v ( ξ, τ ) , then (3.7) is changed into LHS ≡ Z ∗ | ξ | m ( ξ ) m ( ξ ) m ( ξ ) b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . K k h k X , − b k f k X ,b k g k X ,b = K · RHS, (3.8)where Z ∗ = Z ξ ξ ξ, τ + τ = τ dξ dξ dτ dτ , which is corresponding to convolution.Without loss of generality, we may assume further b h δ , ˆ f , ˆ g ∈ L ( R ) are nonnegativefunctions. By symmetry, we may consider only the integration over the region of | ξ | ≥ | ξ | (so, ξ · ξ ≥ | ξ | ≤ | ξ | ), and divide it into the following different parts.Part 1. | ξ | , | ξ | , | ξ | . N ; Part 2. | ξ | . N, | ξ | , | ξ | ≫ N ;Part 3. | ξ | . N, | ξ | , | ξ | ≫ N ; Part 4. | ξ | , | ξ | , | ξ | ≫ N. Part 1 . | ξ | , | ξ | , | ξ | . N .In this part, m ( ξ ) , m ( ξ ) , m ( ξ ) ∈ [ C s ,
1] for some constant C , therefore LHS ∼ Z ∗ | ξ | b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . Here and below
LHS denotes the integral in the left hand side of (3.8) over the corre-sponding part of the integration region.
Subpart (I) . | ξ | . a . By Plancherel’s identity, H¨older’s inequality, (2.9), (2.19) andLemma 2.7(iv), we have LHS . Z ∗ b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ )= Z h δ ( x, t ) f ( x, t ) g ( x, t ) dxdt ≤ k h δ k L xt k f k L xt k g k L xt . k h δ k X , k I ˜ u k X ,
13 + k I ˜ v k X ,
13 + . δ − RHS. ubpart (II) . | ξ | ≫ | ξ | and | ξ | ≫ a . In this subpart we have | ξ | ∼ | ξ | , and | φ ′ ( ξ ) − φ ′ ( ξ ) | ∼ | ξ | . Therefore, by Plancherel’s identity, H¨older’s inequality, (2.11) and Lemma 2.7(iv), wehave
LHS . Z b h δ ( ξ, τ ) \ I ( f, g )( ξ, τ ) dξdτ ≤ k h δ k L xt (cid:13)(cid:13)(cid:13) I ( f, g ) (cid:13)(cid:13)(cid:13) L xt . k h δ k X , k f k X ,b k g k X ,b . δ − RHS,
Subpart (III) . | ξ | ∼ | ξ | and | ξ | ≫ a . We split the integration into three casesaccording to (3.6).(a) | τ − φ ( ξ ) | & | ξ || ξ || ξ | & | ξ | (since | ξ | . | ξ | ). Then by (2.9), Lemma 2.7(iv) and(2.19), LHS . Z ∗ h τ − φ ( ξ ) i b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) ≤ k h δ k X , k f k L xt k g k L xt . k h δ k X , k f k X ,
13 + k g k X ,
13 + . δ − RHS.
For cases (b) and (c), the estimation is similar to (a), and we omit the details.
Part 2 . | ξ | . N , | ξ | , | ξ | ≫ N .In this part, | ξ | ∼ | ξ | ≫ | ξ | . By (1.6), LHS . N s Z ∗ | ξ || ξ | − s b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . We split the integration into three cases according to (3.6).(a) | τ − φ ( ξ ) | & | ξ || ξ || ξ | . Since | φ ′ ( ξ ) − φ ′ ( ξ ) | ∼ | ξ || ξ | , s > −
34 , we have
LHS . N s Z ∗ | ξ | b | ξ | − s +2 b − h τ − φ ( ξ ) i − b b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . N s Z ∗ | ξ | b − | ξ | − s +2 b − h τ − φ ( ξ ) i − b b h δ ( ξ, τ ) | ξ | | ξ | ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . N b − Z h τ − φ ( ξ ) i − b b h δ ( ξ, τ ) \ I ( f, g )( ξ, τ ) dξdτ, (3.9)since − s + 2 b − ≤ b −
12 small enough. Therefore, by (2.11) and Lemma2.7 (iv), (3.9) is controlled by N − + k h δ k X , − b (cid:13)(cid:13)(cid:13) I ( f, g ) (cid:13)(cid:13)(cid:13) L xt . N − + RHS. (b) | τ − φ ( ξ ) | & | ξ || ξ || ξ | . Since | φ ′ ( ξ ) − φ ′ ( ξ ) | ∼ | ξ | in this situation, then by the definition of (2.18), we have LHS . N s Z ∗ | ξ | − b | ξ | − s − b b h δ ( ξ, τ ) h τ − φ ( ξ ) i b ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . N s Z ∗ | ξ | − b | ξ | − s − b − θ h τ − φ ( ξ ) i b ˆ f ( ξ , τ ) | ξ | θ b h δ ( ξ, τ )ˆ g ( ξ , τ ) , (3.10)where θ = ( 54 − b ) − such that 1 − b >
16 + 32 θ and − s − b − θ <
0. Therefore, by (2.17)in the version of (2.18) and Lemma 2.7 (iv), (3.10) is controlled by N − b − θ Z h τ − φ ( ξ ) i b ˆ f ( ξ , τ ) \ I θ ( h δ , g )( ξ , τ ) dξ dτ . N − + k f k X ,b (cid:13)(cid:13) I θ ( h δ , g ) (cid:13)(cid:13) L xt . N − + k f k X ,b k h δ k X , − b k g k X ,b . N − + RHS, where we note that 1 − b − θ = −
32 +.The part (c) is similar to (b), and the details are omitted.
