# High-Speed Transmission in Long-Haul Electrical Systems

Authors: Beatriz Juárez-Campos; Elena I. Kaikina; Pavel I. Naumkin; Héctor Francisco Ruiz-Paredes
Journal: International Journal of Differential Equations
 (2018)

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## Abstract

We study the equations governing the high-speed transmission in long-haul electrical systemsi∂tu-1/3∂x3u=iλ∂xu2u, t,x∈R+×R, u0,x=u0x, x∈R, where λ∈R,∂xα=F-1ξαF, and F is the Fourier transformation. Our purpose in this paper is to obtain the large time asymptotics for the solutions under the nonzero mass condition ∫u0xdx≠0.


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## Body

## 1. Introduction

We study the equations governing the high-speed transmission in long-haul electrical systems(1)i∂tu-13∂x3u=iλ∂xu2u,t,x∈R+×R,u0,x=u0x,x∈R,where λ∈R, ∂x3=F-1ξ3F, and F is the Fourier transformation defined by Fϕ=1/2π∫Re-ixξϕdx. Note that we have the relation u-t,x=u¯t,-x, so we can only consider the case t>0. For the regular solution of (1) we have the conservation law utL2=u0L2. We are interested in the case of nonzero mass condition ∫Ru0xdx≠0. By (1) we get the conservation of the mass ∫Rut,xdx=∫Ru0xdx≠0 for all t>0.This equation arises in the context of high-speed soliton transmission in long-haul optical communication system [1]. Also it can be considered as a particular form of the higher order nonlinear Schrödinger equation introduced by [2] to describe the nonlinear propagation of pulses through optical fibers. This equation also represents the propagation of pulses by taking higher dispersion effects into account than those given by the Schrödinger equation (see [3–11]).The higher order nonlinear Schrödinger equations have been widely studied recently. For the local and global well-posedness of the Cauchy problem we refer to [12–14] and references cited therein. The dispersive blow-up was obtained in [15]. The existence and uniqueness of solutions to (1) were proved in [16–25] and the smoothing properties of solutions were studied in [18–21, 24, 26–31]. The blow-up effect for a special class of slowly decaying solutions of Cauchy problem (1) was found in [32].As far as we know the question of the large time asymptotics for solutions to Cauchy problem (1) is an open problem. We develop here the factorization technique originated in our previous papers [33–38].We denote the Lebesgue space byLp=ϕ∈S′;ϕLp<∞, where the norm ϕLp=∫ϕxpdx1/p for 1≤p<∞ and ϕL∞=supx∈Rϕx. The weighted Sobolev space is Hpm,s=φ∈S′;ϕHpm,s=xsi∂xmϕLp<∞, where m,s∈R, 1≤p≤∞, x=1+x2, and i∂x=1-∂x2. We also use the notations Hm,s=H2m,s, Hm=Hm,0 shortly, if it does not cause any confusion. Let C(I;B) be the space of continuous functions from an interval I to a Banach space B. Different positive constants might be denoted by the same letter C. We denote by Fϕ or ϕ^(ξ)=1/2π∫Re-ixξϕ(x)dx the Fourier transform of the function ϕ, then the inverse Fourier transformation is given by F-1ϕ=1/2π∫Reixξϕ(ξ)dξ.We are now in a position to state our result.Theorem 1.
Assume that the initial datau0∈H1∩H0,1 have a sufficiently small norm u0H1∩H0,1≤ε. Then there exists a unique global solution Fe-it/3∂x3u∈C0,∞;L∞∩H0,1 of Cauchy problem (1). Furthermore the estimate (2)supt>0Fe-it/3∂x3utL∞+t-1/6xe-it/3∂x3utL2+t1/31-1/putLp≤Cε is true, where p>4.Next we prove the existence of the self-similar solutionsvmt,x=t-1/3fmxt-1/3.Theorem 2.
There exists a unique solution of Cauchy problem (1) in the self-similar form vmt,x=t-1/3fmxt-1/3, such that (3)m=12π∫Rvmt,xdx=12π∫Rfmxdx≠0, where m is sufficiently small number and (4)Fe-it/3∂x3vm∈C1,∞;L∞,xe-it/3∂x3vm∈C1,∞;L2.Furthermore the estimate(5)supt>1Fe-it/3∂x3vmtL∞+t-1/6xe-it/3∂x3vmtL2+t1/31-1/pvmtLp≤Cm is true, where p>4.Now we state the stability of solutions to Cauchy problem (1) in the neighborhood of the self-similar solution vmt,x.Theorem 3.
Suppose that(6)12π∫Rfmxdx=12π∫Ru0xdx=m≠0. Let ut,x and vmt,x be the solutions constructed in Theorems 1 and 2, respectively. Then there exists small γ>0 such that the asymptotics(7)ut-vmtL∞≤Ct-1/2+γare true for t≥1.Our approach is based on the factorization techniques. Define the free evolution groupUt=F-1e-it/3ξ3F and write (8)UtF-1ϕ=Dtt2π∫Reitxξ-1/3ξ3ϕξdξ, where Dtϕ=t-1/2ϕx/t is the dilation operator. There is a unique stationary point ξ=μx≡x/xx in the integral ∫Reitxξ-1/3ξ3ϕξdξ, which is defined as the root of the equation ξξ=x for all x∈R. Define the scaling operator Bϕx=ϕμx. Hence we find the following decomposition UtF-1ϕ=DtBMVϕ, where the multiplication factor M=e2it/3η3 and the deformation operator (9)Vtϕ=t2π∫Re-itSξ,ηϕξdξ, where the phase function Sξ,η=1/3ξ3-1/3η3-ηηξ-η. Denote Ak=M¯k1/2tη∂ηMk, k=0,1. We have A1=A0+iη, and also A1V=Viξ, iη,V=-A0V; therefore we obtain the commutator ∂ηV=-2tηiη,V. Since ∂ξSξ,η=ξξ-ηη, then we get itηη,Vϕ=-V∂ξϕ. Also we need the representation for the inverse evolution group FU-tϕ=V∗M¯B-1Dt-1, where the inverse dilation operator Dt-1ϕ=t1/2ϕxt, the inverse scaling operator B-1ϕη=ϕηη, and the inverse deformation operator (10)V∗tϕ=2tπ∫ReitSξ,ηϕηηdη. We have iξV∗ϕ=V∗A1ϕ. Hence the commutator iξ,V∗=V∗A0. Define the new dependent variable φ^=FU-tut. Since FU-tL=∂tFU-t with L=∂t+i/3∂x3, applying the operator FU-t to (1), substituting ut=UtF-1φ^=DtBMVφ^, and using the factorization techniques, we get(11)∂tφ^=FU-tLu=iλξFU-tu2u=iλξV∗M¯B-1Dt-1DtBMVφ^2DtBMVφ^=iλξt-1V∗M¯B-1BMVφ^2BMVφ^=iλξt-1V∗M¯MVφ^2MVφ^=iλξt-1V∗Vϕ2Vϕsince the nonlinearity is gauge invariant. Finally we mention some important identities. The operator J=UtxU-t=x+it∂x∂x plays a crucial role in the large time asymptotic estimates. Note that J commutes with L, that is, J,L=0. To avoid the derivative loss we also use the operator P=3t∂t+∂xx. Note the commutator relation P^,e-it/3ξ3=0 with P^=3t∂t-ξ∂ξ. Thus using ut=UtF-1φ^=DtMVφ^=F-1e-it/3ξ3φ^, we get Pu=UtF-1P^φ^. Also we have the identity J=∂x-1P-3t∂x-1L and L,P=3L holds.


