The Cauchy problem for the Ostrovsky equation with positive dispersion
aa r X i v : . [ m a t h . A P ] J un The Cauchy problem for the Ostrovsky equationwith positive dispersion
Wei Yan a,d , Yongsheng Li b , Jianhua Huang c , Jinqiao Duan da College of Mathematics and Information Science, Henan Normal University,Xinxiang, Henan 453007, China b School of Mathematics, South China University of Technology,Guangzhou, Guangdong 510640, China c College of Science, National University of Defense Technology,Changsha, Hunan 410073, China d Department of Applied Mathematics, Illinois Institute of Technology,Chicago, IL 60616, USA
Abstract.
This paper is devoted to studying the Cauchy problem for the Ostrovskyequation ∂ x (cid:18) u t − β∂ x u + 12 ∂ x ( u ) (cid:19) − γu = 0 , with positive β and γ . This equation describes the propagation of surface waves in arotating oceanic flow. We first prove that the problem is locally well-posed in H − ( R ).Then we reestablish the bilinear estimate, by means of the Strichartz estimates insteadof calculus inequalities and Cauchy-Schwartz inequalities. As a byproduct, this bilinearestimate leads to the proof of the local well-posedness of the problem in H s ( R ) for s > − , with help of a fixed point argument.
1. Introduction
Email: [email protected]: [email protected]: [email protected]: [email protected] e consider the Cauchy problem for the Ostrovsky equation with positive dispersion ∂ x (cid:18) u t − β∂ x u + 12 ∂ x ( u ) (cid:19) − γu = 0 , with β > , γ > , (1.1) u ( x,
0) = u ( x ) , (1.2)where u represents the free surface of a liquid and the positive parameter γ measuresthe effect of rotation. This equation (1.1) was proposed by Ostrovsky [58] as a model forweakly nonlinear long waves in a rotating liquid, by taking into account of the Coriolisforce. It describes the propagation of surface waves in the ocean in a rotating frame ofreference. In fact, β determines the type of dispersion, more precisely, β < β > u t − β∂ x u + 12 ∂ x ( u ) − γ∂ − x u = 0 . (1.3)When γ = 0, it reduces to the Korteweg-de Vries equation which has been investigatedwidely [4–6, 9, 10, 32–36, 55, 57]. By introducing the Fourier restriction norm method,Bourgain [5] proved that the Cauchy problem for the KdV equation is globally well-posedin L for the periodic case and nonperiodic case. By using the Fourier restriction normmethod, Kenig et al. [35, 36] proved that the Cauchy problem for the KdV equation islocally well-posed in H s ( R ) , s > − and ill-posed in H s ( R ) with s < − in the sensethat the data-to-solution map fails to be uniformly continuous as map from H s ( R ) to C t H s ( R ). This means that s = − is the critical regularity index in Sobolev spacesfor KdV equation. By using the I-method and the Fourier restriction norm method,Colliander et al. [10] showed that the Cauchy problem for the KdV equation is globallywell-posed in H s ( R ) with s > − . Guo [20] and Kishimoto [37] proved that the Cauchyproblem for the KdV equation is globally well-posed in H − ( R ) with the aid of the I -method and the modified Besov spaces. Kappeler and Topalov [31] proved that the flowmap extends continuously to H − in the periodic case with the aid of inverse scatteringtransformation. Molinet [49] proved that no well-posedness result can possibly hold2elow s = − H s for s < − H s ( R ) for s < −
