www.elsevier.com/locate/anihpc
Global well-posedness for the KP-II equation on the background of a non-localized solution
Luc Molinet
a, Jean-Claude Saut
b, Nikolay Tzvetkov
c,∗aLaboratoire de Mathématiques et Physique Théorique, UMR CNRS 6083, Faculté des Sciences et Techniques, Université François Rabelais Tours, Parc de Grandmont, 37200 Tours Cedex, France
bUniversité de Paris-Sud et CNRS, UMR de Mathématiques, Bât. 425, 91405 Orsay Cedex, France
cDépartement de Mathématiques, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France Received 15 November 2010; received in revised form 8 April 2011; accepted 19 April 2011
Available online 7 May 2011
Abstract
Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable inR×Tand perturbations that are square integrable inR2. In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.
©2011 Elsevier Masson SAS. All rights reserved.
1. Introduction
1.1. Presentation of the problem
The Kadomtsev–Petviashvili (KP) equations
(ut+uxxx+uux)x±uyy=0 (1)
were introduced in [13] to study the transverse stability of the solitary wave solution of the Korteweg–de Vries equa- tion
ut+ux+uux+
T−1 3
uxxx=0, x∈R, t∈R. (2)
HereT 0 is the Bond number which measures surface tension effects in the context of surface hydrodynamical waves. Actually the (formal) analysis in [13] consists in looking for aweakly transverseperturbation of the transport
* Corresponding author.
E-mail addresses:luc.molinet@lmpt.univ-tours.fr (L. Molinet), jean-claude.saut@math.u-psud.fr (J.-C. Saut), nikolay.tzvetkov@u-cergy.fr (N. Tzvetkov).
0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2011.04.004
equationut+ux=0.This perturbation amounts to adding to the equation a non-local term, namely 12∂x−1uyy. The same formal procedure is applied to the KdV equation (2) yielding the KP equation of the form
ut+ux+uux+
T −1 3
uxxx+1
2∂x−1uyy=0. (3)
By change of frame and scaling, (3) reduces to (1) with the+sign (KP-II) whenT <13 and the−sign (KP-I) when T >13.
As far as thetransverse stabilityof the KdV solitary wave (“1-soliton”)ψc(x−ct, y), where ψc(x, y)=3c
2 cosh−2 √
cx 2
, (4)
is concerned, the natural initial condition associated to (1) should beu0=ψc+v0wherev0is either “localized” inx andy, or localized inxandy-periodic. In any case this rules out initial data in Sobolev spaces likeHs(R2)or their anisotropic versionsHs1,s2(R2), as was for instance the case considered in [4,28,29,9,10,12,27] for the KP-II equation or in [17,14,11,31] for the KP-I equation.
Actually, it was proved in [18] that the Cauchy problem for the KP-I equation is globally well-posed for data which are localized perturbations of arbitrary size of anon-localizedtraveling wave solution such as the KdVN-soliton or the Zaitsev soliton (which is a localized inx and periodic inysolitary wave of the KP-I equation). The same result has been proven in [6] for localized perturbations ofsmallsize using Inverse Scattering techniques.
No such result seems to be known for the KP-II equation for data of arbitrary size. The aim of the present paper is to fix this issue. Observe that the results in [8] concern initial data localized iny and periodic inx, for instance belonging toHs(T×R)which excludes initial data of typeψc+v0as above.
On the other hand, the Inverse Scattering method has been used formally in [1] and rigorously in [2] to study the Cauchy problem for the KP-II equation with non-decaying data along a line, that isu(0, x, y)=u∞(x−vy)+φ(x, y) withφ(x, y)→0 asx2+y2→ ∞andu∞(x)→0 as|x| → ∞. Typically,u∞is the profile of a traveling wave solution with its peak localized on a moving line in the(x, y)plane. It is a particular case of the N-soliton of the KP-II equation discovered by Satsuma [22] (see the derivation and the explicit form whenN=1,2 in the Appendix of [19]). As in all results obtained for KP equations by using the Inverse Scattering method, the initial perturbation of the non-decaying solution is supposed to be small enough in a weightedL1space (see [2, Theorem 13]).
As will be proven in the present paper, PDE techniques allow to consider arbitrary large perturbations of a (smaller) class of non-decaying solutions of the KP-II equation.
