Global well-posedness for KdV in Sobolev Spaces of negative index

5 Pages·0.12 MB·English

by  J. Colliander

GLOBAL WELL-POSEDNESS FOR KDV IN SOBOLEV SPACES
OF NEGATIVE INDEX
1
0
0 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO
2

n
Abstract. The initial value problem for the Korteweg-deVries equation on
a
the line is shown to be globally well-posed for rough data. In particular, we
J showglobalwell-posednessforinitialdatainHs(R)for−130 <s.
1
3

]
P
1. Introduction
A
. Consider the initial value problem for the Korteweg-deVries (KdV) equation
h
t ∂ u+∂3u+ 1∂ (u2)=0, x∈R,
a (1.1) t x 2 x
m (cid:26) u(0)=φ,
[ for rough initial data φ ∈ Hs(R), s < 0. This problem is known [9] to be locally

well-posed provided −3 <s. For s≥0, the local result and L2 norm conservation
1 4
v imply(1.1)isgloballywell-posed[1]. Recently,adirectadaptation[7]ofBourgain’s
1 high-low frequency technique [3], [2] showed (1.1) is globally well-posed for φ ∈
6 Hs∩H˙a for certain s,a < 0. A modification of the high-low frequency technique,
2
first used in [8], is presented in this paper which establishes global well-posedness
1
0 of (1.1) in Hs(R), −130 <s.
1 A subsequent paper [6] will establish that (1.1) is globally well-posed in Hs(R)
0 for −3 <s.The simplicity ofthe argumentpresentedhere may extend moreeasily
4
h/ toothersituations,suchasinourtreatment[5]ofcubicNLS onR2 andNLS with
t derivative in R [4].
a
m The Multiplier operator I
Let s<0 and N ≫1 be fixed. Define the Fourier multiplier operator
:
v
1, |ξ|<N,
Xi (1.2) Iu(ξ)=m(ξ)u(ξ), m(ξ)=(cid:26)N−s|ξ|s, |ξ|≥10N
ar with m smoothcand monotonbe. The operator I (barely) maps Hs(R) 7−→ L2(R).
Observe that on low frequencies {ξ : |ξ| < N}, I is the identity operator. Note
also that I commutes with differential operators. The operator I−1 is the Fourier
multiplier operator with multiplier 1 .
m(ξ)
An almost L2 conservation property of (1.1)
1991 Mathematics Subject Classification. 35Q53,42B35,37K10.
Key words and phrases. Korteweg-de Vries equation, nonlinear dispersiveequations, bilinear
estimates.
J.E.C.issupportedinpartbyanN.S.F.Postdoctoral ResearchFellowship.
M.K.issupportedinpartbyN.S.F.GrantDMS9801558.
G.S.issupportedinpartbyN.S.F.GrantDMS9800879andbyaTermanAward.
T.T. is a Clay Prize Fellow and is supported in part by grants from the Packard and Sloan
Foundations.
1
2 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO
Let φ ∈ Hs(R), −3 < s < 0 in (1.1). There is a δ = δ(kφk ) > 0 such that
4 Hs
(1.1) is well-posed for t ∈ [0,δ]. We observe using the Fundamental Theorem of
Calculus, the equation, and integration by parts that
δ d
kIu(δ)k2 = kIu(0)k2 + (Iu(τ),Iu(τ))dτ,
L2 L2 Z dτ
0
δ
= kIu(0)k2 +2 (Iu˙(τ),Iu(τ))dτ,
L2 Z
0
δ 1
= kIu(0)k2 +2 (I(−u − ∂ [u2])(τ),Iu(τ))dτ
L2 Z xxx 2 x
0
δ
= kIu(0)k2 + (I(−∂ [u2]),Iu)dτ.
L2 Z x
0
Finally, we add 0= δ ∂ (I(u)2)I(u)dτ to observe
0 x
R R
δ
(1.3) kIu(δ)k2 =kIu(0)k2 + ∂ (I(u))2−I(u2) Iu dxdτ.
L2 L2 Z0 Z xn o
Thislaststepenablesustotakeadvantageofsomeinternalcancellation. Weapply
Cauchy-Schwarzas in [10] and bound the integral above by
(1.4) ∂ {(I(u))2−I(u2)} kIuk .
x Xδ
(cid:13)(cid:13) (cid:13)(cid:13)X0δ,−12− 0,12+
(cid:13) (cid:13)
Remark 1. Anefforttofindatermprovidingmorecancellationthan δ ∂ (I(u)2)I(u)dτ