1. Liu [46] established a priori bounds for KdV equation in H − ( R ). Buckmaster and Koch [7] proved the existence of weak solutions to the KdVinitial value problem on the real line with H − initial data and studied the problem oforbital and asymptotic H s stability of solitons for integers s = −
1; and established newa priori H − bound for solutions to the KdV equation.The stability of the solitary waves or soliton solutions of the Ostrovsky equation(1.1) has also been examined [40–42, 47, 48, 60, 63]. Choudhury et al. [8] studiedthe Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovskyequation. Others have studied the Cauchy problem for (1.1); for instance, see [18, 19,21, 22, 25–30, 42, 43, 47, 49, 59, 61–63]. The results in [25, 27, 56] showed that s = − is the critical regularity index for (1.1) in Sobolev spaces. Recently, Coclite and Ruvo[11, 12] have investigated the convergence of the Ostrovsky equation to the Ostrovsky-Hunter equation, and also the dispersive and diffusive limits for the Ostrovsky-Huntertype equation. Moreover, Li et al. [44] proved that the Cauchy problem for the Ostrovskyequation with negative dispersion is locally well-posed in H − ( R ) . However, the well-posedness of the Ostrovsky equation with positive dispersion in H − ( R ) has not yetbeen shown up to now.Observe that if u ( x, t ) is the solution to the Cauchy problem for (1.3), then u λ ( x, t ) = λ − u (cid:0) xλ , tλ (cid:1) , for λ >
0, is the solution to the following equation u λt − β∂ x u λ + 12 ∂ x (( u λ ) ) − γλ − ∂ − x u λ = 0 , (1.4) u λ ( x,
0) = λ − u (cid:16) xλ (cid:17) . (1.5)If u is the solution to (1.3), then v ( x, t ) = β − u ( x, β − t ) is the solution v t − v xxx + ∂ x ( v ) − β − γ∂ − x v = 0. Hence without loss of generality, we can assume that γ = β = 1 in thispaper.In the present paper, we first prove that (1.3) with initial condition (1.2) is locallywell-posed in H − ( R ). More precisely, we establish a bilinear estimate with s = − ,which combines with the fixed point theorem to yield the local well-posedness of the3auchy problem for (1.3) in H − ( R ). Then by using the Strichartz estimates insteadof calculus inequalities and the Cauchy-Schwartz inequality, we reestablish the bilinearestimate for the Ostrovsky equation (1.3) with s > − , which combines with Lemma2.8 (below) and the fixed point theorem to imply the local well-posedness of the Cauchyproblem for (1.3) in H s ( R ) with s > − . Before stating the main results, we introduce some notations. Throughout this paper,we assume that λ ≥ C is a positive constant which may vary from line to line. Let ǫ be a small number with 0 < ǫ < − . Note that a ∼ b means that | b | ≤ | a | ≤ | b | , a ≫ b means that | a | > | b | . Let ψ be a smooth function with support in [ − ,