Part 3 . | ξ | . N , | ξ | , | ξ | ≫ N .In this part, LHS ∼ Z ∗ | ξ | b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . LHS can be controlled by δ − RHS.
Part 4 . | ξ | , | ξ | , | ξ | ≫ N .In this part, LHS . N s Z ∗ | ξ | s | ξ | − s | ξ | − s b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . First, we divide the integral into two subparts, then in each subpart below we split itagain into three subsubparts by (3.6).
Subpart (I) . | ξ | ≪ | ξ | , then | ξ | ∼ | ξ | .(a) | τ − φ ( ξ ) | & | ξ || ξ || ξ | ∼ | ξ || ξ | . Since | φ ′ ( ξ ) − φ ′ ( ξ ) | ∼ | ξ || ξ | , then by (2.11) and Lemma 2.7 (iv), we have LHS . N s Z ∗ | ξ | s + b | ξ | − s +2 b − h τ − φ ( ξ ) i − b b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) ≤ N s Z ∗ | ξ | s + b − | ξ | − s +2 b − h τ − φ ( ξ ) i − b b h δ ( ξ, τ ) | ξ | | ξ | ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . N b − Z h τ − φ ( ξ ) i − b b h δ ( ξ, τ ) \ I ( f, g )( ξ, τ ) dξdτ . N − + k h δ k X , − b (cid:13)(cid:13)(cid:13) I ( f, g ) (cid:13)(cid:13)(cid:13) L xt . N − + RHS, since s + b − ≤ − s + 2 b − ≤ | τ − φ ( ξ ) | & | ξ || ξ || ξ | ∼ | ξ || ξ | . Since | φ ′ ( ξ ) − φ ′ ( ξ ) | ∼ | ξ | in this situation, then by (2.17) in the version of (2.18) and Lemma 2.7 (iv), we have LHS . N s Z ∗ | ξ | s − b | ξ | − s − b b h δ ( ξ, τ ) h τ − φ ( ξ ) i b ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . N − b − θ Z h τ − φ ( ξ ) i b ˆ f ( ξ , τ ) \ I θ ( h δ , g )( ξ , τ ) dξ dτ . N − + k f k X ,b (cid:13)(cid:13) I θ ( h δ , g ) (cid:13)(cid:13) L xt . N − + k f k X ,b k h δ k X , − b k g k X ,b . N − + RHS.
