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

  • * Corresponding author: Andreia Chapouto

    * Corresponding author: Andreia Chapouto

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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  • 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.

    Mathematics Subject Classification: 35Q53.


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