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

  • * Corresponding author: Tadahiro Oh

    * Corresponding author: Tadahiro Oh 

The first author was supported by the European Research Council (grant no. 637995 "ProbDynDispEq" and grant no. 864138 "SingStochDispDyn")

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  • 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^{2,p}_{s}( \mathbb{R}) $ for $ s \ge \frac14 $ and $ 2\leq p < \infty $. For $ s < \frac 14 $, we show that the solution map for mKdV is not locally uniformly continuous in $ M^{2,p}_{s}( \mathbb{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^{2,p}_{s}( \mathbb{R}) $ for $ s \ge \frac14 $ and $ 2\leq p < \infty $.

    Mathematics Subject Classification: Primary: 35Q53.


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