Unconditional uniqueness results for the nonlinear Schrödinger equation

Herr, Sebastian; Sohinger, Vedran

doi:10.1142/s021919971850058x

Cited by 28 publications

(36 citation statements)

References 45 publications

(116 reference statements)

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“…In [22], the first author (with Z. Guo and S. Kwon) proved unconditional uniqueness in H 1 6 (T) and Theorem 1.5 extends this result to the Fourier-Lebesgue setting. We also mention a recent work by Herr-Sohinger [24] where they proved unconditional uniqueness of the cubic NLS (1.1) in L p ([−T, T] × T) for p > 3. The main difference between unconditional uniqueness and uniqueness for sensible weak solutions is that the former does not assume that a solution comes with a sequence of smooth approximating solutions, while, by definition, sensible weak solutions are equipped with smooth approximating solutions.…”

Section: Resultsmentioning

confidence: 94%

Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

Wang

2021

JAMA

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In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp($${\cal F}{L^p}(\mathbb{T})$$ ℱ L p ( T ) ), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp($${\cal F}{L^p}(\mathbb{T})$$ ℱ L p ( T ) ) for $$1 \leq p \leq {3 \over 2}$$ 1 ≤ p ≤ 3 2 .

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Section: Resultsmentioning

confidence: 94%

Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

Wang

2021

JAMA

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“…We should also mention [19] where the authors use a different approach to unconditional uniqueness of the cubic NLS (6) which applies to various spatial domains. The main idea is to exploit the relation of solutions of the cubic NLS to solutions of the Gross-Pitaevskii hierarchy.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We will define many operators, R t 1 , R t 2 , N t 11 ,Q 1,t n , N t 21 , N t 4 , N t 31 and we need to be able to control all of them in the appropriate norms. This will be done in Lemmata 15,16,17,19,20 and 21. To move forward we use the splitting…”

Section: Description Of the Iteration Processmentioning

confidence: 99%

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Klaus¹,

Kunstmann²,

Pattakos³

2021

Czech.Math.J.

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We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with initial data u0 ∈ X, where6 and s2 > 1 2 we show that the Cauchy problem is unconditionally wellposed in X. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work.

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“…To prove Theorem 1.1, we will use the cubic Gross–Pitaevskii (GP) hierarchy on , which is uncommon in the analysis of the NLS and is being explored [24, 35]. Let denote the space of trace class operators on .…”

Section: Introductionmentioning

confidence: 99%

“…Analysis of GP hierarchy did not yield new NLS results with regularity lower than that of NLS analysis until [35, 24]. 9 , 10 In [35], using the scheme in [6], Herr and Sohinger generalised the usual Sobolev multilinear estimates to Besov spaces and obtained new unconditional-uniqueness results regarding equation (1.2) and hence the NLS (equation (1.1)) on . The result has pushed the regularity requirement for the uniqueness of equation (1.1) lower than the number coming from NLS analysis.…”

Section: Introductionmentioning

confidence: 99%

Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

Chen

Holmer

2022

Forum of Mathematics, Pi

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We consider the $\mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.

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Unconditional uniqueness results for the nonlinear Schrödinger equation

Cited by 28 publications

References 45 publications

Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

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