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Abstract
We consider the quadratic derivative nonlinear Schrödinger equation (dNLS) on the circle. In particular, we develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the ColeHopf transformation, we prove unconditional global wellposedness in L^2(핋), and more generally in certain FourierLebesgue spaces FL^{s,p}(핋), under the meanzero and smallness assumptions. As a byproduct, we construct an infinite sequence of quantities that are invariant under the dynamics. We also show the necessity of the smallness assumption by explicitly constructing a finite time blowup solution with nonsmall meanzero initial data.
Original language  English 

Pages (fromto)  12731297 
Number of pages  25 
Journal  Annales de l'Institut Henri Poincaré C 
Volume  34 
Issue number  5 
Early online date  18 Oct 2016 
DOIs  
Publication status  Published  26 Sep 2017 
Keywords
 quadratic derivative nonlinear Schr ̈odinger equation
 normal form
 ColeHopf transform
 FourierLebesgue space
 wellposedness
 finitetime blowup
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 1 Finished

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