## 2. Estimates in the Uniform Norm

### 2.1. Kernels

Define the kernel(12)Ajt,η=t2π∫Re-itSξ,ηΘξη-1ξjdξ for η≠0, where the cutoff function Θz∈C2R is such that Θz=0 for z≤1/3 or z≥3 and Θz=1 for 2/3≤z≤3/2. We change ξ=ηy, then we get Ajt,η=ηηjt/2π∫1/33e-itη3GyΘyyjdy, where Sηy,η=η3Gy and Gy=1/3y+2y-12, y>0. To compute the asymptotics of the kernel Ajt,η for large t we apply the stationary phase method (see [39], p. 110)(13)∫Reizgyfydy=eizgy0fy02πzg′′y0eiπ/4sgng′′y0+Oz-3/2for z→+∞, where the stationary point y0 is defined by the equation g′y0=0. By virtue of formula (13) with gy=-Gy, fy=Θyyj, and y0=1, we get (14)Ajt,η=t1/2ηηj2itη3+Ot1/2η1+jtη3-1 for tη3→∞. Also we have the estimate Ajt,η≤Ct1/2ηj+1tη3-1/2.In the same manner changingη=ξy, we get for the kernel (15)A∗t,ξ=2tπ∫ReitSξ,ηΘηξ-1ηdη=ξ22tπ∫1/33eitξ3G~yΘyydy for ξ≠0, where Sξ,ξy=ξ3G~y with G~y=1/32y+1y-12, y>0. Then by virtue of formula (13) with gy=G~y, fy=Θyy, and y0=1, we obtain (16)A∗t,ξ=2t1/2ξ22itξ3+Ot1/2ξ2tξ3-1 for tξ3→∞. Also we have the estimate A∗t,ξ≤Ct1/2ξ2tξ3-1/2.


### 2.2. Asymptotics for the OperatorV

In the next lemma we estimate the operatorV in the uniform norm. Define the cutoff function χ1z∈C2R such that χ1z=0 for z≥3 and χ1z=1 for z≤2 and χ2z=1-χ1z. Consider two operators (17)Vjtϕ=t2π∫Re-itSξ,ηϕξχjξη-1dξ, so that we have Vtϕ=V1tϕ+V2tϕ for η≠0. Define the norm ϕY=ϕL∞+t-1/6∂ξϕL2.Lemma 4.
The following estimatesV1ξjϕ-Ajt,ηϕη≤Ct1/2ηj+1t1/3η-7/4ϕY if j≥0 and V2ξjϕ≤Ct1/6-j/3t1/3ηj-3/2ϕY if j=0,1 are valid for all t≥1, η≠0.Proof.
We write(18)V1ξjϕ-Ajϕ=t2π∫Re-itSξ,ηϕξ-ϕηΘξη-1ξjdξ+t2π∫Re-itSξ,ηϕξχ1ξη-1-Θξη-1ξjdξ=I1+I2 for η≠0. For the summand I1 we integrate by parts via identity(19)e-itSξ,η=H1∂ξξ-ηe-itSξ,ηwith H1=1-itξ-η∂ξSξ,η-1, to get (20)I1=Ct1/2∫Re-itSξ,ηϕξ-ϕηξ-η∂ξH1Θξη-1ξjdξ+Ct1/2∫Re-itSξ,ηξ-ηH1Θξη-1ξj∂ξϕξdξ. We find the estimates (21)H1Θξη-1ξj+ξ-η∂ξH1Θξη-1ξj≤Cηj1+tηξ-η2 in the domain 1/3<ξ/η<3. Therefore we obtain (22)I1≤Ct1/2ηj∫1/3<ξ/η<3ϕξ-ϕηξ-ηξ-ηdξ1+tηξ-η2+Ct1/2ηj∫1/3<ξ/η<3ξ-η∂ξϕξdξ1+tηξ-η2. By the Hardy inequality ∫1/3<ξ/η<3ϕξ-ϕη2/ξ-η2dξ≤C∂ξϕL22 and by the Cauchy-Schwarz inequality, changing ξ=ηy we find (23)I1≤Ct1/2ηj∂ξϕL2∫1/3<ξ/η<3ξ-η2dξ1+tηξ-η221/2≤Ct1/2ηj+3/2∂ξϕL2∫1/33y-12dy1+tη3y-1221/2≤Ct1/2∂ξϕL2ηj+3/2tη3-3/4.
To estimate the integralI2 we integrate by parts via the identity(24)e-itSξ,η=H2∂ξξe-itSξ,ηwith H2=1-itξ∂ξSξ,η-1, to get (25)I2=Ct1/2ϕ0∫Re-itSξ,ηξ∂ξχ1ξη-1-Θξη-1H2ξjdξ+Ct1/2∫Re-itSξ,ηϕξ-ϕ0ξξ2∂ξχ1ξη-1-Θξη-1H2ξjdξ+Ct1/2∫Re-itSξ,ηχ1ξη-1-Θξη-1H2ξj+1ϕξξdξ. We find the estimates ξ∂ξχ1ξη-1-Θξη-1H2ξj≤Cξj/1+tξη2 and (26)χ1ξη-1-Θξη-1H2ξj+1+ξ2∂ξχ1ξη-1-Θξη-1H2ξj≤Cξ1+j1+tξη2. Then by the Hardy inequality we obtain (27)I2≤Ct1/2ϕ0∫ξ≤3ηξjdξ1+tξη2+Ct1/2∂ξϕL2∫ξ≤3ηξ2+2jdξ1+tξη221/2. We have ∫ξ≤3ηξjdξ/1+tξη2≤Cηj+1∫03yjdy/1+tη3y≤Cηj+1tη3-1log⁡tη3 and ∫ξ≤3ηξ2+2jdξ/1+tξη22≤Cη2j+3∫03y2j+2dy/1+tη3y2≤Cη2j+3tη3-2. Thus we have I2≤Ct1/2ϕ0ηj+1tη3-3/4+Ct1/2∂ξϕL2ηj+3/2tη3-1 for all ≥1, η≠0.
To estimateV2ξjϕ we integrate by parts via identity (24)(28)V2ξjϕ=Ct1/2ϕ0∫Re-itSξ,ηξ∂ξH2χ2ξη-1ξjdξ+Ct1/2∫Re-itSξ,ηϕξ-ϕ0ξξ2∂ξH2χ2ξη-1ξjdξ+Ct1/2∫Re-itSξ,ηξ1+jχ2ξη-1H2ϕξξdξ. We find the estimates ξ∂ξH2χ2ξη-1ξj≤Cξj/1+tξ3 and ξ2∂ξH2χ2ξη-1ξj+ξ1+jχ2ξη-1H2≤Cξ1+j/1+tξ3 in the domain ξ≥3/2η. Then by the Hardy inequality we obtain (29)V2ξjϕ≤Ct1/2ϕ0∫ξ≥3/2ηξjdξ1+tξ3+Ct1/2∂ξϕL2∫ξ≥3/2ηξ2+2jdξ1+tξ321/2. We have ∫ξ≥3/2ηξjdξ/1+tξ3≤Cη1+j∫3/2∞yjdy/1+tη3y3≤Ct-j+1/3tη3j-2/3 and ∫ξ≥3/2ηξ2+2jdξ/1+tξ32≤Cη3+2j∫3/2∞y2+2jdy/1+tη3y32≤Ct-2j/3-1tη32j/3-1 if j=0,1. Thus we have V2ξjϕ≤Cϕ0t1/6-j/3tη3j-2/3+C∂ξϕL2t-j/3tη3j/3-1/2 for all t≥1 if j=0,1. Lemma 13 is proved.


