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Abstract
We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global wellposedness of the renormalized 4NLS in negative Sobolev spaces H^s(T), s > 1/3, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in H^s(T), s > 9/20, via the shorttime Fourier restriction norm method. By following the argument in GuoOh for the cubic NLS, this also leads to nonexistence of solutions for the (nonrenormalized) 4NLS in negative Sobolev spaces. (ii) We then prove enhanced uniqueness in H^s(T), s > 1/3, by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the H^senergy functional, allowing us to introduce an infinite sequence of correction terms to the H^senergy functional in the spirit of the Imethod. In fact, the main novelty of this paper is this reduction of the H^senergy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.
Original language  English 

Pages (fromto)  180 
Number of pages  80 
Journal  Forum of Mathematics, Sigma 
Volume  6 
DOIs  
Publication status  Published  11 May 2018 
Keywords
 fourth order nonlinear Schrödinger equation
 biharmonic nonlinear Schrödinger equation
 shorttime Fourier restriction norm method
 normal form reduction
 enhanced uniqueness
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Dive into the research topics of 'Global wellposedness of the periodic cubic fourth order NLS in negative Sobolev spaces'. 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
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Tadahiro Oh
 School of Mathematics  Personal Chair of Dispersive Equations
Person: Academic: Research Active