We will therefore study here the initial value problem for the Kadomtsev–Petviashvili (KP-II) equation
(ut+uxxx+uux)x+uyy=0. (5)
We will either consider periodic iny solutions or we will suppose thatu=u(t, x, y),(x, y)∈R2,t∈R, with initial data
u(0, x, y)=φ(x, y)+ψ (x, y), (6)
whereψis the profile of a non-localized (i.e. not decaying in all spatial directions) traveling wave of the KP-II equation andφis localized, i.e. belonging to Sobolev spaces onR2. We recall (see [3]) that, contrary to the KP-I equation, the KP-II equation does not possess any localized in both directionstraveling wave solution. The background solution could be for instance the line soliton (1-soliton)ψc(x−ct, y)moving with velocitycof the Korteweg–de Vries (KdV) equation defined by (4). But it could also be the profile of theN-soliton solution of the KdV equation,N2.The KdVN-soliton is of course considered as a two-dimensional (constant iny) object.
Solving the Cauchy problem in both those functional settings can be viewed as a preliminary (and necessary!) step towards the study of the dynamics of the KP-II equation on the background of a non-fully localized solution, in particular towards a PDE proof of the nonlinear stability of the KdV soliton orN-soliton with respect to transversal perturbations governed by the KP-II flow. This has been established in [2, Proposition 17] by Inverse Scattering methods and very recently by Mizumachi and Tzvetkov [15] who proved (by PDE techniques but using the Miura transform for the KP-II equation) theL2stability of the KP-II line soliton with respect to transverse perturbations.
We recall that it is established in [30] by Inverse Scattering methods and in [20,21] by PDE techniques which extend to a large class of (non-integrable) problems, that the KdV 1-soliton is transversally nonlinearly unstable with respect to the KP-I equation.
The advantage of the PDE approach of the present paper compared to an Inverse Scattering one is that it can be straightforwardly applied to non-integrable equations such as the higher order KP-II equations (see [25,26]).
1.2. Statement of the results
As was previously mentioned our main result is that the KP-II equation is globally well-posed for data of arbitrary size in the two aforementioned functional settings.
Theorem 1.1.The Cauchy problem associated with the KP-II equation is globally well-posed inHs(R×T)for any s0. More precisely for everyφ∈Hs(R×T)there is a global strong solution of(5)inC(R;Hs(R×T))with initial datumu(0, x, y)=φ(x, y). The solution is unique inC(R;Hs(R×T))ifs >2. Fors∈ [0,2]the uniqueness holds in the following sense. For everyT >0there is a Banach spaceXT continuously embedded inC([−T , T];Hs(R×T)) and containingC([−T , T];H∞(R×T))such that the solutionubelongs toXT and is unique in this class. Moreover the flow is Lipschitz continuous on bounded sets ofHs(R×T). Namely, for every bounded setBofHs(R×T)and everyT >0there exists a constantC such that for everyφ1, φ2∈Bthe corresponding solutionsu1,u2satisfy
u1−u2L∞([−T ,T];Hs(R×T))Cφ1−φ2Hs(R×T). Finally theL2-norm is conserved by the flow, i.e.
u(t,·)
L2(R×T)= φL2(R×T), ∀t∈R.
We next state our result concerning localized perturbations.
Theorem 1.2.Letψc(x−ct, y)be a solution of the KP-II equation such that for everyσ0,(1−∂x2−∂y2)σ2ψc:R2→ Ris bounded and belongs toL2xL∞y (R2). Lets0be an integer. Then for everyφ∈Hs(R2)there exists a global strong solutionuof (5)with initial dataφ+ψc. The uniqueness holds in the following sense. For everyT >0there is a Banach spaceXT continuously embedded inC([−T , T];Hs(R×T))and containingC([−T , T];H∞(R×T)) such that the solutionubelongs toXT and is unique in this classψc(x−ct, y)+XT.
The spacesXT involved in the statements of the above results are suitable Bourgain spaces defined below.
Our proof is based on the approach by Bourgain to study the fully periodic case. We need however to deal with difficulties linked to several low frequencies cases (see for example Lemma 3.4 below) which do not occur in the purely periodic setting. Moreover in the proof of Theorem 1.2 one needs to make a use of the Kato type smoothing effect for KP-II which was not present in previous works on the low regularity well-posedness of the KP-II equation.
