# Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line

Soonsik Kwon, Tadahiro Oh, Haewon Yoon

Research output: Contribution to journalArticlepeer-review

## Abstract

In this paper, we revisit the infinite iteration scheme of normal form reductions, introduced by the first and second authors (with Z. Guo), in constructing solutions to nonlinear dispersive PDEs. Our main goal is to present a simplified approach to this method. More precisely, we study normal form reductions in an abstract form and reduce multilinear estimates of arbitrarily high degrees to successive applications of basic trilinear estimates. As an application, we prove unconditional well-posedness of canonical nonlinear dispersive equations on the real line. In particular, we implement this simplified approach to an infinite iteration of normal form reductions in the context of the cubic nonlinear Schr\"odinger equation (NLS) and the modified KdV equation (mKdV) on the real line and prove unconditional well-posedness in $H^s(R)$ with (i) $s\geq 1/6$ for the cubic NLS and (ii) $s > 1/4$ for the mKdV. Our normal form approach also allows us to construct weak solutions to the cubic NLS in $H^s(R)$, $0 \leq s < 1/6$, and distributional solutions to the mKdV in $H^{1/4}(R)$ (with some uniqueness statements).
Original language English 649-720 Annales de la Faculte des Sciences de Toulouse 29 3 https://doi.org/10.5802/afst.1643 Published - 16 Nov 2020

## Keywords

• nonlinear Schrödinger equation
• modified KdV equation
• normal form reduction
• unconditional well-posedness
• unconditional uniqueness

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• ### ProbDynDispEq - Probabilistic and Dynamical Study of Nonlinear Dispersive Equations

Oh, T.

EU government bodies

1/03/1529/02/20

Project: Research