Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line

Kwon, Soonsik; Oh, Tadahiro; Yoon, Haewon

doi:10.5802/afst.1643

Cited by 31 publications

(29 citation statements)

References 36 publications

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“…By exploiting the completely integrable structure of the equation, Kappeler-Topalov [21] used the inverse spectral method to show global existence and uniqueness of solutions to the real-valued defocusing mKdV (with the + sign) in H s (T), s ≥ 0. Here, solutions are understood as the unique limit of smooth solutions and it is not required that the equation is satisfied in the sense of distributions (see [21,25,32] for further details).…”

Section: Andreia Chapoutomentioning

confidence: 99%

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

Chapouto¹

2021

DCDS

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We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle. We first consider the real-valued setting and show global well-posedness of the (usual) renormalized mKdV equation in the Fourier-Lebesgue spaces. In the complex-valued setting, we observe that the momentum plays an important role in the well-posedness theory. In particular, we prove that the complex-valued mKdV equation is ill-posed in the sense of non-existence of solutions when the momentum is infinite, in the spirit of the work on the nonlinear Schrödinger equation by Guo-Oh (2018). This nonexistence result motivates the introduction of the second renormalized mKdV equation, which we propose as the correct model in the complex-valued setting outside H 1 2 (T). Furthermore, imposing a new notion of finite momentum for the initial data, at low regularity, we show existence of solutions to the complex-valued mKdV equation. In particular, we require an energy estimate, from which conservation of momentum follows.2020 Mathematics Subject Classification. 35Q53.

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Section: Andreia Chapoutomentioning

confidence: 99%

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

Chapouto¹

2021

DCDS

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“…We say that u and u are regular solutions to (1.2) and (1.9), respectively, if u and u are solutions to (1.2) and (1.9), respectively, such that u ∈ C(R; FL The main idea is to apply a normal form reduction to (1.9), namely integration by parts in (1.11), to exploit the oscillatory nature of the non-resonant contribution. As in [22,29], we implement an infinite iteration of normal form reductions and derive the following normal form equation:…”

Section: Normal Form Equationmentioning

confidence: 99%

“…In a recent paper [14], the first author with Forlano studied the cubic NLS on R. In particular, by implementing an infinite iteration of normal form reductions, they proved analogues of Theorems 1.4, 1.5, and 1.9 in almost critical Fourier-Lebesgue spaces FL p (R), 2 ≤ p < ∞, and almost critical modulation spaces M 2,p (R), 2 ≤ p < ∞. Relevant multilinear estimates were studied based on the idea introduced in [29], namely, successive applications of basic trilinear estimates (called localized modulation estimates).…”

Section: Normal Form Equationmentioning

confidence: 99%

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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“…In [27], Tao gave an alternative proof of the local well-posedness in H 1 4 (R) by using the Fourier restriction norm method. We also mention recent papers [23,22] on unconditional uniqueness of solutions to (1.1) in H s (R), s > 1 4 . Let us now turn our attention to global well-posedness of (1.1).…”

Section: Introductionmentioning

confidence: 99%

On global well-posedness of the modified KdV equation in modulation spaces

Wang²

2021

DCDS

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We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces M 2,p s (R) for s ≥ 1 4 and 2 ≤ p < ∞. For s < 1 4 , we show that the solution map for mKdV is not locally uniformly continuous in M 2,p s (R). By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in M 2,p s (R) for s ≥ 1 4 and 2 ≤ p < ∞.

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Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line

Cited by 31 publications

References 36 publications

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

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

On global well-posedness of the modified KdV equation in modulation spaces

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