Remark 1.3.As was previously noticed, the result of Theorem 1.2 holds (with the same proof) when we replaceψc by the value att=0 of theN-soliton solutionSNof the KdV equation which satisfies similar smoothness and decay properties asψc. This follows from the structure of theSN. For instance (see [5]) the 2-soliton of the KdV equation written in the form
ut−6uux+uxxx=0 is
S2(x, t )= −123+4 cosh(2x−8t )+cosh(4x−64t ) (3 cosh(x−28t )+cosh(3x−36t ))2 .
On the other hand, a priori one cannot take instead ofψc a functionψ which is non-decaying along a line{(x, y)| x−vy=x0},as for instance the line soliton of the KP-II equation which writesψ (x−vy−ct ).However, as was pointed out to us by Herbert Koch the change of variables(X=x+vy−v2t, Y=y−2vt )leaves invariant the KP-II equation and transforms the line soliton to an independent ofyobject. Therefore our analysis applies equally well to the line soliton.
1.3. Organization of the paper
The second section is devoted to the proof of a bilinear estimate for localized functions inR2×Z.It is based on Bourgain’s method in [4] who considers the case of functions inR×Z2.We prove Theorem 1.1 in Section 3 as a consequence of two bilinear estimates in BourgainXb,sspaces. Finally we prove Theorem 1.2 in Section 4 by a fixed point argument in suitable Bourgain spaces, using in a crucial way the dispersive and smoothing properties of the KP-II linear group.
1.4. Notations
We will denote · Lp(resp · Hs) the standard norm in the Lebesgue spaceLp(resp. the Sobolev spaceHs), the domain being clear from the context. The Fourier transform onR2x,y×Rt (resp.Rx) is denotedForˆ(resp.Fx). We will use the “Japanese bracket notation” · =(1+|·|2)1/2.The notationU (t )will stand for the (unitary in allHs(R2) orHs(R×T)Sobolev spaces) KP-II group, that isU (t )=e−t (∂x3+∂−x1∂y2). For(b, s)∈R×R,the norm of the Bourgain space associated toU (t )is, for functions defined onR2x,y×Rt(with the obvious modification that integration inη∈R is replaced by summation inq∈Zfor functions defined onRx×Ty×Rt):uXb,b1,s = U (−t )uHb,b1,s,where
u2Hb,b1,s =
R2ξ,η×Rτ
τ ξ3
2b1
ζ2s τ2bu(τ, ξ, η)ˆ 2dτ dξ dη, ζ =(ξ, η),
that is
u2Xb,b1,s=
R2ξ,η×Rτ
σ (τ, ξ, η) ξ3
2b1
ζ2s σ (τ, ξ, η)2bu(τ, ξ, η)ˆ 2dτ dξ dη,
whereσ (τ, ξ, η)=τ−ξ3+η2/ξ. For anyT >0,the norm in the localized versionXb,bT 1,sis defined as uXb,b1,s
T =inf
wXb,b1,s, w(t )=u(t )on(−T , T ) .
For(b, s)∈R×Rthe spaceZb,s is the space associated to the norm uZb,s=σb−12 ζsuˆ
L2ζL1τ.
We define the restricted spaces ZTb,s similarly to above. The notationis used for C whereC is an irrelevant constant. For a real numbers, s−means “for anyr < sclose enough tos”. IfA, B∈R,we denoteA∧B=min(A, B) and A∨B=max(A, B). A∼B means that there exists c >0 such that 1c|A||B|c|A|. IfS is a Lebesgue measurable set inRn,|S|,or mesS,denotes its Lebesgue measure. #Sdenotes the cardinal of a finite setS.
2. A bilinear estimate for localized functions inRRR2×ZZZ
In this section we will prove the following crucial bilinear estimate for functions having some localizations related to the KP-II dispersion relation inR2×Z. This is essentially a bilinearL4–L2Strichartz estimate associated to the linear KP-II group. We mainly follow the proof of Bourgain [4] that treats the case of functions inR×Z2.
Proposition 2.1.Letu1andu2be two real valuedL2functions defined onR×R×Zwith the following support properties
(τ, ξ, q)∈supp(ui) ⇒ ξ∈Ii, |ξ| ∼Mi, τ −ξ3+q2/ξ
∼Ki, i=1,2.
Then the following estimates hold, u1 u2L2(K1∧K2)1/2
|I1| ∧ |I2|1/2
(K1∨K2)1/4(M1∧M2)1/4
u1L2u2L2 (7) and ifM1∧M21then
u1 u2L2(K1∧K2)1/2(K1∨K2)1/4(M1∧M2)1/4
×
(M1∧M2)1/4
(K1∨K2) (M1∨M2)1−
+(|I1| ∧ |I2|)1/2 (K1∨K2)1/4
u1L2u2L2. (8) 2.1. Three basic lemmas
Before starting the proof of the proposition let us recall the three following basic lemmas that we will use inten- sively.