Subpart (II) . | ξ | ∼ | ξ | .(a) | τ − φ ( ξ ) | & | ξ || ξ || ξ | ∼ | ξ | | ξ | . By Lemma 2.3 we have LHS . N s Z ∗ | ξ | b − | ξ | − s + b − h τ − φ ( ξ ) i − b b h δ ( ξ, τ ) ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . N s Z ∗ | ξ | b − | ξ | − s + b − h τ − φ ( ξ ) i − b b h δ ( ξ, τ ) \ D x P a f ( ξ , τ ) \ D x P a g ( ξ , τ ) . N b − k h δ k X , − b (cid:13)(cid:13)(cid:13) D x P a f (cid:13)(cid:13)(cid:13) L xt (cid:13)(cid:13)(cid:13) D x P a g (cid:13)(cid:13)(cid:13) L xt . N b − k h δ k X , − b k f k X ,
38 + k g k X ,
38 + . N − + δ − RHS, since | ξ | . | ξ | and 3 b − s − ≤ | τ − φ ( ξ ) | & | ξ || ξ || ξ | ∼ | ξ | | ξ | . By Lemma 2.3 we have LHS . N s Z ∗ | ξ | − b | ξ | − s − b b h δ ( ξ, τ ) h τ − φ ( ξ ) i b ˆ f ( ξ , τ ) ˆ g ( ξ , τ ) . N s Z ∗ | ξ | − b | ξ | − s − b − \ D x P a h δ ( ξ, τ ) h τ − φ ( ξ ) i b b f ( ξ , τ ) \ D x P a g ( ξ , τ ) . N − b + (cid:13)(cid:13)(cid:13) D x P a h δ (cid:13)(cid:13)(cid:13) L xt k f k X ,b (cid:13)(cid:13)(cid:13) D x P a g (cid:13)(cid:13)(cid:13) L xt . N − b + k h δ k X ,
38 + k f k X ,b k g k X ,
38 + . N − − δ − RHS,
The part (c) is treated in a very similar manner as (a) and (b), so we omit the details.This completes the proof of the proposition. Now, we are ready to prove Proposition 3.1. For this purpose, we define the operatorΦ δ as Φ δ u ( t ) = ψ ( t ) S ( t ) u + ψ ( t ) Z t S ( t − s ) ψ ( s/δ ) ∂ x ˜ u ( s ) ds, (3.11)where ˜ u is the extension of u | t ∈ [ − δ,δ ] such that k Iu k X δ ,b = k I ˜ u k X ,b ( I = I N,s ). Then(1.1)-(1.2) is locally well-posed if Φ δ has a unique fixed point.15cting the operator I onto both sides of (3.11), taking the X δ ,b -norm for b = 12 +, andemploying Lemma 2.7 and (3.4) we have k I (Φ δ u ) k X δ ,b . k ψ ( t ) S ( t ) Iu k X δ ,b + (cid:13)(cid:13)(cid:13)(cid:13) ψ ( t ) Z t S ( t − s ) ψ ( s/δ ) ∂ x I (˜ u ( s )) ds (cid:13)(cid:13)(cid:13)(cid:13) X δ ,b . k Iu k L + (cid:13)(cid:13) ψ ( t/δ ) ∂ x I (˜ u ) (cid:13)(cid:13) X ,b − ≤ C k Iu k L + C ( δ − + N − − ) k Iu k X δ ,b . Consider B r = { u : Iu ∈ X δ ,b , such that k Iu k X δ ,b ≤ r } , where r = 2 C k Iu k L , some small δ and large N will be decided later. Then B r is acomplete metric space. Observing that if we choose N , δ such that C ( δ − + N − − ) r ≤ , (3.12)then the operator Φ δ maps B r into itself. (3.12) is valid if we choose N, δ such that100 C C N − − · k Iu k L ≤ C C δ − ≤ k Iu k − L . (3.13)Similarly, under the condition (3.13), one has k I (Φ δ ( u ) − Φ δ ( v )) k X δ ,b ≤ k I ( u − v ) k X δ ,b , ∀ u, v ∈ B r , and thus Φ δ is a contraction on B r . Thus by the fixed point theorem, we complete theproof of Proposition 3.1. In this section, we consider the global well-posedness of (1.1)-(1.2) by adopting the argu-ment in [8], which is based on a multilinear correction technique and iteration. Unfortu-nately, the phase function φ ( ξ ) loses some symmetries which brings much convenience in[8] to obtain some pointwise estimates, this makes many difficulties. To overcome thesedifficulties, we use some multiplier decomposition argument to deal with it. More pre-cisely, we also introduce the third version modified energy, but we only use it to cancel apart of “correction term” in the second modified energy. Similar argument were appearedpreviously in [1], [4], [10]. 