The initial value problem for the Korteweg-deVries equation on the line is
shown to be globally well-posed for rough data. In particular, we show global
well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.Comment: 5 pages. Electronic Journal of Differential equations (submitted

Colliander, J.

Keel, M.

Staffilani, G.

Takaoka, H.

Tao, T.

English

arXiv

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in $H^s(\mathbb{R})$ for -3/10 < s

Colliander, James

Keel, Markus

Staffilani, Gigliola

Takaoka, Hideo

Tao, Terence

Caltech Authors - Main

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

Electronic Journal of Differential Equations, Vol. 2001(2001), No. 26, pp. 1–7.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
Global well-posedness for KdV in Sobolev spaces
of negative index ∗
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, & T. Tao
Abstract
The initial value problem for the Korteweg-deVries equation on the
line is shown to be globally well-posed for rough data. In particular, we
show global well-posedness for initial data in Hs(R) for −3/10 < s.
1 Introduction
Consider the initial value problem for the Korteweg-deVries (KdV) equation
∂tu+ ∂3xu+
1
2
∂x(u2) = 0, x ∈ R,
u(0) = φ,
(1.1)
for rough initial data φ ∈ Hs(R), s < 0. The initial data φ and the solution u are
assumed to take values in R. This problem is known [9] to be locally well-posed
provided −3/4 < s. For s ≥ 0, the local result and L2 norm conservation imply
(1.1) is globally well-posed [1]. Recently, a direct adaptation [7] of Bourgain’s
high-low frequency technique [3], [2] showed (1.1) is globally well-posed for φ ∈
Hs∩H˙a for certain s, a < 0. A modification of the high-low frequency technique,
first used in [8], is presented in this paper which establishes global well-posedness
of (1.1) in Hs(R), −3/10 < s.
A subsequent paper [6] will establish that (1.1) is globally well-posed in
Hs(R) for −3/4 < s. The simplicity of the argument presented here may extend
more easily to other situations, such as in our treatment [5] of cubic NLS on
R2 and NLS with derivative in R [4].
The Multiplier operator I
Let s < 0 and N  1 be fixed. Define the Fourier multiplier operator
Îu(ξ) = m(ξ)û(ξ), m(ξ) =
{
1, |ξ| < N,
N−s|ξ|s, |ξ| ≥ 10N (1.2)
∗Mathematics Subject Classifications: 35Q53, 42B35, 37K10.
Key words: Korteweg-de Vries equation, nonlinear dispersive equations,
bilinear estimates.
c©2001 Southwest Texas State University.
Submitted January 31, 2001. Published April 27, 2001.
1
2 Global well-posedness for KdV EJDE–2001/26
with m smooth and monotone. 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 1m(ξ) .
An almost L2 conservation property of (1.1)
Let φ ∈ Hs(R), −3/4 < s < 0 in (1.1). There is a δ = δ(‖φ‖Hs) > 0 such that
(1.1) is well-posed for t ∈ [0, δ]. We observe using the Fundamental Theorem of
Calculus, the equation, and integration by parts that
‖Iu(δ)‖2L2 = ‖Iu(0)‖2L2 +
∫ δ
0
d
dτ
(Iu(τ), Iu(τ))dτ,
= ‖Iu(0)‖2L2 + 2
∫ δ
0
(Iu˙(τ), Iu(τ))dτ,
= ‖Iu(0)‖2L2 + 2
∫ δ
0
(I(−uxxx − 12∂x[u
2])(τ), Iu(τ))dτ
= ‖Iu(0)‖2L2 +
∫ δ
0
(I(−∂x[u2]), Iu)dτ.
Finally, we add 0 =
∫ δ
0
∫
∂x(I(u)
2)I(u)dτ to observe
‖Iu(δ)‖2L2 = ‖Iu(0)‖2L2 +
∫ δ
0
∫
∂x
{
(I(u))2 − I(u2)
}
Iu dxdτ. (1.3)
This last step enables us to take advantage of some internal cancellation. We
apply Cauchy-Schwarz as in [10] and bound the integral above by∥∥∥∂x{(I(u))2 − I(u2)}∥∥∥
Xδ
0,− 12−
‖Iu‖Xδ
0, 12 +
. (1.4)
The space Xδs,b of functions of space-time is defined via the Fourier restriction
norm ‖u‖Xδs,b = inf{‖w‖Xs,b := ‖(1 + |k|)
s(1 + |τ − k3|)bŵ(k, τ)‖L2k,τ : w =
u for t ∈ [0, δ]}.
Remark 1 An effort to find a term providing more cancellation than∫ δ