Lemma 2.2.Consider a setΛ⊂Rξ×Zq. Let the projection ofΛon theξ-axis be contained in a setI ⊂R. Assume in addition that for any fixedξ0∈I the cardinal of the setΛ∩ {(ξ0, q), q∈Z}is bounded by a constantC. Then
|Λ|C|I|.
Lemma 2.3.LetI andJ be two intervals on the real line andf:J→Rbe a smooth function. Then mes
x∈J: f (x)∈I
|I|
infξ∈J|f(ξ )| and
#
q∈J∩Z: f (q)∈I
|I|
infξ∈J|f(ξ )|+1.
Lemma 2.4.Leta=0,b,cbe real numbers andI be an interval on the real line. Then mes
x∈R: ax2+bx+c∈I
|I|1/2
|a|1/2 and
#
q∈Z: aq2+bq+c∈I
|I|1/2
|a|1/2+1.
2.2. Proof of Proposition 2.1
According to [4, p. 320], we can suppose thatξ 0 on the support ofuj(τ, ξ, q)(see also [24, p. 460] for a detailed discussion). We thus have to evaluate
q∈Z
R×R+
q1∈Z
R×R+
χ{ξ−ξ10}u1(τ1, ξ1, q1)u2(τ−τ1, ξ−ξ1, q−q1) dτ1dξ1 2dτ dξ.
By the Cauchy–Schwarz inequality in(τ1, ξ1, q1)we thus get u1 u22L2 sup
(τ,ξ0,η)
|Aτ,ξ,q|u12L2u22L2 (9)
whereAτ,ξ,q⊂R2×Zis the set Aτ,ξ,q:=
(τ1, ξ1, q1): ξ1∈I1, ξ−ξ1∈I2, 0< ξ1∼M1, 0< ξ−ξ1∼M2,
τ1−ξ13+q12 ξ1
∼K1,
τ−τ1−(ξ−ξ1)3+(q−q1)2 ξ−ξ1
∼K2
.
A use of the triangle inequality yields
|Aτ,ξ,q|(K1∧K2)|Bτ,ξ,q| (10)
where
Bτ,ξ,q:=
(ξ1, q1): ξ1∈I1, ξ−ξ1∈I2, 0< ξ1∼M1, 0< ξ−ξ1∼M2,
τ−ξ3+q2
ξ +3ξ ξ1(ξ−ξ1)+(ξ q1−ξ1q)2 ξ ξ1(ξ−ξ1)
(K1∨K2)
.
Let us now first prove (7) by bounding|Bτ,ξ,q|in a direct way. The measure of the projection ofBτ,ξ,q on theξ1- axis is bounded by (|I1| ∧ |I2) and for a fixedξ1, using Lemma 2.4, the cardinal of its q1-section is bounded by c(M1∧M2)1/2(K1∨K2)1/2+1. Therefore Lemma 2.2 and (9)–(10) yield the bound (7).
To prove (8) we divideBτ,ξ,qinto two regions by setting Bτ,ξ,q1 :=
(ξ1, q1)∈Bτ,ξ,q: q1
ξ1 −q−q1 ξ−ξ1
1
(11) and
Bτ,ξ,q2 :=
(ξ1, q1)∈Bτ,ξ,q: q1
ξ1 −q−q1 ξ−ξ1
1
. (12)
In the next lemma we estimate the size of the first region.
Lemma 2.5.ForM1∧M21, the following estimate holds Bτ,ξ,q1 (K1∨K2)1/2(M1∧M2)1/2.
Proof. Recall thatξ1andξ−ξ1are non-negative real numbers and thusξ∼ξ1∨(ξ−ξ1). Note also that (ξ q1−ξ1q)2
ξ ξ1(ξ−ξ1) =ξ1(ξ−ξ1) ξ
q1
ξ1−q−q1 ξ−ξ1
2
. (13)
From the definition ofBτ,ξ,q1 we thus deduce that
τ−ξ3+q2/ξ+3ξ ξ1(ξ−ξ1)K1∨K2+M1∧M2.