16 .1 Modified Energies and I-method First we observe some arithmetic facts. Recall that φ ( ξ ) = − νξ | ξ | + µξ , let α k ≡ i ( φ ( ξ ) + · · · + φ ( ξ k )) . Then α = 0 for ξ + ξ = 0. Similar to (3.5), when ξ + ξ + ξ = 0 with max {| ξ | , | ξ | , | ξ |} ≥ a , then | α | & | ξ || ξ || ξ | . (4.1)Next, we state the definitions of the modified energies and adopt the notations in [8].In this section, let u be the real-valued solution of (1.1)-(1.2). For a given function m ( ξ , · · · , ξ k ) defined on the hyperplaneΓ k = { ( ξ , · · · , ξ k ) : ξ + · · · + ξ k = 0 } , we define Λ k ( m ) = Z Γ k m ( ξ , · · · , ξ k ) k Y j =1 F x u ( ξ j , t ) dξ · · · dξ k − . Denote the modified energy as E I ( t ) ≡ k Iu ( t ) k L = Λ ( m ( ξ ) m ( ξ )) , then by the arithmetic fact above, (1.1) and a direct computation (cf. [8]), one has ddt E I ( t ) = Λ ( m ( ξ ) m ( ξ ) α ) + Λ ( M ) = Λ ( M ) , where M ( ξ , ξ , ξ ) = − i (cid:0) m ( ξ ) ξ + m ( ξ ) ξ + m ( ξ ) ξ (cid:1) . (4.2)Define the second modified energy E I ( t ) by E I ( t ) = Λ ( σ ) + E I ( t ) , (4.3)where σ ( ξ , ξ , ξ ) = − M ( ξ , ξ , ξ ) (cid:14) α ( ξ , ξ , ξ ) , ddt E I ( t ) = Λ ( M ) . (4.4)where M ( ξ , · · · , ξ ) = − i [ σ ( ξ , ξ , ξ + ξ )( ξ + ξ )] sym . (4.5)We denote the sets thatΩ = { ( ξ , ξ , ξ , ξ ) ∈ Γ : | ξ | , · · · , | ξ | & N } , and rewrite (4.4) by ddt E I ( t ) = Λ ( ¯ M ) + Λ ( ˜ M ) , (4.6)where ¯ M = ( χ Γ − χ Ω ) M ; ˜ M = χ Ω M . (4.7)Now, we define the third modified energy E I ( t ) as E I ( t ) = Λ ( σ ) + E I ( t ) , (4.8)where σ ( ξ , · · · , ξ ) = − ˜ M ( ξ , · · · , ξ ) α ( ξ , · · · , ξ ) . Then one has ddt E I ( t ) = Λ ( ¯ M ) + Λ ( M ) . (4.9)where M ( ξ , · · · , ξ ) = − i [ σ ( ξ , ξ , ξ , ξ + ξ )( ξ + ξ )] sym . (4.10) Remark.
The second version of modified energy is not enough to obtain the claim resultin Theorem 1.1 for s >
34 . Indeed, the best estimate (in our opinion) on almost conservedquantity is sup t ∈ [0 ,δ ] k Iu ( t ) k L . k Iu k L + ( N − + δ − + N − ) k Iu k L , which implies that (1.1)-(1.2) is globally well-posed in H s ( R ) when s > − /
2. So one mayneed to introduce the third modified energy. For this purpose, as the general argument,one may defined σ = − M α , and give the definition of M as (4.10). Unfortunately, it is18uch hard to give the pointwise estimates on this M , especially because of the complexityof the phase function φ ( ξ ). We note that the most hand case appears when the third andthe fourth highest values of | ξ | , · · · , | ξ | : | ξ ∗ | , | ξ ∗ | ≪ N , but on the other hand, thiscase behaves well in the estimation of the increment of E I ( t ). This is the reason that wedivided the multiplier M into two parts and use the multiplier decomposition argumentto deal with it. The argument brings much convenient for us in this paper. By mean value theorem, one has an estimate on M : Lemma 4.1