2] and takingthe value 1 in [ − , I ⊂ R , define the indicator function χ I (( ξ, τ )) = 1if ( ξ, τ ) ∈ I ; χ I (( ξ, τ )) = 0 if ( ξ, τ ) / ∈ I . Let F u be the Fourier transformation of u with respect to both space and time variables, and F − u be the corresponding inversetransformation, while F x u denotes the Fourier transformation of u with respect to thespace variable and F − x u denotes the corresponding inverse transformation. Define h·i = 1 + | · | ,D ′ := (cid:26) ( ξ, τ ) ∈ R : | ξ | ≤ , | τ | ≥ | ξ | − (cid:27) ,D := (cid:26) ( ξ, τ ) ∈ R : | ξ | ≤ , | τ | < | ξ | − (cid:27) ,D := (cid:26) ( ξ, τ ) ∈ R : 18 < | ξ | ≤ , | τ | < | ξ | − (cid:27) ,D := (cid:26) ( ξ, τ ) ∈ R : 18 < | ξ | ≤ , | τ | ≥ | ξ | − (cid:27) ,A j := (cid:8) ( ξ, τ ) ∈ R : 2 j ≤ h ξ i < j +1 (cid:9) ,φ λ ( ξ ) = ξ + 1 λ ξ , σ λ = τ + φ λ ( ξ ) , σ λj = τ j + φ λ ( ξ j )( j = 1 , ,B k := (cid:8) ( ξ, τ ) ∈ R : 2 k ≤ (cid:10) σ λ (cid:11) < k +1 (cid:9) ,S λ ( t ) φ = e t ( ∂ x + λ − ∂ − x ) φ = C Z R e − it ( ξ + ξ − λ − ) F x φ ( ξ ) dξ. Obviously, (cid:8) ( ξ, τ ) ∈ R : | ξ | ≤ , τ ∈ R (cid:9) = D ′ ∪ D ∪ D ∪ D . Here j, k are nonnegativeintegers. Space X s, bλ is defined by X s, bλ = (cid:26) u ∈ S ′ ( R ) : k u k X s, bλ = (cid:13)(cid:13)(cid:13) h ξ i s (cid:10) σ λ (cid:11) b F u ( ξ, τ ) (cid:13)(cid:13)(cid:13) L τξ ( R ) < ∞ (cid:27) .X s,bλ was introduced by Rauch and Reed [52], Beals [1], Bourgain [5], Klainerman and4achedon [38], and further developed by Kenig, Ponce and Vega [33]. Space X s, b, λ = n u ∈ S ′ ( R ) : k u k X s, b, λ < ∞ o is equipped with the following norm k u k X s, b, λ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18)(cid:13)(cid:13)(cid:13) h ξ i s (cid:10) σ λ (cid:11) b F u ( ξ, τ ) (cid:13)(cid:13)(cid:13) L τξ ( A j ∩ B k ) (cid:19) j, k ≥ (cid:13)(cid:13)(cid:13)(cid:13) ℓ j ( ℓ k ) ∼ (cid:20)X j js (cid:18)X k bk k F u ( ξ, τ ) k L τξ ( A j ∩ B k ) (cid:19) (cid:21) / . Space X λ is defined by X λ = (cid:26) u ∈ S ′ ( R ) : k u k X λ = (cid:13)(cid:13) F − [ χ D c F u ] (cid:13)(cid:13) X − , , λ + k F − [ χ D ′ F u ] k X − , λ < ∞ (cid:27) and space Y is defined by Y = (cid:26) u ∈ S ′ ( R ) : k u k Y = (cid:13)(cid:13)(cid:13) h ξ i − F u ( ξ, τ ) (cid:13)(cid:13)(cid:13) L ξ L τ < ∞ (cid:27) , where D c ∪ D ′ = R τξ , respectively. Spaces ˆ X λ , ˆ X λs, b, , ˆ X λs, b are equipped with thefollowing norms k f k ˆ X λ = k F − f k X λ , k f k ˆ X λs,b, = k F − f k X s,b, λ , k f k ˆ X λs,b = k F − f k X s,bλ , respectively. The space X λ ,T denotes the restriction of X λ onto the finite time interval[ − T, T ] and is equipped with the norm k u k X λ, T = inf {k w k X λ : w ∈ X λ , u ( t ) = w ( t ) for − T ≤ t ≤ T } . The main results of this paper are stated in Theorem 1.1 and Theorem 1.2.
Theorem 1.1. (Well-posedness in H − ( R ) ) The Cauchy problem for (1.3) is locallywell-posed in H − ( R ) . That is, for u ∈ H − ( R ) , there exists a positive number T such that the solution map u u ( t ) is locally Lipschitz continuous from H − ( R ) into C ([ − T, T ]; H − ( R )) ∩ k u k X λ,T . Moreover, the solution to the Cauchy problem for (1.3)on the time interval [ − T, T ] is unique in the space X λ,T . Remark 1:
Isaza and J. Mej´ıa [27] and Taugawa [56] have proved that the Cauchyproblem (1.3) - (1.2) is locally well-posed in H s ( R ) with s > − . Moreover, Isaza andJ. Mej´ıa [27] have proved that the problem (1.3)- (1.2) is not quantitatively well-posedin H s ( R ) with s < − . Thus, s = − is the critical regularity index in Sobolev spacefor (1.3)- (1.2). 5 emark 2: Inspired by [23, 24, 37] and in view of the structure of the Ostrovskyequation with positive dispersion, we choose the space X λ , which is slightly differentfrom space X of [37]. More precisely, Kishimoto [37] used the set D := (cid:8) ( ξ, τ ) ∈ R , | ξ | ≤ , | τ | ≥ | ξ | − (cid:9) in [37]. However, we need to take D ′ := (cid:26) ( ξ, τ ) ∈ R , | ξ | ≤ , | τ | ≥ | ξ | − (cid:27) . Now we give a specific example to explain the reason why we take D ′ . In proving (vii)of Lemma 3.1 (below) for the case 2 k = 2 k max > (cid:8) k , k (cid:9) and ( τ , ξ ) ∈ D ′ , wehave C | ξ | − ≤ k ∼ | σ + − σ +1 − σ +2 | ∼ k = | ξξ ξ − ξ + ξ ξ + ξ λ ξξ ξ | ∼ C | ξξ ξ | ∼ | ξ | j ,and | ξ | ≥ C − j , which is crucial in establishing (vii) of Lemma 3.1. But on the set D = (cid:8) ( τ, ξ ) ∈ R , | ξ | ≤ , | τ | ≥ | ξ | − (cid:9) , we can not guarantee that | ξ | ≥ C − j . Remark 3:
Now we outline the proof for Theorem 1.1. Comparing with the KdVequation and the Ostrovsky equation with the negative dispersion, the structure of theOstrovsky equation with positive dispersion is much more complicated. More precisely,for λ >
0, note that( ξ + ξ ) + 1 λ ( ξ + ξ ) − ξ − λ ξ − ξ − λ ξ = 3 ξ ξ ( ξ + ξ ) − ξ + ξ ξ + ξ λ ξ ξ ( ξ + ξ ) , (1.6) ξ + 1 λ ξ + ξ + 1 λ ξ − ( ξ + ξ ) − λ ( ξ + ξ )= 34 ( ξ + ξ )( ξ − ξ ) (cid:20) λ ξ ξ ( ξ + ξ ) (cid:21) , (1.7)and ξ λ ξ − (cid:18) ( ξ + ξ ) + 1 λ ( ξ + ξ ) (cid:19) + (cid:18) ξ + 1 λ ξ (cid:19) = 34 ξ (2 ξ + ξ ) (cid:20) − λ ( ξ + ξ ) ξ ξ (cid:21) . (1.8)From (1.6)-(1.8), we know that there exist no positive constants C j ( j = 1 , ,
3) such that (cid:12)(cid:12)(cid:12)(cid:12) ( ξ + ξ ) + 1 λ ( ξ + ξ ) − ξ − λ ξ − ξ − λ ξ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ξ ξ ( ξ + ξ ) − ξ + ξ ξ + ξ λ ξ ξ ( ξ + ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ C | ξ ξ ( ξ + ξ ) | , (1.9)6 (cid:12)(cid:12)(cid:12) ξ + 1 λ ξ + ξ + 1 λ ξ − ( ξ + ξ ) − λ ( ξ + ξ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)
34 ( ξ + ξ )( ξ − ξ ) (cid:20) λ ξ ξ ( ξ + ξ ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C | ξ ξ ( ξ + ξ ) | , (1.10)and (cid:12)(cid:12)(cid:12)(cid:12) ξ λ ξ − (cid:18) ( ξ + ξ ) + 1 λ ( ξ + ξ ) (cid:19) + (cid:18) ξ + 1 λ ξ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ξ (2 ξ + ξ ) (cid:20) − λ ( ξ + ξ ) ξ ξ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≥ C | ξ ξ ( ξ + ξ ) | (1.11)for any ξ , ξ ∈ R . For the KdV equation, we have that (cid:12)(cid:12) ( ξ + ξ ) − ξ − ξ (cid:12)(cid:12) = 3 | ξ ξ ( ξ + ξ ) | , (1.12) (cid:12)(cid:12)(cid:12)(cid:12) ξ + ξ − ( ξ + ξ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)