### 2.3. Asymptotics for the OperatorV∗

We next consider the operatorV∗. Since V∗ϕL∞≤Ct1/2ηϕL1 and V∗ϕL2≤CηϕL2, then by the Riesz interpolation theorem (see [40], p. 52) we have(30)V∗ϕLp≤Ct1/2-1/pη1-1/pϕLp/p-1for 2≤p≤∞. In the next lemma we find the asymptotics of V∗. Denote ξ^=ξt1/3. Also define the norm ϕIα,β=ηαη^-β∂ηϕL2+ηα-1η^-βϕL2.Lemma 5.
Let1/4+2β≤α<5/2-2β, 0≤β<1/2. Then the estimate ξ^βV∗ϕ-A∗ϕL∞≤Ctα-1/3ϕIα,β is valid for all t≥1.Proof.
We write(31)V∗ϕ-A∗ϕ=2tπ∫-∞∞eitSξ,ηϕη-ϕξΘηξ-1ηdη+2tπ∫-∞∞eitSξ,ηϕη1-Θηξ-1ηdη=I1+I2 for ξ≠0. In the integral I1 we use the identity(32)eitSξ,η=H3∂ηη-ξeitSξ,ηwith H3=1+itη-ξ∂ηSξ,η-1, ∂ηSξ,η=-2ηξ-η, and integrate by parts (33)I1=Ct1/2∫-∞∞eitSξ,ηϕη-ϕξη-ξηαη^-βη-ξ2∂ηH3η1-αη^βΘηξ-1dη+Ct1/2∫-∞∞eitSξ,ηη-ξH3η1-αη^βΘηξ-1∂ηηαη^-βϕηdη. Then apply the estimates η-ξH3η1-αη^βΘηξ-1+η-ξ2η^β∂ηH3η1-αΘηξ-1≤Cξ1-αξ^βη-ξ/1+tξη-ξ2 in the domain 1/3≤η/ξ≤3. If κx≤Cκyx for all y∈0,1 then we find the Hardy inequality (34)κxxϕx-ϕ0L2=κxx∫0xϕ′ydtyL2=κx∫01ϕ′xydyL2≤C∫01κxyϕ′xydyL2≤∫01κxyϕ′xyL2dt≤∫01y-1/2dyκϕ′L2≤2κϕ′L2. Hence ∫1/3≤η/ξ≤3ϕη-ϕξ2/η-ξ2η2αη^-2βdη≤CϕIα,β2. Therefore changing η=yξ, we have (35)ξ^βI1≤Ct1/2ξ1-αξ^2βϕIα,β∫1/3≤η/ξ≤3η-ξ2dη1+tξη-ξ221/2≤Ct1/2ξ1-αξ^2βϕIα,β∫1/33ξ31-y2dy1+tξ31-y221/2≤Ct1/2ξ5/2-αξ^2β-9/4ϕIα,β≤Ctα-1/3ϕIα,βif 1/4+2β≤α≤5/2. In the integral I2 using the identity(36)eitSξ,η=H4∂ηηeitSξ,ηwith H4=1+itη∂ηSξ,η-1, ∂ηSξ,η=-2ηξ-η, we integrate by parts (37)I2=Ct1/2∫-∞∞eitSξ,ηηαϕηηη^βη2∂ηH41-Θηξ-1η1-αη^βdη+Ct1/2∫-∞∞eitSξ,ηηH41-Θηξ-1η1-αη^β∂ηηαη^-βϕηdη. Then using the estimates ∂ηSξ,η=Oηξ+η in the domains η/ξ<1/3 and η/ξ≥3, we get (38)η2∂ηH41-Θηξ-1η1-αη^β+ηH41-Θηξ-1η1-αη^β≤Cη2-αη^β1+tη2ξ+η. Therefore by the Hardy inequality (39)ξ^βI2≤Ct1/2ξ^β∂ηηαη^-βϕηL2∫-∞∞η^2βη4-2αdη1+tη2ξ+η21/2≤Ct1/2ϕIα,β∫-∞∞ξ^2βη^2βη4-2αdη1+tη2ξ+η21/2. We have (40)∫-∞∞ξ^2βη^2βη4-2αdη1+tη2ξ+η2≤C∫01ξ^2βη4-2αη^2βdη1+tη2ξ+η2+Ct2β/3-2∫1∞ξ^2βη2β-2αdηξ+η2≤C∫01η4-2αη^2β-6dη+Ct-4β/3∫01η4-4β-2αη^6β-6dη+Ct4β/3-2∫1∞η4β-2α-2dη≤Ct2α-5/3∫0t1/3η4-2αη2β-6dη+Ct2α-5/3∫0t1/3η4-4β-2αη6β-6dη+Ct4β/3-2≤Ct2α-5/3 if 1/4+2β≤α<5/2-2β, 0≤β<1/2. Therefore we get ξ^βI2≤Ctα-1/3ϕIα,β. Lemma 5 is proved.


## 2.1. Kernels

Define the kernel(12)Ajt,η=t2π∫Re-itSξ,ηΘξη-1ξjdξ for η≠0, where the cutoff function Θz∈C2R is such that Θz=0 for z≤1/3 or z≥3 and Θz=1 for 2/3≤z≤3/2. We change ξ=ηy, then we get Ajt,η=ηηjt/2π∫1/33e-itη3GyΘyyjdy, where Sηy,η=η3Gy and Gy=1/3y+2y-12, y>0. To compute the asymptotics of the kernel Ajt,η for large t we apply the stationary phase method (see [39], p. 110)(13)∫Reizgyfydy=eizgy0fy02πzg′′y0eiπ/4sgng′′y0+Oz-3/2for z→+∞, where the stationary point y0 is defined by the equation g′y0=0. By virtue of formula (13) with gy=-Gy, fy=Θyyj, and y0=1, we get (14)Ajt,η=t1/2ηηj2itη3+Ot1/2η1+jtη3-1 for tη3→∞. Also we have the estimate Ajt,η≤Ct1/2ηj+1tη3-1/2.In the same manner changingη=ξy, we get for the kernel (15)A∗t,ξ=2tπ∫ReitSξ,ηΘηξ-1ηdη=ξ22tπ∫1/33eitξ3G~yΘyydy for ξ≠0, where Sξ,ξy=ξ3G~y with G~y=1/32y+1y-12, y>0. Then by virtue of formula (13) with gy=G~y, fy=Θyy, and y0=1, we obtain (16)A∗t,ξ=2t1/2ξ22itξ3+Ot1/2ξ2tξ3-1 for tξ3→∞. Also we have the estimate A∗t,ξ≤Ct1/2ξ2tξ3-1/2.