0 x
used above led to the general procedure described in [6]. R R
Proposition 1. (A variant of local well-posedness) The initial value problem (1.1)
is locally well-posed in the Banach space I−1L2 = {φ ∈ Hs with norm kIφk }.
L2
with existence lifetime δ satisfying
(1.5) δ &kIφk−α, for some α>0,
L2
and moreover
(1.6) kIuk ≤CkIφk .
Xδ L2
0,12+
This proposition is not difficult to prove using the argument in [9]. Using
Duhamel’s formula and X space properties reduces matters to proving the bi-
s,b
linear estimate
(1.7) k∂ I(uv)k ≤CkIuk kIvk
x X0,−21+ X0,21+ X0,21+
to obtain the contraction. The space-time norm bound is then implied by the
contractionestimate. The estimate (1.7)followsfromthe nextpropositionandthe
bilinear estimate of Kenig, Ponce and Vega [9].
Proposition 2. (Extra smoothing) The bilinear estimate
(1.8) k∂x{I(u)I(v)−I(uv)}kXδ ≤CN−34+kIukXδ kIvkXδ .
0,−12− 0,21+ 0,12+
holds.
GLOBAL WELL-POSEDNESS FOR KDV IN SOBOLEV SPACES OF NEGATIVE INDEX 3
Recall the bilinear estimate k∂ (uv)k ≤ Ckuk kvk from [9].
x X0,−12+ X0,12+ X0,21+
Proposition2revealsasmoothingbeyondtherecoveryofthefirstderivativeforthe
particular quadratic expression encountered above in (1.3). We prove Proposition
2 in the next section.
The required pieces are now in place for us to give the proof of global well-
posedness of (1.1) in Hs(R), − 3 <s. Global well-posedness of (1.1) will follow if
10
we show well-posedness on [0,T] for arbitrary T >0. We renormalize things a bit
via scaling. If u solves (1.1) then u (x,t) = (1)2u(x, t ) solves (1.1) with initial
λ λ λ λ3
data φ (x,t) = (1)2φ(x). Note that u exists on [0,T] if and only if u exists on
λ λ λ λ
[0,λ3T]. A calculation shows that
(1.9) kIφλkL2 ≤Cλ−23−sN−skφkHs.
Here N = N(T) will be selected later but we choose λ = λ(N) right now by
requiring
(1.10) Cλ−32−sN−skφk ∼1 =⇒ λ∼N−3+2s2s.
Hs
We now drop the λ subscript on φ by assuming that
(1.11) kIφk =ǫ ≪1
L2 0
and our goal is to construct the solution of (1.1) on the time interval [0,λ3T].
The local well-posedness result of Proposition 1 shows we can construct the
solution for t ∈ [0,1] if we choose ǫ small enough. The almost L2 conservation
0
property shows kIu(1)k22 ≤ kIu(0)k22 +N−43+kIuk3X0,21+. Using (1.6) and (1.11)
gives
kIu(1)k2 ≤ǫ2+N−34+.
2 0
We can iterate this process N43− times before doubling kIu(t)kL2. Therefore, we
advance the solution by taking N43− time steps of size O(1). We now restrict s by
demanding that
(1.12) N34− &λ3T =N3−+62ssT
is ensured for large enough N, so s>− 3 .
10
2. Proof of the bilinear smoothing estimate
This section establishes Proposition 2. We distinguish the very low frequen-
cies {ξ : |ξ| . 1}, the low frequencies {ξ : 1 . |ξ| . 1N} and the high
2
frequencies {ξ : 1N . |ξ|}. Decompose the factor u in the bilinear estimate by
2
writing u = u +u +u with u supported on the low frequencies and similarly
vl l h l
for the very low and high frequency pieces. We decompose v the same way. Since
I is the identity operator on theblow and very low frequencies, we can assume one
of the factors u,v in the estimate to be shown has its Fourier transform supported
in the high frequencies. Symmetry allows us to assume u = u and we need to
h
