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

Oh, Tadahiro; Wang, Yuzhao

doi:10.3934/dcds.2020393

Cited by 11 publications

(5 citation statements)

References 24 publications

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“…For (mKdV), analogous almost-critical results in Fourier–Lebesgue spaces were obtained in [18, 22]. The threshold for analytic well-posedness of (mKdV) in modulation spaces was determined in [8, 50]; however, this still does not coincide with scaling criticality.…”

Section: Introductionmentioning

confidence: 88%

Sharp well-posedness for the cubic NLS and mKdV in

Harrop-Griffiths,

Killip,

Vişan

2024

Forum of Mathematics, Pi

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We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s({{\mathbb {R}}})$ for any regularity $s>-\frac 12$ . Well-posedness has long been known for $s\geq 0$ , see [55], but not previously for any $s<0$ . The scaling-critical value $s=-\frac 12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in $H^s({{\mathbb {R}}})$ for any $s>-\frac 12$ . The best regularity achieved previously was $s\geq \tfrac 14$ (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity.

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Section: Introductionmentioning

confidence: 88%

Sharp well-posedness for the cubic NLS and mKdV in

Harrop-Griffiths,

Killip,

Vişan

2024

Forum of Mathematics, Pi

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“…Therefore, the modulation space is a good space for initial data of the Cauchy problem for nonlinear dispersive equations (see Refs. 3–8). However, as showed by Sugimoto and Tomita, 9 modulation spaces do not have good scaling properties such as

L^{p}

spaces.…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness of generalized KdV and one‐dimensional fourth‐order derivative nonlinear Schrödinger equations for data with an infinite L² norm

2023

Stud Appl Math

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“…Motivated by recent applications of modulation spaces in the context of nonlinear harmonic analysis and its applications, cf. [3,4,6,14,23,39,40,48,55] we focus our attention to boundedness for multiplications and convolutions for elements in such spaces. The basic results in that direction go back to the original contribution [17], and were thereafter reconsidered by many authors in different contexts.…”

Section: Introductionmentioning

confidence: 99%

“…Here we illustrate this approach by considering the nonlinear cubic Schrödinger equation, which appear for example in in Bose-Einstein condensate theory [36]. We also refer to [6,Chapter 7] for an overview of results related to well-posedness of the nonlinear Schrödinger equations in the framework of modulation spaces, see also [5,39,40].…”

Section: Introductionmentioning

confidence: 99%