## 2.2. Asymptotics for the OperatorV

In the next lemma we estimate the operatorV in the uniform norm. Define the cutoff function χ1z∈C2R such that χ1z=0 for z≥3 and χ1z=1 for z≤2 and χ2z=1-χ1z. Consider two operators (17)Vjtϕ=t2π∫Re-itSξ,ηϕξχjξη-1dξ, so that we have Vtϕ=V1tϕ+V2tϕ for η≠0. Define the norm ϕY=ϕL∞+t-1/6∂ξϕL2.Lemma 4.
The following estimatesV1ξjϕ-Ajt,ηϕη≤Ct1/2ηj+1t1/3η-7/4ϕY if j≥0 and V2ξjϕ≤Ct1/6-j/3t1/3ηj-3/2ϕY if j=0,1 are valid for all t≥1, η≠0.Proof.
We write(18)V1ξjϕ-Ajϕ=t2π∫Re-itSξ,ηϕξ-ϕηΘξη-1ξjdξ+t2π∫Re-itSξ,ηϕξχ1ξη-1-Θξη-1ξjdξ=I1+I2 for η≠0. For the summand I1 we integrate by parts via identity(19)e-itSξ,η=H1∂ξξ-ηe-itSξ,ηwith H1=1-itξ-η∂ξSξ,η-1, to get (20)I1=Ct1/2∫Re-itSξ,ηϕξ-ϕηξ-η∂ξH1Θξη-1ξjdξ+Ct1/2∫Re-itSξ,ηξ-ηH1Θξη-1ξj∂ξϕξdξ. We find the estimates (21)H1Θξη-1ξj+ξ-η∂ξH1Θξη-1ξj≤Cηj1+tηξ-η2 in the domain 1/3<ξ/η<3. Therefore we obtain (22)I1≤Ct1/2ηj∫1/3<ξ/η<3ϕξ-ϕηξ-ηξ-ηdξ1+tηξ-η2+Ct1/2ηj∫1/3<ξ/η<3ξ-η∂ξϕξdξ1+tηξ-η2. By the Hardy inequality ∫1/3<ξ/η<3ϕξ-ϕη2/ξ-η2dξ≤C∂ξϕL22 and by the Cauchy-Schwarz inequality, changing ξ=ηy we find (23)I1≤Ct1/2ηj∂ξϕL2∫1/3<ξ/η<3ξ-η2dξ1+tηξ-η221/2≤Ct1/2ηj+3/2∂ξϕL2∫1/33y-12dy1+tη3y-1221/2≤Ct1/2∂ξϕL2ηj+3/2tη3-3/4.
To estimate the integralI2 we integrate by parts via the identity(24)e-itSξ,η=H2∂ξξe-itSξ,ηwith H2=1-itξ∂ξSξ,η-1, to get (25)I2=Ct1/2ϕ0∫Re-itSξ,ηξ∂ξχ1ξη-1-Θξη-1H2ξjdξ+Ct1/2∫Re-itSξ,ηϕξ-ϕ0ξξ2∂ξχ1ξη-1-Θξη-1H2ξjdξ+Ct1/2∫Re-itSξ,ηχ1ξη-1-Θξη-1H2ξj+1ϕξξdξ. We find the estimates ξ∂ξχ1ξη-1-Θξη-1H2ξj≤Cξj/1+tξη2 and (26)χ1ξη-1-Θξη-1H2ξj+1+ξ2∂ξχ1ξη-1-Θξη-1H2ξj≤Cξ1+j1+tξη2. Then by the Hardy inequality we obtain (27)I2≤Ct1/2ϕ0∫ξ≤3ηξjdξ1+tξη2+Ct1/2∂ξϕL2∫ξ≤3ηξ2+2jdξ1+tξη221/2. We have ∫ξ≤3ηξjdξ/1+tξη2≤Cηj+1∫03yjdy/1+tη3y≤Cηj+1tη3-1log⁡tη3 and ∫ξ≤3ηξ2+2jdξ/1+tξη22≤Cη2j+3∫03y2j+2dy/1+tη3y2≤Cη2j+3tη3-2. Thus we have I2≤Ct1/2ϕ0ηj+1tη3-3/4+Ct1/2∂ξϕL2ηj+3/2tη3-1 for all ≥1, η≠0.
To estimateV2ξjϕ we integrate by parts via identity (24)(28)V2ξjϕ=Ct1/2ϕ0∫Re-itSξ,ηξ∂ξH2χ2ξη-1ξjdξ+Ct1/2∫Re-itSξ,ηϕξ-ϕ0ξξ2∂ξH2χ2ξη-1ξjdξ+Ct1/2∫Re-itSξ,ηξ1+jχ2ξη-1H2ϕξξdξ. We find the estimates ξ∂ξH2χ2ξη-1ξj≤Cξj/1+tξ3 and ξ2∂ξH2χ2ξη-1ξj+ξ1+jχ2ξη-1H2≤Cξ1+j/1+tξ3 in the domain ξ≥3/2η. Then by the Hardy inequality we obtain (29)V2ξjϕ≤Ct1/2ϕ0∫ξ≥3/2ηξjdξ1+tξ3+Ct1/2∂ξϕL2∫ξ≥3/2ηξ2+2jdξ1+tξ321/2. We have ∫ξ≥3/2ηξjdξ/1+tξ3≤Cη1+j∫3/2∞yjdy/1+tη3y3≤Ct-j+1/3tη3j-2/3 and ∫ξ≥3/2ηξ2+2jdξ/1+tξ32≤Cη3+2j∫3/2∞y2+2jdy/1+tη3y32≤Ct-2j/3-1tη32j/3-1 if j=0,1. Thus we have V2ξjϕ≤Cϕ0t1/6-j/3tη3j-2/3+C∂ξϕL2t-j/3tη3j/3-1/2 for all t≥1 if j=0,1. Lemma 13 is proved.


