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

Tadahiro Oh; Yuzhao Wang

doi:10.3934/dcds.2020393

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

Tadahiro Oh, Yuzhao Wang

Mathematics

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Abstract

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_s^2,p(R) for s ≥ ¹₄ and 2 ≤ p < ∞. For s < ¹₄, we show that the solution map for mKdV is not locally uniformly continuous in M_s^2,p(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_s^2,p(R) for s ≥ ¹₄ and 2 ≤ p < ∞.

Original language	English
Pages (from-to)	2971-2992
Number of pages	22
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	41
Issue number	6
DOIs	https://doi.org/10.3934/dcds.2020393
Publication status	Published - Jun 2021

Bibliographical note

Funding Information:
The first author was supported by the European Research Council (grant no. 637995 “Prob-DynDispEq” and grant no. 864138 “SingStochDispDyn”). ∗ Corresponding author: Tadahiro Oh.

Publisher Copyright:
© 2021 American Institute of Mathematical Sciences. All rights reserved.

Keywords

Modified KdV equation
Modulation space
Well-posedness

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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Cite this

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title = "On global well-posedness of the modified KdV equation in modulation spaces",

abstract = "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 Ms2,p(R) for s ≥ 14 and 2 ≤ p < ∞. For s < 14, we show that the solution map for mKdV is not locally uniformly continuous in Ms2,p(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 Ms2,p(R) for s ≥ 14 and 2 ≤ p < ∞.",

keywords = "Modified KdV equation, Modulation space, Well-posedness",

author = "Tadahiro Oh and Yuzhao Wang",

note = "Funding Information: The first author was supported by the European Research Council (grant no. 637995 “Prob-DynDispEq” and grant no. 864138 “SingStochDispDyn”). ∗ Corresponding author: Tadahiro Oh. Publisher Copyright: {\textcopyright} 2021 American Institute of Mathematical Sciences. All rights reserved.",

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doi = "10.3934/dcds.2020393",

language = "English",

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AU - Wang, Yuzhao

N1 - Funding Information: The first author was supported by the European Research Council (grant no. 637995 “Prob-DynDispEq” and grant no. 864138 “SingStochDispDyn”). ∗ Corresponding author: Tadahiro Oh. Publisher Copyright: © 2021 American Institute of Mathematical Sciences. All rights reserved.

PY - 2021/6

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N2 - 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 Ms2,p(R) for s ≥ 14 and 2 ≤ p < ∞. For s < 14, we show that the solution map for mKdV is not locally uniformly continuous in Ms2,p(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 Ms2,p(R) for s ≥ 14 and 2 ≤ p < ∞.

AB - 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 Ms2,p(R) for s ≥ 14 and 2 ≤ p < ∞. For s < 14, we show that the solution map for mKdV is not locally uniformly continuous in Ms2,p(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 Ms2,p(R) for s ≥ 14 and 2 ≤ p < ∞.

KW - Modified KdV equation

KW - Modulation space

KW - Well-posedness

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DO - 10.3934/dcds.2020393

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VL - 41

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JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

IS - 6

ER -

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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