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

Andreia Chapouto

doi:10.3934/dcds.2021022

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2021, Volume 41, Issue 8: 3915-3950. Doi: 10.3934/dcds.2021022

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

Andreia Chapouto^,

School of Mathematics, The University of Edinburgh, and Maxwell Institute for Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom

^* Corresponding author: Andreia Chapouto
^* Corresponding author: Andreia Chapouto

Received: June 2020

Revised: December 2020

Early access: January 2021

Published: August 2021

The author is supported by the Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council [grant EP/L016508/01], the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh; and Tadahiro Oh's ERC Starting Grant (grant no. 637995 ProbDynDispEq).

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Abstract

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 non-existence result motivates the introduction of the second renormalized mKdV equation, which we propose as the correct model in the complex-valued setting outside $ H^\frac12(\mathbb{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.

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Mathematics Subject Classification: 35Q53.

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