Let | ξ min | = min {| ξ | , | ξ | , | ξ |} , ξ + ξ + ξ = 0 , then, | M ( ξ , ξ , ξ ) | . m ( ξ min ) | ξ min | , (4.11)Now we state some simple facts. Lemma 4.2
The following estimates hold, (i) α ( ξ , ξ , ξ + ξ ) + α ( ξ , ξ , ξ + ξ ) = α ( ξ , ξ , ξ , ξ ) ; (ii) | α ( ξ , ξ , ξ , ξ ) | ∼ | ξ + ξ || ξ + ξ || ξ + ξ | ; (iii) | m ( ξ ) ξ + m ( ξ ) ξ + m ( ξ ) ξ + m ( ξ ) ξ | . | α ( ξ , ξ , ξ , ξ ) | / | ξ m | . Proof. (i) easily follows from a direct check. For (ii), we may assume that | ξ | ≥ | ξ | ≥| ξ | ≥ | ξ | by symmetry. By the facts that α ( ξ , ξ , ξ , ξ ) = − ν ( ξ | ξ | + ξ | ξ | + ξ | ξ | + ξ | ξ | ) + µ ( ξ + ξ + ξ + ξ )and ξ + ξ + ξ + ξ = ( ξ + ξ )( ξ + ξ )( ξ + ξ ) , we only need to show ξ | ξ | + ξ | ξ | + ξ | ξ | + ξ | ξ | ≪ | ξ + ξ || ξ + ξ || ξ + ξ | . ξ > , ξ > , ξ > , ξ < , ξ <
0; (2) , ξ > , ξ < , ξ < , ξ > , ξ > , ξ < , ξ < , ξ < . (4.12)For (1), (cid:12)(cid:12) ξ | ξ | + ξ | ξ | + ξ | ξ | + ξ | ξ | (cid:12)(cid:12) = | ξ + ξ − ξ − ξ | = | ξ + ξ || ξ + ξ | ≪ | ξ + ξ || ξ + ξ || ξ + ξ | . For (2), (cid:12)(cid:12) ξ | ξ | + ξ | ξ | + ξ | ξ | + ξ | ξ | (cid:12)(cid:12) = | ξ + ξ || ξ + ξ | ≪ | ξ + ξ || ξ + ξ || ξ + ξ | . For (3), on one hand, (cid:12)(cid:12) ξ | ξ | + ξ | ξ | + ξ | ξ | + ξ | ξ | (cid:12)(cid:12) = | ξ − ξ − ξ − ξ |≤ | ξ + ξ || ξ − ξ | + | ξ + ξ | ∼ | ξ + ξ || ξ | ;on the other hand, we note that | ξ | − | ξ | & | ξ | , so | ξ + ξ || ξ + ξ || ξ + ξ | ∼ | ξ + ξ || ξ | . Thus we have the claim (ii).For (iii), we also assume that | ξ | ≥ | ξ | ≥ | ξ | ≥ | ξ | and split it into three cases as(4.12). Then, for (1), we have | ξ | ∼ | ξ | , thus by the double mean value theorem (see [8],for example), | m ( ξ ) ξ + · · · + m ( ξ ) ξ | . | ξ + ξ || ξ + ξ || f ′′ ( ξ ) | . | ξ + ξ || ξ + ξ || ξ + ξ | / | ξ | , where f ( ξ ) = m ( ξ ) ξ . For (2), if | ξ | ∼ | ξ | , it can be show similarly as (1). If | ξ | ≫ | ξ | ,we have | ξ | − | ξ | ∼ | ξ | , thus | ξ + ξ || ξ + ξ || ξ + ξ | ∼ | ξ + ξ || ξ | . | ξ + ξ | ∼ | ξ | , we have, | m ( ξ ) ξ + · · · + m ( ξ ) ξ | . | ξ + ξ | + | ξ | + | ξ | . | ξ + ξ | . | ξ + ξ || ξ + ξ || ξ + ξ | / | ξ | . For (3), we can treat it as (2) when | ξ | ≫ | ξ | . Hence we have the claim by (ii). Thiscompletes the proof of the lemma. Now we establish the following pointwise upper bound on the multiplier M . Lemma 4.3 let the M defined in (4.5) and | ξ max | = max {| ξ | , | ξ | , | ξ |} , we have | M ( ξ , ξ , ξ , ξ ) | . min (cid:26) | ξ max | , | α ( ξ , ξ , ξ , ξ ) || ξ || ξ || ξ || ξ | (cid:27) . (4.13) Proof.