## 2.3. Asymptotics for the OperatorV∗

We next consider the operatorV∗. Since V∗ϕL∞≤Ct1/2ηϕL1 and V∗ϕL2≤CηϕL2, then by the Riesz interpolation theorem (see [40], p. 52) we have(30)V∗ϕLp≤Ct1/2-1/pη1-1/pϕLp/p-1for 2≤p≤∞. In the next lemma we find the asymptotics of V∗. Denote ξ^=ξt1/3. Also define the norm ϕIα,β=ηαη^-β∂ηϕL2+ηα-1η^-βϕL2.Lemma 5.
Let1/4+2β≤α<5/2-2β, 0≤β<1/2. Then the estimate ξ^βV∗ϕ-A∗ϕL∞≤Ctα-1/3ϕIα,β is valid for all t≥1.Proof.
We write(31)V∗ϕ-A∗ϕ=2tπ∫-∞∞eitSξ,ηϕη-ϕξΘηξ-1ηdη+2tπ∫-∞∞eitSξ,ηϕη1-Θηξ-1ηdη=I1+I2 for ξ≠0. In the integral I1 we use the identity(32)eitSξ,η=H3∂ηη-ξeitSξ,ηwith H3=1+itη-ξ∂ηSξ,η-1, ∂ηSξ,η=-2ηξ-η, and integrate by parts (33)I1=Ct1/2∫-∞∞eitSξ,ηϕη-ϕξη-ξηαη^-βη-ξ2∂ηH3η1-αη^βΘηξ-1dη+Ct1/2∫-∞∞eitSξ,ηη-ξH3η1-αη^βΘηξ-1∂ηηαη^-βϕηdη. Then apply the estimates η-ξH3η1-αη^βΘηξ-1+η-ξ2η^β∂ηH3η1-αΘηξ-1≤Cξ1-αξ^βη-ξ/1+tξη-ξ2 in the domain 1/3≤η/ξ≤3. If κx≤Cκyx for all y∈0,1 then we find the Hardy inequality (34)κxxϕx-ϕ0L2=κxx∫0xϕ′ydtyL2=κx∫01ϕ′xydyL2≤C∫01κxyϕ′xydyL2≤∫01κxyϕ′xyL2dt≤∫01y-1/2dyκϕ′L2≤2κϕ′L2. Hence ∫1/3≤η/ξ≤3ϕη-ϕξ2/η-ξ2η2αη^-2βdη≤CϕIα,β2. Therefore changing η=yξ, we have (35)ξ^βI1≤Ct1/2ξ1-αξ^2βϕIα,β∫1/3≤η/ξ≤3η-ξ2dη1+tξη-ξ221/2≤Ct1/2ξ1-αξ^2βϕIα,β∫1/33ξ31-y2dy1+tξ31-y221/2≤Ct1/2ξ5/2-αξ^2β-9/4ϕIα,β≤Ctα-1/3ϕIα,βif 1/4+2β≤α≤5/2. In the integral I2 using the identity(36)eitSξ,η=H4∂ηηeitSξ,ηwith H4=1+itη∂ηSξ,η-1, ∂ηSξ,η=-2ηξ-η, we integrate by parts (37)I2=Ct1/2∫-∞∞eitSξ,ηηαϕηηη^βη2∂ηH41-Θηξ-1η1-αη^βdη+Ct1/2∫-∞∞eitSξ,ηηH41-Θηξ-1η1-αη^β∂ηηαη^-βϕηdη. Then using the estimates ∂ηSξ,η=Oηξ+η in the domains η/ξ<1/3 and η/ξ≥3, we get (38)η2∂ηH41-Θηξ-1η1-αη^β+ηH41-Θηξ-1η1-αη^β≤Cη2-αη^β1+tη2ξ+η. Therefore by the Hardy inequality (39)ξ^βI2≤Ct1/2ξ^β∂ηηαη^-βϕηL2∫-∞∞η^2βη4-2αdη1+tη2ξ+η21/2≤Ct1/2ϕIα,β∫-∞∞ξ^2βη^2βη4-2αdη1+tη2ξ+η21/2. We have (40)∫-∞∞ξ^2βη^2βη4-2αdη1+tη2ξ+η2≤C∫01ξ^2βη4-2αη^2βdη1+tη2ξ+η2+Ct2β/3-2∫1∞ξ^2βη2β-2αdηξ+η2≤C∫01η4-2αη^2β-6dη+Ct-4β/3∫01η4-4β-2αη^6β-6dη+Ct4β/3-2∫1∞η4β-2α-2dη≤Ct2α-5/3∫0t1/3η4-2αη2β-6dη+Ct2α-5/3∫0t1/3η4-4β-2αη6β-6dη+Ct4β/3-2≤Ct2α-5/3 if 1/4+2β≤α<5/2-2β, 0≤β<1/2. Therefore we get ξ^βI2≤Ctα-1/3ϕIα,β. Lemma 5 is proved.