0
∫
∂x(I(u)
2)I(u)dτ used above led to the general procedure described in [6].
Proposition 1 (A variant of local well-posedness) The initial value prob-
lem (1.1) is locally well-posed in the Banach space
I−1L2 = {φ ∈ Hs with norm ‖Iφ‖L2} with existence lifetime δ satisfying
δ & ‖Iφ‖−αL2 , for some α > 0, (1.5)
and moreover
‖Iu‖Xδ
0, 12 +
≤ C‖Iφ‖L2 . (1.6)
EJDE–2001/26 J. Colliander et al. 3
This proposition is not difficult to prove using the argument in [9]. Using
Duhamel’s formula and Xs,b space properties reduces matters to proving the
bilinear estimate
‖∂xI(uv)‖X0,− 12 + ≤ C‖Iu‖X0, 12 +‖Iv‖X0, 12 + (1.7)
to obtain the contraction. The space-time norm bound is then implied by the
contraction estimate. The estimate (1.7) follows from the next proposition and
the bilinear estimate of Kenig, Ponce and Vega [9].
Proposition 2 (Extra smoothing) The bilinear estimate
‖∂x{I(u)I(v)− I(uv)}‖Xδ
0,− 12−
≤ CN− 34 +‖Iu‖Xδ
0, 12 +
‖Iv‖Xδ
0, 12 +
. (1.8)
holds.
Recall the bilinear estimate ‖∂x(uv)‖X0,− 12 + ≤ C‖u‖X0, 12 +‖v‖X0, 12 + from [9].
Proposition 2 reveals a smoothing beyond the recovery of the first derivative
for the 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), −3/10 < s. Global well-posedness of (1.1) will
follow if we show well-posedness on [0, T ] for arbitrary T > 0. We re-normalize
things a bit via scaling. If u solves (1.1) then uλ(x, t) = ( 1λ )
2
u(xλ ,
t
λ3 ) solves
(1.1) with initial data φλ(x, t) = ( 1λ )
2
φ(xλ ). Note that u exists on [0, T ] if and
only if uλ exists on [0, λ3T ]. A calculation shows that
‖Iφλ‖L2 ≤ Cλ−
3
2−sN−s‖φ‖Hs . (1.9)
Here N = N(T ) will be selected later but we choose λ = λ(N) right now by
requiring
Cλ−
3
2−sN−s‖φ‖Hs ∼ 1 =⇒ λ ∼ N−
2s
3+2s . (1.10)
We now drop the λ subscript on φ by assuming that
‖Iφ‖L2 = 0  1 (1.11)
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 0 small enough. The almost L2 conservation
property shows ‖Iu(1)‖22 ≤ ‖Iu(0)‖22 +N−
3
4 +‖Iu‖3X0, 12 + . Using (1.6) and (1.11)
gives
‖Iu(1)‖22 ≤ 20 +N−
3
4 +.
We can iterate this process N
3
4− times before doubling ‖Iu(t)‖L2 . Therefore,
we advance the solution by taking N
3
4− time steps of size O(1). We now restrict
s by demanding that
N
3
4− & λ3T = N
−6s
3+2sT (1.12)
is ensured for large enough N , so s > −3/10.
4 Global well-posedness for KdV EJDE–2001/26
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| . 12N} and the high
frequencies {ξ : 12N . |ξ|}. Decompose the factor u in the bilinear estimate
by writing u = uvl + ul + uh with ûl supported on the low frequencies and
similarly for the very low and high frequency pieces. We decompose v the same
way. Since I is the identity operator on the low and very low frequencies, we can
assume one of the factors u, v in the estimate to be shown has its Fourier trans-
form supported in the high frequencies. Symmetry allows us to assume u = uh
and we need to consider the three possible interactions of uh with vvl, vl and
vh. Finally, since we are considering (weighted) L2 norms, we can replace û and
v̂ by |û| and |v̂|. Assume therefore that û, v̂ ≥ 0.
Very low/high interaction
An explicit calculation shows that
F (∂x{I(uhvvl)− I(uh)vvl}) (ξ) =
∫
ξ=ξ1+ξ2
iξ[m(ξ)−m(ξ1)]ûh(ξ1)v̂vl(ξ2),
(2.1)
where F denotes the Fourier transform. The mean value theorem gives
|m(ξ)−m(ξ1)| ≤ |m′(ξ˜1)||ξ2|,
which may be interpolated with the trivial estimate to give
|m(ξ)−m(ξ1)| ≤ CN−s|ξ1|s|ξ1|−θ|ξ2|θ (2.2)
for 0 ≤ θ ≤ 1. Recall that m was defined to be smooth and monotone in (1.2).
Therefore, upon defining F(∇θf)(ξ) = |ξ|θf̂(ξ), we can write
|F(∂x{I(uhvvl)− I(uh)vvl})(ξ)| ≤ |F(∂x(∇−θI(uh)(∇θvvl))(ξ)|.
