On the existence of periodic solutions to the modified Korteweg–de Vries equation below $$H^{1/2}({\mathbb {T}})$$

Schippa, Robert

doi:10.1007/s00028-019-00538-0

Cited by 7 publications

(24 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the same paper, he showed that mKdV is ill-posed below L 2 (T), in the sense that the data-to-solution map is discontinuous in H s (T) for s < 0. See also the work by Schippa [33]. This ill-posedness result shows the sharpness of the well-posedness theory in L 2 -based Sobolev spaces.…”

Section: Andreia Chapoutosupporting

confidence: 58%

See 1 more Smart Citation

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

Chapouto¹

2021

DCDS

View full text Add to dashboard Cite

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.

show abstract

Section: Andreia Chapoutosupporting

confidence: 58%

“…Consider the case (33) and proceed as in the estimate for R 0 . Assuming that we can associate the time cut-off with the first factor, we have…”

Section: Partmentioning

confidence: 99%

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

Chapouto¹

2021

DCDS

View full text Add to dashboard Cite

show abstract

“…This result also implies that the solutions constructed by Kappeler and Topalov are actually distributional solutions. We also mention that Schippa [37] extended Molinet's result to H s (T) for s > 0.…”

Section: Introductionmentioning

confidence: 61%

A remark on the well-posedness of the modified KdV equation in $L^2$

Forlano¹

2022

Preprint

View full text Add to dashboard Cite

We study the real-valued modified KdV equation on the real line and the circle, in both the focusing and the defocusing case. By employing the method of commuting flows introduced by Killip and Vişan (2019), we prove global well-posedness in H s for 0 ≤ s < 1 2 . On the line, we show how the arguments in the recent paper by Harrop-Griffiths, Killip, and Vişan (2020) may be simplified in the higher regularity regime s ≥ 0. On the circle, we provide an alternative proof of the sharp global well-posedness in L 2 due to Kappeler and Topalov (2005), and also extend this to the large-data focusing case. 2020 Mathematics Subject Classification. 35Q53. Key words and phrases. modified Korteweg-de Vries equation, global well-posedness.

show abstract

“…In a recent breakthrough, Harrop-Griffiths et al [16] showed the optimal global well-posedness of (1.1) in H s (R) for s > − 1 2 by exploiting the complete integrability of the equation. In the periodic setting, M = T, the real-valued mKdV equation (1.2) has garnered more attention than its complex-valued counterpart (1.1) [1,3,5,6,20,21,[26][27][28][29]35,36]. Exploiting the conservation of mass μ u(t) = 1 2π T |u(t)| 2 dx, Bourgain [1] introduced the first renormalized mKdV equation (mKdV1):…”

Section: Modified Korteweg-de Vries Equationmentioning

confidence: 99%

A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier–Lebesgue Spaces

Chapouto

2021

J Dyn Diff Equat

View full text Add to dashboard Cite

We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2021), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside $$H^\frac{1}{2}({\mathbb {T}})$$ H 1 2 ( T ) . Here, we employ the method introduced by Deng et al. (Commun Math Phys 384(1):1061–1107, 2021) to prove local well-posedness of the second renormalized mKdV equation in the Fourier–Lebesgue spaces $${\mathcal {F}}L^{s,p}({\mathbb {T}})$$ F L s , p ( T ) for $$s\ge \frac{1}{2}$$ s ≥ 1 2 and $$1\le p <\infty $$ 1 ≤ p < ∞ . As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier–Lebesgue spaces with infinite momentum.

show abstract

On the existence of periodic solutions to the modified Korteweg–de Vries equation below $$H^{1/2}({\mathbb {T}})$$

Cited by 7 publications

References 30 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

A remark on the well-posedness of the modified KdV equation in $L^2$

A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier–Lebesgue Spaces

Contact Info

Product

Resources

About