## 3. Commutators withV

First we estimate the Fourier type integral(41)Wϕ=t1/2∫Re-itSξ,ηqt,ξ,ηϕξdξ in the L2-norm. In the particular factorized case qt,ξ,η=q1ξq2η, with estimate q2μx≤μx1/2, we find (42)WϕL2≤t1/2M¯q2η∫Reitη2ξe-it/3ξ3q1ξϕξdξL2≤t1/2q2μxμx-1/2∫Reitxξe-it/3ξ3q1ξϕξdξL2≤Ce-it/3ξ3q1ϕL2=Cq1ϕL2. Next we obtain a more general result.Lemma 6.
Suppose thatη∂ηkqt,ξ,η≤Cη^νη^-ν for all ξ,η∈R, k=0,1,2, where ν∈0,1. Then the estimate η1/2WϕL2≤CϕL2 is true for all t≥1.Proof.
We write(43)η1/2WϕL22=Ct∫Rdηη∫Rdξ∫RdζeitSζ,η-Sξ,ηqt,ξ,ηϕξqt,ζ,η¯ϕζ¯=Ct∫Rdξe-it/3ξ3ϕξ∫Rdζeit/3ζ3ϕζ¯Kt,ξ,ζ, where the kernel Kt,ξ,ζ=∫Rdηηeitηηξ-ζqt,ξ,ηqt,ζ,η¯. Integrating two times by parts via the identity eitηηξ-ζ=H∂ηηeitηηξ-ζ, with H=1+2itηηξ-ζ-1 we get (44)Kt,ξ,ζ=∫Reitη2ξ-ζη∂ηHη∂ηHηqt,ξ,ηqt,ζ,η¯dη. Since η∂ηHη∂ηHηqt,ξ,ηqt,ζ,η¯≤Cηη^2νη^-2ν/1+tη2ξ-ζ2 we get (45)Kt,ξ,ζ≤C∫01η^2νη^-2νηdη1+tη2ξ-ζ2+C∫1∞η^-2νηdη1+tη2ξ-ζ2≤Ct-2/3∫01η1+2νdη1+η2t1/3ξ-ζ2+Ct-2/3∫1∞η1-2νdη1+t1/3η2ξ-ζ2+Ct-2ν/3ξ-ζtν-1∫t1/2ξ-ζ1/2∞η1-2νη-4dη≤Ct-2/3ξ-ζt1/3-1-ν+Ct-2/3ξ-ζt1/3ν-1ξ-ζt1/3-1+Ct-2ν/3ξ-ζtν-1ξ-ζt-1-ν. Then by the Cauchy-Schwarz inequality and Young inequality we obtain (46)η1/2WϕL22≤Ct1/3ϕL2∫ξ-ζt1/3-1-νϕζdζL2+Ct1/3ϕL2∫ξ-ζt1/3ν-1ξ-ζt1/3-1ϕζdζL2+Ct1-2ν/3ϕL2∫ξ-ζtν-1ξ-ζt-1-νϕζdζL2≤Ct1/3ξt1/3-1-νL1ϕL22+Ct1/3ξt1/3ν-1ξt1/3-1L1ϕL22+Ct1-2ν/3ξtν-1ξt-1-νL1ϕL22≤CϕL22 if ν∈0,1. Lemma 14 is proved.Next we estimateWϕ0.Lemma 7.
Suppose thatξ∂ξkqt,ξ,η≤Cη+ξ-1 for all ξ,η∈R, k=0,1. Then the estimate η1/2W1L2≤Ct1/6 is true.Proof.
As in the proof of Lemma13 we decompose (47)W1=t1/2∫Re-itSξ,ηqt,ξ,ηdξ=t1/2∫Re-itSξ,ηqt,ξ,ηΘξη-1dξ+t1/2∫Re-itSξ,ηqt,ξ,ηχ1ξη-1-Θξη-1dξ+t1/2∫Re-itSξ,ηqt,ξ,ηχ2ξη-1dξ=I1+I2+I3 for η≠0. In the first summand I1 we integrate by parts via identity (19), to get (48)I1=Ct1/2∫Re-itSξ,ηξ-η∂ξH1qt,ξ,ηΘξη-1dξ. Using the estimate ξ-η∂ξH1qt,ξ,ηΘξη-1≤Cη-1/1+tηξ-η2 then changing ξ=ηy, we obtain (49)I1≤Ct1/2η-1∫1/3<ξ/η<3dξ1+tηξ-η2≤Ct1/2∫1/33dy1+tη3y-12≤Ct1/2η^-3/2. Thus we get η1/2I1L2≤Ct1/2η1/2η^-3/2L2≤Ct1/6. To estimate the integral I2 we integrate by parts via identity (24), to get (50)I2=Ct1/2∫Re-itSξ,ηξ∂ξχ1ξη-1-Θξη-1H2qt,ξ,ηdξ. We find the estimates ξ∂ξχ1ξη-1-Θξη-1H2qt,ξ,η≤Cη-1/1+tξη2, then we obtain (51)I2≤Ct1/2η-1∫ξ≤3ηdξ1+tξη2≤Ct1/2∫03dy1+tη3y≤Ct1/2η^-3/2. Thus as above we get η1/2I2L2≤Ct1/6 for all t≥1 if ν>-1/2. To estimate I3 we integrate by parts via identity (24) (52)I3=Ct1/2∫Re-itSξ,ηξ∂ξH2χ2ξη-1qt,ξ,ηdξ. We find the estimates ξ∂ξH2χ2ξη-1qt,ξ,η≤C/η+ξ1+tξ3 in the domain ξ≥3/2η, and then we obtain (53)I3≤Ct1/2∫3/2η∞dξη+ξ1+tξ3≤Ct1/2∫3/23/2ηη-1dy1+y1+tη3y3+Ct-1/2η-1∫3/2η∞ξ-3dξ≤Ct1/2η^-γη^γ-3+Ct-1/2η-3 if 0<γ<1. Thus we find (54)η1/2I3L2≤Ct1/2η1/2η^-γη^γ-3L2+Ct-1/2η-5/2L2≤Ct1/6 for all t≥1. Lemma 7 is proved.In the next lemma we estimate the commutatorη,V1. Define the norm ϕY=ϕL∞+t-1/6∂ξϕL2.Lemma 8.
Letj≥0, ν∈0,1. Then the estimate η3/2-jη^νη^-νtη,V1ξjϕL2≤Ct1/6ϕY is true for all t≥1.Proof.
Forη≠0 we integrate by parts (55)η1-jη^νη^-νtη,V1ξjϕ=η1-jη^νη^-νt2π∫Re-itSξ,η∂ξη-ξi∂ξSξ,ηχ1ξη-1ξjϕξdξ=t2π∫Re-itSξ,ηq1t,ξ,η∂ξϕξdξ+t2π∫Re-itSξ,ηq2t,ξ,ηϕξ-ϕ0ξdξ+ϕ0t2π∫Re-itSξ,ηq3t,ξ,ηdξ, where q1t,ξ,η=η^νη^-νη1-jη-ξ/i∂ξSξ,ηξjχ1ξη-1, q2t,ξ,η=ξ∂ξq1t,ξ,η, and q3t,ξ,η=∂ξq1t,ξ,η. Since ∂ξSξ,η=ξξ-ηη and η-ξ/∂ξSξ,η=ξ+η/ξ2+η2+ξη+ηξ, we have (56)q1t,ξ,η=η1-jη^νη^-νη-ξξξ-ηηξjχ1ξη-1=Oη^νη^-ν and similarly q2t,ξ,η=ξ∂ξq1ξ,η=Oη^νη^-ν for all ξ,η∈R in the domain ξ≤3η. Hence we have η∂ηkqlt,ξ,η≤Cη^νη^-ν for all ξ,η∈R, k=0,1,2, and l=1,2, where ν∈0,1, and by Lemma 14(57)η1/2t2π∫Re-itSξ,ηq1t,ξ,η∂ξϕξdξL2≤C∂ξϕL2 and by the Hardy inequality and Lemma 14(58)η1/2t2π∫Re-itSξ,ηq2t,ξ,ηϕξ-ϕ0ξdξL2≤Cϕξ-ϕ0ξL2≤C∂ξϕL2. Also we have (59)q3t,ξ,η=η1-jη^νη^-ν∂ξη-ξξξ-ηηξjχ1ξη-1=Oη+ξ-1ηβ+j-1 for all ξ,η∈R, in the domain ξ≤3η. Hence we get ξ∂ξkq3t,ξ,η≤Cη+ξ-1 for all ξ,η∈R, k=0,1. Therefore applying Lemma 7 we obtain η1/2t/2π∫Re-itSξ,ηq3t,ξ,ηdξL2≤C1/6. Lemma 8 is proved.In the next lemma we estimate the operatorV2.Lemma 9.
Letj=0,1,2, ν∈0,1. Then the estimate η5/2-jη^νη^-νtV2ξjϕL2≤Ct1/6ϕY is true for all t≥1.Proof.
We integrate by parts(60)η^νη^-νη2-jtV2ξjϕ=η^νη^-νη2-jt2π∫Re-itSξ,η∂ξξjχ2ξη-1i∂ξSξ,ηϕξdξ=t2π∫Re-itSξ,ηq1t,ξ,η∂ξϕξdξ+t2π∫Re-itSξ,ηq2t,ξ,ηϕξ-ϕ0ξdξ+ϕ0t2π∫Re-itSξ,ηq3t,ξ,ηdξ, where q1t,ξ,η=η^νη^-νη2-jξj/i∂ξSξ,ηχ2ξη-1, q2t,ξ,η=ξ∂ξq1t,ξ,η, and q3t,ξ,η=∂ξq1t,ξ,η. We find (61)q1t,ξ,η=Oη^νη^-νη2-jξj-2=Oη^νη^-ν, and q2t,ξ,η=Oη^νη^-ν for all ξ,η∈R, in the domain ξ≥3/2η if j=0,1,2. Then η∂ηkqlt,ξ,η≤Cη^νη^-ν for all ξ,η∈R, k=0,1,2, and l=1,2. Hence by Lemma 14 we find (62)η1/2t2π∫Re-itSξ,ηq1t,ξ,η∂ξϕξdξL2≤C∂ξϕL2, and by the Hardy inequality (63)η1/2t2π∫Re-itSξ,ηq2t,ξ,ηϕξ-ϕ0ξdξL2≤Cϕξ-ϕ0ξL2≤C∂ξϕL2. Also we have q3t,ξ,η=Oη+ξ-1 for all ξ,η∈R, in the domain ξ≥3/2η, if j=0,1,2. Hence ξ∂ξkq3t,ξ,η≤Cη+ξ-1 for all ξ,η∈R, k=0,1, and by Lemma 7(64)η1/2t2π∫Re-itSξ,ηq3t,ξ,ηdξL2≤Ct1/6. Lemma 9 is proved.In the next lemma, we estimate the derivative∂ηV.Lemma 10.
Letj=0,1, ν∈0,1. Then the estimate η1/2-jη^νη^-ν∂ηVξjϕL2≤Ct1/6ϕY is true for all t≥1.Proof.
SinceA1V=Viξ with A1=A0+iη, A0=1/2tη∂η, then we obtain the commutator ∂ηV=2tηiη,V. Also V=V1+V2. Hence (65)η1/2-jη^νη^-ν∂ηVξjϕL2≤Ctη3/2-jη^νη^-νη,V1ξjϕL2+Ctη3/2-jη^νη^-νV2ξjϕL2+Ctη1/2-jη^νη^-νV2ξj+1ϕL2. By Lemma 8 we find η3/2-jη^νη^-νtη,V1ξjϕL2≤Ct1/6ϕY for all t≥1, if j≥0, ν∈0,1. Also by Lemma 9 we get η5/2-jη^νη^-νtV2ξjϕL2+η3/2-jη^νη^-νtV2ξj+1ϕL2≤Ct1/6ϕY for all t≥1, if j=0,1, ν∈0,1. Hence the result of the lemma follows. Lemma 10 is proved.