34 ( ξ + ξ )( ξ − ξ ) (cid:12)(cid:12)(cid:12)(cid:12) , (1.13) (cid:12)(cid:12)(cid:12)(cid:12) ξ − ( ξ + ξ ) + ξ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ξ (2 ξ + ξ ) (cid:12)(cid:12)(cid:12)(cid:12) , (1.14)respectively. The equalities (1.13)-(1.14) ensure that Lemmas 3.2, 3.3 of [37] are valid.The estimate (1.9) and Lemmas 3.2, 3.3 of [37] are the key tools of proving the mainresult of Theorem 1.2 of [37]. From Lemmas 2.3, 2.6 of [44] and the following inequality( ξ + ξ ) − λ ( ξ + ξ ) − ξ + 1 λ ξ − ξ + 1 λ ξ = 3 ξ ξ ( ξ + ξ ) + ξ + ξ ξ + ξ λ ξ ξ ( ξ + ξ ) , (1.15)we know that the proof of Theorem 1.1 of [44] is similar to the proof of Theorem 1.2 of[37].From (1.6)-(1.11), we know that we cannot completely follow the method of [37, 44]to establish the local well-posedness of the Cauchy problem for the Ostrovsky equationwith positive dispersion in H − ( R ) . Thus, for λ ≥ (cid:12)(cid:12)(cid:12)(cid:12) ξ ξ ( ξ + ξ ) − ξ + ξ ξ + ξ λ ξ ξ ( ξ + ξ ) (cid:12)(cid:12)(cid:12)(cid:12) < | ξ ξ ( ξ + ξ ) | , (1.16) (cid:12)(cid:12)(cid:12)(cid:12) ξ ξ ( ξ + ξ ) − ξ + ξ ξ + ξ λ ξ ξ ( ξ + ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ | ξ ξ ( ξ + ξ ) | , (1.17) (cid:12)(cid:12)(cid:12)(cid:12) λ ξ ξ ( ξ + ξ ) (cid:12)(cid:12)(cid:12)(cid:12) < , (1.18) (cid:12)(cid:12)(cid:12)(cid:12) λ ξ ξ ( ξ + ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ , (1.19)7nd (cid:12)(cid:12)(cid:12)(cid:12) − λ ( ξ + ξ ) ξ ξ (cid:12)(cid:12)(cid:12)(cid:12) < , (1.20) (cid:12)(cid:12)(cid:12)(cid:12) − λ ( ξ + ξ ) ξ ξ (cid:12)(cid:12)(cid:12)(cid:12) ≥ . (1.21)Only when (1.17), (1.19) and (1.21) are valid simultaneously, can we follow the methodof [37]. Other cases have to be handled with the aid of techniques introduced in thepresent paper.Our second main result is stated in Theorem 1.2. Theorem 1.2. (Bilinear estimate for s > − )Let s > − and β > and γ > . Then the following bilinear estimate holds: k ∂ x ( u u ) k X s, −
12 +2 ǫλ ≤ C Y j =1 k u j k X s,
12 + ǫλ . (1.22) Remark 4:
By using the calculus inequalities and Cauchy-Schwartz inequalities, Isazaand Mej´ıa [25, 27] and Tsugawa [56] have already established Theorem 1.2. But in thispaper, we use the Strichartz estimates to present an alternative proof of Theorem 1.2.As a byproduct, Theorem 1.2, combining with Lemma 2.8 (below) and a fixed pointargument, leads to the well-posedness of the Cauchy problem for the Ostrovsky equationwith positive dispersion in H s ( R ) with s > − . Remark 5:
Now we outline the proof of Theorem 1.2. Isaza and Mej´ıa [25, 27] haveused certain calculus inequalities and Cauchy-Schwartz inequality to establish this crucialbilinear estimate, which plays the key role in proving the local well-posedness of theCauchy problem for the Ostrovsky equation with positive dispersion in H s ( R ) with s > − . In this paper, we use the Strichartz estimates instead of calculus inequalities andCauchy-Schwartz inequality to reestablish the same bilinear estimate for the Ostrovsky8quation with positive dispersion. From Lemma 2.9 (below), we know that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ξ = P j =1 ξ j τ = P j =1 τ j | ξ − ξ | s Y j =1 F u j ( ξ j , τ j ) dξ dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ξτ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ξ = P j =1 ξ j τ = P j =1 τ j (cid:12)(cid:12) ξ − ξ (cid:12)(cid:12) s (cid:12)(cid:12)(cid:12)(cid:12) λ ξ ξ (cid:12)(cid:12)(cid:12)(cid:12) s F u ( ξ , τ ) F u ( ξ , τ ) dξ dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ξτ ≤ C Y j =1 k u j k X , s ǫλ , < s ≤ , (1.23)which is useful in establishing the bilinear estimate. When (1.17) is valid, we can easilyestablish the bilinear estimate by using (1.23). When (1.16) is valid, we have | ξ min | ∼ λ − | ξ max | − , (1.24)where | ξ min | := min {| ξ | , | ξ | , | ξ |} and | ξ max | := max {| ξ | , | ξ | , | ξ |} . We will combine(1.24) and (1.23), together with a suitable splitting of region, to establish the bilinearestimate. Remark 6:
Levandosky and Liu [41] studied the stability of the generalized Ostrovskyequation (cid:2) u t − βu xxx + ( u k ) x (cid:3) x = γu, k ≥ , k ∈ N, γ > , β = 0 . (1.25)The techniques used in proving Theorem 1.2 are conducive to establish the multilinearestimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ x k Y j =1 ( u j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X s, −
12 +2 ǫλ ≤ C k Y j =1 k u j k X s,
12 + ǫλ (1.26)for s = with k = 3 and s = k − k with k ≥
4. As a byproduct, (1.26) in combinationwith Lemma 2.8 yields a result on the optimal regularity for (1.25) in Sobolev spaces.The rest of this paper is arranged as follows. After presenting some preliminaries inthe next section, we establish bilinear estimate for s = − in Lemmas 3.1-3.2 in Section3. Then, we prove local well-posedness (Theorem 1.1) for s = − , and bilinear estimate(Theorem 1.2) for s > − , in Sections 4 and 5, respectively.9 . Preliminaries In this section, we establish Lemmas 2.1-2.9 which play an important role in estab-lishing Lemma 3.1 and Theorems 1.1 and 1.2. More precisely, we establish Lemmas 2.1,2.2, 2.3 and 2.6. These lemmas will be used to prove Lemma 3.1 which implies Lemma3.2 immediately with the help of a suitable decomposition. Lemma 3.2 and Lemmas 2.4- 2.5, in combination with a fixed point argument, yield Theorem 1.1. Lemmas 2.7-2.9are used to establish Theorem 1.2.
Lemma 2.1.
Assume that f k ( k = 1 , ∈ S ′ ( R ) with supp f k ⊂ A j k ( k = 1 , , and K := inf {| ξ − ξ | : ∃ τ , τ , s.t. ( ξ k , τ k ) ∈ supp f k ( k = 1 , } > . If ( ξ , τ ) ∈ supp f , ( ξ , τ ) ∈ supp f , ξ ξ > , (2.1) or ( ξ k , τ k ) ∈ supp f k ( k = 1 , , ξ ξ < , (cid:12)(cid:12)(cid:12)(cid:12) λ ξ ξ ξ (cid:12)(cid:12)(cid:12)(cid:12) > , (2.2) then there exists a positive constant C such that the following inequalities hold k| ξ | f ∗ f k L ( R ) ≤ C Y k =1 k f k k ˆ X λ , , , (2.3) k| ξ | f ∗ f k L ( R ) ≤ CK − Y k =1 k f k k ˆ X λ , , . (2.4) Proof.
Since τ = τ + τ , ξ = ξ + ξ , by a direct computation, we have that τ + ξ λ ξ − (cid:18) τ + ξ + 1 λ ξ (cid:19) − (cid:18) τ + ξ + 1 λ ξ (cid:19) = − ξ ( ξ − ξ ) (cid:20) λ ξ ξ ξ (cid:21) . (2.5)By using (2.5) and a proof similar to that for Lemma 2.1 of [44], we conclude that(2.3)-(2.4) are valid.This completes the proof of Lemma 2.1. Lemma 2.2.
