a r X i v :m a t h /0110049v 3 [m a t h .A P ] 14 J u l 2006
MULTILINEAR ESTIMATES FOR PERIODIC KDV EQUATIONS,
AND APPLICATIONS
J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,AND T.TAO
Abstract.We prove an endpoint multilinear estimate for the X s,b spaces associated to the periodic Airy equation.As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations,as well as some global well-posedness results below the energy norm.
Contents
1.Introduction 21.1.Main Results
31.2.Remarks and possible extensions 41.3.Outline
52.A counter-example 63.Linear Estimates
74.Reduction to a multiplier bound 105.The non-endpoint case
126.Some elementary number theory 137.The proof of (4.4)in the endpoint case.
15
2J.COLLIANDER,M.KEEL,G.STAFFILANI,H.TAKAOKA,AND T.TAO
8.The proof of(4.5)in the endpoint case.17
9.Proof of Proposition121
10.Local well-posedness for small data24
rge periods25
12.An interpolation lemma29
13.Global well-posedness31
14.Proof of Lemma13.133
References38
1.Introduction
This paper studies the Cauchy problem for periodic generalized KdV equations of the form
(1.1) ∂t u+1
4π2is convenient in order to make the
dispersion relationτ=ξ3,but it is inessential and we recommend that the reader ignore all powers of2πwhich appear in the sequel.We can assume that F has no constant or linear term since these can be removed by a Gallilean transformation.
The main result established in this paper is a sharp multilinear estimate which allows us to show the initial value problem(1.1)is locally well-posed in H s(T)for s≥1
MULTILINEAR ESTIMATES AND APPLICATIONS 3
The low-regularity study of the equation (1.1)on T has been based around
iteration in the spaces X s,
1
2
+ ξ s ˆu L 2ξL 1τ
.We shall also need the companion spaces Z s defined by
(1.3) u Z s := u s,−1 τ−ξ3 L 2ξL 1
τ
.
1.1.Main Results.The main technical result of this paper is the following mul-tilinear estimate.
Theorem 1.For any s ≥
1
2
k
i =1
u i Y s .
This improves on the results in [29],where this estimate was proven for s ≥1.By combining this estimate with an estimate of Kenig,Ponce,and Vega [24]we
shall obtain
Proposition 1.For any s ≥
1
2.
In [24]the restriction s ≥
1
2
for k ≥4,and also for the k =3case if one allows the mean of the
u i to be non-zero.In the k =3case with mean zero the counter-example is a little trickier,and is discussed in Section 2.
The sharp estimate (1.5)of Proposition 1will be useful in studying the dynami-cal behavior of solutions of polynomial generalizations of the KdV equation.In this paper,we use Proposition 1to obtain some local and global well-posedness results for the Cauchy problem (1.1).In [29](see also [5],[22]),such equations were shown to be globally well-posed for H 1
data.
