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

Tadahiro Oh; Yuzhao Wang

doi:10.3934/dcds.2020393

Article Contents

2021, Volume 41, Issue 6: 2971-2992. Doi: 10.3934/dcds.2020393 open access

This issue Previous Article Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems Next Article Study of fractional Poincaré inequalities on unbounded domains

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

Tadahiro Oh^1, , and
Yuzhao Wang^2,

1.
School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
2.
School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom

^* Corresponding author: Tadahiro Oh
^* Corresponding author: Tadahiro Oh

Received: January 2020

Revised: October 2020

Early access: January 2021

Published: June 2021

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

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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^{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 $.

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Mathematics Subject Classification: Primary: 35Q53.

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