Assume that f ∈ S ′ ( R ) , g ∈ S ( R ) , with supp f ⊂ A j for some j ≥ and Ω ⊂ R with positive measure. Let K := inf {| ξ + ξ | : ∃ τ, τ s.t. ( ξ, τ ) ∈ Ω , ( ξ , τ ) ∈ A j } > . f ( ξ, τ ) ∈ Ω , ( ξ , τ ) ∈ supp f, ξξ < , (2.6) or ( ξ, τ ) ∈ Ω , ( ξ , τ ) ∈ supp f, ξξ > , (cid:12)(cid:12)(cid:12)(cid:12) − λ ξξ ξ (cid:12)(cid:12)(cid:12)(cid:12) > , (2.7) then, for every k ≥ , there exists a positive constant C > such that the followinginequalities hold k f ∗ g k L ( B k ) ≤ C k k f k ˆ X λ , , k| ξ | − g k L ( R ) , (2.8) k f ∗ g k L (Ω ∩ B k ) ≤ C k K − k f k ˆ X λ , , k | ξ | − g k L ( R ) . (2.9) Proof.
Combining the identity τ + ξ λ ξ − (cid:18) τ + ξ + 1 λ ξ (cid:19) + (cid:18) τ + ξ + 1 λ ξ (cid:19) = 34 ξ (2 ξ − ξ ) (cid:20) − λ ξξ ξ (cid:21) with a proof similar to Lemma 2.4 of [44], we obtain the estimates in Lemma 2.2.This completes the proof of Lemma 2.2. Lemma 2.3.
The space ˆ X λ has the following properties:(i) For every b > / , there exists C > such that k f k ˆ X λ ≤ C k f k ˆ X λ − , b . (2.10) (ii) For p = 2 , , there exists C > such that kh ξ i − f k L ξ L pτ ≤ C k f k ˆ X λ − , − ǫ , (2.11) kh ξ i − f k ˆ X λ − , −
12 + ǫ ≤ C kh ξ i − f k L ξ L τ , (2.12) kh ξ i − f k L ξ L τ ≤ C k f k ˆ X λ − , , . (2.13) Proof.
We first prove (i). By using the Cauchy-Schwartz inequality, since b > , we11ave k u k X λ − , , = X j − j X k k k u k L ξτ ( A j ∩ B k ) ! ≤ X j − j X k bk ( − b ) k k u k L ξτ ( A j ∩ B k ) ! ≤ X j − j X k bk k u k L ξτ ( A j ∩ B k ) ! X k (1 − b ) k ! ≤ X j − j X k bk k u k L ξτ ( A j ∩ B k ) ! = k u k X λ − ,b . (2.14)Note that k u k ˆ X λ − , ≤ C k u k ˆ X λ − ,b . (2.15)Combining (2.14) with (2.15), we have k u k ˆ X λ = k F − u k X λ = (cid:13)(cid:13) F − [ χ D c F u ] (cid:13)(cid:13) X − , , λ + k F − [ χ D F u ] k X − , λ ≤ C k u k ˆ X λ − ,b . The completes the proof of (i).Now we prove (ii). The inequality (2.13) is in [2]. We only prove the case p = of(2.11), as the case p = 2 of (2.11) is easily checked. By the H¨older inequality, we have k f k L ξ L τ = Z R (cid:18)Z R | f | dτ (cid:19) dξ = Z R (cid:18)Z R h σ λ i − (1 − ǫ ) h σ λ i (1 − ǫ ) | f | dτ (cid:19) dξ ≤ " sup ξ ∈ R (cid:18)Z R h σ λ i − − ǫ ) dτ (cid:19) R h σ λ i (1 − ǫ ) | f | dτ dξ (cid:19) ≤ C Z R h σ λ i (1 − ǫ ) | f | dτ dξ. Therefore, from the above inequality, we have kh ξ i − f k L ξ L τ = "Z R h ξ i − (cid:18)Z R | f | dτ (cid:19) dξ ≤ C (cid:18)Z R h ξ i − h σ λ i (1 − ǫ ) | f | dτ dξ (cid:19) = C (cid:13)(cid:13)(cid:13) h ξ i − f (cid:13)(cid:13)(cid:13) ˆ X λ − , − ǫ . (2.16)By duality, from (2.16) we derive that (2.12) is valid. This completes the proof of (ii).This completes the proof of Lemma 2.3.12 emma 2.4. Let e − t ( − ∂ x − λ − ∂ − x ) v ( x, be the solution to the linear version (discard thenonlinear part) of equation (1.3). Then we have the following estimate (cid:13)(cid:13)(cid:13) ψ ( t ) e − t ( − ∂ x − λ − ∂ − x ) v ( x, (cid:13)(cid:13)(cid:13) X λ ,T + k ψ ( t ) e − t ( − ∂ x − λ − ∂ − x ) v ( x, k H − x ( R ) ≤ C k v ( x, k H − x ( R ) . Proof.