The first term | M | . | ξ max | easily follows from (4.1) and (4.11). Now we turnto prove the second term. Rewrite M as M ( ξ , ξ , ξ , ξ ) = C (cid:20) M ( ξ , ξ , ξ + ξ )( ξ + ξ ) α ( ξ , ξ , ξ + ξ ) + M ( ξ , ξ , ξ + ξ )( ξ + ξ ) α ( ξ , ξ , ξ + ξ ) (cid:21) sym = C (cid:20)(cid:18) M ( ξ , ξ , ξ + ξ ) α ( ξ , ξ , ξ + ξ ) − M ( ξ , ξ , ξ + ξ ) α ( ξ , ξ , ξ + ξ ) (cid:19) ( ξ + ξ ) (cid:21) sym . (4.14)Moveover, it is easy to see that M ( ξ , ξ , ξ + ξ ) α ( ξ , ξ , ξ + ξ ) − M ( ξ , ξ , ξ + ξ ) α ( ξ , ξ , ξ + ξ ) = I + I , for the I , I defined as I = M ( ξ , ξ , ξ + ξ ) · (cid:18) α ( ξ , ξ , ξ + ξ ) + 1 α ( ξ , ξ , ξ + ξ ) (cid:19) ; I = − m ( ξ ) ξ + m ( ξ ) ξ + m ( ξ ) ξ + m ( ξ ) ξ α ( ξ , ξ , ξ + ξ ) . For I , by (4.1), (4.11) and Lemma 4.2(i), we have | I | . | ξ + ξ | | α ( ξ , ξ , ξ + ξ ) + α ( ξ , ξ , ξ + ξ ) || α ( ξ , ξ , ξ + ξ ) α ( ξ , ξ , ξ + ξ ) | . | α ( ξ , ξ , ξ , ξ ) || ξ || ξ || ξ || ξ || ξ + ξ | . For I , , by Lemma 4.2(iii) and (4.1), we also have | I | . | α ( ξ , ξ , ξ , ξ ) | / | ξ m | | α ( ξ , ξ , ξ + ξ ) | . | α ( ξ , ξ , ξ , ξ ) || ξ || ξ || ξ || ξ || ξ + ξ | . Now we turn to give the pointwise upper bound on the multiplier M . It directlyfollows from Lemma 4.3. Lemma 4.4
For the M defined in (4.10), we have | M ( ξ , ξ , ξ , ξ ) | . χ Ω | ξ || ξ || ξ | . (4.15) where Ω = { ( ξ , ξ , ξ , ξ , ξ ) : ξ + ξ + ξ + ξ + ξ = 0 , | ξ | , | ξ | , | ξ | , | ξ + ξ | & N } . First we give the comparison between E I ( t ) and E I ( t ). Lemma 4.5
Let I = I N,s for s ≥ − , then | E I ( t ) − E I ( t ) | . N − k Iu ( t ) k L + N − k Iu ( t ) k L . (4.16) Proof.
By the definitions (4.3) and (4.8), we need to show | Λ ( σ ) | . N − k Iu ( t ) k L ; (4.17) | Λ ( σ ) | . N − k Iu ( t ) k L . (4.18)We may assume that F x u ( ξ, t ) is nonnegative. For (4.17), since ξ + ξ + ξ = 0, bysymmetry we may assume again that | ξ | ∼ | ξ | ≥ | ξ | . Note that σ vanishes when | ξ j | ≤ N for j = 1 , ,
3, so we may assume further that | ξ | , | ξ | & N .Set △ ≡ | σ | m ( ξ ) m ( ξ ) m ( ξ ) = 2 | m ( ξ ) ξ + m ( ξ ) ξ + m ( ξ ) ξ | | α ( ξ , ξ , ξ ) | m ( ξ ) m ( ξ ) m ( ξ ) , then (4.16) follows if we show | Λ ( △ ) | . N − k u k L . By (4.1), (4.11) and s ≥ −
34 , we have △ . | ξ ξ | m ( ξ ) m ( ξ ) ∼ N s | ξ | − − s | ξ | − − s . N − | ξ | − | ξ | − . | Λ ( △ ) | . N − | Λ ( | ξ | − | ξ | − ) | . N − (cid:13)(cid:13)(cid:13) D − x u (cid:13)(cid:13)(cid:13) L · k u k L . N − k u k L . Now we turn to (4.18). Set e △ ≡ | σ | Q j =1 m ( ξ j ) = | ˜ M ( ξ , · · · , ξ ) || α ( ξ , · · · , ξ ) | Q j =1 m ( ξ j ) , then (4.18) suffices if we show | Λ ( e △ ) | . N − k u ( t ) k L . Since | ξ j | & N in Ω and s ≥ −
34 , by Lemma 4.3, we have e △ . χ Ω4 Q j =1 m ( ξ j ) | ξ j | . N − Y j =1 | ξ j | − . Therefore, we have | Λ ( e △ ) | . N − (cid:13)(cid:13)(cid:13) D − u ( t ) (cid:13)(cid:13)(cid:13) L x . N − k u ( t ) k L by Sobolev’s inequality. Next lemmas are the key estimates related to the almost conservation of E I ( t ). Lemma 4.6
Let
I, s be as Lemma 4.5, then (cid:12)(cid:12)(cid:12)(cid:12)Z δ Λ ( ¯ M ) dt (cid:12)(cid:12)(cid:12)(cid:12) . N − k Iu k X δ ,
12 + . (4.19) Proof.
