Electronic Journal of Differential Equations, Vol. (), No. 6, pp. 7. ISSN: 7-669. 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 H s (R) for 3/ < s. Introduction Consider the initial value problem for the Korteweg-deVries (KdV) equation t u 3 xu x(u ) =, x R, u() = φ, (.) for rough initial data φ H s (R), s <. 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, the local result and L norm conservation imply (.) is globally well-posed []. Recently, a direct adaptation [7] of Bourgain s high-low frequency technique [3], [] showed (.) is globally well-posed for φ H s Ḣa for certain s, a <. A modification of the high-low frequency technique, first used in [], is presented in this paper which establishes global well-posedness of (.) in H s (R), 3/ < s. A subsequent paper [6] will establish that (.) is globally well-posed in H s (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 R and NLS with derivative in R [4]. The Multiplier operator I Let s < and N be fixed. Define the Fourier multiplier operator Îu(ξ) = m(ξ)û(ξ), m(ξ) = {, ξ < N, N s ξ s, ξ N Mathematics Subject Classifications: 35Q53, 4B35, 37K. Key words: Korteweg-de Vries equation, nonlinear dispersive equations, bilinear estimates. c Southwest Texas State University. Submitted January 3,. Published April 7,. (.)
Global well-posedness for KdV EJDE /6 with m smooth and monotone. The operator I (barely) maps H s (R) L (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 is the Fourier multiplier operator with multiplier m(ξ). An almost L conservation property of (.) Let φ H s (R), 3/4 < s < in (.). There is a δ = δ( φ H s) > such that (.) is well-posed for t [, δ]. We observe using the Fundamental Theorem of Calculus, the equation, and integration by parts that Iu(δ) L = Iu() L δ d (Iu(τ), Iu(τ))dτ, dτ δ = Iu() L (I u(τ), Iu(τ))dτ, δ = Iu() L (I( u xxx x[u ])(τ), Iu(τ))dτ δ = Iu() L (I( x [u ]), Iu)dτ. Finally, we add = δ x (I(u) )I(u)dτ to observe δ } Iu(δ) L = Iu() L x {(I(u)) I(u ) Iu dxdτ. (.3) This last step enables us to take advantage of some internal cancellation. We apply Cauchy-Schwarz as in [] and bound the integral above by x {(I(u)) I(u )} Iu X δ X δ. (.4),, The space Xs,b δ of functions of space-time is defined via the Fourier restriction norm u X δ = inf{ w s,b Xs,b := ( k ) s ( τ k 3 ) b ŵ(k, τ) L : w = k,τ u for t [, δ]}. Remark An effort to find a term providing more cancellation than δ x (I(u) )I(u)dτ used above led to the general procedure described in [6]. Proposition (A variant of local well-posedness) The initial value problem (.) is locally well-posed in the Banach space I L = {φ H s with norm Iφ L } with existence lifetime δ satisfying and moreover δ Iφ α L, for some α >, (.5) Iu X δ, C Iφ L. (.6)
EJDE /6 J. Colliander et al. 3 This proposition is not difficult to prove using the argument in [9]. Using Duhamel s formula and X s,b space properties reduces matters to proving the bilinear estimate x I(uv) X, C Iu X, Iv X, (.7) to obtain the contraction. The space-time norm bound is then implied by the contraction estimate. The estimate (.7) follows from the next proposition and the bilinear estimate of Kenig, Ponce and Vega [9]. Proposition (Extra smoothing) The bilinear estimate holds. x {I(u)I(v) I(uv)} X δ, CN 3 4 Iu X δ, Iv X δ,. (.) Recall the bilinear estimate x (uv) X, C u X, v X, from [9]. Proposition reveals a smoothing beyond the recovery of the first derivative for the particular quadratic expression encountered above in (.3). We prove Proposition in the next section. The required pieces are now in place for us to give the proof of global wellposedness of (.) in H s (R), 3/ < s. Global well-posedness of (.) will follow if we show well-posedness on [, T ] for arbitrary T >. We re-normalize things a bit via scaling. If u solves (.) then u λ (x, t) = ( λ ) u( x λ, t λ ) solves 3 (.) with initial data φ λ (x, t) = ( λ ) φ( x λ ). Note that u exists on [, T ] if and only if u λ exists on [, λ 3 T ]. A calculation shows that Iφ λ L Cλ 3 s N s φ H s. (.9) Here N = N(T ) will be selected later but we choose λ = λ(n) right now by requiring Cλ 3 s N s φ H s = λ N s 3s. (.) We now drop the λ subscript on φ by assuming that Iφ L = ɛ (.) and our goal is to construct the solution of (.) on the time interval [, λ 3 T ]. The local well-posedness result of Proposition shows we can construct the solution for t [, ] if we choose ɛ small enough. The almost L conservation property shows Iu() Iu() N 3 4 Iu 3 X,. Using (.6) and (.) gives Iu() ɛ N 3 4. We can iterate this process N 3 4 times before doubling Iu(t) L. Therefore, we advance the solution by taking N 3 4 time steps of size O(). We now restrict s by demanding that N 3 4 λ 3 T = N 6s 3s T (.) is ensured for large enough N, so s > 3/.