According to Lemma 2.4, the projection ofBτ,ξ,q1 on theξ1-axis is thus bounded by (K1∨K2+M1∧M2)1/2
(M1∨M2)1/2 (K1∨K2)1/2
(M1∨M2)1/2+(M1∧M2)1/2
(M1∨M2)1/2. (14)
We separate two cases:
(1)M1∧M2K1∨K2. Then from the definition ofBτ,ξ,q1 and Lemma 2.3 we infer that forξ1fixed, the cardinal of theq1-section ofBτ,ξ,q1 is bounded from above by M1∧M2 ∼M1∧M2. We thus get by Lemma 2.2 that in the considered case
Bτ,ξ,q1 (K1∨K2)1/2(M1∧M2)
(M1∨M2)1/2 (K1∨K2)1/2(M1∧M2)1/2. (2)M1∧M2K1∨K2. In this case we subdivide once more:
(2.1)|qξ1
1 −qξ−−qξ1
1|(M(K1∨K2)1/2
1∧M2)1/2. Then again by (14), Lemmas 2.2 and 2.3, we have that in the considered case Bτ,ξ,q1 (M1∧M2)1/2
(M1∨M2)1/2
(K1∨K2)1/2(M1∧M2) (M1∨M2)1/2
(K1∨K2)1/2(M1∧M2)1/2. (2.2) 1|qξ1
1 −qξ−−qξ1
1|(M(K1∨K2)1/2
1∧M2)1/2. Since ∂
∂q1
τ−ξ3+q2/ξ+3ξ ξ1(ξ−ξ1)+ξ1(ξ−ξ1) ξ
q1
ξ1−q−q1 ξ−ξ1
2 =2
q1
ξ1−q−q1 ξ−ξ1
(15)
it follows from Lemma 2.3 that the cardinal of theq1-section at fixedξ1is bounded by 2(K1∨K2)(M1∧M2)1/2
(K1∨K2)1/2 +1(K1∨K2)1/2(M1∧M2)1/2.
According to (14), the projection of this region on theξ1-axis is of measure less than a uniform constant and thus in the considered case
Bτ,ξ,q1 (K1∨K2)1/2(M1∧M2)1/2. This completes the proof of Lemma 2.5. 2
We now divideBτ,ξ,q2 into three regions by setting Bτ,ξ,q2,1 :=
(ξ1, q1)∈Bτ,ξ,q2 : q1
ξ1 −q−q1
ξ−ξ1
< (K1∨K2)1/2 (M1∧M2)1/2
,
Bτ,ξ,q2,2 :=
(ξ1, q1)∈Bτ,ξ,q2 : (K1∨K2)1/2 (M1∧M2)1/2
q1
ξ1 −q−q1
ξ−ξ1
<(M1∨M2)1/2 (M1∧M2)1/2
and
Bτ,ξ,q2,3 :=
(ξ1, q1)∈Bτ,ξ,q2 : max
(K1∨K2)1/2
(M1∧M2)1/2,(M1∨M2)1/2 (M1∧M2)1/2
q1
ξ1 −q−q1 ξ−ξ1
.
Note thatBτ,ξ,q2,1 andBτ,ξ,q2,2 may be empty.
Lemma 2.6.The following estimates hold wheneverM1∧M21 Bτ,ξ,q2,1 (K1∨K2)(M1∧M2)1/2
(M1∨M2)1/2 (16)
and Bτ,ξ,q2,2 (K1∨K2)1/2(M1∧M2)1/2. (17)
Proof. Assuming thatBτ,ξ,q2,1 is not empty, it follows from (13) and Lemma 2.4 that the measure of its projection on theξ1-axis can be estimated as
(K1∨K2)1/2 (M1∨M2)1/2.
