Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS

Guo, Zihua; Kwon, Soonsik; Oh, Tadahiro

doi:10.1007/s00220-013-1755-5

Cited by 69 publications

(240 citation statements)

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“…Note that our enhanced uniqueness does not assert unconditional uniqueness in C(R; H s (T)), since we do assume that solutions with smooth approximating solutions have some extra regularity so that the cubic nonlinearity makes sense. (By slightly modifying the presentation in [18], one can easily prove unconditional uniqueness of (1.1) and (1.5) in C(R; H s (T)) for s 1 6 . Clearly, the threshold s 1 6 is sharp in view of the embedding:…”

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confidence: 99%

“…In fact, this process basically corresponds to the Poincaré-Dulac normal form reductions. See the introduction in [18].) In [18], the first author with Guo and Kwon proved unconditional well-posedness of the cubic NLS on T in low regularity by performing normal form reductions infinitely many times.…”

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confidence: 99%

“…See the introduction in [18].) In [18], the first author with Guo and Kwon proved unconditional well-posedness of the cubic NLS on T in low regularity by performing normal form reductions infinitely many times. See also [11].…”

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confidence: 99%

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Global Well-Posedness of the Periodic Cubic Fourth Order NLS in Negative Sobolev Spaces

Wang

2018

Forum of Mathematics, Sigma

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We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces H s (T), s > − , by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the H s -energy functional, allowing us to introduce an infinite sequence of correction terms to the H s -energy functional in the spirit of the I -method. In fact, the main novelty of this paper is this reduction of the H s -energy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.

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Global Well-Posedness of the Periodic Cubic Fourth Order NLS in Negative Sobolev Spaces

Wang

2018

Forum of Mathematics, Sigma

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“…In [23], we constructed weak solutions in the extended sense in L 2 (T) without any auxiliary function space. Moreover, the result in [23] yields unconditional uniqueness in H s (T) for s ≥ with respect to the canonical Gaussian measures on H s (T). In Theorem 1.2, we claim only existence of solutions to (1.3) in negatives Sobolev spaces, i.e.…”

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“…For simplicity of notations, we use a n (t) to denote a n (t) in the following. The use of the interaction representation allows us to illustrate the connection between the Poincaré-Dulac normal form reduction discussed in [23] and the method of adding correction terms in the I-method [15,16]. With this notation, (1.3) can be written as…”

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Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$

Guo

2016

Int Math Res Notices

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Abstract. We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to H s (T) \ L 2 (T) for some s ∈ (− 1 8, 0). The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short-time Fourier restriction norm method. where u is a complex-valued function. The Cauchy problem (1.1) has been studied extensively from both theoretical and applied points of view. It is known to be one of the simplest partial differential equations (PDEs) with complete integrability [1,2,19]. In the following, however, we only discuss analytical aspects of (1.1) without using the complete integrable structure of the equation. It is well known that (1.1) enjoys several symmetries. In the following, we concentrate on the scaling symmetry and the Galilean symmetry. The scaling symmetry states that if u(x, t) is a solution to (1.1) on R with initial condition u 0 , then u λ (x, t) = λ −1 u(λ −1 x, λ −2 t) is also a solution to (1.1) with the λ-scaled initial condition u λ 0 (x) = λ −1 u 0 (λ −1 x). Associated to the scaling symmetry, there is a scaling-critical Sobolev index s c such that the homogeneousḢ sc -norm is invariant under the dilation symmetry. It is commonly conjectured that a PDE is ill-posed in H s for s < s c . In the case of the one-dimensional cubic NLS, the scaling-critical Sobolev index is s c = − 1 While there is no dilation symmetry in the periodic setting, the heuristics provided by the scaling argument plays an important role even in the periodic setting. For example, a norm inflation result for (1.1) on T analogous to [12] also holds for s < s c = − x u 0 (x). This symmetry also holds on the circle for β ∈ 2Z. Note that the L 2 -norm is invariant under the Galilean symmetry.2 This induces another critical regularity s ∞ crit = 0. Indeed, there is a strong dichotomy in the behavior of solutions for s ≥ 0 and s < 0.Bourgain [3] introduced the so-called Fourier restriction norm method via the X s,bspaces (see (3.1) below) and proved local well-posedness of (1.1) in L 2 (T). Thanks to the L 2 -conservation, this result was immediately extended to global well-posedness in L 2 (T). On the other hand, (1.1) is known to be ill-posed below L 2 (T). Burq-Gérard-Tzvetkov [5] showed that the solution map Φ t : u 0 ∈ H s → u(t) ∈ H s is not locally uniformly continuous if s < 0. See also [10]. Moreover, Christ-Colliander-Tao [11] and Molinet [31] proved that the solution map is indeed discontinuous if s < 0. Our main result in this paper states an even stronger form of ill-posedness holds true for (1.1) in negative Sobolev spaces. Theorem 1.1 (Non-existence of solutions for the cubic NLS). Let s ∈ (− 1 8 , 0) andNote that the condition (ii) is a natural condition to impose since it would follow from the continuity of the solution map :, which is one of the essential components in the well-posedness theory of evolution equations. On the one hand, the previous ill-posedness results [5,10,11,12,31] ...

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Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

et al. 2019

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We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space M s p,q (R) where 1 ≤ q ≤ 2, 2 ≤ p < 10q ′ q ′ +6 and s ≥ 0. Moreover, for either 1 ≤ q ≤ 3 2 , s ≥ 0 and 2 ≤ p ≤ 3 or 3 2 < q ≤ 18 11 , s > 2 3 − 1 q and 2 ≤ p ≤ 3 or 18 11 < q ≤ 2, s > 2 3 − 1 q and 2 ≤ p < 10q ′ q ′ +6 we show that the Cauchy problem is unconditionally wellposed in M s p,q (R). This improves [9], where the case p = 2 was considered and the differentiation by parts technique was introduced to a problem with continuous Fourier variable. Here the same technique is used, but more delicate estimates are necessary for p = 2. 1 NLS, Differentiation by Parts and M s p,q (R) 37 leonid chaichenets, department of mathematics, institute for analysis, karlsruhe institute of technology, 76128 karlsruhe, germany dirk hundertmark, department of mathematics, institute for analysis, karlsruhe institute of technology, 76128 karlsruhe, germany E-mail address: dirk.hundertmark@kit.edu peer kunstmann, department of mathematics, institute for analysis, karlsruhe institute of technology, 76128 karlsruhe, germany

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Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS

Cited by 69 publications

References 21 publications

Global Well-Posedness of the Periodic Cubic Fourth Order NLS in Negative Sobolev Spaces

Global Well-Posedness of the Periodic Cubic Fourth Order NLS in Negative Sobolev Spaces

Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$

Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

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