## 4. A Priori Estimates

Local existence and uniqueness of solutions to Cauchy problem (1) were shown in [19, 20] when u0∈H1. By using the local existence result, we have the following.Theorem 11.
Assume that the initial datau0∈H1∩H0,1. Then there exists a unique local solution u of Cauchy problem (1) such that U-tu∈C0,T;H1∩H0,1.We can takeT>1 if the data are small in H1∩H0,1 and we may assume that uX1≤ε. To get the desired results, we prove a priori estimates of solutions uniformly in time. Define the following norm (66)uXT=supt∈1,Tφ^tL∞+t-1/6JutL2+t1/31-1/putLp, where J=UtxU-t, φ^t=FU-tut, and p>4.First we obtain the large time asymptotic behavior of the nonlinearityFU-t∂xu2u.Lemma 12.
The asymptoticstFU-t∂xu2u=iξ^ξ^5ξ^-6φ^2φ^+Oξ^ξ^-νφ^Y3 is true for all t≥1 and ξ∈R, where φ^t=FU-tut and ν>0 is small.Proof.
In view of factorization formula (11) we find tFU-t∂xu2u=iξV∗tψ2ψ=V∗tA1tψ2ψ, where ψ=Vφ^. Then by Lemma 5 with α=1/2+ν, β=2ν, and ν>0 small, we get (67)tFU-t∂xu2u=iξV∗tψ2ψ=iξA∗ψ2ψ+Ot-1/2+ν/3ξ^η1/2+νη^-2ν∂ηψ2ψL2+η-1/2+νη^-2νψ2ψL2 in the case of ξ<t-1/3 and (68)tFU-t∂xu2u=V∗A1ψ2ψ=A∗A1ψ2ψ+Otν/3-1/6ξ^-2νη1/2+νη^-2ν∂ηA1ψ2ψL2+η-1/2+νη^-2νA1ψ2ψL2 in the case of ξ>t-1/3. Via identity ην=t-ν/3η^νη^ν, we consider the remainder terms (69)η1/2+νη^-2ν∂ηψ2ψL2≤Cη1/2+νη^-2νψ2∂ηψL2≤Cηνη^-νη^-νψ2L∞η1/2η^νη^-ν∂ηψL2≤Ct1/6-ν/3ϕYψ2L∞,η-1/2+νη^-2ν∂ηA1ψ2ψL2≤Cη1+νη^-νη^-νψ2L∞η-1/2η^νη^-ν∂ηψ1L2+Cηνη^-νη^-νψψ1L∞η1/2η^νη^-ν∂ηψL2≤Ct1/6-ν/3ϕYηψ2L∞+ψψ1L∞, where ψ1=Viξφ^. By Lemma 10 with j=0,1, we have (70)η1/2η^νη^-ν∂ηψL2+η-1/2η^νη^-ν∂ηψ1L2≤Ct1/6φ^Y. Using Lemma 13 we get ψ≤Ct1/2ηη^-3/2+t1/6η^-3/2ϕY≤Ct1/6η^-1/2φ^Y and ψ1≤Ct1/2η2η^-3/2+t-1/6η^-1/2ϕY≤Cη1/2+t-1/6φ^Y. Hence ψ2L∞≤Ct1/3φ^Y2 and ηψ2L∞+ψψ1L∞≤Cφ^Y2. Also we find η-1/2+νη^-2νψ2ψL2≤Ct1/2φ^Y3η-1/2+νη^-2ν-3/2L2≤Ct1/2-ν/3φ^Y3 and η-1/2+νη^-2νA1ψ2ψL2≤Ct1/6φ^Y3η-1/2+νη^-2ν-1/2L2≤Ct1/6-ν/3φ^Y3. Therefore we obtain tFU-t∂xu2u=iξA∗ψ2ψ+Oξ^ξ^-2νφ^Y3 for ξ<t-1/3 and tFU-t∂xu2u=A∗A1ψ2ψ+Oξ^ξ^-2νφ^Y3 for ξ>t-1/3. Next by Lemma 13 we have ψjt,ξ=t1/2ξiξj/2itξ3φ^ξ+Ot1/6-j/3ξ^j-3/4φ^Y for j=0,1. Then we get iξA∗ψ2ψ=iξ^ξ^5ξ^-6φ^2φ^+Oξ^ξ^-νφ^Y3 and A∗A1ψ2ψ=iξ^ξ^5ξ^-6φ^2φ^+Oξ^ξ^-νφ^Y3. Lemma 12 is proved.Next we estimate the solutions in the normXT.Lemma 13.
Assume thatuX1≤ε holds. Then there exists ε>0 such that the estimate uXT<Cε is true for all T>1.Proof.
By continuity of the normuXT with respect to T, arguing by the contradiction we can find the first time T>0 such that uXT=Cε. To prove the estimate for the norm supt∈1,Tφ^L∞<Cε we use (11). Then in view of Lemma 12, we get (71)∂tφ^=λFU-t∂xu2u=iλt-1ξ^ξ^5ξ^-6φ^2φ^+Oε3t-1ξ^ξ^-ν. For the case of ξ<t-1/3 we can integrate φ^t,ξ≤φ^1,ξ+Cξφ^Y3∫1tτ-2/3dτ≤ε+Cξt1/3ε3≤ε+Cε3. For the case of ξ≥t-1/3 multiplying by φ^¯ and taking the real part of the result we obtain ∂tφ^t,ξ2=Ot-1ξ^ξ^-νφ^Y4. Integrating in time we obtain (72)φ^t,ξ2≤φ^ξ-3,ξ2+Cφ^Y4∫ξ-3tξτ1/3ξτ1/3-νdττ≤ε2+Cε4∫1ξt1/3y-1-νdy≤ε2+Cε4. Therefore FU-tutL∞<Cε. Applying estimate of Lemma 4 we find ∂xjut,x≤Cεxt-1/3j/2-1/4t-1/3-j/3, ut,x∂xut,x≤Cε2t-1, and(73)uLp≤Ct-1/3uXTxt-1/3-1/4Lp≤Cεt-1/31-1/pif p>4. Consider a priori estimates for JutL2. Using the identity ∂x-1Pu-Ju=3t∂x-1Lu, we get JuL2≤C∂x-1PuL2+CtuL63≤C∂x-1PuL2+Cε3t1/6. Applying the operator ∂x-1P to (1), in view of the commutators L,P=3L, P,∂x=-∂x, we get L∂x-1Pu=∂x-1P+3Lu=λ∂x-1P+3∂xu2u=λP+2u2u. Then by the energy method we obtain (74)ddt∂x-1PuL22≤CuuxL∞∂x-1PuL22+uL63∂x-1PuL2≤Cε2t-1∂x-1PuL22+Cε3t-5/6∂x-1PuL2 from which it follows that ∂x-1PuL2≤εtCε2+Cε3t1/6. Therefore JuL2≤C∂x-1PuL2+CtuL63≤Cε3t1/6 for all t∈1,T. Thus we obtain uXT<Cε. Lemma 13 is proved.