We now estimate the left side of the bilinear estimate in this interaction by∥∥∂x(∇−θI(uh))(∇θvvl)∥∥X0, 12 + (2.3)
and by the bilinear estimate of Kenig, Ponce and Vega
≤ C∥∥∇−θI(uh)∥∥X0, 12 +∥∥∇θvvl∥∥X0, 12 + . (2.4)
The frequency support of vvl shows that
∥∥∇θvvl∥∥X0, 12 + . ‖vvl‖X0, 12 + . A mo-
ments thought shows∥∥∇−θI(uh)∥∥X0, 12 + ≤ N−θ‖I(uh)‖X0, 12 + (2.5)
and the claim of the Proposition follows for the (very low)(high) interaction by
choosing θ > 3/4.
EJDE–2001/26 J. Colliander et al. 5
Low/high interaction
The preceding calculations reduce matters to controlling∥∥∂x∇−θI(uh)∇θvl∥∥X0, 12 + (2.6)
and we know that ûh and v̂l are supported outside the very low frequencies.
Lemma 1 Assume û and v̂ are supported outside {|ξ| < 1}. Then
‖∂x(uv)‖X
α,− 12 +
≤ C‖u‖X−γ1, 12 +‖v‖X−γ2, 12 + (2.7)
provided
α− (γ1 + γ2) < 34 ,
α− γi < 12 , i = 1, 2.
We will apply the lemma momentarily with α = 0, γ1 = γ2 = −3/8+.
The proof of the lemma is contained in the proof of Theorem 2 in [7]. In
particular, the support properties on û, v̂ 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− (γ1 + γ2) < 3/4
arises in Case A.4.c.ii of [7] which is the region containing the counterexample
of [9]. Case B.4.b of [7] requires the other condition α− γi < 12 .
The lemma applied to (2.6) gives
≤ C∥∥∇−θI(uh)∥∥X− 38 +, 12 +∥∥∇θvl∥∥X− 38 +, 12 + .
Setting θ = 38− leaves
C
∥∥∥∇− 34 +I(uh)∥∥∥
X0, 12 +
‖vl‖X0, 12 + ≤ CN
− 34 +‖I(uh)‖X0, 12 +‖vl‖X0, 12 +
which was to be shown.
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
‖∂x{I(uh)I(vh)}‖X0,− 12 + + ‖∂x{I(uhvh)}‖X0,− 12 + .
For the first contribution we use the lemma to get
‖I(uh)‖X− 38 +, 12 +‖I(vh)‖X− 38 +, 12 + ≤ N
− 34 +‖I(uh)‖X0, 12 +‖I(vh)‖X0, 12 + . (2.8)
6 Global well-posedness for KdV EJDE–2001/26
The second contribution is bounded by throwing away I and applying the
lemma,
‖∂x{uhvh}‖X0,− 12 + ≤ ‖uh‖X− 38 +, 12 +‖uh‖X− 38 +, 12 +
≤ N− 38 +s+‖uh‖X
s, 12 +
N−
3
8 +s+‖vh‖X
s, 12 +
≤ N− 34 +‖uh‖X0, 12 +‖vh‖X0, 12 + .
Acknowledgments J.E.C. is supported in part by an N.S.F. Postdoctoral
Research Fellowship. M.K. is supported in part by N.S.F. Grant DMS 9801558.
G.S. is supported in part by N.S.F. Grant DMS 9800879 and by a Terman
Award. T.T. is a Clay Prize Fellow and is supported in part by grants from the
Packard and Sloan Foundations.
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James Colliander
Department of Mathematics
University of California
Berkeley, California, 94720-3840 USA
e-mail: colliand@math.berkeley.edu
Markus Keel
Department of Mathematics
Caltech
Pasadena, California, 91125, USA
e-mail: keel@cco.caltech.edu
Gigliola Staffilani
Department of Mathematics
Stanford University
Stanford, California, 94305, USA
e-mail: gigliola@math.stanford.edu
Hideo Takaoka
Division of Mathematics
Graduate School of Science
Hokkaido University
Sapporo, 060-0810, Japan.
e-mail: takaoka@math.sci.hokudai.ac.jp
Terence Tao
Department of Mathematics
University of California,
Los Angeles, California, 90095-1596, USA
e-mail:tao@math.ucla.edu


A bilinear estimate with applications to the KdV equation.

Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I,II.

Global solutions of nonlinear Schro¨dinger equations.

Global wellposedness for Schro¨dinger equations with derivative.

Global wellposedness of 2d NLS.

Local and Global Well-Posedness of Wave Maps on R1+1 for Rough Data.

On the growth of high Sobolev norms of solutions for KdV and Schro¨dinger

Refinements of Strichartz’ inequality and applications to 2DNLS with critical nonlinearity.

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

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

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