Multilinear estimates for periodic KdV equations, and applications

By James E. Colliander, Markus Keel, Gigliola Staffilani, Hideo Takaoka, Terence Tao. 2001

208 citations recorded by the source corpus.

We prove an endpoint multilinear estimate for the Xs,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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f58326352e5226bc2c65bf9d2c5248117e4dc93

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Global well-posedness for Schrödinger equation with derivative in H 1 2 ( R ) (2009, background): “1 in [12], it suffices to prove the multilinear estimates, ‖w1∂xw2w3‖Zs .”
Sharp ill-posedness results for the KdV and mKdV equations on the torus (2011, methodology): “As written in [7], (dξ)λ is the counting measure on the integers when λ = 1 and converges weakly to the Lebesgue measure when λ → ∞.”
Sharp global well-posedness for KdV and modified KdV on ℝ and (2003, background): “Our results here and elsewhere [16], [14], [ 17 ] suggest that local well-posedness implies global well-posedness in subcritical dispersive initial value problems.”
Improvement of the energy method for strongly non resonant dispersive equations and applications (2014, methodology): “We use the same notations as in [Colliander et al. 2004] to deal with Fourier transform of space-periodic functions with a large period 2 .”
On the regularization mechanism for the periodic Korteweg–de Vries equation (2009, result): “20) (similar results can be found in [7], [8]) can be proven for 0 ≥ s > −1 using the boundedness in negative spaces of linear operator L v0 and multilinear operators which enter (6.”
Lower Bounds on the Radius of Spatial Analyticity for the KdV Equation (2015, unknown): “cessary properties are contained in the following lemmas; proofs of the ﬁrst two lemmas can be found in section 2.6 of [19], whereas the third lemma follows by the argument used to prove Lemma 3.1 of [5]. Lemma3. Let σ ≥ 0, s ∈ Rand b > 1/2. Then Xσ,s,b ⊂ C(R,Gσ,s) and sup t∈R ku(t)kGσ,s ≤ CkukXσ”
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ℝ3 (2003, methodology): “conservation law approach used here with the argument in [3, 4]. See [22, 21, 11, 14] for further applications of almost conservation laws; and [15, 13 , 12] for instances where the inclusion of correction terms in the almost conserved energy leads to sharp results.”
Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $S^1$ (2010, methodology): “A similar fact was proved in [36], but in slightly different spaces.”
Global well-posedness in L2 for the periodic Benjamin-Ono equation (2006, methodology): “( dξ ) λ will be the renormalized counting measure on λ − 1 Z : As written in [8], ( dξ ) λ is the counting measure on the integers when λ = 1 and converges weakly to the Lebesgue measure when λ → ∞ .”
KdV is well-posed in H–1 (2018, background): “ local well-posedness for (KdV) in Hs(R) for s≥ −3/4 and in Hs(R/Z) for s≥ −1/2. Excepting the endpoints, this was proved by Kenig–Ponce– Vega in [37]. For a discussion of the endpoints, see [10] and [12, 13, 14]. These ranges of sare sharp if one requires the data to solution map to be uniformly co”