## 5. Proof of Theorem1

By Lemma13 we see that a priori estimate uXT≤Cε is true for all T>0. Therefore the global existence of solutions of Cauchy problem (1) satisfying the estimate uX∞≤Cε follows by a standard continuation argument by local existence Theorem 11.


## 6. Proof of Theorem2

In this section we prove the existence of a unique self-similar solutionvmt,x≡t-1/3fmxt-1/3 for (1), which is uniquely determined by the total mass condition m=1/2π∫Rvmt,xdx≠0. Define the operators (75)V~ϕ=12π∫Re-iSξ,ηϕξdξ,V~∗ϕ=2π∫ReiSξ,ηϕηηdη.Then for the self-similar solutions vmt,x=t-1/3fmxt-1/3=DtBMVφm^, where φm^t,ξ≡FU-tvmt, we find that φm^ have a self-similar form, that is, φm^t,ξ≡ϕmξ^ with ξ^=ξt1/3. Using the relation ∂tφm^t,ξ=1/3t-1ηϕm′η by factorization formula (11) we get 1/3η∂ηϕmη=iληV~∗V~ϕm2V~ϕm. Therefore ∂ηϕmη=3iλV~∗V~ϕm2V~ϕm≡Fϕm. Note that Fϕm is not in L2. Therefore we need the approximate equation. Define Θη=1 for η≤1 and Θη=0 for η>2, and denote ΘRη=Θη/R. Also define the approximate equation(76)∂ηϕm,Rη=3iλV~∗ΘRV~ΘRϕm,R2V~ΘRϕm,R≡FRϕm,Rη.Let us show a priori estimate ϕm,RZ=ϕm,RL∞+∂ηϕm,RL2≤3m uniformly in R. Applying Lemma 12 with t=1 we get (77)∂ηϕm,Rη=3iλη5η-6ΘRϕm,Rη2ΘRϕm,Rη+Oη-1-νϕm,RZ3. Integrating with respect to η, we obtain ∂ηϕm,RL2≤Cϕm,RZ3. Also multiplying by ϕm,R¯ and integrating with respect to η we get ϕm,Rη=m+O∫0ηη-1-γϕm,RZ3dη. Hence ϕm,RL∞≤m+Cϕm,RZ3. Thus we obtain ϕm,RZ≤2m+Cϕm,RZ3≤3m for some small m. Taking the limit R→∞, we find that there exists a unique solution ϕm of equation ∂ηϕmη=3iλV~∗V~ϕm2V~ϕm in Z. By the definition of ϕmη, we obtain ∂ξφm^L2=t1/6∂ηϕmL2≤Cmt1/6 and φm^L∞≤3m. In the same way as in the proof of (73) we have Lp estimate of vm for p>4.


## 7. Proof of Theorem3

Define the normuYT=supt∈1,Tt1/2-γuL∞+t-γJuL2 with a small γ>0.Lemma 14.
Suppose thatujXT≤Cε, j=1,2, where ε is sufficiently small. Let φ1^t,0=φ2^t,0 for j=1,2, t≥1, where φj^t,ξ=FU-tujt. Let u2=t-1/3fxt-1/3 be a self-similar solution. Then the estimate u1-u2YT<Cε is true for all T>1.Proof.
By the continuity of the normu1-u2YT with respect to T, arguing by the contradiction we can find for the first time T>0 such that u1-u2YT=Cε. We denote w^=φ1^-φ2^ and y=u1-u2. Applying estimate of Lemma 4 we find(78)yt,x≤Ct-1/3x1/2t-1/6-1/2w^t,x1/2t-1/2+Ct-1/2x1/2t-1/6-3/4∂ξw^L2≤Ct-1/2∂ξw^L2.Thus we need to estimate the norm ∂ξw^L2=JyL2. For the difference y we get from (1) L∂x-1Py=λP+2u12u1-u22u2. Hence by the energy method (79)ddt∂x-1PyL22=4Re∫R∂x-1Py¯u12Pu1-u22Pu2dx+2Re∫R∂x-1Py¯u12Pu1¯-u22Pu2¯dx+4Re∫R∂x-1Py¯u12u1-u22u2dx. Next we get (80)4Re∫R∂x-1Py¯u12Pu1-u22Pu2dx=4Re∫Ru12∂x-1Py¯Pydx+4Re∫Ru12-u22Pu2∂x-1Pydx=4Re∫R∂xu12∂x-1Py2dx+4Re∫Ru12-u22Pu2∂x-1Py¯dx≤Cu1u1xL∞∂x-1PyL22+C∂x-1PyL2u12-u22Pu2L2. In the same manner (81)2Re∫R∂x-1Py¯u12Pu1¯-u22Pu2¯dx=2Re∫Ru12∂x-1Py¯Py¯dx+2Re∫Ru12-u22Pu2¯∂x-1Py¯dx=2Re∫R∂xu12∂x-1Py¯2dx+2Re∫Ru12-u22Pu2¯∂x-1Py¯dx≤Cu1u1xL∞∂x-1PyL22+C∂x-1PyL2u12-u22Pu2L2. Note that Pu2=∂xxt-1/3fxt-1/3+3t∂tt-1/3fxt-1/3=0 for the case of self-similar solution u2=t-1/3fxt-1/3. Hence(82)ddt∂x-1PyL22≤Cu1u1xL∞∂x-1PyL22+C∂x-1PyL2u12u1-u22u2L2.By (78) we have x1/2t-1/61/2ut,x≤Cεt-1/3. To estimate u12u1-u22u2L2 we use the above estimates to get(83)u12u1-u22u2L2≤C∑j=12x1/2t-1/61/2ujL∞2x1/2t-1/6-1yL2≤Cε2t-2/3x1/2t-1/6-1yL2.In view of Lemma 4(84)x1/2t-1/6-1yL2≤Ct-1/3x1/2t-1/6-3/2w^t,x1/2t-1/2Lx2+Ct-1/2x1/2t-1/6-7/4L2∂ξw^L2. Since w^0=0, we get by the Hardy inequality x1/2t-1/6-3/2w^t,x1/2t-1/2Lx2≤C∂ξw^L2 and by a direct calculation x1/2t-1/6-7/4L2≤Ct1/6. Hence x1/2t-1/6-1yL2≤Ct-1/3∂ξw^L2. Therefore by (83) u12u1-u22u2L2≤Cε2t-1∂ξw^L2≤Cε3t-1+γ. Thus we obtain from (82) d/dt∂x-1PyL2≤Cε3t-1+γ which implies ∂x-1PyL2≤Cε3tγ. Therefore JyL2≤∂x-1PyL2+Ctu12u1-u22u2L2≤Cε3tγ. Lemma 14 is proved.Now we turn to the proof of asymptotic formula (7) for the solutions u of Cauchy problem (1). Let vmt,x be the self-similar solution with the total mass condition m=1/2π∫Ru0xdx=1/2π∫Rvmt,xdx≠0. Note that vmX∞≤Cε by Theorem 2 and uX∞≤Cε by Theorem 1. Also m=φ^t,0=vm^t,0 for t≥1. Then by Lemma 14 we find ut,x=vmt,x+Oεt-1/2+γ. Thus asymptotics (7) follows. Theorem 3 is proved.



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*Source: 8236942-2018-04-18.xml*