consider the three possible interactions of u with v , v and v . Finally, since we
h vl l h
areconsidering(weighted)L2 norms,wecanreplaceuandvby|u|and|v|. Assume
therefore that u,v ≥0.
Very low/high interaction b b b b
b b
4 J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO
An explicit calculation shows that
(2.1) F(∂ {I(u v )−I(u )v })(ξ)= iξ[m(ξ)−m(ξ )]u (ξ )v (ξ ),
x h vl h vl 1 h 1 vl 2
Z
ξ=ξ1+ξ2
c c
where F denotes the Fourier transform. The mean value theorem gives
|m(ξ)−m(ξ )|≤|m′(ξ˜)||ξ |,
1 1 2
which may be interpolated with the trivial estimate to give
(2.2) |m(ξ)−m(ξ )|≤CN−s|ξ |s|ξ |−θ|ξ |θ
1 1 1 2
for 0≤θ ≤1. Recall that m was defined to be smooth and monotone in (1.2).
Therefore, upon defining F(∇θf)(ξ)=|ξ|θf(ξ), we can write
|F(∂ {I(u v )−I(u )v })(ξ)|≤b|F(∂ (∇−θI(u )(∇θv ))(ξ)|.
x h vl h vl x h vl
We now estimate the left side of the bilinear estimate in this interaction by
(2.3) ∂ (∇−θI(u ))(∇θv )
(cid:13) x h vl (cid:13)X0,12+
(cid:13) (cid:13)
and by the bilinear estimate of Kenig, Ponce and Vega
(2.4) ≤C ∇−θI(u ) ∇θv .
(cid:13) h (cid:13)X0,21+(cid:13) vl(cid:13)X0,12+
(cid:13) (cid:13) (cid:13) (cid:13)
The frequency support of v shows that ∇θv . kv k . A moments
vl (cid:13) vl(cid:13)X0,21+ vl X0,12+
thought shows (cid:13) (cid:13)
(2.5) ∇−θI(u ) ≤N−θkI(u )k
(cid:13) h (cid:13)X0,21+ h X0,21+
(cid:13) (cid:13)
and the claim of the Proposition follows for the (very low)(high) interaction by
choosing θ > 3.
4
Low/high interaction
The preceding calculations reduce matters to controlling
(2.6) ∂ ∇−θI(u )∇θv
(cid:13) x h l(cid:13)X0,21+
(cid:13) (cid:13)
and we know that u and v are supported outside the very low frequencies.
h l
Lemma 1. Assumce u andbv are supported outside {|ξ|<1}. Then
(2.7) k∂ (uv)k ≤Ckuk kvk
bx bXα,−21+ X−γ1,12+ X−γ2,21+
provided
3
α−(γ +γ ) < ,
1 2
4
1
α−γ < , i=1,2.
i
2
We will apply the lemma momentarily with α=0,γ =γ =−3+.
1 2 8
The proof of the lemma is containedin the proofof Theorem2 in [7]. In partic-
ular, the support properties on u, v reduce matters to considering Cases A.3, A.4,
A.6, B.3, B.4, B.5 and B.6 in [7]. The restriction α−(γ +γ )< 3 arises in Case
1 2 4
A.4.c.ii of [7] which is the regionbcobntaining the counterexample of [9]. Case B.4.b
of [7] requires the other condition α−γ < 1.
i 2
GLOBAL WELL-POSEDNESS FOR KDV IN SOBOLEV SPACES OF NEGATIVE INDEX 5
The lemma applied to (2.6) gives
≤C ∇−θI(u ) ∇θv .
(cid:13) h (cid:13)X−83+,21+(cid:13) l(cid:13)X−38+,21+
Setting θ = 3− leaves (cid:13) (cid:13) (cid:13) (cid:13)
8
C(cid:13)(cid:13)∇−43+I(uh)(cid:13)(cid:13)X0,12+kvlkX0,12+ ≤CN−43+kI(uh)kX0,12+kvlkX0,12+
which was to(cid:13)be shown. (cid:13)
High/high interaction
In this region of the interaction, we do not take advantage of any cancellation
and estimate the difference with the triangle inequality
k∂ {I(u )I(v )}k +k∂ {I(u v )}k .
x h h X0,−12+ x h h X0,−21+
For the first contribution we use the lemma to get
(2.8) kI(uh)kX−38+,21+kI(vh)kX−38+,21+ ≤N−43+kI(uh)kX0,12+kI(vh)kX0,21+.
The second contribution is bounded by throwing away I and applying the lemma,
k∂ {u v }k ≤ ku k ku k
x h h X0,−21+ h X−38+,21+ h X−38+,21+
≤ N−83+s+kuhkXs,12+N−38+s+kvhkXs,21+
≤ N−43+kuhkX0,21+kvhkX0,12+.
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