4 Global well-posedness for KdV EJDE /6 Proof of the bilinear smoothing estimate This section establishes Proposition. We distinguish the very low frequencies {ξ : ξ }, the low frequencies {ξ : ξ N} and the high frequencies {ξ : N ξ }. Decompose the factor u in the bilinear estimate by writing u = u vl u l u h with û l 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 transform supported in the high frequencies. Symmetry allows us to assume u = u h and we need to consider the three possible interactions of u h with v vl, v l and v h. Finally, since we are considering (weighted) L norms, we can replace û and v by û and v. Assume therefore that û, v. Very low/high interaction An explicit calculation shows that F ( x {I(u h v vl ) I(u h )v vl }) (ξ) = iξ[m(ξ) m(ξ )]û h (ξ ) v vl (ξ ), ξ=ξ ξ where F denotes the Fourier transform. The mean value theorem gives m(ξ) m(ξ ) m ( ξ ) ξ, which may be interpolated with the trivial estimate to give (.) m(ξ) m(ξ ) CN s ξ s ξ θ ξ θ (.) for θ. Recall that m was defined to be smooth and monotone in (.). Therefore, upon defining F( θ f)(ξ) = ξ θ f(ξ), we can write F( x {I(u h v vl ) I(u h )v vl })(ξ) F( x ( θ I(u h )( θ v vl ))(ξ). We now estimate the left side of the bilinear estimate in this interaction by x ( θ I(u h ))( θ v vl ) X, (.3) and by the bilinear estimate of Kenig, Ponce and Vega C θ I(u h ) X, θ v X, vl. (.4) The frequency support of v vl shows that θ v X, vl v vl X, A moments thought shows θ I(u h ) X,. N θ I(u h ) X, (.5) and the claim of the Proposition follows for the (very low)(high) interaction by choosing θ > 3/4.
EJDE /6 J. Colliander et al. 5 Low/high interaction The preceding calculations reduce matters to controlling x θ I(u h ) θ v l X, (.6) and we know that û h and v l are supported outside the very low frequencies. Lemma Assume û and v are supported outside { ξ < }. Then provided x (uv) Xα, C u X γ, v X γ, (.7) α (γ γ ) < 3 4, α γ i <, i =,. We will apply the lemma momentarily with α =, γ = γ = 3/. The proof of the lemma is contained in the proof of Theorem in [7]. In particular, the support properties on û, 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 ) < 3/4 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 γ i <. The lemma applied to (.6) gives C θ I(u h ) X 3, θ v l X 3,. Setting θ = 3 leaves C 3 4 I(u h ) which was to be shown. High/high interaction X, v l X, CN 3 4 I(u h ) X, v l X, In this region of the interaction, we do not take advantage of any cancellation and estimate the difference with the triangle inequality x {I(u h )I(v h )} X, x {I(u h v h )} X,. For the first contribution we use the lemma to get I(u h ) X I(v 3 h ), X N 3 4 I(u 3 h ), X, I(v h ) X, (.).
6 Global well-posedness for KdV EJDE /6 The second contribution is bounded by throwing away I and applying the lemma, x {u h v h } X, u h X 3, u h X 3, N 3 s u h Xs, N 3 s v h Xs, N 3 4 u h X, v h X,. 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 955. G.S. is supported in part by N.S.F. Grant DMS 979 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. References [] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I,II. Geom. Funct. Anal., 3:7 56, 9 6, 993. [] J. Bourgain. Refinements of Strichartz inequality and applications to D- NLS with critical nonlinearity. International Mathematical Research Notices, 5:53 3, 99. [3] J. Bourgain. Global solutions of nonlinear Schrödinger equations. American Mathematical Society, Providence, RI, 999. [4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Global wellposedness for Schrödinger equations with derivative. Submitted to SIAM J. Math. Anal.,. [5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Global wellposedness of d NLS. (in preparation),. [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Global wellposedness of KdV and modified KdV on R and T. (preprint),. [7] J. E. Colliander, G. Staffilani, and H. Takaoka. Global wellposedness of KdV below L. Mathematical Research Letters, 6(5,6):755 77, 999. [] M. Keel and T. Tao. Local and Global Well-Posedness of Wave Maps on R for Rough Data. International Mathematical Research Notices, :7 56, 99. [9] C. Kenig, G. Ponce, and L. Vega. A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc., 9:573 63, 996.
EJDE /6 J. Colliander et al. 7 [] G. Staffilani. On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations. Duke Math. J., 6():9 4, 997. James Colliander University of California Berkeley, California, 947-34 USA e-mail: colliand@math.berkeley.edu Markus Keel Caltech Pasadena, California, 95, USA e-mail: keel@cco.caltech.edu Gigliola Staffilani Stanford University Stanford, California, 9435, USA e-mail: gigliola@math.stanford.edu Hideo Takaoka Division of Mathematics Graduate School of Science Hokkaido University Sapporo, 6-, Japan. e-mail: takaoka@math.sci.hokudai.ac.jp Terence Tao University of California, Los Angeles, California, 995-596, USA e-mail:tao@math.ucla.edu