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

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 $H^s(\mathbb R)$ with (i) $s\geq \frac 16$ for the cubic NLS and (ii) $s > \frac 14$ for the mKdV. Our normal form approach also allows us to construct weak solutions to the cubic NLS in $H^s(\mathbb R)$, $0 \leq s < \frac 16$, and distributional solutions to the mKdV in $H^\frac{1}{4}(\mathbb R)$ (with some uniqueness statements).