By symmetry we may assume again that | ξ | ∼ | ξ | ≥ | ξ | . Since ¯ M = 0, when | ξ | , · · · , | ξ | ≤ N , we may assume again that | ξ | ∼ | ξ | & N . We always have | ξ | ≪ N in Γ / Ω. To extend the integration domain from [0 , δ ] to R , we may need to borrow | ξ | − from the multiplier (see [7], for the argument), but this will not be mentioned since it willonly be recorded by N at the end. Therefore, by Plancherel’s identity, we only need toshow Z ∗ ¯ M ( ξ , · · · , ξ ) b f ( ξ , τ ) · · · b f ( ξ , τ ) m ( ξ ) · · · m ( ξ ) . N − k f k X ,
12 + · · · k f k X ,
12 + , (4.20)23here Z ∗ = Z ξ ··· + ξ , τ + ··· + τ =0 dξ dξ dξ dτ dτ dτ .First, we note that | ξ | − | ξ | ∼ | ξ | . Otherwise, if | ξ | = | ξ | + o ( | ξ | ) , then | ξ | = | ξ | + o ( | ξ | ) , and ξ · ξ < , ξ · ξ >
0. Thus we have | ξ | = | ξ + ξ | + o ( | ξ | ) = | ξ | + | ξ | + o ( | ξ | ) = 2 | ξ | + o ( | ξ | ) , but it doesn’t happen. Therefore, we have | φ ′ ( ξ ) − φ ′ ( ξ ) | ∼ | ξ | , | φ ′ ( ξ ) − φ ′ ( ξ ) | ∼ | ξ | . Thus, by Lemma 4.3 and using (2.11) two times, the left-hand side of (4.20) is controlledby N s Z ∗ | ξ | − − s m ( ξ ) · | φ ′ ( ξ ) − φ ′ ( ξ ) | | φ ′ ( ξ ) − φ ′ ( ξ ) | b f ( ξ , τ ) · · · b f ( ξ , τ ) . N − Z ∗ | φ ′ ( ξ ) − φ ′ ( ξ ) | | φ ′ ( ξ ) − φ ′ ( ξ ) | f ( ξ , τ ) · · · f ( ξ , τ )= N − Z I ( f , f )( x, t ) I ( f , f )( x, t ) dxdt . N − (cid:13)(cid:13)(cid:13) I ( f , f ) (cid:13)(cid:13)(cid:13) L xt (cid:13)(cid:13)(cid:13) I ( f , f ) (cid:13)(cid:13)(cid:13) L xt . N − k f k X ,
12 + · · · k f k X ,
12 + . This completes the proof of the lemma. Lemma 4.7
Let
I, s be as Lemma 4.5, then (cid:12)(cid:12)(cid:12)(cid:12)Z δ Λ ( M ) dt (cid:12)(cid:12)(cid:12)(cid:12) . N − + k Iu k X δ ,
12 + . (4.21) Proof.