Without loss of generality, we assume that T = 1. By the embedding X − , + ǫλ ֒ → X λ and Lemma 3.1 of [33], we have (cid:13)(cid:13)(cid:13) ψ ( t ) e − t ( − ∂ x − λ − ∂ − x ) v ( x, (cid:13)(cid:13)(cid:13) X λ , ≤ C (cid:13)(cid:13)(cid:13) ψ ( t ) e − t ( − ∂ x − λ − ∂ − x ) v ( x, (cid:13)(cid:13)(cid:13) X − ,
12 + ǫλ ≤ C k v ( x, k H − x ( R ) . Thus, (cid:13)(cid:13)(cid:13) ψ ( t ) e − t ( − ∂ x − λ − ∂ − x ) v ( x, (cid:13)(cid:13)(cid:13) H − x ( R ) ≤ C k v ( x, k H − x ( R ) . This completes the proof of Lemma 2.4.
Lemma 2.5.
The following estimate holds: (cid:13)(cid:13)(cid:13)(cid:13) ψ ( t ) Z t e − ( t − s )( − ∂ x − λ − ∂ − x ) F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) X λ ,T + (cid:13)(cid:13)(cid:13)(cid:13) ψ ( t ) Z t e − ( t − s )( − ∂ x − λ − ∂ − x ) F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) H − x ( R ) ≤ C (cid:13)(cid:13)(cid:13) F − (cid:16)(cid:10) σ λ (cid:11) − F F (cid:17)(cid:13)(cid:13)(cid:13) X λ + (cid:13)(cid:13)(cid:13) F − (cid:16)(cid:10) σ λ (cid:11) − F F (cid:17)(cid:13)(cid:13)(cid:13) Y . Proof.
Without loss of generality, we assume that T = 1. Note that (cid:13)(cid:13)(cid:13)(cid:13) ψ ( t ) Z t e − ( t − s )( − ∂ x − λ − ∂ − x ) F ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) H − x ( R ) ≤ C (cid:13)(cid:13)(cid:13) F − (cid:16)(cid:10) σ λ (cid:11) − F F (cid:17)(cid:13)(cid:13)(cid:13) Y . By using the Fourier transformation with respect to the space variable, we have that F x (cid:20)Z t e − ( t − s )( − ∂ x − λ − ∂ − x ) F ( s ) ds (cid:21) ( ξ ) = Ce − it ( ξ + λ ξ ) Z R e it ( τ + ξ + λ ξ ) − τ + ξ + λ ξ F ( τ, ξ ) dτ = I + I + I , where I = Ce − it ( ξ + λ ξ ) Z R e it ( τ + ξ + λ ξ ) − τ + ξ + λ ξ ( I B F ) ( τ, ξ ) dτ,I = C Z R e itτ τ + ξ + λ ξ ( I B > F ) ( τ, ξ ) dτ,I = C Z R e − it ( ξ + λ ξ ) τ + ξ + λ ξ ( I B > F ) ( τ, ξ ) dτ.
13y using a proof similar to [15] and Lemma 4.1 of [37], we obtain that k I + I + I k X λ ≤ C (cid:13)(cid:13)(cid:13) F − (cid:16)(cid:10) σ λ (cid:11) − F F (cid:17)(cid:13)(cid:13)(cid:13) X λ + (cid:13)(cid:13)(cid:13) F − (cid:16)(cid:10) σ λ (cid:11) − F F (cid:17)(cid:13)(cid:13)(cid:13) Y . This completes the proof of Lemma 2.5.
Lemma 2.6.
The following embeddings are true: X , , λ ֒ → X , λ , X , , λ ֒ → C ( R ; L ( R )) . Proof.