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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 wellposedness 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 wellposedness 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 

Pages (fromto)  649720 
Journal  Annales de la Faculte des Sciences de Toulouse 
Volume  29 
Issue number  3 
DOIs  
Publication status  Published  16 Nov 2020 
Keywords
 nonlinear Schrödinger equation
 modified KdV equation
 normal form reduction
 unconditional wellposedness
 unconditional uniqueness
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Dive into the research topics of 'Normal form approach to unconditional wellposedness of nonlinear dispersive PDEs on the real line'. Together they form a unique fingerprint.Projects
 1 Finished

ProbDynDispEq  Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20
Project: Research