By the argument at the beginning of the proof of Lemma 4.6, we may usePlancherel’s identity and turn to show Z ∗ M ( ξ , · · · , ξ ) b f ( ξ , τ ) · · · b f ( ξ , τ ) m ( ξ ) · · · m ( ξ ) . N − + k f k X ,
12 + · · · k f k X ,
12 + , (4.22)where Z ∗ = Z ξ ··· + ξ , τ + ··· + τ =0 dξ · · · dξ dτ · · · dτ . By the definition of ˜ M , we have: | ξ | , | ξ | , | ξ | , | ξ + ξ | & N . We may assume that | ξ | ≥ | ξ | by symmetry. Now we split it into twocases to analysis: Case 1, | ξ | , | ξ | & N ; Case 2, | ξ | & N ≫ | ξ | .24ase 1, | ξ | , | ξ | & N . By Lemma 4.4, M ( ξ , · · · , ξ ) m ( ξ ) · · · m ( ξ ) . N s | ξ | − − s | ξ | − − s | ξ | − − s | ξ | − s | ξ | − s Then, by Lemma 2.3 and note that s ≥
34 , the left-hand side of (4.22) is bounded by N s Z ∗ | ξ | − − s | ξ | − − s | ξ | − − s | ξ | − s | ξ | − s b f ( ξ , τ ) · · · b f ( ξ , τ ) . N − (cid:13)(cid:13)(cid:13) D − x P a f (cid:13)(cid:13)(cid:13) L x L ∞ t · · · (cid:13)(cid:13)(cid:13) D − x P a f (cid:13)(cid:13)(cid:13) L x L ∞ t (cid:13)(cid:13)(cid:13) D x P a f (cid:13)(cid:13)(cid:13) L x L t (cid:13)(cid:13)(cid:13) D x P a f (cid:13)(cid:13)(cid:13) L x L t . N − k f k X ,
12 + · · · k f k X ,
12 + . Case 2, | ξ | & N ≫ | ξ | . In this case, M ( ξ , · · · , ξ ) m ( ξ ) · · · m ( ξ ) . N s | ξ | − − s | ξ | − − s | ξ | − − s | ξ | − s . Then, by Lemma 2.4 and Lemma 2.5, the left-hand side of (4.22) is bounded by N s Z ∗ | ξ | − − s | ξ | − − s | ξ | − − s | ξ | − s b f ( ξ , τ ) · · · b f ( ξ , τ ) . N − k f k L x L t · · · k f k L x L t (cid:13)(cid:13)(cid:13) I ( f , f ) (cid:13)(cid:13)(cid:13) L x L t . N − k f k X ,
12 + · · · k f k X ,
12 + . This completes the proof of the lemma. Now we are ready to prove Theorem 1.1 by iteration.Fix N large and depending on k u k H s . First of all, by Proposition 3.1, (1.1)-(1.2) iswell-posed on [0 , δ ] in H s ( R ) with δ ∼ k I N,s u k − − L & N s − . Next, we turn to estimate E I ( δ ) ≡ k I N,s u ( δ ) k L . By (4.4), Lemma 4.6 and Lemma 4.7,we have E I ( t ) . E I (0) + N − k Iu k X δ ,
12 + + N − + k Iu k X δ ,
12 + , t ∈ [0 , δ ] . (4.23)By Lemma 4.5, we have E I ( t ) . E I ( t ) + N − k Iu ( t ) k L + N − k Iu ( t ) k L , (4.24)25nd for t = 0, E I (0) . k Iu k L + N − k Iu k L + N − k Iu k L . (4.25)Therefore, using (4.23) ∼ (4.25), (3.3) and (1.7), we have E I ( t ) . k Iu k L + N − k Iu k L + N − k Iu k L + N − k Iu k X δ ,
12 + + N − + k Iu k X δ ,
12 + + N − k Iu ( t ) k L + N − k Iu ( t ) k L . k Iu k L + N − k Iu k L + N − k Iu k L + N − k Iu k L + N − + k Iu k L + N − k Iu ( t ) k L + N − k Iu ( t ) k L ≤ C N − s + C ( N − N − s + N − + N − s )+ C (cid:16) N − k Iu ( t ) k L + N − k Iu ( t ) k L (cid:17) for t ∈ [0 , δ ], where C is the constant such that k Iu k L ≤ C N − s . Therefore, E I ( t ) ≤ C N − s + C (cid:16) N − k Iu ( t ) k L + N − k Iu ( t ) k L (cid:17) provided C ( N − N − s + N − + N − s ) ≤ C N − s . (4.26)So, for large N , it’s easy to see that E I ( t ) ≤ C N − s , t ∈ [0 , δ ] . In particular, E I ( δ ) ≤ C N − s . Therefore, by taking u ( δ ) as a new initial data andemploying Proposition 3.1, we can extend the solution to [0 , δ ] under the condition(4.26).Repeating this process k times, then E I ( kδ ) ≤ C N − s provided kC ( N − N − s + N − + N − s ) ≤ C N − s . (4.27)Set T = kδ , then (4.27) becomes T · C δ − ( N − N − s + N − + N − s ) ≤ C N − s . (4.28)26herefore, for a given T >
0, the solution can be extended to [0 , T ] if (4.28) holds.Choosing N & T , (4.28) becomes δ − ( N − N − s + N − + N − s ) . N − s . Since δ − . N − s + , the above inequality amounts − s − − s < − s, − s − − s < − s, that is, s > −
34 . This completes the proof of Theorem 1.1.
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