On the other hand, it follows from Lemma 2.3 that for a fixedξ1the cardinal of itsq1-section is bounded by
(M1∧M2)1/2(K1∨K2)1/2+1(M1∧M2)1/2(K1∨K2)1/2. (18)
This proves (16) in view of Lemma 2.2. Now coming back to (13) and using Lemma 2.4 we infer that the projection ofBτ,ξ,q2,2 on theξ1-axis is bounded by
1 (M1∨M2)1/2
(K1∨K2)+(M1∨M2)1/2
C
sinceBτ,ξ,q2,2 is empty as soon asK1∨K2M1∨M2. On the other hand, it follows from (15) and Lemma 2.2 that for a fixedξ1 the cardinal of itsq1-section is also bounded by (18). This leads to (17) thanks to Lemma 2.2. This completes the proof of Lemma 2.6. 2
We finally estimate the size ofBτ,ξ,q2,3 . Lemma 2.7.For any0< ε1it holds
Bτ,ξ,q2,3 Cε
(M1∧M2)1/2+ε/2
(M1∨M2)1/2−ε/2(K1∨K2)+ |I1| ∧ |I2|. (19)
Proof. We subdivideBτ,ξ,q2,3 by setting Bτ,ξ,q2,3 (+∞):=
(ξ1, q1)∈Bτ,ξ,q2,3 : q1
ξ1 −q−q1
ξ−ξ1
|K1∨K2|
and
Bτ,ξ,q2,3 (L):=
(ξ1, q1)∈Bτ,ξ,q2,3 : q1
ξ1−q−q1 ξ−ξ1
∼L
whereL=2l,l0ll1, with l0∼ln
max
M1∨M2
M1∧M2, K1∨K2 M1∧M2
, l1∼ln(K1∨K2).
In view of (15), for a fixedξ1, theq1-section ofBτ,ξ,q2,3 (+∞)contains at most two elements and thus by Lemma 2.2,
Bτ,ξ,q2,3 (+∞)|I1| ∧ |I2|. (20)
Now, inBτ,ξ,q2,3 (L)it holds
τ−ξ3+q2/ξ+3ξ ξ1(ξ−ξ1)K1∨K1+(M1∧M2)L2
and thus from Lemma 2.4 we infer that the measure of the projection of this region on theξ1-axis is bounded by (K1∨K2)1/2
(M1∨M2)1/2 +(M1∧M2)1/2 (M1∨M2)1/2L.
Interpolating with the crude boundM1∧M2we obtain the bound (K1∨K2)1/2
(M1∨M2)1/2+(M1∧M2)1/2 (M1∨M2)1/2L
1−ε
(M1∧M2)ε.
On the other hand, for a fixedξ1, by using again (15) and Lemma 2.3, we obtain that the cardinal of itsq1-section is bounded by
K1∨K2
L +1K1∨K2
L .
We thus get by Lemma 2.2, Bτ,ξ,q2,3 (L)
(K1∨K2)1/2
(M1∨M2)1/2+(M1∧M2)1/2 (M1∨M2)1/2L
1−ε
(M1∧M2)ε(K1∨K2) L
(M1∧M2)1/2+ε/2(K1∨K2) (M1∨M2)1/2−ε/2Lε ,
where we used that in the considered case L (M(K1∨K2)1/2
1∧M2)1/2. A summation overL yields the claimed bound. This completes the proof of Lemma 2.7. 2
Now, using Lemmas 2.5–2.7, we get
|Bτ,ξ,q|(K1∨K2)1/2(M1∧M2)1/2+Cε(M1∧M2)1/2(K1∨K2)
(M1∨M2)1/2−ε + |I1| ∧ |I2|, and thus according to (9)–(10),
u1 u2L2Cε(K1∧K2)1/2(K1∨K2)1/4
×
(M1∧M2)1/4
(K1∨K2)1/4 (M1∨M2)1/4−ε
+(|I1| ∧ |I2|)1/2 (K1∨K2)1/4
u1L2u2L2. This completes the proof of Proposition 2.1.
Corollary 2.8.Let u1 and u2 be two real valued L2 functions defined onR×R×Z with the following support properties
(τ, ξ, q)∈supp(ui) ⇒ |ξ| ∼Mi, τ−ξ3+q2/ξ
∼Ki, i=1,2.
Then the following estimates hold,
u1 u2L2(|ξ|∼M)(K1∧K2)1/2(M1∧M2∧M)1/2
× (K1∨K2)1/4(M1∧M2)1/4
u1L2u2L2 (21) and ifM1∧M21then
u1 u2L2(|ξ|∼M)(K1∧K2)1/2(K1∨K2)1/4
×
(M1∧M2)1/4
(K1∨K2) (M1∨M2)1−
1/4
+(M1∧M2∧M)1/2 (K1∨K2)1/4
u1L2u2L2. (22) Proof. Rewriting the functionsui asui=
k∈Zui,kwith ui,k=uiχ{kMξ <(k+1)M},
we easily obtain by support considerations and the Minkowsky inequality that u1 u2L2(|ξ|∼M)=
k∈Z
0|q|1
u1,k u2,−k+q
L2(|ξ|∼M)
0q1
k∈Z
u1,k u2,−k+qL2.
The desired result follows by applying Proposition 2.1 with|I1| = |I2| =Mfor eachk∈Z, and then Cauchy–Schwarz ink. 2
A rough interpolation between (7) and (8) on one side and between (21) and (22) on the other side gives the following bilinear estimates that we will use intensively in the next section.
Corollary 2.9.There existsβ0<1/2such that the following holds true. Letu1andu2be two real valuedL2functions defined onR×R×Zwith the following support properties
(τ, ξ, q)∈supp(ui) ⇒ |ξ| ∼Mi1, τ−ξ3+q2/ξ
∼Ki, i=1,2.
Then the following estimates hold,
u1 u2L2(K1K2)β0(M1∧M2)1/2u1L2u2L2 (23) and
u1 u2L2(|ξ|∼M)(K1K2)β0
(M1∧M2)1/4+(M1∧M2∧M)1/2
u1L2u2L2. (24) Proof. Let us first prove (23). IfM1∧M21 then we can easily conclude by only using (7). IfM1∧M21 then (8) gives
u1 u2L2(K1∧K2)1/2(K1∨K2)1/4(M1∧M2)1/4
×
1+ (K1∨K2)1/4
(M1∨M2)1/4−+(M1∧M2)1/4 (K1∨K2)1/4
u1L2u2L2.
We now distinguish cases according to which terms dominate in the sum 1+ (K1∨K2)1/4
(M1∨M2)1/4−+(M1∧M2)1/4 (K1∨K2)1/4.
If the first or the third term dominates then we get directly the needed bound. The only case we really need an interpolation between (7) and (8) is when the second term dominates. In this case we interpolate with weightθon the bound (7) and weight 1−θon the second bound (8) and the conditions onθto get the needed estimate are
θ >0, 3 4θ <1
2
which of course can be achieved. Let us now turn to the proof of (24). Again, if M1∧M21 then we can easily conclude by only using (21). IfM1∧M21 then (22) gives
u1 u2L2(|ξ|∼M)(K1∧K2)1/2(K1∨K2)1/4
×
(M1∧M2)1/4
1+ (K1∨K2)1/4 (M1∨M2)1/4−
+(M1∧M2∧M)1/2 (K1∨K2)1/4
u1L2u2L2. Again, we distinguish cases according to which terms dominate in the sum
(M1∧M2)1/4+(M1∧M2)1/4(K1∨K2)1/4
(M1∨M2)1/4− +(M1∧M2∧M)1/2 (K1∨K2)1/4 .
If the first or the third term dominates then we get directly the needed bound. If the second term dominates than we interpolate with weightθ on the bound (21) and weight 1−θon the bound (22) and the conditions onθ to get the needed estimate are
θ >0, θ <1 2,
the first assumption being imposed in order to ensure the K1, K2conditions and the second one for theM,M1and M2conditions. This completes the proof of Corollary 2.9. 2
3. Global well-posedness onRRR×TTT 3.1. Two bilinear estimates inXb,b1,s spaces
The local well-posedness result is a direct consequence of the following bilinear estimates combined with a stan- dard fixed point argument inXb,b1,s spaces.
Proposition 3.1.There exist β <1/2 and1/4< b1<3/8 such that for allu, v∈X1/2,b1,s, the following bilinear estimates hold
∂x(uv)
X−1/2,b1,s uX1/2,b1,svXβ,0,s+ uXβ,0,svX1/2,b1,s (25) and ∂x(uv)
Z−1/2,s uX1/2,b1,svXβ,0,s+ uXβ,0,svX1/2,b1,s, (26)
provideds0.
To prove this bilinear estimate we first note that by symmetry it suffices to consider ∂xΛ(u, v) whereΛ(·,·)is defined by
Fx
Λ(u, v) :=
R
χ|ξ1||ξ−ξ1|(Fxu)(ξ1)(Fxv)(ξ−ξ1) dξ1.
The following resonance relation (see [4]) is crucial for our analysis:
|σ−σ1−σ2| =
3ξ ξ1(ξ−ξ1)+(ξ q1−ξ1q)2 ξ ξ1(ξ−ξ1)
3ξ ξ1(ξ−ξ1), (27) where
σ:=σ (τ, ξ, q):=τ−ξ3+q2/ξ, σ1:=σ (τ1, ξ1, q1),
